Parametric Learning and Monte Carlo Optimization
Keywords: Monte Carlo Optimization, Black-box Optimization, Parametric Learning,
Automated Annealing, Bias-variance-covariance
Parametric Learning and Monte Carlo Optimization
David H. Wolpert dhw@email.arc.nasa.gov
MS 269-1, Ames Research Center
Moffett Field, CA 94035.
Dev G. Rajnayaran dgorur@stanford.edu
Department of Aeronautics and Astronautics,
Durand Rm. 158, 496 Lomita Mall, Stanford, CA 94305.
Abstract
This paper uncovers and explores the close relationship between Monte Carlo Optimization
of a parametrized integral (MCO), Parametric machine-Learning (PL), and ‘blackbox’ or
‘oracle’-based optimization (BO). We make four contributions. First, we prove that MCO is
mathematically identical to a broad class of PL problems. This identity potentially provides
a new application domain for all broadly applicable PL techniques: MCO. Second, we
introduce immediate sampling, a new version of the Probability Collectives (PC) algorithm
for blackbox optimization. Immediate sampling transforms the original BO problem into
an MCO problem. Accordingly, by combining these first two contributions, we can apply
all PL techniques to BO. In our third contribution we validate this way of improving BO
by demonstrating that cross-validation and bagging improve immediate sampling. Finally,
conventional MC and MCO procedures ignore the relationship between the sample point
locations and the associated values of the integrand; only the values of the integrand at
those locations are considered. We demonstrate that one can exploit the sample location
information using PL techniques, for example by forming a fit of the sample locations to the
associated values of the integrand. This provides an additional way to apply PL techniques
to improve MCO.
1. Introduction
This paper uncovers and explores some aspects of the close relationship between Monte
Carlo Optimization of a parametrized integral (MCO), Parametric machine Learning (PL),
and ‘blackbox’ or ‘oracle-based’ optimization (BO). We make four primary contributions.
First, we establish a mathematical identity equating MCO with PL. This identity poten-
tially provides a new application domain for all broadly-applicable PL techniques, viz.,
Our second contribution is the introduction of immediate sampling. This is a new
version of the Probability Collectives (PC) approach to blackbox optimization. PC encom-
passes Estimation of Distribution Algorithms (EDAs)(De Bonet et al., 1997; Larraaga and
Lozano, 2001; Lozano et al., 2005) and the Cross Entropy (CE) method (Rubinstein and
Kroese, 2004) as special cases. However PC is broader and more fully motivated. This
means it uncovers (and overcomes) formal shortcomings in those other approaches.
In the immediate sampling version of PC the original BO problem is transformed into an
MCO problem. In light of our first contribution, this means we can apply PL to immediate
sampling. In this way all PL techniques — including cross-validation, bagging, boosting,
active learning, stacking, and others — can be applied to blackbox optimization.
In our third contribution we experimentally explore the power of this identity between
MCO and PL. In these experiments we demonstrate that cross-validation and bagging
improve the performance of immediate sampling blackbox optimization. In particular, in
these experiments we show that cross-validation can be used to adaptively set an ‘annealing
schedule’ for blackbox optimization using immediate sampling without any extra calls to
the oracle. In some cases, we show that this adaptively formed annealing schedule results
in better optimization performance than any exponential annealing schedule.1
Finally, conventional MC and MCO procedures ignore the relationship between the
sample point locations and the associated values of the integrand. (Only the values of
the integrand at the sample locations are considered by such algorithms.) We end by
exploring ways to use PL techniques to exploit the information in the sample locations, for
instance, by Bayesian fitting of a surface from the sample locations to the associated values
of the integrand. This constitutes yet another way of applying PL to MCO in general, and
therefore to BO in particular.
1.1 Background on PL, MCO, Blackbox Optimization, and PC
We begin by sketching the four disciplines discussed in this paper:
1. A large number of parametric machine-learning problems share the following two properties.
First, the goal in these problems is to find a set of parameters, θ, that minimizes an integral of
a function that is parametrized by θ. Second, to help us find that θ we are are given samples
of the integrand. These problems reduce to a sample-based search for the θ that we predict
minimizes the integral. We will refer to problems of this class as Parametric Learning (PL)
problems.
An example of PL is parametric supervised learning, where we want to find an optimal
predictor or regressor zθ that minimizes the associated expected loss,
dx dy P (x)P (y |
x)L[y, zθ(x)], where x’s are inputs and y’s are outputs. We do not, however, know P (x)P (y |
x). Instead, we are provided a training set of samples of P (x)P (y | x). The associated PL
problem is to use those samples to estimate the optimal θ.
2. MCO is a technique for solving problems of the form argminφ
dw U(w, φ) (see Ermoliev and
Norkin, 1998). MCO starts by replacing that integral with an importance-sample generated
estimate of it. That estimate is a sum parametrized by φ. In MCO one searches for the value
φ that minimizes this sum; the result of this search is one’s estimate of the φ that optimizes
the original integral.
3. Blackbox optimization algorithms are ways to minimize functions of the form G : X → R when
one does not actually know the function G. Such algorithms work by an iterative process in
which they first select a query x ∈ X, and then an ‘oracle’ returns to the algorithm a (poten-
tially noise-corrupted) value G(x), and no other information, in particular, no gradient infor-
mation. The difference between one blackbox optimization algorithm and another is how they
select each successive query based on the earlier responses of the oracle. Examples of blackbox
optimization algorithms are genetic algorithms (Mitchell, 1996), simulated annealing (Kirk-
patrick et al., 1983), hill-climbing algorithms, Response-Surface Methods (RSMs) (Myers and
Montgomery, 2002), and some forms of Sequential Quadratic Programming (SQP) (Gill et al.,
1981; Nocedal and Wright, 1999), Estimation of Distribution Algorithms (EDAs)(De Bonet
et al., 1997; Larraaga and Lozano, 2001; Lozano et al., 2005), tabu search, the Cross Entropy
(CE) method (Rubinstein and Kroese, 2004), and others.
1. Since they are special cases of PC, presumably we could similarly apply PL techniques to improve EDA’s
or the CE method.
4. PC is a set of techniques that can be used for blackbox optimization. Broadly speaking,
PC works by transforming a search for the best value of a variable x into a search for the
best probability distribution over the variable, q(x) (see Wolpert et al., 2006; Macready and
Wolpert, 2005; Wolpert, 2003, 2004a; Bieniawski and Wolpert, 2004; Antoine et al., 2004; Lee
and Wolpert, 2004). Once one solves for the optimal q(x), inversion to get the optimal x for
the original search problem is stochastic; one simply samples q. As described below, PC has
many practical strengths, and is related to RSMs, EDAs, and the CE method.
1.2 Roadmap of This Paper
We make four primary contributions:
1. Sec. 2 begins with a detailed review of MCO and PL. Conventional analysis of Monte Carlo
estimation involves a bias-variance decomposition of the error of the estimator of a particular
integral. We show that for MCO, a full analysis requires more than simply extending such
bias-variance analysis separately to each of the estimators given by the separate φ’s. Moments
coupling the errors of the estimators for the separate φ’s must also be taken into account. How
should we do that?
To answer this, we note that in a different context, the techniques of PL take such coupling
moments into account, albeit implicitly. This leads us to explore the relation between MCO
and PL. This in turn leads to our first major contribution, the proof that MCO is identical
to PL. This contribution means that one can apply all PL techniques, for instance, cross-
validation, bagging, boosting, stacking, active learning and others, to MCO. Such PL-based
MCO (PLMCO) provides a new way of minimizing potentially high-dimensional parametrized
integrals.
Experimentally testing the utility of applying PL to MCO requires an MCO application do-
main. Here we choose the domain of blackbox optimization. To establish how blackbox
optimization is an application domain for MCO requires our second contribution, as follows.
2. We start in Sec. 3 by presenting an overview of previous versions of the blackbox optimization
approach of PC. We then make our second contribution in the following section, where we
introduce immediate sampling, a new version of PC that overcomes some of the limitations of
previous versions.
These first two contributions are combined by the fact that immediate sampling is a special
case of MCO. The resultant identity between PL and immediate sampling means that, in
principle, any PL technique can be applied to blackbox optimization. In particular, regular-
ization, cross-validation, bagging, active learning, boosting, stacking, kernel machines, and
others, can be ‘cut and paste’ to do blackbox optimization. This use of PL for blackbox
optimization constitutes a new application domain for PL.
In Sec. 4.5 we present some concrete instances of how to modify immediate sampling to use
PLMCO rather than conventional MCO. It is important to note that when applied (via im-
mediate sampling) to blackbox optimization, these PL techniques do not require additional
calls to the oracle. For example, using cross-validation to set regularization parameters in
immediate sampling (the equivalent of an annealing schedule in SA) does not involve running
the entire blackbox optimization algorithm with different regularization schedules. As an-
other example, using bagging in immediate sampling does not mean running the optimization
algorithm multiple times based on different subsets of the sample points found so far.
3. Our third contribution is to experimentally demonstrate in Sec. 5 that PLMCO substantially
outperforms conventional MCO when used this way for blackbox optimization. We are partic-
ularly interested in blackbox optimization problems where calls to the oracle are the primary
expense. Accordingly, non-oracle, ‘offline’ computation is considered free. So in our experi-
ments we compare algorithms based on the values of G found by the algorithms versus the
associated number of calls to the oracle2. In particular, we show that bagging and cross-
validation leads to faster blackbox optimization on two well-known benchmark problems for
continuous nonconvex optimization.
It should be emphasized that these experiments are not intended to investigate whether
PLMCO applied to immediate sampling is superior to other blackbox optimization algorithms.
Rather their purpose is to investigate whether one can indeed leverage the formal connection
between PL and MCO to improve immediate sampling. Accordingly, these experiments are
on toy domains, and we do not compare performance with other blackbox optimization algo-
rithms. We leave such comparisons to future papers.
4. In estimating the value of an integral based on random samples of its integrand, conventional
MC and MCO techniques ignore how the locations of the sample points are related to the
associated values of the integrand. Such techniques concentrate exclusively on those sample
values of the integrand that are returned by the oracle. However, one can use the sample
locations and associated integrand values to form a supervised learning fit to the integrand.
In principle, such a fit can then be used to improve the overall estimate of the integral.
In ‘fit-based’ MC and MCO one uses all the data at hand to fit the integrand and then uses
that fit to improve the algorithm. In this paper, we concentrate on situations where the data at
hand consist only of sample locations and the associated values of the integrand, but in other
situations the data at hand may also include information like the gradient of the integrand at
the sample points. In their most general form, fit-based MC and MCO include techniques to
exploit such information.
One natural Bayesian approach to fit-based MC uses Gaussian processes. Work adopting this
approach, for the case where the data only contain sample locations and associated integrand
values, is reviewed in Rasmussen and Gharamani (2003). In Sec. 6 we generalize that work
on fit-based MC, e.g., to allow non-Bayesian approaches. In that section, we also consider
fit-based MCO in general, and fit-based immediate sampling in particular.
One of the ways cross-validation is used in these experiments is to set a regularization
parameter. In immediate sampling, the regularization parameter plays the same role as
the temperature does in simulated annealing. So intuitively speaking, our results show
how to use cross-validation to set an annealing schedule adaptively for blackbox optimiza-
tion, without extra calls to the oracle. We show in particular that such auto-annealing
outperforms the best-fit exponential annealing schedule.
There are more topics involving the connection between MCO, immediate sampling
and PL, than can be explored in this single paper. One such topic is how to incorporate
constraints on x in immediate sampling. Another important topic involves a derivation
from first principles of the objective function used in immediate sampling. These two
topics are briefly discussed in the appendices. Some other topics are mentioned, albeit
even more briefly, in the conclusion.
1.3 Notation
As a point of notation, we will use the term ‘distribution’ to refer either to a probability
distribution or a density function, with the associated Borel field implicitly fixing the
meaning. Similarly, we will write integrals even when we mean sums; the measure of the
2. See Wolpert and Macready (1997); Droste et al. (2002); Wolpert and Macready (2005); Corne and
Knowles (2003); Igel and Toussaint (2004); Schumacher et al. (2001) for a discussion of the mathematics
relating algorithms under such performance measures.
integral is implicitly taken to be the one appropriate for the the argument. We will use Θ
to indicate the Heaviside or indicator function, which is 1 if its argument is positive, and
0 otherwise.
We will use P to mean the set of all distributions over X. We are primarily interested
in X’s that are too large to permit computations involving all members of P. Accordingly,
we will will work with parametrized subsets Q ⊂ P. We generically write that (possibly
vector-valued)parameter as θ, and write the element of Q specified by θ as qθ. We use E
to indicate the expectation of a random variable. Subscripts on E are sometimes used to
indicate the distribution(s) defining the expectation.
We take any oracle G to be an x-indexed set of independent stochastic processes, and
use the symbol g to indicate the generic output of the oracle in response to any query. With
some abuse of notation, we denote the output of the oracle for query x as P (g | x,G ). For
a noise-free, or single-valued oracle, we write P (g | x,G ) = δ(g −G(x)) for some function
G implicitly specified by G , where δ(.) is the Dirac delta function.
When there is both a factual version of a random variable and a posterior distribution
over counter-factual values of that variable, they must be distinguished. In general this
requires extending the conventional Bayesian formalism (see Wolpert, 1997, 1996). Here,
though, it suffices for us to indicate counter-factual values by a subscript c. Say there
is a factual oracle G , and we are provided a data set D formed by sampling G . We use
superscripts to denote different samples in that data set. Then D in turn induces a posterior
over oracles, and we write that posterior as P (Gc | D).
2. MCO and PL
In this section we review PL and MCO show that they are mathematically identical.
2.1 Overview of PL
A broad class of parametric machine learning problems try to find
(P1) : argminξ
dx P (x)Rx(ξ).
For subsequent purposes, it will be useful to write x as a subscript and ξ as an argument
of R, even though x is the integration variable and ξ is the parameter being optimized.
To perform this minimization, we have a set of function values D ≡ {Rxi(ξ)}, where we
typically assume that the samples xi, i = 1, . . . , N were formed by IID sampling of P (x).
The maximum likelihood approach to this minimization first makes the approximation∫
dx P (x)Rx(ξ) ≈
Rxi(ξ),
Ri(ξ). (1)
One then solves for the ξ minimizing the sum, and uses this as an approximation to the
solution to P1. In practice, though, this procedure is seldom used directly: although
the approximation in Eq. 1 is unbiased for any fixed ξ, minξ
i(ξ) is not an unbiased
estimate of minξ
dx P (x)Rx(ξ). Therefore, when this approximation is exploited, it is
modified to incorporate bias-reduction techniques.
Example: Parametric Supervised Learning:
Let X,Y be input and output spaces, respectively. Let L(y1, y2) : Y × Y → R be a loss
function, and zξ : X → Y be a ξ-parametrized set of functions. In parametric machine
learning with IID error our goal is to solve
argminξ
dx P (x)
dy P (y | x)L(y, zξ(x)),
argminξ
dx P (x)Rx(ξ). (2)
Intuitively, Rx(ξ) is the expected loss at x for the ‘fit’ zξ(x) to the x-indexed set of distri-
butions P (y | x).
To perform this minimization we have a training set of pairs D ≡ {xi, yi}, i =
1, . . . , N , that we assume were formed by IID sampling of P (x)P (y | x). The maximum
likelihood approach to this minimization first makes the approximation∫
dx P (x)
dy P (y | x)L(y, zξ(x)) ≈
L(yi, zξ(x
Ri(ξ).
One then solves for the ξ minimizing the sum, and uses this as an approximation to the
solution to (P1). As discussed above, in practice, this minimization is rarely used directly,
and is usually combined with a bias-reducing technique like cross-validation.
2.2 Overview of MCO
Consider the problem
(P2) : argminφ∈Φ
dw U(w, φ).
For now, we do not impose constraints on φ, nor restrict Φ. Monte Carlo Optimiza-
tion (Ermoliev and Norkin, 1998) is a way to search for the solution of (P2). In MCO we
use importance sampling to rewrite the integral in (P2) as∫
dw U(w, φ) =
dw v(w)
U(w, φ)
dw v(w)rv,U,w(φ), (3)
for some sampling distribution v. Following the usual importance sampling procedure, we
IID sample v to form a sample set {U(wi, .) : i = 1, . . . N}, which specifies a set of N
sample functions
ri(φ) , rv,U,wi(φ).
It is implicitly assumed that for any w, we can evaluate v(w) up to an overall normalization
constant.
In MCO, these N functions are used in combination with any prior information to
estimate the solution to (P2). Conventionally, this is done by approximating the solution
to (P2) with the solution to the problem
(P3) : argminφ
ri(φ).
We define
LU (φ) ,
dw U(w, φ),
L̂v,U,{wi}(φ) ,
rv,U,wi(φ),
φ̂v,U,{wi}(φ) , argmin[L̂v,U,{wi}(φ)].
For notational simplicity, the subscripts will usually be omitted in these expressions. We
will use the term naive MCO to refer to solving (P3) by minimizing L̂ (φ).
2.3 Statistical Analysis of MCO
The statistical analysis of MC estimation of integrals is a relatively mature field (see Robert
and Casella, 2004; Fishman, 1996). We now show that when such MC estimation is com-
bined with parameter optimization in MCO, the analysis becomes much more involved.
2.3.1 Review: MC Estimation
First consider MC estimation, with no mention of MCO. We first need to to specify a loss
function L(., .) that will couple our mathematics with real-world costs. The first argument
of such an L is the output of the estimation algorithm under consideration. The second
argument is the quantity statistically sampled by that algorithm. The associated value of
L is the cost if the algorithm produces the output specified in that first argument, using
the quantity specified in the second argument.
As an example, consider importance-sampled MC estimation of an integral. Using the
MCO notation just introduced, we use L̂ (φ) as an estimate of L (φ) for some fixed φ. The
quantity being sampled is the function U(., φ), and the output of the algorithm is L̂ (φ).
Accordingly, these are the arguments of the loss function.
The most popular loss function in statistical analysis of MC integral estimation is
quadratic loss, given below.
L(L̂ (φ), U(., φ)) , [L̂ (φ)−
dw U(w, φ)]2.
Unless explicitly stated otherwise, we will henceforth use the term ‘expected loss’ to refer
to the average of this loss function over sample sets. Since L̂ (φ) is an unbiased estimate
of L (φ), the expected loss is the sample variance,
Var(L̂ (φ)) = E([
U(wj , φ)
Nv(wj)
]2) − [E(
U(wj , φ)
Nv(wj)
dw v(w)[
U(w, φ)
]2 − [
dw v(w)
U(w, φ)
dw v(w)[
U(w, φ)
]2 − [L (φ)]2}.
This expansion for the sample variance is quite useful. For example, one can solve for the
v that minimizes this variance (and therefore minimizes expected loss) as a function of
U(., φ). For nowhere-negative U , that optimal v is given by (see Robert and Casella, 2004)
v(w) ,
U(w, φ)∫
dw′U(w′, φ)
Given the formula for the optimal v, one can estimate it from a current sample set, and
then use the estimated optimal v for future sampling. This is what is done in the VEGAS
Algorithm (Lepage, 1978, 1980). Consideration of the sample variance has also led to
algorithms that partition X and then run importance sampling on each partition element
separately, for instance, stratified sampling (Fishman, 1996). MC estimators that do not
use strict importance sampling may introduce bias. However, if the variance is sufficiently
reduced, expected quadratic error is reduced. This can be exploited to tradeoff bias and
variance.
2.3.2 From MC to MCO
When we combine MC with parameter optimization in MCO, quantities like Var(L̂ (φ))
for one particular φ are not the main objects of interest. Instead, we are interested in
expected loss of our iterated MCO algorithm, which involves multiple φ’s. So what is the
appropriate loss function for analyzing MCO? From the very definition of (P2), it is clear
that we want L(φ,U) to be minimized by the φ that minimizes
dw U(w, φ). The simplest
approach to doing this, which will be assumed from now on, stipulates that
L(φ,U) = L (φ) =
dw U(w, φ), (4)
the same integral appearing in (P2). If we can solve (P2) exactly, then we will have
produced the φ with minimal value of this loss function.
Given this choice of loss function, expected loss in naive MCO is
E(L | U, v) =
dw1 . . . dwN
v(wi)L (argminφ[L̂v,U,{wi}(φ)])
dw1 . . . dwN
v(wi)
dw′ U(w′, argminφ[
U(wi, φ)
v(wi)
]). (5)
The optimal v for naive MCO is the one that minimizes E(L | U, v). There is no direct
relation between this v and the one that minimizes loss for some single φ. In stark contrast
to the MC analysis in Sec. 2.3.1, in addition to the sample variance Var(L̂ (φ)) for any single
φ, the expected loss E(L | U, v) now also depends on moments coupling the distributions
of L̂ (φ) for different φ’s. Loosely speaking, the bias-variance tradeoff in Sec. 2.3.1 now
becomes a more complicated bias-variance-covariance tradeoff. Now, setting w is more
involved, but we can approach it as follows.
Expressing the expected loss slightly differently gives us an important insight. Note
that each sample set {wi} gives rise to an associated set of estimates for all φ ∈ Φ. Call this
(possibly infinite dimensional) vector of estimates ~l, each of whose components is indexed
by φ and is an estimate for that particular φ. Now, instead of computing expected loss by
averaging over all possible sample sets, we average over all possible vectors ~l. In order to
do this, we need to specify the probability of each vector ~l. Define
πv,U,Φ(~l) , Pr({wi} : L̂v,U,{wi}(φ) = lφ ∀φ ∈ Φ).
So, π
v,U,Φ(~l) is the probability of a set of sample points {wi} such that for each φ ∈ Φ, the
associated empirical estimate
i rv,U,wi(φ) equals the corresponding component of
~l. For
notational simplicity the subscripts of π
v,U,Φ will sometimes be omitted. We can now write
Eq. 5 succinctly as
E(L | U, v) =
d~l π(~l) L (argminφ[lφ]) (6)
where ‘argminφ[lφ]’ means the index φ of the smallest component of ~l. The risk is the
difference between this expected loss and the lowest possible loss. We can write that risk
d~l πv,U,Φ(~l) [L (argminφ[lφ]) − minφ[L (φ)]]. (7)
Our sample set constitutes a set of samples of πv,U,Φ occurring in Eq. 6,This fact can
potentially be exploited to dynamically modify v and/or Φ to reduce E(L | U, v). Indeed,
for the simpler case of MC estimation, this is essentially the kind of computation done in
the VEGAS algorithm mentioned above. As a practical issue, it may be difficult to update
v and/or Φ using the full formula Eq. 5. Instead, one could approximate that formula
E(L | U, v) near a single φ of interest, e.g., about a current estimate for the optimal φ.
Intuitively though, one would expect that for a fixed set of φ’s, everything else being
equal, it would be advantageous to have small variances of unbiased estimators and large
covariances between them. Such considerations based on the second moments may help
one choose quantities like the sampling distribution v.
Such considerations may also help one choose the set of candidate φ’s, Φ. For example,
one way to have large covariances between the φ ∈ Φ is to have the associated functions
over w, {U(., φ) : φ ∈ Φ}, all lie close to one another in an appropriate function space (e.g.,
according to an l∞ norm comparing such functions). However, choosing such a Φ will tend
to mean there is a small ‘coverage’ of that set of functions, {U(., φ) : φ ∈ Φ}. More precisely,
it will tend to prevent the best of those φ’s from being very good; minφ∈Φ[
dw U(w, φ)]
will not be very low.
This illustrates that, in choosing Φ, there will be a tradeoff between two quantities:
The first quantity is the best possible performance with any of the φ ∈ Φ. The second
quantity is the risk, that is, how close a given MCO algorithm operating on Φ is likely
to come to that best possible performance of a member of Φ. Choosing Φ to have large
covariances of the (MC estimators based on the) members of Φ, and in particular to have
large covariances with the truly optimal φ, argminφ∈ΦL (φ), will tend to result in low risk.
But it will also tend to result in poor best-possible performance over all φ ∈ Φ.
Similarly, one would expect that as the size of Φ increases, there would be a greater
chance that a sample set for one of the suboptimal φ ∈ Φ would have low expected loss
‘by luck’. This would then mislead one into choosing that suboptimal φ. So increasing the
size of Φ may increase risk. However increasing Φ’s size should also improve best possible
performance. So again, we get a tradeoff.
It may be that such considerations involving the size of Φ and the covariances of its
members can be encapsulated in a single number, giving an ‘effective size’ of Φ (somewhat
analogously to the VC dimension of a set of functions). Such tradeoffs are specific to the
use of MCO, and do not arise in plain (single-φ) MC. They are in addition to the usual
bias-variance tradeoffs, which still apply to each of the separate MC estimators.
An illustrative example of the foregoing is provided in App. C. A more complete sta-
tistical analysis of risk in MCO, including Bayesian considerations, is in Sec. 6.
MCO PL
v(w) P (x)
rv,w(φ) Rx(ξ)
riv(φ) R
Table 1: Correspondence between PL and MCO.
2.4 PL Equals MCO
In MCO, we have to extrapolate from the sample set of w values to perform the integral
minimization in Eq. 3. As discussed above, this can recast as having a set of sample
functions φ→ ri(φ) that we want to use to estimate the φ that achieves that minimization.
Similarly, in PL, we have to extrapolate from a training set of functions Ri(ξ) to minimize
the integral
dx P (x)Rx(ξ). Though not usually viewed this way, at the root of this
extrapolation problem is the problem of using the sample functions ξ → Ri(ξ) to estimate
the minimizer of Eq. 2.
In addition, the analysis of Sec. 2.3 is closely related to the PL field of uniform conver-
gence theory. That field can be cast in the terms of the current discussion as considering
a broad class of U ’s, U . Its starting point is the establishment of bounds on how
maxv,U∈U [
d~l πv,U,Φ(~l) Θ(L (argminφ[lφ]) − minφ[L (φ)] − κ) (8)
depends3 on κ. Of particular interest is how the function taking κ to the associated bound
varies with characteristics of U and Φ (see Vapnik, 1982, 1995). Eq. 8 should be compared
with Eq. 7.
All of this suggests that the general MCO problem of extrapolation from a sample set
of empirical functions to minimize the integral of Eq. 3, is, in fact, identical to the general
PL problem of extrapolation from a training set of empirical functions to minimize the
integral of Eq. 2. This is indeed the case. As shown in Table 1, identify ξ ↔ φ, x ↔
w,P (x) ↔ v(w), Rx(ξ) ↔ rv,w(φ), riv(φ) ↔ Ri(ξ). Then the integrals in Eq. 3 and (P1)
become identical. So the MCO expected loss function in Eq. 4 becomes identical to the PL
expected loss. Similarly, the sample functions for MCO and PL become identical.
In particular, in supervised learning, when there is no noise, P (y | x) becomes a single-
valued function y(x), and the parametric supervised learning problem becomes
argminξ
dx P (x)[L(y(x), zξ(x))]
This should be compared to the MCO problem as formulated in Eq. 3. For the same reasons
that direct minimization of Eq. 1 is seldom used in PL, we now see that naive MCO will
be biased, and should preferably not be used directly.
Note that most sampling theory analysis of PL does not directly consider the biases and
variances of the separate Monte Carlo estimators for each ξ, nor does it directly consider the
moments that couple the distributions of those estimates. Rather, it considers a different
3. As an example, rewrite w → x, φ → α, v(x) → P (x). Also choose U to be all functions of the form
U(w, φ) = U(x, α) ,
dy P (y | x)(y − F (x, α))2 for any function F and distribution P (y | x). Under
this substitution, Eq. 8 becomes the archetypal uniform convergence theory problem for regression with
quadratic loss.
type of bias and variance — the bias and variance of an entire algorithm that chooses a ξ
based on associated MC estimates of expected loss (Wolpert, 1997). In this sense, such PL
analysis bypasses the issues considered in Sec. 2.3. The bias-variance-covariance approach
described in this section might have important implications on PL analysis of learning
algorithms, but for the moment, in our exploration of the identity between MCO and PL,
we simply use PL-based techniques to reduce the bias or variance of our algorithms.
3. Review of PC
This section cursorily reviews the previously investigated type of PC. It then briefly dis-
cusses the advantages of PC for blackbox optimization and its relation to other blackbox
optimization techniques.
3.1 Introduction to PC
To introduce PC, consider the general (not necessarily blackbox) optimization problem
(P4) : argminx∈XE(g | x,G ).
For now, we ignore constraints on x. In PC we transform (P4) into the problem
(P5) : argminqθ∈QFG (qθ),
for some appropriate function FG . After solving (P5) we stochastically invert qθ to get an
x (the ultimate object of interest), by sampling qθ. This type of “randomizing transform”
contrasts with conventional transform techniques, where inversion is deterministic.
Ideally, FG should be chosen in a first-principles manner, based on exactly how qθ will
be sampled and how those samples used (see Sec. 6). In practice though, computational
considerations might lead one to choose FG heuristically. Intuitively, such considerations
might compel us to choose FG both so that (P5) is readily easy to solve, and so that any
solution qθ to (P5) is concentrated about the solutions of (P4). Taking the parametrization
to be implicit, we often abbreviate FG (qθ) as just FG (θ).
In many variants of PC explored to date, FG (θ) is an integral transform4 over X,
FG (θ) ,
dx dg P (g | x,G )F (g, qθ(x)). (9)
dx rP (g|x,G )(x, θ) (10)
As an example of such an integral transform, consider optimization with a noise-free (single-
valued) oracle, P (g | x,G ) = δ(g −G(x)), where the transformed objective is the expected
value of (g|x) under x ∼ qθ. In other words, FG = Eqθ [G(x)]. In addition, suppose that
Q = P, that is, qθ can be any distribution. Under fairly weak assumptions, it can be shown
that one solution to (P5) is given by the point-wise limit of Boltzmann distributions,
p?(x) = lim
pβ(x), where pβ(x) ∝ exp[−βG(x)].
In the case where Q ⊂P, we could choose FG (θ) to be a measure of the dissimilarity
between such a Boltzmann (or other) ‘target’ distribution, and a given qθ. For instance,
4. An instance where this is not the case is with the elite objective function, described in App. B.
we could use a Kullback-Leibler (KL) divergence between qθ and pβ , which we refer to as
“pq” KL distance:
FG (θ) = KL(p
β || qθ)
dx pβ(x)ln[
qθ(x)
pβ(x)
In terms of the quantities in Eq. 9, F (g, qθ(x)) ∝ e−βgln[qθ(x)], up to an overall additive
constant. So rP (g|x,G )(x, θ) in Eq. 10 is the contribution to the KL distance between pβ
and qθ given by the argument x.
To see why this choice of FG (θ) is reasonable, first note that pβ(x) is large where G(x)
is small. Indeed, as β →∞, pβ becomes a delta function about the x(s) minimizing G(x),
that is, about the solution(s) to (P4). Now, suppose that Q is a broad enough class that it
can approximate any sufficiently peaked distribution. That means that there is a qθ ∈ Q
for which KL(pβ || qθ) is small for large β. In such a situation, the qθ solving (P5) will be
highly peaked about the x(s) solving (P4). Accordingly, if we can solve (P5) for large β,
sampling the resultant qθ will result in an x with a low E(g | x,G ).
3.2 Review of Delayed Sampling
We now present a review of conventional, delayed-sampling PC. In this type of PC we
exploit characteristics of the parametrization of qθ, and pursue the algebraic solution of
(P5) as far as possible, in closed form. At some point, if there remain quantities in this
algebraic expression that we cannot evaluate closed-form, we estimate them using Monte
Carlo sampling.
As an example, consider a noise-free oracle, and instead of pq Kullback-Leibler distance,
choose FG (qθ) to be the expected value returned by the oracle under qθ,
Eqθ,G (g) ,
dx dg gP (g | x,G )qθ(x)
dx G(x)qθ(x) (11)
where the second equality reflects the fact that we are assuming a noise-free oracle. To
emphasize the fact that we’re considering noise-free oracles5, we will sometimes write
Eqθ,G (g) = Eqθ,G(g). While Eqθ,G (g) is a linear function of qθ, in general it will not be
a linear function of θ. Accordingly, finding the θ minimizing Eqθ,G (g) may be a non-trivial
optimization problem.
Since qθ must be a probability distribution, (P5) is actually a constrained optimization
problem, involving |X| inequality constraints {qθ(x) ≥ 0 ∀x}, and one equality constraint,∫
dx qθ(x) = 1. As discussed by Wolpert et al. (2006), such a constrained optimization
problem can be converted into one with no inequality constraints by the use of barrier
function methods. These methods transform the original optimization problem into a
sequence of new optimization problems, {(P5)i}, each of which is easier to solve than the
original problem (P5). Solving those problems in sequence leads to a solution to the original
problem (P5).
Consider applying this method with an entropic barrier for the case where FG (qθ) =
Eqθ,G(g). Then, it turns out that up to additive constants, each problem (P5)
i is again
5. Even though it is noise-free, the oracle G may be a random variable — we may not know G, and may
attempt to predict it probabilistically from data, in a Bayesian fashion. In such a situation, notation
like ‘E(G )’ refers to the expected oracle under our prior distribution over oracles. So, we use Eqθ,G(g)
rather than Eqθ (G), even though the latter is the notation we used in previous work on PC.
of the form of (P5). However the FG (qθ) of each problem (P5)i is the ‘qp’ KL distance,
KL(qθ || pβi), where βi is the value of the ‘barrier parameter’ specifying problem (P5)i. In
other words, up to irrelevant additive constants, each (P5)i is the problem of finding the θ
that minimizes
FG,βi(qθ) = Eqθ,G(g)− βi
−1S(qθ)
where S(.) is conventional Shannon entropy6. In this case the barrier function method
directs us to iterate the following process: Solve for the qθ that minimizes KL(qθ || pβi),
and then update βi. At the end of this process we will have a local solution to (P5).
In the case where X is a Cartesian product, we often use distributions parametrized as
a product distribution, qθ =
i qi(xi). Under this parametrization each problem (P5)
i can
be solved by gradient descent, where the gradient components of FG,βi(qθ) are given by
∂FG,βi(qθ)
∂qi(xi)
= Eqθ,G(g | xi) + βi
−1ln[qi(xi)] + λi
where the Lagrange parameters λi enforce normalization of each qi.
There are many better alternatives7 to simple gradient descent for minimizing each
FG,βi(qθ), involving Newton’s method, block relaxation, and related techniques (Wolpert
et al., 2006). In all such schemes investigated to date, we need to repeatedly evaluate
terms like Eqθ,G(g | xi). Sometimes that evaluation can be done closed form (Macready
and Wolpert, 2005, 2004). In blackbox optimization though, this is not possible.
Typically, when we cannot evaluate the terms Eqθ,G(g | xi) in closed-form we use
MC to estimate them. Since we have a product distribution, we can generate samples
of the joint distribution qθ(x) by sampling each of the marginals qi(xi) separately. One
can use those sample x’s as queries to the oracle. Then, by appropriately averaging the
oracle’s responses to those queries, one can estimate each term Eqθ,G(g | xi) (Wolpert and
Bieniawski, 2004). The product factorization implies that our iterative procedure can be
performed in a completely decentralized manner, with a separate program controlling each
component xi, and communicating only with the oracle8.
In this scheme, once qθ is modified, samples of the oracle that were generated from
preceding qθ’s can no longer be directly used to estimate the terms Eqθ,G(g | xi). However,
there are several ‘data-aging’ heuristics one can employ to reuse such old data by down-
weighting it.
In all these schemes, while we ultimately use Monte Carlo in the PC, it is delayed as
long as possible in the course of solving (P5). This is the basis for calling this variant of
PC ‘delayed sampling’.
3.3 Advantages of PC
The PC transformation can substantially alter the optimization landscape. For a noise-
free oracle G(x), (P4) reduces to the problem of finding the x that minimizes G(x). In
contrast, (P5) is the problem of finding the θ that minimizes FG (θ). The characteristics
6. This qp distance is just the free energy of qθ for Hamiltonian function G and inverse temperature βi.
This gives a novel derivation of the physics injunction to minimize the free energy of a system.
7. One of these alternatives can be cast as a corrected version of the replicator dynamics of evolution-
ary game theory (Wolpert, 2004b). This may have interesting implications for GAs, which presume
evolutionary processes.
8. Each such program may be thought of as an ‘agent’ who updates his probability distribution, and this
‘collective’ of agents performs optimization in a decentralized manner. This led to the name ‘Probability
Collective’.
of the problems of minimizing G(x) and minimizing FG (θ) can be vastly different. For
example, suppose qθ is log-concave in its parameters, and FG is pq KL distance. In this
case, regardless of the function G(x), FG (θ) is a convex function of θ, over Q. So the
PC transformation converts a problem with potentially many local minima into a problem
with none. See Wolpert et al. (2006) for a discussion of the geometry of the surface
FG (θ) : θ → R.
Since it works directly on distributions, PC can handle arbitrary data types. X can be
categorical, real-valued, integer-valued, or a mixture of all of these, but in each case, the
distribution over X is parametrized by a vector of real numbers. This means that all such
problems ‘look the same’ to much of the mathematics of PC. Moreover, PC can exploit
extremely well-understood techniques (like gradient descent) for optimization of continuous
functions of real-valued vectors, and apply them to problems in these arbitrary spaces.
Optimizing over distributions can give sensitivity information: The distribution qθ pro-
duced in PC will typically be tightly peaked along certain directions, while being relatively
flat along other directions. This tells us the relative importance of getting the value of x
along those different directions precisely correct.
We can set the initial distribution for PC to be a sum of broad peaks, each centered on a
solution produced by some other optimization algorithm. Then, as that initial distribution
gets updated in the PC algorithm, the set of solutions provided by those other optimization
algorithms are in essence combined, to produce a solution that should be superior to any
of them individually.
Yet another advantage to optimizing a distribution is that a distribution can easily
provide multiple solutions to the optimization problem, potentially far apart in X. Those
solutions can then be compared by the analyst in a qualitative fashion.
As discussed later, there are other advantages that accrue specifically if one uses the
immediate-sampling variant of PC. These include the ability to reuse all old data, the ability
to exploit prior knowledge concerning the oracle, and the ability to leverage PL techniques.
See Wolpert et al. (2006) for a discussion of other advantages of PC, in particular in the
context of distributed control.
3.4 Relation to Other Work
PC is related to several other optimization techniques. Consider, for instance, Response
Surface Models (RSM)s Jones et al. (1998). In these techniques, one uses Design of Ex-
periments (DOE) to evaluate the objective function at a set of points. Then, a low-order
parametric function, often a quadratic, is fitted to these function values. Optimization of
this ‘response surface’ or ‘surrogate model’ is considered trivial compared to the original
optimization. The result of this surrogate optimization is then used to get more samples
of the true objective at a different set of points. This procedure is then iterated using
some heuristics, often in conjunction with trust-region methods to ensure validity of the
low-order approximation. We note the similarities with PC in Table 2.
As another example, some variants of PC exploit MC techniques as discussed above,
and thus stochastically generate populations of samples. In their use of random populations
these variants of PC are similar to simulated annealing (Kirkpatrick et al., 1983), and even
more so to techniques like EDA’s Larraaga and Lozano (2001); De Bonet et al. (1997);
Lozano et al. (2005) and the CE method (Rubinstein and Kroese, 2004). However, these
other approaches do not explicitly pursue the optimization of the underlying distribution
qθ, as in (P5). Accordingly, those approaches cannot exploit situations in which (P5) can be
solved without using a stochastically generated population (Macready and Wolpert, 2005,
2004). See Macready and Wolpert (2005) for a more extensive discussion of the relation of
PC to other techniques.
RSM PC
Fit parametric function to Fit parametric distribution to
objective function values target distribution
Heuristics to grow trust region Cross-validation for regularization
DOE for sample points Random sampling for sample points
Axis alignment of stencil matters Parametrization can address axis alignment
Surrogate minimization not always easy Implicit, probabilistic ‘minimization’ of surrogate
Table 2: Relation to RSM.
4. Immediate sampling
This section introduces a new PC technique called immediate sampling, and cursorily
compares it to delayed sampling. As we have just described, in delayed sampling, we use
algebra for as long as possible in our solution of (P5). When closed-form expressions can
no longer be evaluated, we resort to MC techniques. In immediate sampling, we form
an MC sample immediately, rather than delaying it as long as possible. That sample gives
us an approximation to our objective, FG (θ) for all θ ∈ Q. We then search for the θ that
optimizes that sample-based approximate objective.
4.1 The General Immediate-sampling Algorithm
We begin with an illustrative example. Consider an integral transform FG (qθ), and use
importance sampling to rewrite it as∫
dx h1(x)
dg P (g | x,G )F (g, qθ(x))
h1(x)
dxdg h1(x)P (g | x,G )
F (g, qθ(x))
h1(x)
, (12)
dx h1(x)rP (g|x,G ),h1(θ). (13)
where we call h1(x) the sampling distribution. Note that the r in Eq. 13 differs from
the one defined in Eq. 10, as indicated by the extra subscript. This new r is used in the
next section.
We form a sample set of N pairs D1 ≡ {xi, gi} by IID sampling the distribution
h1(x)P (g | x,G ) in the integrand of Eq. 12. That sampling is the ‘immediate’ Monte Carlo
process. D1 is equivalent to a set of N sample functions
rih1(x
i, θ) ,
F (gi, qθ(xi))
h1(xi)
: i = 1, . . . N.
In the simplest version of immediate sampling, we would now use the functions ri
(xi, θ),
together with our prior knowledge (if any), to estimate the θ that minimizes FG (qθ). As
an example, not using any prior knowledge, we could estimate FG (qθ) for any θ as∫
dx h1(x)rP (g|x,G ),h1(θ) ≈
(xi, θ)
. (14)
This estimate is both an unbiased estimate of FG (qθ) and the maximum likelihood estimate
of FG (qθ). Moreover, it has these attributes for all θ. (This is the advantage of estimating
FG (qθ) using importance sampling with a proposal distribution h that doesn’t vary with
θ. ) Accordingly, to estimate the θ that minimizes FG (qθ) we could simply search for the
θ that minimizes
(xi, θ).9
Once again, even though the average in Eq. 14 is an unbiased estimate of FG (qθ) for any
fixed θ, its minimizer is not an unbiased estimate of minθFG (qθ). This is because searching
for the minimizing θ introduces bias. Therefore, one should use some other technique than
directly minimizing the righthand side of Eq. 14 to estimate argminθFG (qθ).
4.2 Immediate Sampling with Multiple Sample Sets
In general, we will not end the algorithm after forming a single sample set D1. Instead we
will use a map η that takes D1 to a new sampling distribution, h2. We then generate new
(x, g) pairs using h2, giving us a new sample set D2. We then iterate this process until we
decide to end the algorithm, at which point we use all our samples sets together to estimate
argminθFG (qθ).
To illustrate this we first present an example of a θ-estimation procedure we could run
at the end of the immediate sampling algorithm. This example is just the extension of the
maximum likelihood θ-estimation procedure introduced above to accommodate multiple
sample sets. Let N be the total number of samples, drawn from M sample sets, with
Nj samples in the j’th sample set. Let hj be sample distribution for the j’th sample set,
and ri,j the sample function value for the i’th element of the j’th sample set. Also define
Dj ≡ {xi,j , gi,j : i = 1, . . . Nj}. Then
i=1 r
(xi,j , θ)/Nj is an unbiased estimate of
FG (qθ) for any sample set j. Accordingly, any weighted average of these estimates is an
unbiased estimate of FG (qθ):
FG (qθ) ≈
(xi,j , θ)
. (15)
Modulo unbiasedness concerns, we could then use the minimizer of Eq.15 as our estimate
of argminθFG (qθ).
Say we have fixed on some such θ-estimation procedure to run at the end of the algo-
rithm. The final step of each iteration of immediate sampling is to run η, the map taking the
samples generated so far to a new h. Ideally, we want to use the η that, when repeatedly run
during the algorithm, maximizes the expected accuracy of the final θ-estimation. However
even for a simple θ-estimation procedure, determining this optimal η can be quite difficult.
As discussed later, it is identical to the active learning problem in machine learning.
In this paper we adopt a two-step heuristic for setting η. In the first step, at the end
of each iteration, we estimate the optimal qθ based on all the sample sets generated so
far, using Eq. 15. In the second step, we complete η by setting the new h to the current
estimate of the optimal qθ. At that point, the new h is used to generate a new sample set,
and the process repeats.
4.3 Immediate Sampling with MCMC
For certain types of FG , it is possible to form samples using other sampling methods
like Markov Chain Monte Carlo (MCMC) (see Mackay, 2003; Bernardo and Smith, 2000;
9. Since qθ is normalized, so is
dx F (G(x), qθ(x)) =
dx F (G(x),qθ(x))R
dx qθ(x)
. In Eq. 14 we fix the denominator
integral to 1. In practice though, it may make sense to replace both of the integrals in this ratio with
importance sample estimates of them. That means dividing the sum in Eq. 15 by
q(xi)/h(xi) and
then finding the θ that optimizes that ratio of sums, rather than the θ that just optimizes the numerator
term (see Robert and Casella, 2004). For example, this can be helpful when one uses cross-validation to
set β, as described below.
Berger, 1985). For example, if FG (θ) is pq distance from the Boltzmann distribution pβ to
qθ, then we can use MCMC to form a sample set of pβ (not of qθ). We can then use that
sample set to form an unbiased estimate of FG (θ) for any θ. But if β were to change, these
old samples cannot be used directly. One would have to resort to additional techniques
like rejection sampling in order to reuse these samples. The advantage of using importance
sampling is that all previous samples can be reused by the simple expedient of modifying
their likelihood ratios. Therefore, in this paper, we only consider sampling distributions h
that can be sampled directly, without any need for techniques like MCMC.
4.4 Advantages of Immediate-Sampling PC
In contrast to delayed sampling, immediate sampling usually presents no difficulty with
reusing old data, as shown above; all (xi,j , gi,j) pairs can be used directly. Note that we
can also reuse data that was generated when F was different, for instance, data generated
under a differerent βi during a KL distance minimization procedure. As long as we store
hj(xi,j) in addition to gi,j and xi,j for every sample, we can always evaluate ri,j
(xi,j , θ) for
any F .
Indeed, we can even comment on optimal ways of reusing this old data. Since each
(xi,j , θ) is an unbiased estimate of the integral FG (θ), any weighted average of the
(xi,j , θ)’s is also an unbiased estimate. This can be exploited in the θ-estimation pro-
cedure. For instance, consider the estimator of Eq. 15. If we have good estimates of the
variances of the individual ri,j
(xi,j , θ), we can weight the terms ri,j
(xi,j , θ) to minimize the
variance of the associated weighted average estimator. Those weights are proportional to
the inverses of the variances (see Macready and Wolpert, 2005; Lepage, 1978, 1980). As
discussed in Sec. 2.3, the accuracy of the associated MCO algorithm could be expected to
improve under such weighting.
We can also shed light on how to go about gathering new data. As in the VEGAS
Algorithm (Lepage, 1978, 1980), one could incorporate bias-variance considerations into
the operator η that sets the next sampling distribution. To give an example, let Ξ be
the range of η, and fix θ. Given Ξ and θ, one can ask what proposal distribution h ∈ Ξ
would minimize the sample variance of the estimator in Eq. 14. Intuitively, this is akin to
asking how best to do active learning. In general, the answer to this question, the optimal
sampling distribution h(x), will be set by the function rP (g|x,G ),h(θ), viewed as mapping
X → R. Accordingly, for any fixed θ, one can use the MC samples generated so far to
estimate the x-dependence of rP (g|x,G ),h(θ), and thereby estimate the optimal h ∈ Ξ. One
then uses that estimate as the next sampling distribution h.
Another advantage of immediate sampling over delayed sampling is that the analysis in
delayed sampling relies crucially on the parametrization of the q’s; some such parametriza-
tions will permit the closed-form calculations of delayed sampling, and others will not. In
immediate sampling, this problem disappears.
4.5 Implications of the Identity Between MCO and PL
For the case where FG (θ) is an integral transform like Eq. 9, the PC optimization problem
(P5) becomes a special case of minimizing a parametrized integral, the problem (P2).
Formally, the equivalence is made by equating x with the parameter φ, g with w, and
g × P (g | x,G ) with U(w, φ). In particular, immediate sampling is a special case of MCO.
This identity means that we can exploit the extremely well-researched field of PL to improve
many aspects of immediate sampling. In particular:
• PL techniques like boosting (Schapire and Singer, 1999) and bagging (Breiman, 1994) can be
used in (re)using old samples before forming new ones.
• Variants of active learning10 can be used to set and update h. Some aspect of this are discussed
in Sec. 6 below.
• Cross-validation is directly applicable in many ways: In our context, the curse of dimen-
sionality arises if Q is very large. We can address this the conventional PL way, by adding
a regularization function of qθ to the objective function. The parameters controlling this
regularization can be updated dynamically, as new data is generated, using cross-validation
To use cross-validation this way, one forms multiple partitions of the current data. For each
such partition, one calculates the optimal qθ for the training subset of that partition. One
then examines error on the validation subset of that partition. More precisely, one calculates
the unregularized objective value on the held-out data.
• More generally, we can use cross-validation to dynamically update any parameters of the
immediate sampling algorithm. For example, we can update the ‘temperature’ parameter β
of the Boltzmann distribution, arising in both qp and pq KL distance, this way.
Note that doing this does not involve making more calls to the oracle.
• We can also use cross-validation to choose the best model class (parametrization) for qθ, among
several candidates.
• As an alternative to all these uses of cross-validation, one can use stacking to dynamically
combine differrent temperatures, different parametrized density functions, and so on.
• One may also be able to apply kernel methods to do the density estimation (see Macready,
2005).
5. Experiments
In this section, we demonstrate the application of PL and immediate-sampling PC tech-
niques to the unconstrained optimization of continuous functions, both deterministic and
nondeterministic. We first describe our choice of FG , in this case pq KL distance. Next,
as an illustrative example, we apply immediate sampling to the simplest of optimization
problems, where the objective is a 2-D quadratic. Subsequently, we apply it to determinis-
tic and stochastic versions of two well-known unconstrained optimization benchmarks, the
Rosenbrock function and the Woods function.
We highlight the use of PL techniques to enhance optimizer performance on these
benchmark problems. In particular, we show that cross-validation for regularization yields
a performance improvement of an order of magnitude. We then show that cross-validation
for model-selection results in improved performance, especially in the early stages of the
algorithm. We also show that bagging can yield significant improvements in performance.
5.1 Minimizing pq KL Distance
Recall that the integral form of pq KL distance is
KL(p‖q) =
dx p(x) ln
It is easy to show that when there are no restrictions on q being a parametrized density,
pq KL distance is minimized if p = q. However, owing to sampling considerations, we
10. Active learning in the precise machine learning sense uses current data to decide on a new query x to
feed to the oracle. We use the term more loosely here, to refer to any scheme for using current data to
dynamically modify a process for generating for future queries.
usually choose q to be some parametric distribution qθ. In this case, we want to find the
parameter vector θ that minimizes KL(p‖qθ). Since the target distribution p is derived
purely from G and is independent of qθ, minimizing pq KL distance is equivalent to the
following cross-entropy minimization problem.
minimize −
dx p(x) ln (q(x)) ,
subject to
dx q(x) = 1,
q(x) ≥ 0 ∀x.
5.1.1 Gaussian Densities
If q is log-concave in its parameters θ, the minimization problem (16) is a convex opti-
mization problem. In particular, consider the case where X = Rn, and qθ is a multivariate
Gaussian density, with mean µ and covariance Σ, parametrized as follows,
qµ,Σ(x) =
(2π)n/2|Σ|1/2
(x− µ)TΣ−1(x− µ)
then the optimal parameters are given by matching first and second moments of p and qθ.
dxx p(x),
dx (x− µ?)(x− µ?)T p(x).
It is easy to generalize this to the case where X ⊂ Rn, by making a suitable modification
to the definition of p. This is described in Sec. 5.2.1.
5.1.2 Immediate Sampling with a Single Gaussian
Using importance sampling, we can convert the cross-entropy integral in Eq. 16 to a sum
over data points, as follows.
p(xi)
h(xi)
where D is the data set {(xi, gi)}, i = 1, . . . , N . This sets up the minimization problem for
immediate sampling for pq KL distance.
minimize −
p(xi)
h(xi)
. (17)
Denote the likelihood ratios by si = p(xi)/h(xi). Differentiating Eq. 17 with respect to the
parameters µ and Σ−1 and setting them to zero yields11
i(xi − µ?)(xi − µ?)T∑
11. Remarks:
1. As expected, these formulæ, in the infinite-data limit, are identical to the moment-matching results
for the full-blown integral case.
2. The formulæ resemble those for MAP density estimation, often used in supervised learning to find
the MAP parameters of a distribution from a set of samples. The difference in this case is that each
sample point is weighted by the likelihood ratio si, and is equivalent to ‘converting’ samples from h
to samples from p.
5.1.3 Mixture Models
The single Gaussian is a fairly restrictive class of models. Mixture models can significantly
improve flexibility, but at the cost of convexity of the KL distance minimization problem.
However, a plethora of techniques from supervized learning, in particular the Expectation
Maximization (EM) algorithm, can be applied with minor modifications.
Suppose qθ is a mixture of M Gaussians, that is, θ = (µ,Σ, φ) where φ is the mixing
p.m.f, we can view the problem as one where a hidden variable z decides which mixture
component each sample is drawn from. We then have the optimization problem
minimize −
p(xi)
h(xi)
i, zi)
Following the standard EM procedure, we multiply and divide the quantity inside the
logarithm by some Qi(zi), where Qi is a distribution over the possible values of zi. As
before, let si be the likelihood ratio of the i’th sample.
minimize −
si ln
qθ(xi, zi)
Qi(zi)
Then using Jensen’s inequality, we can take Qi outside the logarithm to get a lower bound.
To make this lower bound tight, choose Qi(zi) to be the constant p(zi|xi). Finally, differ-
entiating with respect to µj ,Σ
j and φj gives us the EM-like algorithm:
E-step: For each i, set Qi(zi) = p(zi|xi),
that is, wij = qµ,Σ,φ(z
i = j|xi), j = 1, . . . ,M.
M-step: Set µj =
i xi∑
i (xi − µj)(xi − µj)T∑
Since this is a nonconvex problem, one typically runs the algorithm multiple times with
random initializations of the parameters.
5.2 Implementation Details
In this section we describe the implementation details of an iterative immediate-sampling
PC algorithm that uses the Gaussian mixture models described in the previous section to
minimize pq KL distance to a Boltzmann target parametrized by β. We also describe the
modification of a variety of techniques from parmetric learning that significantly improve
performance of this algorithm. An overview of the procedure is presented in Algorithm 1.
5.2.1 Example: Quadratic G(x)
Consider the 2-D box X = {x ∈ R2 | ‖x‖∞ < 1}. Consider a simple quadratic on X,
GQ(x) = x
1 + x
2 + x1x2, x ∈ X.
The surface and contours of this simple quadratic on X are shown in Fig. 1. Also shown
are the corresponding Boltzmann target distributions pβ on X, for β = 2, 10 and 50. As
Algorithm 1 Overview of pq minimization using Gaussian mixtures
1: Draw uniform random samples on X
2: Initialize regularization parameter β
3: Compute G(x) values for those samples
4: repeat
5: Find a mixture distribution qθ to minimize sampled pq KL distance
6: Sample from qθ
7: Compute G(x) for those samples
8: Update β
9: until Termination
10: Sample final qθ to get solution(s).
can be seen, as β increases, pβ places increasing probability mass near the optimum of
G(x), leading to progressively lower EpβG(x). Also note that since G(x) is a quadratic,
pβ(x) ∝ exp(−βG(x)) is a Gaussian, restricted to X and renormalized. We now ‘fit’ a
Gaussian density qθ to the Boltzmann pβ by minimizing KL(pβ‖qθ), for a sequence of
increasing values of β. Note that qθ is a distribution over R2, and GQ is not defined
everywhere in R2. Therefore, we extend the definition of GQ to all of R2 as follows.
GQ(x) =
x21 + x
2 + x1x2, x ∈ X.
∞ otherwise.
Now pβ = 0 for all x /∈ X, and the integral for KL distance can be reduced to an integral
over X. This means that samples outside X are not considered in our computations.
5.2.2 Constant β
Figure 1: Quadratic G(x)
and associated
Gaussian targets
First, we fix β = 5, and run a few iterations of the PC
algorithm. To start with, we draw Nj = 30 samples from
the uniform distribution on X. The best-fit Gaussian is
computed using the immediate sampling procedure out-
lined in the preceding section. At each successive itera-
tion, Nj = 30 more samples are drawn from the current qθ
and the algorithm proceeds. A total of 6 such iterations
are performed. The 90% confidence ellipsoids correspond-
ing to pβ (heavy line) and the iterates of qθ (thin line) are
shown in Fig. 2. Also shown are the corresponding values
of EqθG(x), computed using the sample mean of GQ(x)
for 1000 samples of x drawn from each qθ, and KL(pβ‖qθ),
computed as the sample mean of ln(pβ(x)/qθ(x)) for 1000
samples of x drawn according to pβ .
5.2.3 Varying β
Next, we change β between iterations, in the ‘update β’
step shown in algorithm(1). With the same algorithm pa-
rameters, we start with β = 10, and at each iteration, we
use a multiplicative update rule β ← kββ, for some constant kβ > 1, in this case, 1.5. As
the algorithm progresses, the increasing β causes the target density pβ to place increasing
probability mass on regions with low G(x), as shown in Fig. 1. Since the distributions qθ
(a) Constant β: Confidence ellipsoids (b) Constant β: KL distance and expected G
(c) Varying β: Confidence ellipsoids (d) Varying β: KL distance and expected G
Figure 2: PC iterations for quadratic G(x).
are best-fits to p, successive iterations will generate lower EqθG(x). The 90% confidence
ellipsoids and evolution of EqθG(x) and KL distance are shown in Fig. 2.
5.2.4 Cross-validation to Schedule β
In more complex problems, it may be difficult to find a good value for the β update ratio
kβ . However, we note that the objective KL(pβ‖qθ) can be viewed as a regularized version
of the original objective, Eqθ [G(x)]. Therefore, we use the standard PL technique of cross-
validation to pick the regularization parameter β from some set {β}. At each iteration,
we partition the data set D into training and test data sets DT and DV . Then, for each
β ∈ {β}, we find the optimal parameters θ?(β) using only the training data DT . Next, we
test the associated qθ?(β) on the test data DV using the following performance measure.
ĝ(θ) =
qθ(xi)G(xi)
h(xi)∑
qθ(xi)
h(xi)
, (18)
The objective ĝ(θ) is an estimate12 of the unregularized objective Eqθ [G(x)]. Finally, we
set β? = arg minβ∈{β} ĝ(θ?(β)), and compute θ?(β?) using all the data D. Note that the
whole cross-validation procedure is carried out without any more calls to the oracle G .
We demonstrate the functioning of cross-validation on the well-known Rosenbrock prob-
lem in two dimensions, given by
GR(x) = 100(x2 − x21)
2 + (1− x1)2,
over the region X = {x ∈ R2 | ‖x‖∞ < 4}. The optimum value of 0 is achieved at x = (1, 1).
The details of the cross-validation algorithm used are presented in Algorithm 2. For this
Algorithm 2 Cross-validation for β.
Initialize interval extension count extIter = 0, and maxExtIter and β0..
repeat
At β = β0, consider the interval ∆β = [k1β0, k2β0].
Choose {β} be a set of nβ equally-spaced points in ∆β.
Partition the data into K random disjoint subsets.
for each fold k, do
Training data is the union of all but the kth data partitions.
Test data is the kth partition.
for βi in {β}, do
Use training data to compute optimal parameters θ?(βi, DTk).
Use test data to compute held-out performance ĝ(θ?(βi, DVk)), from Eq. 18.
end for
end for
Compute average held-out performance, g(βi), of ĝ(θ?(βi, DVk)).
Fit a quadratic Q(β) in a least-squares sense to the data (βi, g(βi)).
if Q is convex then
Set optimum regularization parameter β? = arg minβ∈∆β Q(β).
Fit a line L(β) in a least-squares sense to the data (βi, g(βi)).
Choose β? = arg minβ∈∆β L(β).
end if
Increment extIter
Update β0 ← β?
until extIter > maxExtIter or Q is convex.
experiment, we choose
maxExtIter = 4, k1 = 0.5, k2 = 2,
Nj = 10, nβ = 5, K = 10.
The histories of EqG(x) and β are shown in Fig. 3. Also shown are plots of the fitted
Q(β) at iterations 8 and 15. As can be seen, the value of β sometimes decreases from one
12. The reason for dividing by the sum of q(xi)/h(xi) is as follows. If the training data is such that no
probability mass is placed on the test data, the numerator of bgqθ is 0, regardless of the parameters of
qθ. In order to avoid this peculiar problem, we divide by the sum of q(x
i)/h(xi), as desribed by Robert
and Casella (2004).
(a) EqG(x) history. (b) β history.
(c) Fit Q(β), iteration 8. (d) Fit Q(β), iteration 15.
Figure 3: Cross-validation for β: 2-D Rosenbrock G(x).
iteration to the next, which can never happen in any fixed multiplicative update scheme.
We now demonstrate the need for an automated regularization scheme, on another
well-known test problem in R4, the Woods problem, given by
Gwoods(x) = 100(x2 − x1)2 + (1− x1)2 + 90(x4 − x23)
2 + (1− x3)2
+10.1[(1− x2)2 + (1− x4)2] + 19.8(1− x2)(1− x4).
The optimum value of 0 is achieved at x = (1, 1, 1, 1). We run the PC algorithm 50 times
with cross-validation for regularization. For this experiment, we used a single Gaussian q,
and set
maxExtIter = 4, k1 = 0.5, k2 = 3,
Nj = 20, nβ = 5, K = 10.
From these results, we then attempt to find the best-fit multiplicative update rule for
β, only to find that the average β variation is not at all well-approximated by any fixed
update β ← kββ. This poor fit is shown in Fig. 4, where we show a least-squares fit to
both β and log(β). In the fit to log(β) the final β is off by over 100%, and in the fit to
(a) Least-squares fit to β: (b) Least-squares fit to log(β):
Figure 4: Best-fit β update rule.
β, the initial β is off by several orders of magnitude. We then compare the performance
of cross-validation to that of PC algorithm using the fixed update rule derived from the
best least-squares fit to log(β). From a comparison over 50 runs, we see that using this
best-fit update rule performs extremely poorly - cross-validation yields an improvement in
final EqθG(x) by over an order of magnitude, as shown in Fig. 5.
(a) log(EqG) history.
Figure 5: Cross-validation beats best-fit fixed β update: 4-D Woods G(x).
5.2.5 Bagging
While regularization is a method to decrease bias, bagging is a well-known variance-reducing
technique. Bagging is easily incorporated in our algorithm. Suppose, at some stage in the
algorithm, we have N samples (xi, gi), we resample our existing data set exactly N times
with replacement. This gives us a different set of data set D′, which also contains some
duplicates. We compute optimal parameters θ?(D′). We repeat this resampling process kb
times and uniformly average the resulting optimal densities qθ?(D′
), k = 1, . . . , kb.
We demonstrate this procedure, using the Rosenbrock function and a single Gaussian
qθ. In this experiment, we also demonstrate the ability of PC to handle non-deterministic
oracles by adding uniform random noise to every function evaluation, that is, (g | x,G) ∼
U [−0.25, 0.25]. For this experiment, Nj = 20, kb = 5. The β update is performed using the
same cross-validation algorithm described above. Fig. 6 shows the results of 50 runs of the
PC algorithm with and without bagging. We see that bagging finds better solutions, and
Figure 6: Bagging improves performance: Noisy 2-D Rosenbrock.
moreover, it reduces the variance between runs. Note that the way we use bagging, we are
only assured of improved variance for a single MC estimation at a given θ, and not over
the whole MCO process of searching over θ.
5.2.6 Cross-validation for Regularization and Model Selection
In many problems like the Rosenbrock, a single Gaussian is a poor fit to pβ for many values
of β. In these cases, we can use a mixture of Gaussians to obtain a better fit to pβ . We
now describe the use of cross-validation to pick the number of components in the mixture
model. We use an algorithm very similar to the one described for regularization. In these
experiments, we use a greedy algorithm to search over the joint space of β and models:
1. We first pick the regularization parameter β, using Algorithm 2.
2. For that β, we use Algorithm 3 to pick the number of mixture components.
For this experiment, the details are the same as the preceding section, but without bagging.
The set of models {{φ}} is the set of Gaussian mixtures with one, two or three mixing
components. Fig. 7 shows the variation of Eq(G) vs. iteration. The mixture model is much
quicker to yield lower expected G, because the Boltzmann at many values of β is better
approximated by a mixture of Gaussians. However, note that the mixture models performs
poorly towards the end of the run. The reason for this is as follows: No shape regularization
was used during the EM procedure. This means that the algorithm often samples from
nearly degenerate Gaussians. These ‘strange’ sample sets hurt the subsequent performance
of importance sampling, and hence of the associated MCO problem. This can be alleviated
by using some form of shape regularization in the EM algorithm.
Algorithm 3 Cross-validation for model selection.
Initialize set {{φ}} of model classes {φ} to search over.
Partition the data into K disjoint subsets.
for each fold k, do
Training data is all but the kth data partitions.
Test data is the kth data partition.
for {φi} in {{φ}} do
Compute the optimal parameter set θ?(DTk) ∈ {φi}
Compute held-out performance ĝ(θ?(DVk))
end for
Compute the sample held-out performance, g({φi}), from Eq. 18.
end for
Choose best model class {φ?} = arg minφi g({φi}).
Figure 7: Cross-validation for regularization and model-selection: 2-D penalty function
G(x).
6. Fit-based Monte Carlo
Thus far, we have not exploited the locations of the samples in constructing esimates. In
this section, we discuss the incorporation of sample locations to improve both MC and
6.1 Fit-based MC Estimation of Integrals
We first consider MC estimation of an integral, presented at the beginning of Sec. 2.3.
Recall from that discussion, that to accord with MCO notation, we write the integral to
be estimated as L (φ) =
dw U(w, φ) for some fixed φ. In this notation the sampling of v
provides a sample set {(wi, U(wi, φ)) : i = 1, . . . N). The associated sum L̂{(wi,U(wi,φ))} ≡
L̂ (φ) then serves as our estimate of the integral L ≡ L (φ).
In forming the estimate L̂{(wi,U(wi,φ))} we do not exploit the relationships between the
locations of the sample points and the associated values of the integrand. Indeed, since
those locations {wi} do not appear directly in that estimate, L̂{(wi,U(wi,φ))} is unchanged
even if those locations changed in such a way that the values {ri} stayed the same.
The idea behind fit-based (FB) Monte Carlo is to leverage the location data to replace
L̂{(wi,U(wi,φ))} with a more accurate estimate of L . The most straightforward FB MC
treats the sample pairs {(wi, U(wi, φ)) : i = 1, . . . N)} as a training set for a supervised-
learning algorithm. Running such an algorithm produces a fit Ũ(., φ) taking w’s into R.
This fit is an estimate of the actual oracle U(, φ), and this fit defines an estimate for the
full integral,
L̃{wi,U(wi,φ)} ≡
dw Ũ(w, φ) (19)
We will sometimes omit the subscript and just write L̃ (φ) or even just L̃ . In this most
straightforward version of FB MC, we use L̃ as our estimate of L rather than L̂ .
In some circumstances one can evaluate L̃ in closed form. A recent paper reviewing
some work on how to do this with Gaussian processes is given by Rasmussen and Gharamani
(2003). In other circumstances one can form low-order approximations to L , for example
using Laplace approximations (see Robert and Casella, 2004). Alternatively, conventional
deterministic grid-based approximation of the integral L can be cast as a degenerate
version of closed-form fit-based estimation13 of L .
More generally, one can form an approximation to the integral L̃ (φ) by MC sampling
of Ũ(., φ). Generating these fictitious samples of Ũ(., φ) does not incur the expense of
calling the actual oracle U(., φ). So, in this approach, we run MC twice. The first time,
we generate the factual samples {(wi, U(wi, φ)) : i = 1, . . . N)}. Given those samples, we
form the fit to them, Ũ(., φ). We then run a second MC process using the fictitious oracle
Ũ(., φ).
Note that in all of these approaches, the original sampling distribution v does not
directly arise, that is, the values {v(wi)} do not arise. In particular, if one were to
change those values without changing the factual sample locations {wi}, then the esti-
mate L̃{(wi,U(wi,φ))} would not change. Of course, a different v would result in a different
sample set, and thereby a different estimate, but given a sample set, the sampling distribu-
tion is immaterial. This is typically the case with FB MC estimators, and it contrasts with
the estimator L̂{(wi,U(wi,φ))}, which would change if v were changed without changing the
factual sample locations.
Note also that the factual samples underlying the fit Ũ(., φ) are exact samples of the
factual oracle, U(., φ). In contrast, since in general Ũ(., φ) 6= U(., φ), the fictitious samples
will be erroneous, if viewed as samples of U(., φ). Since we are ultimately concerned with an
integral of U(., φ), this suggests that the fictitious samples should be weighted less than the
factual samples. This might be the case even if one had infinitely many fictitious samples.
In fact, even if one could evaluate L̃ in closed form, it might make sense not to use it
directly as our final estimate of L . Instead, combining it with the importance-sampling
estimate L̂ might improve the estimate.
13. To see this, modify the MC process to be sampling without replacement. Choose the proposal distribution
v for this process to be a sum of delta functions. The centers of those delta functions give the points on
a regular grid of points in the space of allowed x’s. Have the number of samples equal the number of
such grid points. Finally, have our fit to the samples, Ũ(., φ), be a sum of step-wise constant functions,
going through the sample points. The closed form integral of that fit given by Eq. 19 is just the Reimann
approximation to the original integral,
dw U(w, φ).
6.2 Bayesian Fit-based MCO
We now extend the discussion to MCO by allowing φ to vary. As with the case of FB MC
estimation of an integral, the most straightforward version of FB MCO uses L̃ (φ) rather
than L̂ (φ) as our estimate of L (φ). This means that we use argminφ[L̃ (φ)] rather than
argminφ[L̂ (φ)] as our estimate of the φ optimizing L (φ).
L̂ (φ) is a sum, whereas L̃ (φ) is an integral. This means that different algorithms are
required to find the φ optimizing them. Indeed, optimizing L̃ (φ) is formally the same
type of problem as optimizing L (φ); both functions of φ are parametrized integrals over
w. So if needed, we can use PLMCO techniques to optimize L̃ (φ). Again, as with MC, the
integrand of L̃ (φ) is not the factual oracle. So, minimizing L̃ (φ) using PLMCO sampling
techniques will not require making additional calls to the factual oracle.
Consider a Bayesian approach to forming the fit. Our problem is to solve the MCO
problem (P2), given that the factual oracle U is not known, and we only can generate
samples of U . Adopting a fully Bayesian perspective, since U is not known, we must
treat it as a random variable. So we have a posterior distribution over all possible oracles,
reflecting all the data we have concerning the factual oracle. We then use that posterior to
try to solve (P2).
Say that in the usual way that our data contains the w’s and the associated functions
U(w, .) of a sample set that was generated by importance sampling U (see Sec. 2). More
generally, we may have additional data, for instance, the gradients of U at the sample points.
For simplicity though, we restrict attention to the case where the provided information is
only the sample set of functions, {wi, ri(.)}, together with v.
We will use D to refer to a sample set for MC or MCO, and for immediate sampling in
particular. So we write our posterior over oracles14 as P (Uc | D). Using this notation, the
goal in Bayesian FB MCO is to exploit P (Uc | D) to improve our estimate of the φ that
minimizes
dw U(w, φ).
How should we use P (Uc | D) to estimate the solution to (P2)? Bayesian decision
theory tells us to minimize posterior expected loss,
dUc P (Uc | D)L(φ,Uc). Given the
loss function of Eq. 4, that means we wish to solve
(P6): min
dUc P (Uc | D)L(φ,Uc) = min
dwcdUc P (Uc | D)Uc(wc, φ).
To avoid confusion, the variable of integration is written as wc, to distinguish it from w’s in
the integral
dw U(w, φ). The solution to (P6) is our best possible guess of the φ solving
problem (P2), given the sample set D. Finding that solution is a problem of minimizing a
parametrized integral.
Sometimes we may be able to solve (P6) in closed form, even when we cannot solve
(P2) in closed form. Performing the integral over Uc may simplify the remaining integral
over wc. More generally, we can address (P6) using MCO techniques, and in particular
using PLMCO.15
To solve (P6) with PLMCO one generates fictitious samples by sampling one distri-
bution over Uc’s and one over wc’s. This MC process does not involve calls to the actual
oracle U , but samples a new distribution over Uc’s, to generate counter-factual Uc’s, and
then samples those Uc’s.
14. Practically, when running a computer experiment, U is the actual oracle generating D according to a
likelihood P (D | U). On the other hand, the posterior P (Uc | D) reflects both that likelihood and a
prior P (Uc) assumed by the algorithm. So, Uc is a random variable, whereas U refers to the single true
factual oracle.
15. To see that explicitly, rewrite the integral in (P2) as
dz V (z, φ), and identify values of z with pairs
(wc, Uc), while taking V (z, φ) = V (wc, Uc, φ) = P (Uc | D)Uc(wc, φ).
6.3 Example: Fit-based Immediate Sampling
To illustrate the foregoing we consider the variant of MCO given by immediate sampling
with a noise-free oracle. In the simple version of MCO considered just above, the estimate
we make for φ has no effect on what points are chosen for any future calls we might make to
the oracle. For simplicity, we restrict attention to the analogous formulation of immediate
sampling. Using immediate sampling terminology, this means that we only consider the
issue of how best to estimate θ after the immediate sampling algorithm has exhausted all
its calls to the oracle. We do not consider the more general active learning issue, of how
best to estimate θ when this estimate will be affect further calls to the oracle. However see
Sec. 6.6 below.
Recall that in immediate sampling, identifying wc with xc and φ with θ, Uc(wc, φ)
becomes F (Gc(xc), θ). Given a sample set D of (x,G(x)) pairs generated from a noise-free
factual oracle, our Bayesian optimization problem in immediate sampling is to find the θ
that minimizes
E(FGc(θ) | D) = EP (Gc|D)[
dxcF (Gc(xc), qθ(xc))]
dxcdGc P (Gc | D)F (Gc(xc), qθ(xc))
, F̃D(θ). (20)
Contrast this with Eq. 11. In particular, the inner integral in Eq. 20 runs over fictitious
oracles Gc that are generated according to P (Gc | D), whereas in Eq. 11, G is the factual
oracle.
In some circumstances one can evaluate the integral in Eq. 20 algebraically, to give
a closed form function of θ. In other cases, we can algebraically evaluate an accurate
low-order approximation, to again give a closed form function of θ. For the rest of this
subsection, however, we consider the situation where neither of these possibilities hold.
To address this situation, we approximate the integral in Eq. 20 using importance
sampling. However, to do this, we may have to importance-sample over two domains. The
first sampling is over sample locations xc, using some sampling distribution hc. (As an
example, we can simply choose hc = h.) The second sampling is over possible oracles Gc,
using some sampling distribution H. More precisely, write
E(FGc(θ) | D) =
dxc dGc
hc(xc)H(Gc)
] P (Gc | D)
hc(xc)H(Gc)
F (Gc(xc), qθ(xc)). (21)
To approximate this integral generate NT locations, {xic}, by sampling hc. This gives us
NT integrals
TD,{xic}(θ) ,
hc(xic)
dGc H(Gc)
P (Gc | D)
H(Gc)
F (Gc(x
c), qθ(x
c)). (22)
Sometimes, these integrals also can be evaluated algebraically, giving closed form sample
functions of θ. As an example, suppose we sample oracles according to the posterior, that
is, take H(Gc) = P (Gc | D), so that
TD,{xic}(θ) =
hc(xic)
dGc P (Gc | D)F (Gc(xic), qθ(x
c)). (23)
Next, say that we have a Gaussian process prior over oracles, P (Gc) (Rasmussen and
Williams, 2006), with a Gaussian covariance kernel. For this choice, and for some F ’s,
we can compute TD,{xic}(θ) exactly for any θ, provided D is not too large. For example,
this is the case for most F ’s whose dependence on their first argument lies in the expo-
nential family.16 In other situations, while we cannot evaluate them exactly, the integrals
TD,{xic}(θ) can be accurately approximated algebraically. This again reduces them to closed
form sample functions of θ.
In either of these cases, there is no need to sample H; samples of hc suffice. More
generally, we may not be able to evaluate the NT functions TD,{xic}(θ) algebraically, and
also cannot form accurate low-order algebraic approximations to them. In this situation,
for each xic, we should generate one sample of Gc from H(Gc).
17 This provides us a total
of NT sample functions
TD,{xic,Gic}(θ) ,
P (Gic | D)F (Gic(xic), qθ(xic))
hc(xic)H(Gic)
. (24)
As an example, say we believe firmly that a particular posterior P (Gc | D) governs our
problem, and can sample that posterior. Then we can take H(Gc) to equal P (Gc | D).
Now Eq. 24 only requires that we have values of our sampled Gc’s at the {xic}, that is,
we only need to have values Gic(x
c), where G
c ∼ H(Gc). So we only need to sample the
NT separate one-dimensional distributions {P (Gc(xic) | D)}. In particular, we might be
able to use Gaussian Process techniques to generate those values. Alternatively, as a rough
approximation, one could simply fit a regression to the data in D, ω(x), and then add noise
to the vector of values {ω(xic)} to get {Gic(xic)}. If we wanted to have multiple Gc’s for each
xic, then we would simply generate more samples of each distribution {P (Gc(xic) | D)}.18
However we sample H, the resultant sample functions provide the estimate
FD,{xic,Gic}(θ) ,
TD,{xic,Gic}(θ)
≈ E(F̃D,{xic,Gic}(θ) | D) (25)
which we sometimes abbreviate as ˆ̃FD(θ). For the situation where we can evaluate the
integral over Gc algebraically (or at least approximate it that way), we instead define
FD(θ) ,
TD,{xic}(θ) (26)
and rely on the context to decide which of the definitions of ˆ̃FD(θ) is meant. So for either
case we can write F̃D(θ) ≈
FD(θ).19
16. Strictly speaking, since the oracle is noise-free, the likelihood P (D | Gc) is a delta function about
having D lie exactly on the function Gc(x). In practice, this may make the computation be ill-behaved
numerically. Typically such problems are addressed modeling the fictitious oracles as though the values
they returned had a small amount of Gaussian noise added.
17. One could have the number of samples of H(Gc) not match the number of samples of hc(x). To avoid
the associated notational overhead, here we just match up the two types of samples, one-to-one.
18. As an alternative, we could reverse the sampling order and sample P (Gc | D) first and hc second.
Practically, this would mean generating NT samples of hc, and then sampling the single NT -dimensional
distribution P (Gc(x
c), . . . Gc(x
c ) | D). (This contrasts to the case considered in the text in which H
is sampled second, so one instead samples NT separate one-dimensional distributions {P (Gc(xic) | D)}.)
If we do this using a ‘rough approximation’ based on a fit ω to D, the noise values added to the values
of the fit, {ω(xic)}, would have to be correlated with each other, since they reflect the same Gc.
19. As a practical issue, we may want to divide the sum in Eq. 26 by the empirical average
P (Gic|D)
Similarly, if we cannot evaluate the integral over Gc algebraically, we may want to modify the estimate
In both cases, under naive MCO, we search for the θ that minimizes ˆ̃FD(θ). We
then use that θ as our estimate for the solution to (P6). More generally, rather than use
naive MCO we can exploit our sample functions with PLMCO. For example, rather than
minimizing ˆ̃FD(θ), we could minimize a sum of {F̃D(θ) and a regularization penalty term.
However we arrive at our (estimated) optimal qθ, most simplistically, we can update h
to equal that new qθ. In a more sophisticated approach, we could set h from the sample
functions using active learning (see Sec. 6.6 below). Once we have that new h we can form
samples of it to generate new factual sample locations x. These in turn are fed to the
factual oracle G to augment our data set D. Then the process repeats.
Note that unlike with non-FB immediate sampling, with FB immediate sampling we
need to evaluate P (Gc | D) (or sample it, if we choose to have H(Gc) , P (Gc | D)).
This may be non-trivial. On the other hand, that very same distribution P (Gc | D) that
may cause difficulty also gives the major advantage of the fit-based approach; it allows any
insights we have into how to fit a curve Gc to the data points D to be exploited.
6.4 Exploiting FB Immediate Sampling
To illustrate how fitting might improve immediate sampling, consider the case where
FG(qθ) is qp KL distance. Say that G(x) is a high-dimensional convex paraboloid in-
side a hypercube, and zero outside of that hypercube. Suppose as well that we have a
single factual sampling distribution h, which is concentrated on one side of the paraboloid.
For example, if the peak of the paraboloid is at the origin, h might be a Gaussian (masked
by the hypercube) whose mean lies several sigmas away from the origin.
To start, consider importance sampling MC estimation of the integral FG(qθ) for one
particular θ, without any concern about choosing among θ’s. Say that the factual sample
D isn’t too large. Then it is likely that no elements of D are in regions where G reaches
its lowest values. For such a D, the associated factual estimate
F̂D(θ) ,
i, θ)
is larger than the actual value, FG(qθ) (cf. Eq. 14). So straightforward importance-
sampling integral estimation is likely to be badly off.20
Intuitively, the problem is that as far as the factual estimate F̂D(θ) is concerned, G
could just as well be a sum of delta functions centered at the x’s in D, with low associated
oracle sample values, as a paraboloid. If G were in fact such a sum, then F̂D would be
correct. However by looking at the (x,G(x)) pairs in D, all of which lie on the same
paraboloid, such an inference of G appears quite unreasonable. It makes sense to instead
infer that G is a paraboloid.
Fitting is a way to formalize (and exploit) such D-based insights. As an example,
consider using a Bayesian PL algorithm to do the fitting. Typical choices for the prior
P (Gc) used in PL would result in a posterior P (Gc | D) that would be far more tightly
concentrated about the actual G’s paraboloid shape than about the sum of delta functions.
Fitting would automatically reflect this, and thereby produce a better estimate of FG(θ)
than F̂D(θ).21
in Eq. 25 by dividing by
P (Gic|D)
H(Gic)hc(x
. Such divisions would accord with the analogous division we
do in our non-FB immediate sampling experiments.
20. Since importance sampling is unbiased, this means its variance is likely to be large.
21. It might be objected that in a different problem G actually would be the sum of delta functions, not the
paraboloid. In that case the FB estimate is the one that would be in error. However this possibility is
Now we aren’t directly concerned with the accuracy of our estimate FG(qθ) for any
single θ. We aren’t even concerned with the overall accuracy of that estimate for a set of
θ’s. Rather we are concerned with the accuracy of the ranking of the θ’s given by those
estimates. For example, consider naive MCO, under which we choose the θ minimizing
F̃D(θ). Even if all of our estimates (one for each θ) were far from the associated actual
values, if their signed errors were identical, the naive MCO would perform perfectly.
In other words, ultimately we are interested in correlations between errors of our esti-
mates of FG(qθ) for different θ’s. (See Sec. 2.3.) Nonetheless, we might expect that if we
tend to have large error in our estimates of FG(qθ) for the θ’s, then everything else being
equal, we would be likely to have large error in the associated estimate of an optimal θ.
In Sec. 4.5 we exploited the equivalence between PL and MCO to improve upon naive
MCO. However the parameter in MCO doesn’t specify a functional fit to a data set. Ac-
cordingly, the incorporation of PL into MCO considered in Sec. 4.5 doesn’t involve fitting
a function to D. This is why those PLMCO techniques don’t address the issue raised in
this example; fitting does that. So in full FB MCO, we may use PL in two separate parts
of the algorithm, both to form the fit to D, and then to use those fits to choose among the
6.5 Statistical Analysis of FB MC
Before analyzing expected performance of FB MCO, we start with the simpler case of FB
MC introduced at the beginning of this section. For simplicity we assume that the integral
L̃D can be calculated exactly for any D, so that no fictitious samples arise.
As discussed in Sec. 2.3, two important properties of an MC estimator of an integral
L (φ) =
dw U(w, φ) are the sample bias and the sample variance of that estimator.
Together, these give the expected loss of the estimator under a quadratic loss function,
conditioned on a fixed oracle U(., φ).
This is just as true for a Bayesian fitting algorithm as for any other. For quadratic loss,
for sample set D ≡ {wi, U(wi, φ)}, the Bayesian FB MC prediction for L is the posterior
mean,
L̃D =
dw′ dUc(., φ) Uc(w
′, φ)P (Uc(., φ) | D). (28)
Accordingly, the expected quadratic loss of Bayesian FBMC is∫
dD P (D | v, U(., φ))[L − L̃D]2 =∫
dw1 . . . dwN
v(wi)[
dw′ U(w′, φ) −
dw′ dUc(, φ) Uc(w
′, φ)P (Uc(., φ) | D)]2 (29)
where v is the proposal distribution that is IID sampled N times to generate the sample
In the usual way one can re-express this expected quadratic loss using a bias-variance
decomposition. Whereas a conventional importance sample estimator of
dw U(w, φ) is
unbiased, the Bayesian estimator is biased in general; typically∫
dD P (D | v, U(., φ)) L̃D 6= L . (30)
exactly what the prior P (Gc) addresses; if in fact you have reason to believe that a G that is a sum of
delta functions is a priori just as likely as a paraboloid G, then that should be reflected in P (Gc). Doing
so would in turn make the FB estimate more closely track the non-FB estimate.
This bias is a general characteristic of Bayesian estimators. Furthermore, for some functions
U(., φ), the Bayesian estimator will both be biased (unlike the factual sample estimator)
and have higher variance than the factual sample estimator. So for those U(., φ), the
Bayesian estimator has worse bias plus variance.
In conventional importance sampling estimation of an integral, the sampling distribu-
tion v is used twice. First it is used to form the sample set. Then, when the sample set
has been formed, v is used again, to set the denominator values in the ratios giving the
MC estimate of the integral (cf. Sec. 2.2). In contrast, Bayesian FB MC doesn’t care what
v is. P (Uc(., φ) | D) is independent of the values v(wi). As mentioned at the beginning of
this section, this is a typical feature of FB MC estimators.
This feature does not mean that the sampling distribution is immaterial in FB MC
however. Even though it does not arise in making the estimate, as Eq. 29 shows, v helps
determine what the expected loss will be. Indeed, in principle at least, Eq. 29 can be used
to guide the choice of the sampling distribution for Bayesian FB MC. It can even be used
this way dynamically, at a midpoint of the sampling process, when one already has some
samples of U(., φ). Such a procedure for using Eq. 29 to set v dynamically amounts to
what is called ‘active learning’ in the PL literature (see Freund et al., 1997; Dasgupta and
Kalai, 2005).
We now generalize the foregoing to the case of a non-quadratic loss function L. The
Bayesian estimator produces the estimate
L̃D , argminρ∈R[
dUc(., φ) P (Uc(., φ) | D)L[L̃D, LUc ]] (31)
Given that the factual oracle is U(., φ), the expected loss with that Bayesian estimator is
dw1 . . . dwN
v(wi)L[L̃{wi,U(wi,φ)},
dw′ U(w′, φ)]. (32)
The expected loss in Eq. 32 is an average over data with the oracle held fixed. This
contrasts with the analogous quantity typically considered in Bayesian analysis, which is
an average over oracles with the data held fixed. That quantity is the posterior expected
loss, ∫
dU(., φ)P (U(., φ) | D)L[L̃D,L U (φ)] (33)
In general, different U(., φ)’s will give different risks for the same estimator. So we
can adapt any measure concerning loss in which U(., φ) varies, to concern risk instead. In
particular, the posterior expected risk is∫
dU(., φ)P (U(., φ) | D) {L[L̃D,LU (φ)] − minρ∈R[L[ρ,LU (φ)]]}. (34)
Often the lower bound on loss is always 0, so that minρ∈R[L[ρ,LU (φ)]] = 0 ∀ U(., φ). In
this case posterior expected risk just equals posterior expected loss.
We can combine the non-Bayesian and Bayesian analyses, involving expected loss and
posterior expected loss respectively. To do this we consider the prior-averaged expected
loss, given by
dU(., φ) P (U(., φ))
dw1 . . . dwN
v(wi)L[L̃{(wi,U(wi,φ))},
dw′ U(w′, φ)]. (35)
where P (U(., φ)) is a prior distribution over oracles.
Note that the prior-averaged expected loss is an average over both oracles and sample
sets. It reflects the following experimental test of our FB MCO algorithm: Multiple times a
factual oracle U(., φ) is generated by sampling P (U(., φ)). For each such U(., φ), many times
a factual sample set D is generated by sampling the likelihood P (D | U(., φ), v). That D
is then used by the FMCO algorithm to calculate LD. In performing that calculation, the
algorithm assumes the same likelihood as was used to generate D, but its prior P (Uc(., φ))
may not be the same function of Uc(., φ) as P (U(., φ)) is of U(., φ). Then the loss between
LD and LU is calculated. The quantity in Eq. 35 is the average of that loss.
Say that P (U(., φ)) is the same function of U(., φ) as P (Uc(., φ)) is of Uc(., φ). Then
the Bayesian estimator is based on the actual prior. In this case, the Bayesian estimator
L̃D will minimize the prior-averaged expected loss of Eq. 35.22 In general though, there
is no reason to suppose that these two priors are the same. In the real world where those
priors differ, expected loss for a Bayesian estimator is given by an inner product between the
posterior used by that estimator, P (Uc(., φ) | D), and the true posterior, P (U(., φ) | D) (see
Wolpert, 1997, 1996).23
As before, since U(.φ) varies in the integrand of prior-averaged expected loss, we can
can adapt it to get a prior-averaged expected risk. This is given by
dU(., φ) P (U(., φ))
dw1 . . . dwN
v(wi) ×
{L[L̃{(wi,U(wi,φ))}, L (φ)] − minρ∈R[L[ρ, L (φ)]]}. (36)
As before, if the minimal loss is always 0, then prior-averaged expected risk just equals
prior-averaged expected loss.
Broadly speaking, in Bayesian approaches to Monte Carlo problems, the sampling
distribution that generated the samples is immaterial once one those samples have been
generated (see Rasmussen and Gharamani, 2003) and references therein). So what differ-
ence does the choice of a sampling distribution like v make to a Bayesian? The answer is
that v determines how likely it is that we will generate a D with a high posterior variance
of the quantity of interest. For example, say one wishes to form an importance sampling
estimate of L =
dx U(x) using sampling distribution v to generate sample set D. Then
if one changes v, one changes the likelihoods of the possible D. Moreover, each D has its
own posterior variance, Var(Lc | D). So what a good choice of v means is that a D with
poor Var(Lc | D) is unlikely to be formed, that is, that
dD P (D | v)Var(Lc | D) is low.
22. To see this, replace L̃D with some arbitrary function of D, f(D). Our task it to solve for the optimal
f . First interchange the integrals over data and over oracles in Eq. 35. Next consider the integrand of
the outer (data) integral,
dU(., φ) P (U(., φ))
)L[f(D),
, φ)]].
Since we are considering a noise-free oracle, we can write this asZ
dU(., φ) P (U(., φ))P (D | v, U(., φ))L[f(D),LU (φ)].
Since P (U(., φ))P (D | v, U(., φ)) ∝ P (U(., φ) | v,D) = P (U(., φ) | D), this integral is minimized by
setting f(D) = L̃D. QED.
23. It is in recognition of the fact that those functions might differ that we have been referring to ‘Bayesian’
rather than ‘Bayes-optimal’ estimators.
6.6 Statistical Analysis of FB MCO
We can extend the statistical analysis of FB MC to the case of FB MCO by allowing φ to
vary. The Bayesian choice of φ is the one that minimizes posterior expected loss,
φ̃D , argminφ[
dUc P (Uc | D)L(φ,Uc)]. (37)
Since P (Uc | v,D) = P (Uc | D), this estimator is independent of v. The same is true for
the posterior expected loss of this Bayesian estimator,∫
dU P (U | D)L(φ̃D, U). (38)
On the other hand, the expected loss associated with this estimator,∫
dw1 . . . dwN
v(wi)L(φ̃{wi,U(wi)}, U), (39)
explicitly depends on v. So does the prior-averaged expected loss,∫
dU P (U)
dw1 . . . dwN
v(wi)L(φ̃{wi,U(wi)}, U). (40)
Next, the posterior expected risk is∫
dU P (U | D){L(φ̃D, U) − minφ′ [L(φ′, U)]} (41)
where φ′ runs over the (implicit) set of all possible φ. In general minφ′ [L(φ′, U)] varies
with U . (For example, this is the case with the loss function LU (φ) of Eq. 4.) Accordingly,
unlike in Bayesian FB MC, typically in Bayesian FB MCO the posterior expected risk does
not equal the posterior expected loss.
Finally, the prior-averaged expected risk is∫
dUdw1 . . . dwN P (U)
v(wi){L(φ̃{wi,U(wi)}, U) − minφ′ [L(φ′, U)]}. (42)
Again, since minφ′ [L(φ′, U)] typically varies with U , in general this prior-averaged expected
risk does not equal the prior-averaged expected loss. However the estimator that minimizes
prior-averaged expected loss — φ̃D — is the same as the estimator that minimizes prior-
averaged expected risk.24
For any particular fitting algorithm, our equations tell us how performance of the asso-
ciated FB MCO depends on v and either P (U(., φ)) or the pair P (U(., φ)) and P (Uc(., φ)),
depending on which equation we consider. So if we fix those prior(s), our equations tell us,
formally, what the optimal v is.
One can consider estimating that optimal v at a mid-way point of the algorithm, based
on the algorithm’s behavior up to that point. One can then set v to that estimate for the
remainder of the algorithm.25 Doing this essentially amounts to a type of active learning.
24. This follows from the fact that the prior-averaged lowest possible risk, the term subtracted in Eq. 42, is
independent of the choice of the estimator.
25. Note though that if one intends to update v more than once, then strictly speaking the first update to
v should take into account the fact that the future update will occur. That means the equations above
for expected loss, prior-averaged expected loss, etc., no longer apply.
As with Bayesian FBMC, we can analyze the effects of having P (U) not be the same
function of U as P (Uc) is of Uc. Since PL and MCO are formally the same, such an
analysis applies to parametric machine learning in addition to FB MCO. In particular,
the analysis gives a Bayesian correction to the bias-variance decomposition of supervised
learning. This correction holds even if the fitting algorithm in the supervised learning
cannot be cast as Bayes-optimal for some assumed prior P (Uc). Intuitively speaking, the
correction means that the bias-variance decomposition gets replaced by a bias-variance-
covariance decomposition. That covariance is between the posterior distribution over target
functions on the one hand, and the posterior distribution over fits produced by the fitting
algorithm on the other (see Wolpert, 1997).
6.7 Combining FB and Non-FB Estimates in FB MCO
Return now to the example in Sec. 6.6, where the factual sample is formed by importance
sampling the factual oracle and we form a fictitious sample set using fictitious oracles.Then
using only D, our estimate of FG(θ) would be the factual estimate, F̂D(θ) =
k=1 r
Using only our fictitious samples would instead give us the estimate ˆ̃FD(θ).
On the one extreme, say we firmly believe that distribution we use for the posterior
P (Gc | D) is correct. (So in particular we firmly believe that the factual oracle G was
generated by sampling the prior P (Gc).) Then in the limit NT →∞, ∀θ our importance-
sample estimate of F̃D will be exactly correct. So Bayesian decision theory would direct
us to use the associated estimate ˆ̃FD(θ), and ignore F̂D(θ). At the other extreme, say
that NT = 1, while N , the number of factual samples, is quite large. In such a situation,
even if we believe our posterior is correct, it would clearly be wrong to use ˆ̃FD(θ) as our
estimate, ignoring F̂D(θ).
How should we combine the estimates in this latter situation? More generally, even
when we believe our posterior is correct, unless the number of fictitious samples is far greater
than the number of factual samples, we should combine the two associated estimates. How
best to do that? Does the fact that ˆ̃FD(θ) is estimated via importance sampling over a
much larger space than F̂D(θ) affect how we should combine them? More generally, say we
don’t presume that our P (Gc | D) is exactly correct; how should we combine the estimates
then?
One is tempted to invoke Bayesian reasoning to determine how best to combine the
two estimates. While that might be possible in certain situations, often determining the
optimal Bayesian combination would necessitate yet more Monte Carlo sampling of some
new integrals. It would be nice if some other approach could be used.
One potential such approach is stacking (Wolpert, 1992; Breiman, 1996; Smyth and
Wolpert, 1999). In this approach, one many times partitions the factual sample D into two
parts, a ‘training set’ D1, and a ‘validation set’ D2. We write the values of w and U in
D1 as {D1w(i)} and {D1U (i, φ)} respectively, and similarly for D
2. For each such partition
one would run both the non-FB MCO algorithm and the FB MCO algorithm on D1. That
generates the estimates φ̂v,U,D1 and
φD1 , respectively.
Those two φ’s give us two associated error values on the validation set,
U (j, φ̂v,U,D1)
U (j,
φD1), respectively. More generally, we can evaluate the error on the the val-
idation set of any φ, in addition to the errors of φ̂v,U,D1 and
φD1 . Moreover, we can do
this for the validation set of any of the partitions of D. Note, however, that only factual
samples are used for cross-validation.
This is what stacking exploits. In the most straightforward use of stacking, one searches
for a function mapping the φ’s produced by our two algorithms to a composite φ. The goal
is to find such a composite φ that will have as small validation set error (when averaged
over all partitions) as possible.
For example, if φ is a Euclidean vector, one could perform a regularized search for the
weighted sum of φ’s that gives minimal partition-averaged validation set error. Let the
weights produced by that search be bFB and bnon−FB . Then to find the final estimate for
φ, one would use those weights to sum the outputs of the algorithms when run on all of D:
bnon−FBφ̂v,U,D + bFBφ̃D.
7. Conclusion
In this paper we explored the relationship between Monte Carlo Optimization of a parametrized
integral, parametric machine learning, and ‘blackbox’ or ‘oracle’-based optimization. We
made four contributions.
First, we proved that MCO is identical to a broad class of parametric machine learning
problems. This should open a new application domain for previously investigated para-
metric machine learning techniques, to the problem of MCO.
To test the use of PL in MCO one needs an MCO problem domain. The one we used
was based on our second contribution, which was the introduction of immediate sampling.
Immediate sampling is a way to transform an arbitrary blackbox optimization problem into
an MCO problem. Accordingly, it provides us a way to test the use of PL to improve MCO,
but testing whether it can improve blackbox optimization.
In our third contribution we validated this way of improving blackbox optimization. In
particular, we demonstratied that cross-validation and bagging improve immediate sam-
pling.
Conventional Monte Carlo and MCO procedures ignore some features of the sample
data. In particular, they ignore the relationship between the sample point locations and
the associated values of the integrand; only the values of the integrand at those locations
are considered. We ended by presenting fit-based MCO, which is a way to exploit the
information in the sample locations.
There are many PL techniques that should be applicable to immediate sampling but
that are not experimentally tested in this paper. These include density estimation active
learning, stacking, kernel-based methods, boosting, etc. Current and future work involves
experimental tests of the ability of such techniques to improve MCO in general and imme-
diate sampling in particular.
Other future work is to conduct experimental investigations of the three techniques
that we presented in this paper but did not test. One of these is fit-based MCO (and
fit-based immediate sampling in particular). The other two are the techniques described in
the appendices: immediate sampling for constrained optimization problems, and immediate
sampling with elite objective functions.
There are also many potential application domains for immediate sampling PC for
blackbox optimization that we intend to explore. Some of these exploit the ability of such
PC to handle arbitrary (mixed) data types of x’s. In particular, one such data type is the
full trajectory of a system through a space; for optimizing a problem over such a space,
PC becomes a form of reinforcement learning.
A. Constrained Optimization
Under the PC transform we replace an optimization problem over X with one over Q. As
discussed at the beginning of Sec. 3.3, the characteristics of the transformed objective can
be very different from those of the original objective.
Similarly, characteristics of any constraints on X in the original problem can also change
significantly under this transformation. More precisely, say we add to (P4) equality and
inequality constraints restricting x ∈ X to a feasible region. Then to satisfy those X-
constraints we need to modify (P5) to ensure that the support of the solution qθ(.) is a
subset of the feasible region in X.
This appendix considers some ways of modifying PC to do this. For earlier work on
this topic in the context of delayed sampling, see Wolpert et al. (2006); Bieniawski and
Wolpert (2004); Bieniawski et al. (2004); Macready and Wolpert (2005).
A.1 Guaranteeing Constraints
Say we have a set of equality and inequality constraints over X. Indicate the feasible region
by a feasibility indicator function
Φ(x) =
1, x is feasible,
0, otherwise.
For simplicity, we assume that for any x, we can evaluate Φ(x) essentially ‘for free’.
The transformed version of this constrained optimization problem is
(P5c) : minimize FG (qθ),
subject to qθ(x)Φ(x) = 0.
We now present a parametrization for q that ensures that it has zero support over infeasible
x. First, let q̃ be any parametrized distribution overX, for instance, a mixture of Gaussians.
Then using Φ(x ∈ X) as a ‘masking funtion’ we parametrize qθ(x) as
qθ(x) ,
q̃θ(x)Φ(x)∫
dx′ q̃θ(x′)Φ(x′)
, q̃Φ,θ(x).
This qθ automatically meets the constraints; it places zero probability mass at infeasible
x’s. It transforms the constrained problem (P5)c into the unconstrained problem
(P5uc) : minimize FG (q̃Φ,θ).
Now consider the case where FG is an integral over X. Typically in this case we are
only concerned with the values of the associated integrand at feasible x’s. For example,
when Eqθ (G) is of interest, it’s usually because our ultimate goal is to find a feasible x
with as good a G(x) as possible. In this situation it makes no sense to choose between
two candidate qθ’s based on differences in (the G values at) the regions of infeasible x
that they emphasize. More formally, our choice between them should be independent of
their respective values of
dx [1 − Φ(x)]qθ(x)G(x). We can enforce this by replacing the
objective Eqθ (G) =
dx qθ(x)G(x) with
dx Φ(x)qθ(x)G(x). If we then use the barrier
function approach outlined above, our final objective becomes qp KL distance with the
integral restricted to feasible x’s.
Generalizing this, when we are not interested in behavior at infeasible x we can reduce
the optimization problem further from (P5uc), by restricting the integral to only run over
feasible x’s. More precisely, write the original problem (P5c) as the minimization of∫
dgP (g | x,G )]F (g, qθ(x)) ,
dx µ(x, qθ(x)),
subject to the constraints on the support of qθ. By using the q̃ construction we can replace
this constrained optimization problem with the unconstrained problem
(A1): argminq
dx Φ(x)µ[x, qθ(x)] = argminθ
dx Φ(x)µ[x, q̃Φ,θ(x)],
= argminθ
dx Φ(x)µ[x,
Φ(x)q̃(x)∫
dx′Φ(x′)q̃(x′)
As an example, say our original objective function is pq KL distance. Define ZβΦ ≡∫
dx pβ(x)Φ(x). Then our new optimization problem is to minimize over θ
KL(pβΦ || q̃Φ,θ) = KL(
|| q̃Φ)
Φ(x)pβ(x)
q̃θ(x)Φ(x)∫
dx′q̃θ(x′)Φ(x′)
Φ(x)pβ(x)
{ln[q̃θ(x)] + ln[Φ(x)]− ln[
dx′q̃θ(x
′)Φ(x′)]}.
The q̃θ minimizing this is the same as the one that maximizes∫
Φ(x)pβ(x)
ln[q̃θ(x)] − ln[
dx′ q̃θ(x
′)Φ(x′)]. (43)
We can estimate ZβΦ using MC techniques. We can then apply MCO to estimate the θ that
maximizes the integral difference26 in Eq. 43.
To generate a sample of sample qθ(x) = q̃θ(x)Φ(x) we can subsample27 q̃θ according to
Φ. In some cases though, this can be very inefficient (that is, one may get many rejections
before getting a feasible x). To deal with such cases, we can first run a density estimator
on the samples of feasible x’s we have so far, getting a distribution π. (Note that no extra
calls to the feasibility oracle are needed to do this.) Next write qθ(x) = π(x)[qθ(x)/π(x)].
This identity justifies the generation of samples of qθ by first sampling π(x) and then
subsampling according to qθ(x)/π(x) = q̃θ(x)Φ(x)/π(x).
In an obvious modification to the foregoing, we can replace the hard restriction that
supp(q) contain only feasible x’s, with a ‘soft’ constraint that q(x) ≤ κ ∀ infeasible x.
A similar alternative is to ‘soften’ Φ(x) by replacing it with κ for all infeasible x, for
some κ > 0. For either alternative we anneal κ down to 0, as usual, perhaps using cross-
validation.
26. Note that in general this difference of integrals will not be convex in q̃θ for product distributions, unlike
Φ || q̃θ). See the discussion at the end of Sec. 3.1 on product distributions and pq distance.
27. Say we want to sample a distribution A(x) ∝ B(x)C(x) where B is a distribution and C is non-negative
definite, with c some upper bound on C. To generate such a sample by ‘subsampling B according to C’
we first generate a random sample of B(.), getting x′. We then toss a coin with bias C(x′)/c. If that coin
comes up heads, we keep x′ as our sample of A. Otherwise we repeat the process (see Wolpert et al.,
2006; Robert and Casella, 2004).
A.2 Alternative FG
Since we’re maximizing our expression over q̃θ, the second, correcting integral in Eq. 43 will
tend to push q̃θ to have probability mass away from feasible regions. To understand this
intuitively, say that q̃θ is a Gaussian and that the feasible region is ‘spiky’, resembling a
multi-dimensional star-fish with a large central region and long, thin legs. For this situation,
if we over-concentrate on keeping most of q̃θ’s mass restricted to feasible x, our Gaussian
will be pushed away from any of the spikes of the feasible x’s, and concentrate on the
center. If the solution to our original optimization problem is in one of those spikes, such
over-concentration is a fatal flaw. The second integral in Eq. 43 corrects for this potential
problem.
More broadly, consider typical case behavior when one applies some particular con-
strained optimization algorithm to any of the problems in a particular class of optimization
problems. As a practical matter, there is a spectrum of such problem classes, indexed by
how difficult it is just to find feasible solutions on typical problems of the class. On the
one side of this spectrum are problem classes where it is exceedingly difficult to find such a
solution, e.g., high-dimensional satisfiability problems with a performance measure G su-
perimposed to compare potential solutions. On the other end are “simple” problem classes
where it is reasonable to expect to find a feasible solution. The ‘starfish’ optimization
problem is an example of a problem of the former type.28
For problems on the first side of the spectrum, where just getting a substantial amount
of probability mass into the feasible region is very difficult, we may want to leave out the
second integral in Eq. 43. In other words, we may want to minimize KL(pβΦ || q̃θ) rather
than KL(pβΦ || q̃Φ,θ). The reason to make this change is so that q̃θ won’t get pushed away
from the feasible region. (As an aside, another potential benefit of this change is that if we
make it, then for product distribution q̃θ, FG (.) is convex.)
Even if we do make this change, when we sample the resultant q̃θ we may not get a
feasible x. If this happens, a natural approach is to repeatedly sample q̃θ until we do get a
feasible x. However the resultant distribution of x’s is the same as that formed by sampling
q̃Φ,θ for the same θ. So under this ’natural approach’ we work to optimize a distribution
(q̃θ) different from the one we ultimately sample (q̃Φ,θ). This means that this approach
may not properly balance our two conflicting needs for q̃θ: that it have most of its support
in the feasible region, and that it be peaked about x’s with high pβ(x).
To illustrate this issue differently, take q̃θ to be normalized, and to avoid multiplying
and dividing by zero, modify Φ(x) to equal some very small non-zero value κ for infeasible x
(as discussed above). Then under this ‘compound procedure’, we ultimately sample q̃Φ,θ(.).
However we do not choose FG (q̃Φ,θ) = KL(p
Φ || q̃Φ,θ) as the function of θ that we want to
minimize. Instead we choose
FG (q̃Φ,θ) = KL(p
Φ || q̃Φ,θ) − ln[
q̃Φ,θ(x′)
Φ(x′)
]. (44)
A.3 Using Constraints for Unconstrained Optimization
Return now to unconstrained optimization problems. Say that we have reason to expect
that over a particular region R, the distribution pβ(x) has values approximately κ times as
small as its value over X \R. It would be nice to reflect this insight in our parametrization
of q, that is, to parametrize q in a way that makes it easy to match it to pβ(x) accurately.
We can do this using a binary-valued function Φ and the approaches presented above.
28. Note that since we are discussing typical-case behavior, computational complexity considerations do not
apply.
To illustrate this, define q̃Φ,θ as above and choose the objective function FG (θ) =
KL(pβ || q̃Φ,θ), where Φ(x) = κ over R, and equals 1 over X \R.29 Then working through
the algebra, the qθ that minimizes this objective is given by the q̃θ that minimizes
dxpβ(x)ln[q̃θ(x)] + ln[
dx′q̃θ(x
′)Φ(x′)] = KL(pβ || q̃θ) + ln[
dx′q̃θ(x
′)Φ(x′)]
= KL(pβ || q̃θ) +
dx′R κq̃θ(x
dx′X\R q̃θ(x
The logarithm on the right-hand side is a ‘correction’ to pq distance from pβ to q̃θ, a correc-
tion that pushes q̃θ away from regions where Φ(x) = 1 (assuming κ < 1). To use immediate
sampling with this parametrization scheme, once we find the q̃θ that minimizes the sum
of pq distance plus that correction term, we would set h to the distribution (proportional
to) q̃θ(x)Φ(x). So we would generate our new samples from q̃θ(x)Φ(x), for example by
subsampling.30
B. The Elite Objective Function
Not all PC objectives can be cast as an integral transform. Properly speaking, the choice
of objective should be set by how the final qθ will be used. For instance, the concept of
expected improvement suggested by Mockus et al. (1978), and used by Jones et al. (1998),
considers an objective (to be maximized) given by max(Gbc − G(x), 0), where Gbc is the
best of all the current samples, mini{G(xi)}. This means that at each step we will take
a single sample, and want to maximize the improvement. This is a simplification; even
though the next sample may yield any improvement, it may be informative, so that we get
a good sample ten steps later. A less simplistic objective is the following:
In blackbox optimization, no matter how many calls to the (factual) oracle we make,
we will ultimately choose the best x (as far as the associated G value is concerned) out of
all the ones that were fed to the oracle during the course of the entire run. Our true goal
in BO is to have the G associated with that best x be as small as possible. For a discussion
of distributions of extremal values, see Leadbetter et al. (1983); Resnick (1987).
Given that qθ varies over the run in a way that we do not know beforehand, how can
one approximate this goal as minimizing an objective function that is well-defined at all
points during the run? One way to do this is to assume that there is some integer N such
that, simultaneously,
1. It is likely that the best x will be one of the final N calls to the oracle during the run;
2. It is likely that qθ will not vary much during the generation of those final N samples.
Under (2) we can approximate the qθ’s that are used to generate the final N calls as all
being equal to some canonical qθ. Under (1), our goal then becomes finding the canonical
29. Note that we use pβ in this FG , not p
Φ, which is what we used for constrained optimization. This is
because our goal now is simply to find a qθ that matches p
β(x). There are no additional aspects to the
problem involving feasibility regions that have no a priori relation to G(x).
30. In practice, Φ(x) for this unconstrained case would not be provided by an oracle. Instead we would
typically have to estimate it. We could do that for example by using a regression to form a fit to samples
of pβ(x) and then use that regression to define the region R.
qθ that, when sampled N times, produces a set of x’s whose best element is as good as
possible. 31
In this appendix we make some cursory comments about this objective function, which
we call the elite objective function. We focus on the use of Bayesian FB techniques
with this objective. For a noise-free oracle the CDF for the elite objective is
CDF(k) , 1−
dx1 . . . dxN
[qθ(x
i)Θ(G(xi)− k)]. (46)
So the associated density function is
f(k) =
d CDF(k)
= Nqθ(k)[
dx qθ(x)Θ(G(x)− k)](N−1). (47)
The associated expectation value,
dk kf(k), is not linear in qθ.
Writing it out, the posterior expected best-of-K value returned by the oracle when
queries are generated by sampling qθ is
dx1 . . . dxK
dG P (G | D)
dg1 . . . dgK
P (gk | xk, G)mink{gk} (48)
We want the θ minimizing this. Say we knew the exact posterior P (G | D) and could
evaluate the associated integral in Eq. 48 closed-form. In this case there would be need
for the parametric machine learning techniques used in the text. In particular there would
be no need for regularization — an analogous role is played by the prior P (G) underlying
P (G | D).
When we cannot evaluate the integral in closed form we must approximate it. To
illustrate this, as in Sec. 6, for simplicity consider a single-valued oracle G. This reduces
Eq. 48 to
dx1 . . . dxK
dG P (G | D)mink{G(xk)}. (49)
(The analogous FB MCO equation for objective functions involving a single integral Eq. 20;
here the single ‘x’ in that equation is replaced with a set of K x’s sampled from qθ.) To
approximate this integral we draw NT sample-vectors of K x’s each, using a sampling
distribution hc(x) to do so. At the same time we draw NT fictitious oracles from some
sampling distribution H over oracles.
31. An obvious variant of this reasoning is to have N vary across the run of the entire algorithm, at any
iteration t being only the number of remaining calls to the oracle that we presume will be made. In
this variant, one would modify the elite objective function to only involve the N(t) remaining samples
whose G value is better than the best found by iteration t. For the case N = 1, this is analogous to the
expected improvement idea in Jones et al. (1998). Note that this variant objective function will change
during the run, which may cause stability problems.
To simplify notation, let ~x indicate such a K-tuple of x’s. (So for multidimensional X,
~x is actually a matrix.) Also write
hc(~x) ,
qθ(~x) ,
G(~x) , (G(x1), . . . , G(xK)). (50)
With this notation, the estimate based on fictitious samples introduced in Sec. 6 becomes32
P (Gi | D)
H(Gi)
mink=1,...K{Gi(xki )}. (51)
As discussed in Sec. 6, it is often good to set H(G) to be as close to P (G | D) as
possible. So for example if we assume a Gaussian process model, typically we can set
H(Gi) = P (Gi | D), and then directly sample H to get the values of one Gi at the K
separate points xji . Alternatively, we can first form a fit φ(x) to the data in D. Next,
for each of N samples ~xi, sample a colored (correlated) noise process over the K points
{~xi} to get K real numbers. Finally, add those K numbers to the corresponding values
{φ(~xji ) : j = 1, . . . ,K}. This gives our desired sample of {Gi(~xi)}.
To illustrate the foregoing, suppose K = 1, and that we have no regularization on qθ.
Then, in general, the sum in Eq. 51 is minimized by a qθ that is a delta function about
that data point x1i with the best associated value Gi(x
i )/hc(x
i ). However for K > 1, even
without regularization, the optimal qθ is not a delta function, in general.33 In addition
to the regularization-based argument in the text, this gives a more formal reason why the
optimal qθ should not be infinitely peaked.
When K > 1, the peakedness of qθ parallels the peakedness of another non-negative
function over x’s, namely P (G : G(x) is minimized at x | D). However, if we run a few iter-
ations of FB MCO with the elite objective, thenD grows, and so P (G : G(x) is minimized at x |
D) gets increasingly peaked over x’s. (Intuitively, the larger D is, the more confident we are
about G, and consequently the more confident we are about what regions of x’s minimize
G.) Accordingly, qθ gets increasingly peaked as the algorithm progresses.
Note that this happens even though there is no external annealing schedule. This
reflects the fact that the elite objective has no hyperparameter or regularization parameter
like the β that appears in both the pq and qp objective functions.
C. Gaussian Example for Risk Analysis
The following example illustrates the foregoing for the case of Gaussian π, where only
moments of π up to order 2 matter.
To illustrate the foregoing, consider the simple case where there are only two φ’s, φ1
and φ2. Suppose that U and X are such that π is a two-dimensional Gaussian. Write π’s
mean as µ. Say that one of π’s principal axes is parallel to the diagonal line, l1 = l2 (that
32. In practice there might be more efficient sampling procedures than Eq. 51. For example, one could form
NK samples of hc(x) and N samples of H(G), to get two sets, which one then subsamples many times,
to get pairs [~x,G(~x)].
33. This suboptimality of a delta function qθ is similar to the suboptimality of having all K pulls in a
multi-armed bandit problem be pulls of the same arm.
is, one of the eigenvectors of π’s covariance matrix is parallel to the diagonal, and one is
orthogonal to the diagonal). Write the standard deviation of π along that diagonal axis as
σA, and write the standard deviation along the other, orthogonal axis as σB .
Since π’s covariance matrix has identical diagonal entries, and since the trace of that
matrix is preserved under rotations, those entries are both 1
[σ2A + σ
B ]. Since the determi-
nant is preserved, and since σA is the variance parallel to the diagonal, this in turn means
that π’s (identical) off-diagonal entries are 1
[σ2A − σ
B ]. The probability that MCO will
choose φ1 is the integral of π over the half-plane where φ1 ≤ φ2:
Pr(L̂ (φ2) > L̂ (φ1)) = erf(
µ2 − µ1
). (52)
Next, define
∆L ≡ L (φ1)−L (φ2),
∆b ≡ [µ1 −L (φ1)] − [µ2 −L (φ2)]
= [µ1 − µ2]−∆L. (53)
So the difference in the value of the loss function between the two φ’s is ∆L, and the the
difference in the biases of the two estimators L̂ (φ1) and L̂ (φ2) is ∆b. Note also that the
variances of the two estimators are the same,
Var[L (φ1)] = Var[L (φ2)] =
σ2A + σ
. (54)
So if we shrink the variance of either of the estimators, then we shrink an upper bound on
For this case of a fixed set of φ’s, it is illuminating to consider the difference between
expected loss under a particular MCO algorithm and minimal expected loss over all φ’s,
that is, the risk of the MCO algorithm. Assuming ∆L < 0, it is given by
[Pr[L̂ (φ2) > L̂ (φ1)] − Θ[L (φ2)−L (φ1)]] × [L (φ1) − L (φ2)]
[erf(
µ2 − µ1
)−Θ(∆b+ µ2 − µ1)] × [µ1 − µ2 −∆b]. (55)
Say that ∆b = 0. Then Eq. 55 shows that so long as µ1 6= µ2, as σB → 0 risk goes to
its minimal possible value of zero. So everything else being equal, shrinking the variance
of either estimator reduces risk, essentially minimizing it. Alternatively, if we leave the
variances of the two estimators unchanged, but increase their covariance, 1
[σ2A−σ
B ], then
σA will increase, while σB must shrink. So again, the risk will get reduced. For the more
general, non-Gaussian case, the high order moments may also come into play.
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	Introduction
	Background on PL, MCO, Blackbox Optimization, and PC
	Roadmap of This Paper
	Notation
	MCO and PL
	Overview of PL
	Overview of MCO
	Statistical Analysis of MCO
	Review: MC Estimation
	From MC to MCO
	PL Equals MCO
	Review of PC
	Introduction to PC
	Review of Delayed Sampling
	Advantages of PC
	Relation to Other Work
	Immediate sampling
	The General Immediate-sampling Algorithm
	Immediate Sampling with Multiple Sample Sets
	Immediate Sampling with MCMC
	Advantages of Immediate-Sampling PC
	Implications of the Identity Between MCO and PL
	Experiments
	Minimizing pq KL Distance
	Gaussian Densities
	Immediate Sampling with a Single Gaussian
	Mixture Models
	Implementation Details
	Example: Quadratic G(x)
	Constant 
	Varying 
	Cross-validation to Schedule 
	Bagging
	Cross-validation for Regularization and Model Selection
	Fit-based Monte Carlo
	Fit-based MC Estimation of Integrals
	Bayesian Fit-based MCO
	Example: Fit-based Immediate Sampling
	Exploiting FB Immediate Sampling
	Statistical Analysis of FB MC
	Statistical Analysis of FB MCO
	Combining FB and Non-FB Estimates in FB MCO
	Conclusion
	Constrained Optimization
	Guaranteeing Constraints
	Alternative FG
	Using Constraints for Unconstrained Optimization
	The Elite Objective Function
	Gaussian Example for Risk Analysis
