The use of Markov chain Monte Carlo (MCMC) methods for Dirichlet process mixtures has made inference tractable Neal (2000).

these methods can exhibit slow convergence and their convergence can be tough to detect.


[1] proposed a trust region method for CMDP called constrained policy optimization (CPO), which can deal with high-dimensional policy classes such as neural networks and claim to maintain feasibility if started with a feasible solution.

we found in Section 5 that feasibility is rarely guaranteed during learning in practice, possibly due to errors in gradient and Hessian matrix estimation.


This makes our approach generic and applicable to any MRF problem.

as we establish a general framework, specific application features can be straightforwardly added in future work.


However, the proposed algorithm requires matrix (tensor) decompositions because of exact moment matching [24], which is hard to scale to higher order moment matching.

by giving up exact moment matching, MMD GAN can match high-order moments with kernel tricks.


It is also very efficient, because it uses a cascade of detectors and very simple image features.

the AdaBoost algorithm requires exact positions of objects to learn.


We leverage fast FFT-based algorithms for GP kernel operations for points on a regular grid [17][18][19].

these require a stationary kernel, which due to the station-specific geometry, ours is not.


For the second approach, this is due to the fact that small perturbations to b do not drastically change the condition number, as shown in [5].

this also means that we cannot expect a good condition number when perturbing b in the hard case.


Anytime A* does not control , and in this experiment it apparently performs a lot of computations that result in a large decrease of at the end.

it does reach the optimal solution first this way.


A key issue with the CSM approach is the computational complexity, where gradients cost O(NT C 3 ) (using approximations), and is infeasible with the larger number of electrodes in EEG data.

the proposed GP adapter requires only a single matrix inversion shared by most data points, which is ).


It is clear from Figure 4.a that LARS is informative about native features for most proteins.

we cannot rely wholly on its predictions.


When the ideal hypothesis performs poorly, we cannot expect to learn a good target classifier by minimizing source error.

2 for the kinds of tasks mentioned in Section 1, we expect λ to be small.


Strong theoretical and empirical results have been achieved by spectral clustering, which uses the singular value decomposition of the network followed by a clustering step on the eigenvectors to determine the latent clusters.

singular value decomposition can be expensive, particularly for (a) large graphs, when (b) many eigenvectors are desired.


On Market-1501 samples, the quality of the images generated by both methods decreases because of the more challenging setting.

the two-stage model is still able to generate better results than the one-stage method.


Thus, the results we present here can be directly applied to classification problems in Σ .

specializing our results to this particular setting may yield several improvements.


Identifying the best decision d ∈ D then amounts to identifying which equivalence class H d contains the realized vector of outcomes, which is an instance of ECD.

one common approach to this problem is to myopically pick tests maximizing the decision-theoretic value of information (VoI): The VoI of a test t is the expected reduction in the expected loss of the best decision due to the observation of x t . we can show there are instances in which such a policy pays Ω(n/ log(n)) times the optimal cost, even under a uniform prior on (h, θ) and with | supp(Θ)| = 2.


The stated upper bounds for k-MCV and k avg -MCV in Table 1 can be obtained directly from our k-AS and k avg -AS algorithms.

all the lower-bound arguments for k-AS and k avg -AS-which crucially rely on the samples being i.i.d.-break down for the two MCV problems, due to the history-awareness property explained in Section 1.1.


The greedy algorithm terminates in 5 seconds.

the integer programming solution takes 128 seconds using the Gurobi solver [8].


This contrasts with existing filtering approaches for deep dynamical models, which use inference models that do not explicitly account for prior predictions during inference.

using iterative inference models [32], which overcome this limitation, we develop a computationally efficient implementation of the variational filtering EM algorithm, which we refer to as amortized variational filtering (AVF).


In practical terms, the simple selections α(γ) = 1 and β(γ) = 0 will provably satisfy the monotone cell update property (see proof details in the supplementary).

for additional flexibility, α and β could be selected to implement various forms of momentum, ultimately leading to cell updates akin to the popular FISTA [4] or monotonic FISTA [3] algorithms.


kernel (2) such that the Gaussians in the kernel expansion (3) are highly localized.

highly localized Gaussians are not well suited for interpolation -the implicit surface decomposes into several components.


Further, for some models, their variance can be bounded (Fan et al., 2015).

reparameterization is not as generic as .


[2015] propose an adaptive algorithm based on the optimistic mirror descent [Rakhlin and Sridharan, 2013], such that the dynamic regret is given in terms of all the three regularities (P * T , F T , and G T ).

it relies on the assumption that the learner can calculate each regularity incrementally.


The tighter BP/OMP bound from [1,4,5] scales as O N −1/2 , although this latter bound is much more general in that it is independent of the magnitudes of the nonzero weights.

neither BP or OMP satisfy a comparable result; in both cases, simple 3D counter examples suffice to illustrate this point.


On the one hand, it needs to explore the space Y in the sense that a sufficient number of labels need to be obtained as the correct label for the true parameters (this is indeed used in our consistency proofs).

p(x) needs to be sufficiently sensitive close to the decision boundaries so that the true parameters can be inferred.


It is claimed in [26] that this leads to faster convergence.

our experimental results show that this heuristic leads to more accurate, but not faster, results.


Exhaustively searching the chain for variables of positive reduced cost by iterating over all settings of all edges would be as expensive as exact max-product message-passing.

our oracle search strategy is efficient because it prunes these away using precomputed bounds on the transitions.


This explains why it is possible to still distinguish left from right sides in most bilaterally-symmetric objects despite symmetries and motions.

this is not the case for other types of symmetries such as radial.


We can see from the results in Figure 3 that designing a good sequential decomposition for the SMC sampler is important.

the intuitive and fairly simple choice L-R does give very good results comparable to that of AIS.


there are no repeating elements, Note that k r is independent of the type of model to be fitted, thus it is applicable to generic statistical model fitting problems.

we concentrate on motion subspaces in this paper.


We can see that clearly the latent subspace representations discovered by the largemargin based MMH show a strong grouping pattern for the images belonging to the same category, while images from different categories tend to be separated from each other on the 2D embedding space.

the latent subspace representations discovered by the likelihood-based unsupervised DWH and supervised TWH do not show a clear grouping pattern, except for the first category.


A challenge with this approach is that marginalization of f will introduce dependencies across time for the state variables that lead to the loss of the Markovian structure of the state-process.

recently developed inference methods, combining sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) allow us to tackle this problem.


Therefore, multi-view latent space MNs do not increase the complexity on testing if our task depends solely on the latent representation (i.e., expectation of H), such as information retrieval [26], classification, clustering etc.

the complexity of parameter estimation and inferring the posterior distribution of each view (e.g., X) will be increased, depending on the structure on the view.


More focused methods can also be applied when more structure of the optimization problem is known [Martens and Grosse, 2015].

communities who focus on sparsity tend to favor very different approaches [Donoho, 2006, Bach et al., 2012.


Momentum angular information for each observed particle is also available to the network, but is not shown, as the one-dimensional projections have little information.

particles B and C, we could calculate the invariant mass of the short-lived intermediate state A in the process A → B + C, via: finite measurement resolution and escaping neutrinos (which are invisible to the detectors) make it impossible to calculate the intermediate state mass precisely.


5 Note that the optimal variational posterior of q(U) with respect to the lower bound can be computed in closed-form.

the computational complexity of the closed-form solution is O(N DM 2 X M 2 H ).


In order to increase the readability we state all results for the linear case.

our results generalise to the case where the x i and z j live in an RKHS spanned by some kernel.


Like other Markov random fields (MRFs), CCMRFs define a joint distribution over a collection of random variables and capture local dependencies through potential functions.

unlike many popular discrete MRFs which are defined over binary random variables, CCMRFs are defined over continuous random variables.


S14) shows that both methods often agree on the features that a neuron has learned to detect.

overall DGN-AM produces more realistic-looking color and texture, despite not requiring optimization to be seeded with averages of real images, thus improving our ability to learn what feature each hidden neuron has learned.


At each round of the game, adversary samples θ(t) ∈ [0, 1] n according to p, with E θ(t)∼p [θ(t)] = θ * p . The learner chooses x(t) ∈ X and gets reward r(x(t), θ(t)).

the learner might not get to know either θ(t) (as in a full information game) or r(x(t), θ(t)) (as in a bandit game).


Notice that it is reasonable to meet the lower bound Ω(n 3/2 ) even given features, since in an extreme case, X, Y could be random matrices and have no correlation to R, and thus the given information is as same as that in standard matrix completion.

in many applications, features will be far from random, and our result provides a theoretical insight to show that features can be useful even if they are imperfect.


As with the Metropolis-Hastings algorithm, we start at some point q 0 ∈ X .

we use an analogue of the laws of physics to tell us where to go for future steps.


The full conditional distribution p(θ uk,t |θ uk,t−1 , θ uk,t+1 ) is conjugate.

it is not possible to attain a positive correlation between θ uk,t and θ uk,t−1 since E[θ uk,t ] = 1/θ uk,t−1 .


With some extra work, one can check that if, for instance, γ = 1 and |µ − 1| ≤ cn −1/2 for some constant c, then the estimator σ has also a risk of the order of sn −1/2 .

the price to pay for being adaptive with respect to the noise level is the presence of Mβ * 1 in the bound on σ, which deteriorates the quality of estimation in the case of large signal-to-noise ratio.


Valid configurations have a joint probability with the factor a∈F exp(−y a ) while AZ configurations do not.

theorem 2 states that, if exp(w i ) 1, AZ configurations have negligible probability.


That common value of η must be small enough that all three algorithms converge; if we take η 1 M N 2 , the learning curves become We began by saying that, because of the credit assignment problem of choosing updates to many parameters based on a single reward signal, intuition is that a stochastic gradientfollowing approach should learn more slowly than a direct one.

for equal small η, the average error for all three algorithms converges at the same rate.


Of course it is generally difficult to get a constant 1 in the batch setting (see Arlot [2007]), and similarly this is a difficult task in our online setting too: If � is chosen to be small, then the error with respect to the optimal allocation is small.

since P n is expressed as an increasing function of �, this implies that the minimax bound on the loss for partition P increases also with �.


It also restricts the kinds of operations that can be batched.

our method identifies and aggregates computation graph nodes that can be executed in a batched fashion for a given graph.


A choice of small λ C translates to a small worst-case budget required by the mechanism.

it means that the market prices are sensitive in that the same update df t changes the prices much more quickly.


We found that standard forms of recurrence (vanilla RNNs and LSTMs) do not perform well within deep CNNs on the ImageNet task.

novel cells that incorporated two structural features, bypassing and gating, were able to boost task accuracy substantially.


With this kind of approach, it is possible to obtain more powerful descriptors than by application of hand-crafted methods.

the binary descriptors are trained using pair-wise learning methods, which substantially limits their applicability to new tasks.


Accordingly, a classification method that is based on both features has better chance to handle trials that are contaminated by one kind of those artifacts.

it might increase the risk of using non-CNS activity for classification which would not be conform with the BCI idea, [1].


Although the mixture distributions have good approximation properties and naturally lead to adaptivity to very high smoothness levels, they may fail to detect or characterize the local features.

by learning a partition of the sample space, the partition based approaches can provide an informative summary of the structure, and allow us to examine the density at different resolutions [14,21].


A related class of relatively tractable models are distributions P (x) = 1 Z exp(−g(x)) for some submodular function g; Djolonga and Krause [10] showed variational inference methods for those models.


Ill-conditioning was long considered their undoing.

careful analysis [7] has shown that poor conditioning is not the problem-rather, it is a deficiency in the search direction.


2) The choice of regularization parameters λ β and λ e can make explicitly: assuming w is a Gaussian random vector whose entries are N (0, σ 2 ) and the design matrix has unit-normed columns, it is clear that with high probability, X * w ∞ ≤ 2 σ 2 log p and w ∞ ≤ 2 σ 2 log n. Thus, it is sufficient to select λ β ≥ 4 γ σ 2 log p and λ e ≥ 4 σ 2 log n. 3) At the first glance, the parameter γ does not seem to have any meaningful interpretation and the γ = 1 seems to be the best selection due to the smallest estimation error it can produce.

this parameter actually control the sparsity level of the regression vector with respect to the fraction of corruption.


Observation 1: MC-b and MC-1 are the best for feature imputation, as Table 2 shows.

the imputations are not perfect, because in these particular parameter settings the ratio between the number of observed entries over the degrees of freedom needed to describe the feature matrix (i.e., r(d + n − r)) is below the necessary condition for perfect matrix completion [2], and because there is some feature noise.


the sparse representation problem: Some typical choices of p include p = ∞ and p = 2 [25].

solving (P0) requires enumerating all subsets of the set {1, 2, • • • , d}, which is essentially a combinatorial optimization problem and is in general NP-hard [5].


Known challenges include the definition of locality (e.g., for structured data [2]), identifiability [12] and computational cost (with some of these methods requiring a full-fledged optimization subroutine [24]).

point-wise interpretations generally do not compare explanations obtained for nearby inputs, leading to unstable and often contradicting explanations [1].


Based on our initial dataset of 100,000 MCTS moves, CAT and TPT have similar performance in the task of predicting the move selected by MCTS, with average top-1 prediction errors of 47.0% and 47.7%, and top-3 prediction errors of 65.4% and 65.7%, respectively.

despite the very similar prediction errors, the TPT network is 50 ± 13 Elo stronger than the CAT network, suggesting that the cost-awareness of TPT indeed gives a performance improvement.


Specifically, we assign triangular motifs to nodes in a natural manner, and then subsample motifs only from nodes with too many of them.

mMSB and latent factor models rely on distributions over 0/1edges (i.e.


At a high-level, the typical analysis to the UCB algorithm for the standard multi-armed bandit problem (Auer et al., 2002) is the following: (1) at some finite time T , the number of pulls of a bad arm i is O with high probability, and (2) the regret suffered by any such pull is O(D i ).

the contribution of arm we analyzed the UCB algorithm in Section 2 by observing that the expected regret suffered on round t is bounded by the difference between the empirical mean estimator and the true mean for the payoff of arm I t . Of course by design this quantity is almost certainly (w.p.


Since RKS is fully randomized and data-independent, it has the smallest training time.

in order to compare the time cost of LKRF, EERF, and our work, we need additional details as the comparison may not be immediate.


Further, even though different heuristics have been developed to address the unbalanced problem, these methods suffer from the drawback that they have to make several decisions prior to reaching a final category, which leads to error propagation and thus a decrease in accuracy.

label embedding approaches [11,5,19] first project the label-matrix into a low-dimensional linear subspace and then use an OVA classifier.


The naive approach would consist in transmitting the 'missing' particles over the network.

even if basic optimizations are used (for example sending particles with multiplicities only once), we show in Section 4 that this transfer can be slow in practice.


Figure 4D shows the Fisher information after Gaussian noise has been added to the monocular inputs.

even in this high intrinsic noise regime (noise variance of the same order as tuning curve amplitude) the model with phase disparity carries significantly more total Fisher information.


We remark that similar conditions are required in [4] in order to enable statistical computations, e.g., constraining the support of the persistence landscapes.

it will be interesting to investigate which properties of the kernel remain valid when lifting these restrictions.


However, since in general exp(A 1 ) exp(A 2 ) = exp(A 1 + A 2 ), the path ( 12) is not contained in the manifold M defined in (11).

it is tempting to try to construct a new manifold M such that Span and X 0 ∈ M , ensuring the gradient flow remains in M . since A i 's do not commute, by combining infinitesimal steps along different directions, it is possible to move (very slowly) in directions that are not of the form A * (s)X + XA * (s) for any s ∈ R m . The possible directions of movements indeed corresponds to the Lie algebra defined by the closure of {A i } m i=1 under the commutator operator [A i , A j ] := A i A j − A j A i .


Due to the relative comparison constraint, LMNN does not satisfy the condition of Lemma 2.

as we already mentioned LMNN satisfies the condition of Lemma 3 so we get the ML h -MKL P variant of the optimization problem for LMNN which we denote by LMNN-MKL P . The resulting optimization problem is: where the matrix Y, Y ij ∈ {0, 1}, indicates if the class labels y i and y j are the same (Y ij = 1) or different (Y ij = 0).


Complexity of one iteration of QI is O(nk + lk), i.e., linear in the number of options, exactly like Greedy.

in practice it is much faster than Greedy since typically k << l. While we have no theoretical results that limit the iterations required by QI to converge, in practice, a fixed point is reached in very quickly (see below).


Bayesian ideal point models are usually fit with Gibbs sampling [2,3,5,18].

fast Gibbs samplers are unavailable for our model because the conditionals needed are not analytically computable.


Because this is the quantity that Vanilla-SC approximates, it is not surprising that it finds partitions with a smaller conductance.

figure 5b shows that if the conductance is computed using only edges in the testing-set, then sometimes the vanilla sets have no internal edges (φ(• , G) = 1).


Most of these works focused on scenarios where the tasks are batch learning problems, in which a training set is directly provided to the learner.

the online learning setting (Cesa-Bianchi and Lugosi, 2006), where the learner is presented with samples in a sequential fashion, has been rarely considered (see Mann and Choe (2012); Taylor (2009) for examples in RL and Sec.


Its solution follows directly from (1): ( ) ( I ) with fJ the current setting of the parameters.

in complex probability models P(hle; fJ) can be difficult and even intractable to compute, mainly because of the normalization in the denominator.


The streams have the same volume, so the optimal policy splits time uniformly between the NW-straight and EW-straight phases.

an additional stream of cars is generated in the south-west corner, at Intersection H, and travels diagonally east to Intersection E. Two equally short routes are available by going straight, or turning east at Intersection F. cars that turn east to join the northbound traffic flow must then turn east again at Intersection D, forcing the controller of that intersection to devote time to a third NS-right phase and forcing the main volume of traffic to pause.


Previous work in audio synthesis on the waveform focused on classification losses [17,24,4].

their computational cost needs to be mitigated by quantization, which inherently limits the resolution of the predictions, and ultimately increasing the number of classes is likely necessary to achieve the optimal accuracy.


If optimal α 1 = α 2 = C, our proposed algorithm can trace shortest path to optimal point from the origin (left plot).

single incremental algorithm moves one coordinate at a time (right plot).


Hierarchical Bayesian inference has been proposed as a solution to this problem, with particle filtering and belief propagation as possible algorithms implemented by the brain [6].

few studies have proposed neural models that are capable of performing hierarchical Bayesian inference (although see [24]).


With these regions, we have the set-valued classifier Those points in the first two sets are classified to a single class as by traditional classifiers.

those in the overlap receive a decision of {−1, 1}, hence may belong to either class.


With probability at least Furthermore, if ℓ is classification-calibrated, there exists a nondecreasing function The term on the right hand side involves both approximation error (that is small if F is large) and estimation error (that is small if F is small).

by appropriately increasing the richness of the class F with sample size, we can ensure that the misclassification probability of f approaches the Bayes risk of the true distribution.


Reinforcement learning (RL) has recently been very successful in solving hard problems in domains with well-specified reward functions (Mnih et al., 2015(Mnih et al., , 2016.

many tasks of interest involve goals that are poorly defined or hard to specify as a hard-coded reward.


We note that for typical values of δ, Algorithm R2 is better than Algorithm R1.

algorithm R1 is always better than the algorithm of [5] for the centralized variant of the problem and matches their number of samples in the personalized variant, so we present both algorithms in this section.


In addition, graphical models have been used to study brain connectivity, such as structural equation models [14]- [15], dynamic causal models [16], and Granger causality.

most of these approaches are confirmative, rather than exploratory, in the sense that they require a prior model of brain connectivity to begin with.


In addition, we also tested our algorithm on an artificially-generated set consisting of 15,000 points in R 10 from five mixtures of Gaussians (MG).

this task is made challenging by roughly merging the five Gaussians so that it is more likely to discover fewer clusters.


The vectors emanating from x and reaching these points represent For x, x ′ ∈ R 1 , this illustration shows that d β (x, x ′ ) ∈ cl cone(R − x) for any β ≥ 0.

figure 1(c) considers x ′ ∈ R 2 , while x ∈ R 1 .


One of the most straightforward explanations is that the observed word order frequencies may be the consequence of genetically encoded biases toward particular orders, as part of the universal grammar hypothesis; this possibility is considered in [4].

this can be only a proximate explanation: why does our genetic endowment happen to bias us in the particular way that it does?


It is important to notice that the rotation planes of the Givens rotations are selected from the whole L-dimensional space whereas the objective function only uses information of correlations among the first R signal components u i . Successive application of Givens rotations Q µν leads to the final rotation matrix Q which is in the ideal case such that , but it is not clear if the final minimum is also the global one.

in various simulations no local minima have been found.


Notice that the VB-EM approach has produced a correlation matrix that is automatically regularized (due to the Ψ AA term) and can be used for localization in its current form.

computing it from the signal estimates obtained by other methods, such as PCA or ICA, yields a poorly conditioned matrix that requires post-processing.


Also, for a fixed , as increases, we put more more emphasis on dynamic compatibility of representatives, hence, the dynamic cost decreases.

the diversity score decreases for smaller , as we select more representatives which become more redundant.


To compare (2) and (3), note that our assumptions on the bounded entries of M ensures that: In particular, we see that the worst case bound for (3) is √ nm which is always lower than the worst case bound for (2): , our bound is only larger by a logarithmic term in m compared to [AM07].

the algorithm proposed in [AM07] requires to store O(δmn) entries of M whereas SLA needs O(n) memory space.


Figures 5(c) and (d) corresponding to the case of large uncertainty are similar to Figures 5(a) and (b), respectively.

the Wasserstein distributionally robust filter now significantly outperforms the classical Kalman filter and, to a lesser extent, the KL distributionally robust filter.


Recently, owing to the increased use of machine learning (ML) to assist in consequential decisions, the topic of quantifying and mitigating ML-based discrimination has attracted interest in both policy and ML.

while the existing legal doctrine offers qualitative ideas, intervention in an MLbased system requires more concrete formalism.


Furthermore, when ĉ − c = 2 and g = 1 in Versatile v2-Model 3, the number of convolution filters will be further reduced.

since the number of filters is very small, the representability of this network is also lower.


However, it deterriorates rather rapidly when the decoding temperature increases.

selective freezing maintains a more steady performance, especially when f = 0.9.


On the one hand, it alleviates the weight transport problem by maintaining a separate set of connections that, under certain conditions, lead to synchronized learning of the network.

similar to backpropagation, FA transports signed error information in the backward pass, which may be problematic to implement as a plausible neural computation.


For all i = 1, . . . , n it holds that Notice that the expression on the right-hand side might be quite large for a particular function f i ; indeed, uniform stability does not hold in our case.

as we show below, the average of these expressions is small.


In this experiment, our learning algorithm is closely related to logistic regression.

since the SSL strategies are based the predicate predict(X, Y), one could replace this learner with any other learning algorithm-in particular one could replace it with a non-declarative specified prediction rule as well, using the same programming mechanism we used to define entropy.


In the above example, a simple uniform distribution was used.

one might use the empirical distribution of the training data, which is equivalent to contrastive divergence [5].


For example, heads-up limit Texas hold'em, a relatively simple form of poker with 10 13 decision points, was essentially solved without decomposition [2].

this approach cannot extend to larger games, such as heads-up no-limit Texas hold'em-the primary benchmark in imperfect-information game solving-which has 10 161 decision points [16].


A potential advantage of systems which fix the grammar and only perform parameter search is that they do compare complete grammars against each other, and are therefore able to detect which give rise to systematically compatible parses.

although early work showed that small, artificial CFGs could be induced with the EM algorithm [12], studies with large natural language grammars have generally suggested that completely unsupervised EM over PCFGs is ineffective for grammar acquisition.


Deep structured models are widely used for tasks like semantic segmentation, where explicit correlations between variables provide important prior information which generally helps to reduce the data needs of deep nets.

current deep structured models are restricted by oftentimes very local neighborhood structure, which cannot be increased for computational complexity reasons, and by the fact that the output configuration, or a representation thereof, cannot be transformed further.


This convergence is distribution dependent.

intuitively, slow convergence is not always the case, for example when the support of the distributions lies in a lower dimensional manifold of the space.


In analyzing the data for the Space-Shuttle, we noted a small number of mid-range real anomalies and a larger number of mid-range predicted anomalies.

when we analyzed similar data for Sine, we found it had many narrow-range real and predicted anomalies that appear close together.


While this results suggests that the sampling algorithm works in practice, it is not surprising and novel since similar results for collective classification have been produced before using other approaches in statistical relational learning (e.g., compare [13]).

the marginal distributions we obtain provide additional information beyond the simple point estimate of its expected value.


One may consider one channel as an important one if the final loss L f would sharply increase without it.

it is not practical when the network is very deep.


Even if the image collection does not contain any labels, it has been shown to help tasks such as image completion and exploration [6,21], image colorization [22], and 3D surface layout estimation [5].

as noted by several authors and illustrated in Figure 1, the major stumbling block of all the scene-matching approaches is that, despite the large quantities of data, for many types of images the quality of the matches is still not very good.


The trade-off between exploration and exploitation has been an ever-present trope in the online learning literature.

this paper studies the trade-off between two different kinds of pure exploration: breadth versus depth.


The middle panel of this Figure suggests that the norm of the learned networks can indeed be predictive of their generalization behavior.

we again observe that sharpness has a poor behavior in these experiments.


Several learning techniques naturally generalize to an arbitrary number of classes, such as density estimation, or logistic regression.

when comparing the reported performance of these systems with the de-facto standard of using two-class techniques in combination with simple, fixed output codes to solve multi-class problems, they often lack in terms of performance, ease of optimization, and/or run-time behavior.


For instance, we notice some small overfitting for the DST-NN starting with hidden layer size 32, while the NN starts to show some small overfitting only at hidden layer size 128.

the RNN does not show any sign of overfitting in 100 epochs of training, regardless the hidden layer size in the tested range, and the performance in training is somewhat equivalent to the performance in testing.


A common approach is to solve (1) for several values of λ, then use holdout data (or theoretical approaches, like AIC or SRM) to select a good value.

if we view the regularized optimization problem as a family of problems, parameterized by the regularization parameter λ, it allows us to define the "path" of optimal solutions { β(λ) : 0 ≤ λ ≤ ∞}, which is a 1-dimensional curve through R p . Path following methods attempt to utilize the mathematical properties of this curve to devise efficient procedures for "following" it and generating the full set of regularized solutions with a (relatively) small computational cost.


However, because of the particular form of the online update, positive-definiteness still must be carefully enforced, which impacts bound quality and empirical performance, making it undesirable for both theoretical and practical purposes.

our proposed algorithm has strong bounds, requires no extra work for enforcing positive definiteness, and can be implemented efficiently.


In [11] a related dynamics reversed is proposed.

the singularities resulting from incorrectly treating p(v t+1:T |h t , s t ) as a density are heuristically finessed.


The second polytope is the upper submodular polyhedron of f [17], defined as Unfortunately, U(f ) is not as tractable as B(f ): even checking membership in U(f ) is hard [17].

we can still succinctly describe specific elements of U(f ).


(3.1) A greedy policy π (i) can be obtained at each iteration i by ∀s : For any u > 0, it can be shown that if one can approximate u and run the above value iteration algorithm using these approximated values, then after Θ((1 ) iterations, the final iteration gives an value function that is u-optimal ([Ber13]).

a u-optimal value function only yields a u/(1 − γ)-optimal greedy policy (in the worst case), even if (3.2) is precisely computed.


From inspection of (3), it is clear that each of the elements of Y * is a linear combination of the sample averages Ȳ .

a stronger statement can be made: Nt < ∞ for all t, then the MTA estimates {Y * t } given in (3) are a convex combination of the task sample averages { Ȳt }.


Thus, this "jump" implements a relative refractory period.

g decays as it tends to converge back to its stable level g stable = log r on r off .


For two different kinds of hyperplane splits, if alternate split is invoked the same number of times in the tree, the difference in their worst-case guarantees for both the trees would again be governed by their worstcase quantization performance (β * ).

for any fixed η, a harder question is whether one type of hyperplane partition violates the inlier condition more often than another type of partition, resulting in more alternate partitions.


However, the differences between model parameters, which can have a high sample complexity when there are few of them, are often pointed out only in passing [4,6,14].

in many situations we might be interested in actually understanding and identifying the differences between elements of the support.


In the existing literature this is usually formulated as finding the mode of the DPP (the so-called MAP set).

the MAP objective inherently assumes that the DPP models "optimal" recommendation sets, and yet obtaining such a DPP is nontrivial when there is no ready source of example optimal sets.


However, [27] focused mostly on the comparison of approaches for tensor factorization and their stability properties, with brief experiments using a latent variable model related but not equivalent to LDA for community detection.

we provide a detailed experimental comparison in the context of LDA in this paper, as well as propose a novel cumulant-based estimator.


Algorithms in [12,26] are unlikely to be practical for most vision problems, which typically have tens of thousands of variables.

in our case we need to minimize a bisubmodular function which has a special structure: it is represented as a sum of low-order bisubmodular terms.


Alternatively, KNN, MF, and MICE perform much better than mean imputation.

these methods demonstrate unstable performances in different tasks.


However, then the dictionary size m affects the permissible radius of perturbation and the stability bound of the sparse coding stability.

the factor of m does not appear in Theorem 2, and thus, the result is effective even for a large m. In addition, whereas ∥x∥ ≤ 1 is assumed in Mehta and Gray (2013), Theorem 2 does not assume that ∥x∥ ≤ 1 and clarifies the dependency for the norm ∥x∥.


A notable exception are approaches that try to infer the scene layout of indoor scenes in the form of 3D bounding boxes [13,22].

these approaches can only cope with limited amounts of clutter (e.g., beds), and rely on the fact that indoor scenes satisfy very closely the manhattan world assumption, i.e., walls (and often objects) are aligned with the three dominant vanishing points.


As long as σ 2 is constant, the auxiliary variable ξ will be able to automatically find its mean value ξ on the fly.

with a parameter-dependent covariance matrix Σ(θ), the SGNHT method (8) would not produce the required target distribution (10).


Unfortunately, due to the r t term in the expression of Ẋ, the function (X, Z, t) → ( Ẋ, Ż) is not Lipschitz at t = 0, and we cannot directly apply the Cauchy-Lipschitz existence and uniqueness theorem.

one can work around it by considering a sequence of approximating ODEs, similarly to the argument used in [28].


They are characterized by a notion of diversity, as shown in Figure 1; points in the plane sampled from a DPP (center) are more spread out than those sampled independently (left).

in many cases we would like to make use of the most likely configuration (MAP inference, right), which involves finding the largest principal minor of a positive semidefinite matrix.


Finding the global minimum of loss function is a NP-hard problem [1,17].

lloyd's algorithm [15] is a simple and elegant method to obtain a locally optimal clustering of the data, corresponding to a local minimum of the loss function.


Many of the existing algorithms for node classification [18,9,10,5,8,6] can in principle be run on signed graphs.

the computational cost may not always be preserved.


As seen from the F1 score, the IB method performs better than the MC method in the first model.

if in an application we know that the parameters are not likely to change much (as in model 2), the MC method also has high performance.


Our preliminary studies have shown the effectiveness of Kernel DisKmeans λ in learning the kernel Gram matrix.

the SDP formulation is limited to small-sized problems.


The resulting iterates are sparse and there is a guaranteed sublinear convergence rate of the objective function to its minimum.

the linear minimization subroutine is hard to perform in general : it is for instance NP-hard for neural networks with homogeneous activations [4].


These hardness results focus on proper learning, where the goal is to find a nearly-optimal predictor with a fixed network architecture A.

if our goal is to find a good predictor, there is no reason to limit ourselves to predictors with one particular architecture.


Since ∇ t = − rt xt,rt , its ∞ -norm is large only when there is a stock with high reward r t,i while the learner puts a small weight x t,i on it.

the reason that the weight x t,i is small is because the learner finds it performing poorly prior to round t, which means that the learner had better choices and actually should have performed better than stock i previously (that is, negative regret against stock i).


Further, they do not transfer to the stochastic setting.

the approach presented here grows the truncation as a natural consequence of the inference algorithm and is easily adapted to stochastic inference.


In contrast to these approaches, our results hold both for rank-k leverage scores and projection distances, for any matrix sketching algorithm-not just FJLT or Frequent Directions-and our space requirement can be as small as log(d) for average case guarantees.

clarkson and Woodruff [39] and Drineas et al.


We note that both of these settings correspond to only a single query repeatedly asked on a dynamic database, precluding meaningful adaptive analysis.

we consider the much richer class of linear queries, including 2 |X | counting queries, allowing for adaptive analysis of a dynamically growing database.


If α = 0, this imposes no constraint on the noise: take c = 1 to see that every probability measure P satisfies (3).

it is easy to verify that α = 1 if and only if |2η(X) − 1| ≥ 1/c a.s. [P X ].


The computational complexity of COMBEXP is determined by the structure of Co(M) and COMB-EXP has O(T log(T )) time complexity due to the efficiency of interior-point methods.

the computational complexity of COMBAND depends on the complexity of sampling from M. COMBAND may have a time complexity that is super-linear in T (see [16, page 217]).


This is akin to the agglomerative clustering sense which is usually done post-hoc.

our model discovers the factor hierarchies as part of the inference procedure itself.


On one hand, if ω is always positive, then the feature space will be totally connected and every two features can interact with each other.

if ω is chosen to be a Dirac-delta function δ ij , namely ω(X i , X j ) = 1 for i = j and is 0 otherwise, then the nonlocal block gets localized and will act like a simplified residual block.


If τ is too small, the tracker cannot adapt well to appearance changes.

if τ is too large, even an occluding object or the background may be mis-treated as the tracked object and hence leads to drifting of the target.


The DD-margin bounds are still always, but less significantly, sharper than the DImargin bounds.

on these small datasets, it is difficult to tell if the bounds select good models.


Also, more expensive versions of the estimator exist with lower variance.

it is still prone to overestimate test probabilities: If 1/ PHME (v) is the Harmonic Mean Estimator in (2), Jensen's inequality gives Similarly log P (v) will be overestimated in expectation.


In contrast, the nonparametric Gaussian process (GP) model [19] offers a flexible and powerful alternative.

a major drawback of GP models is that the computational cost of learning is about O(n 3 ), and the cost of making a single prediction is O(n), where n is the number of training examples.


[14]) could also be used instead of [23].

while this approach would simplify the analysis, it produces looser bounds, as described in Section 7.


For a fixed integer T , the four methods go through the dataset (in terms of multiplying A with column vectors) the same number of times.

since LazySVD does not need block orthogonalization (as needed in PM and Krylov) and does not need a (T k)dimensional SVD computation in the end (as needed in Krylov), the running time of LazySVD is clearly much faster for a fixed value T . We therefore compare the performances of the four methods in terms of running time rather than T . We programmed the four algorithms using the same programming language with the same sparsematrix implementation.


For our purposes, the main disadvantage of the hinge loss is that it is unbounded, and therefore cannot distinguish a single very bad example from say, 10 slightly bad ones, making it ill-suited for constraints on rates.

for the ramp loss the contribution of any single datum is bounded, no matter how far it is from the decision boundary.


One standard method of converting systematic scan into a homogeneous Markov chain is to consider each full scan as one step of a Markov chain.

this makes it difficult to compare with random scan because it completely changes which states are connected by single steps of the transition matrix.


Naturally, such an increase in the learning rate also increases the mean steps E [∆w].

we found that this effect is negligible since E [∆w] is typically orders of magnitude lower than the standard deviation.


Whenever tasks are somehow linearly dependent, even if the weight matrix W * is dense and s ≈ d, the rank s * can be small, thus guaranteeing a dramatic improvement over GL-FQI.

learning a new representation comes at the cost of a worse dependency on d. In fact, the term log(d)/ √ T in GL-FQI, becomes d/T , implying that many more tasks are needed for FL-FQI to construct a suitable representation.


Trivially, we can appeal to this result in the linear optimization case to obtain a |D|T regret by setting K to be the size of D. However, the regret could have a very poor n dependence, as |D| could be exponential in n (or worse).

note that in the full information case, we could appeal to the K-arm case to obtain O( T log |D|) regret, which in many cases is acceptable (such as when D is exponential in n).


In other words, we will add a singleton tree.

in practice our algorithm converges in a small number (≪ m) of iterations and provides an accurate mixture of trees.


[3]) has emphasized that the utility of a VAE hinges on its ability to capture useful information in the latent representation z, complementary to that in powerful decoder networks p θ (x|z).

renewed interest in programmatic representations in AI have sought modular, generalized, compositional representations for high-level concepts [4,5,6,7,8].


This knowledge may be incorporated by a weight w t which indicates the strength of a prior belief about whether the hypothesis is null or not-typically, a larger w t > 1 can be interpreted as a greater likelihood of being a non-null, indicating that the algorithm may be more aggressive in deciding whether to reject H t . Such p-value weighting was first suggested in the offline FDR context by [6], though earlier work employed it in the context of FWER control.

as with penalty weights in the offline setting, offline prior weights are also usually rescaled to have unit mean, and then existing offline algorithms simply replace the p-value P t by the weighted p-value P t /w t . it is not obvious how to incorporate prior weights in the online setting.


In this case, one obtains at each iteration ω (t) i = λ for all i.

when β < ∞, our algorithm adaptively updates the weights so that to penalize less heavily higher singular values.


If the target environment is arbitrary, or drastically different from the study environment, no causal relations can be learned and scientific progress will come to a standstill.

the fact that scientific experimentation continues to provide useful information about our world suggests that certain environments share common characteristics and that, owed to these commonalities, causal claims would be valid even where experiments have never been performed.


In addition, let us also remark that this complexity can be further decreased if the number of distinct values of s y with non-zero probability mass is smaller than m. Moreover, the MEUF framework will not deliver an exact F-maximizer if the assumption of independence is violated.

mEUF relies on a smaller number of parameters (m values representing marginal probabilities).


Thus the clustering-based VSUMM method is prone to select key frames from the first half of the video, collapsing the latter part.

our seqDPP copes with time-varying diversity very well.


In the similarity-based partition, similar results are observed as it also leads to balanced local costs.

when the local sites have significantly different costs (as in the weighted and degree-based partitions), our algorithm outperforms COMBINE.


( 8) over each variational parameter can be derived analytically given other parameters fixed, thus a coordinate ascent can be applied as an efficient message passing algorithm [27,11].

it is not directly applicable to our model due to the non-conjugate observation likelihood p(O|X; γ).


Recent works have shown that a wide class of graphical models can be estimated efficiently in high dimensions [1][2][3].

frequently, graphical models may not suffice to explain all the characteristics of the observed data.


We conjecture that this is because translation mistakes generate noisy features which PSI JapEng circumvents.

we also considered combining PSI JapEng with TFIDF or PSI EngEng using a mixing parameter α and this provided further gains at the expense of requiring a machine translation tool.


AdaBoost.MH is a popular multiclass boosting algorithm that is based on the one-against-all reduction [19].

we show that its implicit demands on the weak classifier space is too strong.


We find that OPT-ADMM [2] also needs to decrease the step size with the iteration.

its step size decreasing rate is O(k 3 2 ) and is faster than ours.


When closing a loop, it always picks the most likely data association.

it also provides a mechanism to undo and redo past data association decisions.


For the functions of unitation, no improvement over standard PBIL is expected, since the scaling using mutation is worse, requiring a < 1/ L rather than a < 1/..fL.

with tuning of the mutation rate, the range of a's with which the optimum was always found could be increased over standard PBIL.


In fact, for the binary loss this result cannot be significantly improved (see Section 3).

when the loss is not binary it may be possible to obtain a tighter bound on the variance, which will lead to a tighter bound on the loss than the PAC-Bayes-kl inequality.


This could violate the exchangeability property and make explicit Bayesian modelling difficult.

we do not have to explicitly represent the posterior to ensure exchangeability.


In other words, by optimizing for the oracle loss, a multiple-prediction learner can respond to ambiguity much like a human does, by making multiple guesses that capture multi-modal beliefs.

a single-prediction learner is forced to produce a solution with low expected loss in the face of ambiguity.


notice that if Pr(S 2 |S 1 ) ≥ 1 − , then Pr(S 2 |S 1 ) ≤ , which implies by definition of weak dependence that Pr(S 2 ) ≤ /α and therefore Pr(S 2 ) ≥ 1 − /α.


Another type of methods are based on MCMC (Friedman & Koller, 2003;Eaton & Murphy, 2007;Grzegorczyk & Husmeier, 2008;Niinimäki & Koivisto, 2013;Su & Borsuk, 2016) where graphs are sampled from the posterior distribution.

there is no guarantee on the quality of the approximation in finite runs (may not mix well and converge in finite runs).


Similar empirical observations were made by the CryptoNets work [8].

to ensure accuracy the values in the network must lie in the range [− p−1 2 , p−1 2 ]; this influences the choice of the prime p. we note that large primes increase the verifier and prover run-time because of the higher cost of performing modular additions and multiplications.


Our inferred TF profile and reconstructed target gene profiles are similar to those obtained in [13].

for certain genes, our model provides a better fit to the gene profile.


As in the general theory of histogram estimators, the approximation error depends on the choice of bandwidth h n . Generally speaking, if the bandwidth is small, then the histogram bias term will be small.

a smaller bandwidth leads to a larger number of cells and hence more Laplacian noises.


Under the assumption that the number of outputs is O(n) this model has computational complexity of O(n).

exact algorithms for the problems we are dealing with are more costly.


An input-dependent noise term is very common in regression tasks with continuous output values y n ∈ R, resulting in heteroscedastic regression models, which have been proven more flexible in numerous applications as already touched upon in the section on related work.

to our knowledge, there is no prior work on heteroscedastic classification models.


The first stage usually takes at least O(dN 2 ) time, which is prohibitive when N is large.

several methods such as K-subspaces clustering [18], K-SVD [1] and online subspace clustering [25] do have linear time complexity O(rdN ) per iteration, however, there are no global or local convergence guarantees.


On one hand, the collaborative nature makes SLFA capture deeper statistical structure hidden in the data set, compared to the matrix factorization problem with the Tikhonov regularization S 2 F or sparse coding with the sparsity-inducing regularization such as S 1 .

sLFA encourages sparse interactions which is very different from previous works such as correlated topic Model [2] and latent Gaussian model [18], where the latent factors are densely related.


Techniques based on gradient matching offer a computationally appealing and successful shortcut for parameter inference but are difficult to extend to include unobserved variables in the model descripton or are unable to keep their performance level from fully observed systems.

most real world applications are only partially observed.


We do not have yet a good understanding of how one should pick the parameters of our model to achieve good performance.

the other papers which have up to now discussed global optimality for neural networks [11,8] have not included any results on real datasets.


Thus, they are not concerned with the generative process that produces the graph, nor with the interpretability and statistical properties of their algorithms.

we view the nodes of a directed graph as a finite sample from a manifold in Euclidean space, and the edges as macroscopic observations of a diffusion kernel between neighboring points on the manifold.


Furthermore the computational cost of evaluating the polynomial does not depend on the number of data points which is a crucial difference with existing nonparametric methods such as nearest neighbors or kernel based methods [3].

this computation requires the inversion of a matrix whose size depends on the dimension of the problem (see Section 3).


In our main application, each function evaluation corresponds to inference on a large neural network and can take a few seconds.

streaming algorithms make a small number of passes (often only one) over the data and have sublinear space complexity, and thus, are ideal for tasks of the above kind.


Second order methods, like the Newton method, are designed to rapidly descend plateaus surrounding local minima by multiplying the gradient steps with the inverse of the Hessian matrix.

the Newton method does not treat saddle points appropriately; as argued below, saddle-points instead become attractive under the Newton dynamics.


This minibatch estimator is biased, since its expectation is a lower bound 3 .

computing it does not require any additional hyperparameters.


These cells show, both in spike and membrane potential responses, a receptive field growth of a factor of 2 (top) and 3 (bottom) at low contrast.

for both cells the spatial summation extent of excitation at low contrast is one aperture less than at high contrast.


IBM Model-4 already integrates the fertility and distortion submodels on top of HMM, which further narrows the word-alignment choices.

iBM Model-4 does not have a scheme to adjust its lexicon probabilities specific to document topicalcontext as in HM-BiTAM.


Each Σ (k) is of size d k × d k and the full covariance is of size k d k × k d k . Thus TGP is projected onto the entire tensor with respect to the latent factors U, including all zero and nonzero elements, rather than a (meaningful) subset of them.

the real-world tensor data are usually extremely sparse, with a huge number of zero entries and a tiny portion of nonzero entries.


If features are observed in the sparse basis, then conjugate gradient solvers can be used for regularized value function approximation.

cBEBF seems to have better performance with smaller sample sizes and provably works under any observation basis.


Several approaches [5,6,12,23] have been proposed to discover mid-level visual elements, that are both 1) representative, i.e., frequently occurring within a visual dataset, and 2) visually discriminative.

the current approaches are rather ad hoc and difficult to analyze and evaluate.


There has been tremendous recent progress on equilibrium-finding algorithms for extensive-form zero-sum games.

there has been a vexing gap between the theory and practice of equilibriumfinding algorithms.


If not, it is possible that the resulting π j (s)'s will not be consistent with any global distribution even though they are locally consistent.

the running intersection property can be enforced by introducing certain additional local distributions in the set of π j (s)'s.


It is for their scalability that these models are frequently a method of choice for learning on large scale data sets (e.g., the implementation of linear SVM [13] has won the large scale learning challenge at ICML 2008 and KDD CUP 2010).

as the set of linear hypotheses defined on a small or moderate number of input features is usually of low capacity these methods often learn inaccurate descriptions of target concepts.


This is the case where gap dependent guarantees of classical analysis break down.

a q or, for Block Krylov Iteration, some q-degree polynomial in our Krylov subspace, also significantly separates singular values > σ k+1 from those < (1 − )σ k+1 .


If the dependency structure across a neural population is relatively homogeneous, then these copulas may be useful in that they can be estimated using far less data than required, e.g., for a full covariance matrix (which has O(n 2 ) parameters).

if the dependencies within a population vary markedly for different pairs of neurons (as in the data set examined here), such copulas will lack the flexibility to capture the complicated dependencies within a full population.


In higher dimensions, it is possible to describe more complex joint distributions, with different marginals along different projections.

although the model distribution for individual coefficients is similar to the fixed sparse priors of ICA and sparse coding, the model is fundamentally non-linear and might yield a different lower-level representation; the coefficients u are no longer mutually independent, and the optimal set of basis functions must account for this.


The analysis of this linear program is considerably more difficult than of the previous one.

we will show that the ν-trick still works in a generalized manner with some subtleties.


Since the main source of smoothing is the base measure, UBM's ability to extrapolate is limited to repeatedly observed words.

uBM is capable of adjusting the probabilities of the most frequent words to focus on fitting the regularities of small probability events.


The first component of the KL divergence term is the entropy of a Gaussian mixture which is not analytically tractable.

a lower bound of this entropy can be obtained using Jensen's inequality (see e.g.


Determining the optimal maintenance dose of Warfarin presents a challenge as the appropriate dose varies significantly from patient to patient and is potentially affected by many factors including age, gender, weight, health history, and genetics.

this is a crucial task because a dose that is too low or too high can put the patient at risk for clotting or bleeding.


A potential solution is constructing features of event sequences via parametric [22] or nonparametric [18] methods.

these feature-based methods have a high risk of overfitting because of the large number of parameters.


Our algorithm appears to incur the largest errors on images which contain very irregular trees, in which most of the 3-D structure in the image is dominated by the shapes of the leaves and branches.

arguably even human-level performance would be poor on these images.


This idea has been used before in the semiparametric statistics literature [3,30].

most studies are restricted to functionals of one distribution and have focused on a "data-split" approach which splits the samples for density estimation and functional estimation.


These strategies are beneficial for increasing the training speed.

their evaluation does not show any improvement in speed.


For 30% corruptions for a small number of samples ULURU outperforms the other subsampling methods.

as the number of samples increases, influence based methods start to outperform OLS.


We note that the graph partitioning methods used in [3] are typically much faster than the time needed by SDP solvers (e.g., SeDuMi [15] and SDPT3) for comparable problem sizes.

given the increasing interest in SDP in the last few years, we may expect the development of new algorithms, and faster/more efficient software tools.


As a result, the sigmoid based function did not perform well on the large dataset.

sigsoftmax achieved lower perplexities than softmax on PTB and achieves the lowest perplexities on WT2.


The Bethe approach (and its generalizations) can be understood as consisting of two steps: (a) replacing the exact entropy −Φ * with a tree (or hypertree) approximation; and (b) replacing the marginal polytope MARG(G; φ) with constraint sets defined by tree (or hypertree) consistency conditions.

since the (hyper)tree approximations used do not bound the exact entropy, the optimal values of Bethe-type variational problems do not provide a bound on the value of the log partition function Φ(θ).


The unlabeled examples have first a beneficial effect on test error, then have a detrimental effect when they overwhelm the number of labeled examples.

the diagnosis models behave smoothly as in the previous case, and the minimum entropy criterion performance improves.


Consequently it does not immediately appear that Γ = Φ is likely to provide any tangible benefit.


From the table, in the early stage the population evolves dominantly based on SGD since the offsprings are worse than almost all the parents.

in late generations the number of elite offsprings increases.


Some existing work [28,18] propose tree-based autoencoder architectures.

in these models, the decoder can only access to a single hidden vector representing the source tree, thus they are not performant on the translation task.


In this case, no single day contains enough observations to be confident about the details of the rate function, so each individually-estimated rate function appears relatively smooth.

when information from other days is included, the rate function begins to resolve into a clearly bi-modal shape for weekdays.


More specifically, the iterations are as follows: x k+1 = arg min (5) If the prior R is appropriately chosen, such as x 1 , a closed-form solution for (3), i.e., a soft thresholding solution is naturally desirable.

for some more complicated regularizations, e.g., a patch based prior [8], solving (3) is nontrivial, and may require iterative methods.


We conjecture that the integration time of the LINs in the first reservoir should ideally be long enough to capture the co-articulations between successive phonemes emerging from the dynamical constraints of the articulatory system.

it has to remain short enough to avoid that information pointing to the presence of a short phoneme is too much blurred by the left phonetic context.


It is clear that in general no single member A ∈ A can contain all other members of A , since adding a null set (i.e., a set of measure zero) to A results in a larger set A ′ which is nevertheless still a member of A .

we can find a member A * ∈ A which contains all but a null set of every other set in A .


The first subproblem is an integer program, and the second one has non-convex orthogonal constraints.

in each iteration the global optimum can be obtained for each sub-problem as discussed above.


An interesting recent paper, "The Case for Learned Index Structures" [7], argues that standard index structures and related structures, such as Bloom filters, could be improved by using machine learning to develop what the authors dub learned index structures.

this paper did not provide a suitable mathematical model for judging the performance of such structures.


Specifically, the dependencies are described by a weighted similarity graph and incorporated through a Laplacian regularizer [18].

graph-based regularization fails in cases where there are negative correlations between two time points.


The only challenge here is to find a way of efficiently handing missing features (since the unacquired features by # k 1 are missing to the classifier ✓ k ).

solving (1) with respect to # k given ✓ k 1 is not trivial.


This phenomenon has been observed by others as well-for example, Cutler and Zhao [7] demonstrate that their PERT classifier (which is similar to batch version of MF) achieves test accuracy comparable to Breiman-RF on many real world datasets.

in the presence of irrelevant features, methods which choose splits independent of labels (MF, ERT-1) perform worse than Breiman-RF and ERT-k (but still better than ORF-Saffari) as indicated by the results on the dna dataset.


A comparison to MMMF in that regard is difficult, as it is implemented in MATLAB and COFI RANK in C++.

cOFI RANK is more than ten times faster than MMMF while using far less memory.


In practice, the prior is typically easy to sample from, but it is typically infeasible to generate exact posterior samples.

in models where we can tractably sample from the joint distribution p(✓, z, y), we can generate exact posterior samples for simulated observations using the elementary fact that p(y) p(✓, z|y) = p(✓, z, y) = p(✓, z) p(y|✓, z).


Independently, [14] presented an approximated structured SVMs program and a message passing algorithm, which reduce to Theorem 1 and Lemma 1 with = 0 and c α = 1, c v = 1.

in this algorithm the messages are not guaranteed to be bounded.


[24] proposed Convolutional Oriented Boundaries (COB), where features from different layers are fused to compute oriented contours and region hierarchies.

these works combine the multi-scale representations from different layers adopting concatenation and weighted averaging schemes while not considering the dependency between the features.


The range of this activation function (as a function of x) is a connected set on the (−1, 1) interval.

the whole (−1, 1) range is not always available, depending on the parameters.


Several researchers (see for instance [4] and [9]) have proposed statistical tests suited for 2-class classification tasks where the performance is measured in terms of the classification error (ratio of the number of errors and the number of examples), which enables the use of assumptions based on the fact that the error can be seen as a sum of random variables over the evaluation examples.

various research domains prefer to measure the performance of their models using different indicators, such as the F 1 measure, used in information retrieval [11], described in Section 2.1.


based algorithm exhibits the exceptional performance for clustering multivariate Gaussian data or subspace data.

it is not suitable for manifold-valued data.


In a classical learning context, model accuracy is ideally tested on a held-out test data set.

to exploit the full available data for model selection and algorithm comparison, cross validation methods have become a standard method.


By limiting the program's length to t, we are able to directly represent the mapping f by a feed forward neural network.

this comes at a cost of limiting the program's length by a fixed maximal length.


Event coreference was also tried by using the semantic types of an ontology [17].

the features used by these approaches are hard to select and require the design of domain specific constraints.


Our RSDN architecture resembles the one proposed in [12] for dimensionality reduction of handwritten digits via learning a nonlinear embedding.

ours distinguishes from theirs in the use of different building blocks in our DAs, loss functions and motivations.


It can be easily shown that the priors (7r d,i can be easily updated and the means (f..Ld,i can be entirely determined by the linear transform A.

updating the linear transform A and the variances (lTd,i) does not have closed form solution and has to be solved iteratively by numerical methods.


A detailed comparison between their result and ours for such a structured formulation will be discussed in Section 4.

under the assumption T . such an assumption holds only if HSGD selects at least O k 2 samples at the k-th iteration due to In this way, their result under WRS is of the same order as ours under WoRS.


More specifically, whenever Z is representative of the instances of the other class that we expect to see in the future, then a binary classification is certainly preferable.

there can be situations where Z is not representative for the other class, for instance due to nonstationarity.


In recent years, distributed representations of inputs have led to performance gains in many applications by allowing statistical information to be shared across inputs.

the predicted outputs (labels, and more generally structures) are still treated as discrete objects even though outputs are often not discrete units of meaning.


They are invariant to translations and rotations.

these kernels only involve local interactions, and regression results thus degrade in presence of large-scale interactions.


Of course, since we have not used the low-rank structure of P , we may be tempted to "smooth by factoring", by performing a low-rank approximation of Pn .

this will not eliminate the zero problem, since a whole column may be zero.


This means that, in the repair example above, the original discriminatory model is a proxy for a protected demographic attribute, indicating the presence of discriminatory behavior in the "repaired" model.

prior treatment of proxy use has been limited to classification models, so regression models remain out of reach of these techniques.


[24] defined a sequence of intermediate target densities as 1 Since F T = F, it follows that the condition (3) is satisfied.

even though this is a valid choice of target distributions, leading to a consistent SMC algorithm, the resulting sampler can have poor performance.


For instance, a naive approach to make predictions about conjunctive queries would be the following: First, one would run an edge prediction model on all possible pairs of nodes, and-after obtaining these edge likelihoods-one would enumerate and score all candidate subgraphs that might satisfy a query.

this naive enumeration approach could require computation time that is exponential in the number of existentially quantified (i.e., bound) variables in the query [12].


In particular, the O(1/ √ T ) and O(1/λT ) bounds apply for all T ∈ {1, 2, . . . , m}, namely up to one full pass over a random permutation of the entire data.

our techniques are also applicable to a constant (> 1) number of passes, by randomly reshuffling the data after every pass.


We could potentially merge such duplicates to form a graph instead.

for simplicity, we choose not to merge these duplicates and keep a tree, which could make the planning problem harder.


For general kernels, the complexity of initialization is O(n 3 ), while the complexity of each iteration is O(n 2 ).

using an incomplete Cholesky decomposition [5] makes all costs linear in n. After the convex optimization, we obtain a low-rank matrix M ∈ C k which is pointwise nonnegative with unit diagonal, of the form U U ⊤ where U ∈ R n×m .


Theorem 2 states that the regret is upper bounded in this case by Θ(∆ −1 * log T ).

it is not possible to know if this bound is comparable to that of the i.i.d case due to the Θ.


For example, the top-6 retrieval images of class "Indigo Bunting" are all from their ground truth class since their visual features are similar.

the query "Black-billed Cuckoo" retrieves some instances from its affinal class "Yellow-billed Cuckoo" since their visual features are too similar to distinguish.


5.1] for the BC and attributed to the manner in which the length-scale is selected.


it is still possible to bound the error of the tug-of-war value function learned with SARSA(1) in terms of the full-vector value function.


Classic sequence analysis [10,11] also addresses the problem of optimal stopping.

it assumes that the samples are drawn i.i.d.


Therefore, we can apply the optimization method in Section 3.2 to minimize J .

when the size of T is large, to find T is still time consuming and requires a very large space.


Dual-decomposition provides a plausible approach for tackling large-scale problems by dividing the task into many small sub-problems [20].

solving a large number of smaller systems is still a bottleneck, and decoding the optimal solution from the sub-problems might be difficult.


One class of latent-feature models for social networks has been developed by [9,10,11], who proposed real-valued vectors as latent representations of the entities in the network where depending on the model, either the distance, inner product, or weighted combination of the vectors corresponding to two entities affects the likelihood of there being a link between them.

extending our high school student example, we might hope that instead of having arbitrary realvalued features (which are still useful for visualization), we would infer binary features where each feature could correspond to an attribute like "male" or "athlete."


In analogy with LDA or DCA, if our data consisted of word counts, with f indexing words and n indexing documents, then the columns of W would describe topics and c f kn would denote the number of occurrences of word f stemming from topic k in document n. In our setting W is considered a free deterministic parameter to be estimated by maximum likelihood.

h is treated as a nonnegative random latent variable over which we will integrate.


This allows us, in theory, to apply a classifier trained on 2D images to 3D volumes.

differences in appearance and geometry of the structures may potentially adversely affect classifier accuracy when 2D-trained ones are applied to 3D stacks, which motivates domain adaptation.


It is possible to define a group structure satisfying properties (i) and (ii) in which each edge of G * appears as a 2-node group; but then, a graph G generated from a group-induced model with a very large exponent α will contain all edges of G * with high probability, and a decentralized search algorithm will be able to follow these edges directly to construct a short path to the target.

a lower bound for the case α > 1 can be obtained if we place one additional restriction on the group structure.


Recent work in generative adversarial networks (GANs) [33], variational autoencoders (VAEs) [34], and other generative models have achieved promising results on various semi-supervised classification tasks [35,36].

we find that these models are not as well-suited for generic regression tasks such as in the UCI repository as for audio-visual tasks.


We have seen that multisensory perception is a suitable tool for studying causal inference.

the causal inference model also has the potential to quantitatively explain a number of other perceptual phenomena, including perceptual grouping and binding, as well as within-modality cue combination [27,28].


3 (b) still can be seen as an independent convolution filter with complex structure and obvious magnitude change.


This means that any adjective placed on " will tend to be reconstructed correctly, and that its reconstruction is independent of the noun we have in position ¥ .

the contributions from nouns and verbs in positions and £ are non-negligible, and notice how those given by words belonging to the subsets are almost symmetric to those given by words in the " subsets.


A good search strategy should find a good incumbent solution early and identify non-promising nodes before they are expanded.

naively applying a single heuristic through the whole process ignores the dynamic structure of the B&B tree.


As can be seen from the examples listed in Table 3, our model is more consistent with the grammatical structure and semantic meaning of the source sentence.

their model achieves sentiment change by generating an entirely new sentence which has little overlap with the original.


Before about 18 months of age, children seem to assume that everyone likes the same things as themselves, having difficulty understanding the subjective nature of preferences (e.g., [6]).

shortly after coming to recognize that different agents can maintain different preferences, children demonstrate a remarkably sophisticated ability to draw conclusions about the preferences of others from their behavior.


Furthermore, the recently proposed Google Knowledge Vault (KV) [5] considers a combination of PRA and a neural network model related to RESCAL for learning from large multi-relational datasets.

in KV both models are trained separately and combined only later in a separate fusion step, whereas ARE learns both models jointly what leads to the desired rank-reduction effect.


Memory must be allocated to hold these examples.

each iteration of the online algorithm only involves one random example which can then be discarded.


Both RNNs and CNNs can stabilize learning via weight sharing, therefore they are less prone to these perturbations.

fNNs trained with normalization techniques suffer from these perturbations and have high variance in the training error (see Figure 1).


Typical values for k are between 10 and 20.

as the "relevant" images that the user wishes to retrieve may not fit perfectly with the similarity metric designed for the search engine, the user may be interested in exploring other regions of the feature space.


The idea of mean-IMM is commonly used in shallow networks [19,20].

the contribution of this paper is to discover when and how mean-IMM can be applied in modern deep neural networks and to show it can performs better with other transfer techniques.


This is similar to QRNN with a filter size of k > 1.


In a separable convolution, the most expensive part is the pointwise convolution, so it has been proposed that this operation could also be grouped over sets of channels.

to maintain some connections between channels, it is helpful to add an operation mixing the channels together (Zhang et al., 2018).


Such interactions may explain the sometimes inconsistent findings in the literature; depending upon the task and the individual items in the stimuli, different levels of consistency of mapping can affect the results.

each of these points represent an average over a set of networks with one average consistency value.


This formulation leads to a linear forward model.

the ill-posed nature of the problem leads to non-unique solutions which must be distinguished by prior information, usually in the form of assumptions regarding the correlation between the sources.


A flash-convolution processor [4] allows a single 5¢5-pixel convolution operation in a single clock cycle by introducing a subtle memory access scheme.

for an N¢M-pixel image, it takes N¢M clock cycles to complete the processing.


This is harmless in some games, such as Pong where no reward is ever higher than 1 or lower than 1, but it is not satisfactory as this heuristic introduces specific domain knowledge that optimizing reward frequencies is approximately is useful as optimizing the total score.

the clipping makes the DQN algorithm blind to differences between certain actions, such as the difference in reward between eating a ghost (reward >= 100) and eating a pellet (reward = 25) in Ms. Pac-Man.


Their memory cells are associated with weights to allow continuous representation of the stack, in order to train it with continuous optimization scheme.

our solution is closer to a standard RNN with special connectivities which simulate a stack with unbounded capacity.


Note that the acceleration values are not smooth and are, therefore, least aided by the dynamics of the model.

adding acceleration to the model improves the prediction of position.


During training, the classifiers f i in the chain are trained to reduce the errors E(y i , ŷi ) by enriching the input vectors x with the corresponding previous true targets y <i as additional features.

at test time, f i generates samples ŷi or estimates P (ŷ i |x, ŷ<i ) where ŷ<i are obtained from the preceding classifiers Another key limitation of PCCs is that the classifiers f i are trained independently according to a fixed label order, so that each classifier is only able to make predictions with respect to a single label in a chain of labels.


Classical text line segmentation algorithms are mostly based on image processing techniques and heuristics.

some methods were devised using statistical models and machine learning techniques such as hidden Markov models [8], conditional random fields [21], or neural networks [24,31,32].


Further, it can be extended to learn the tasks' structures when they are unknown [3,11,12,13,14,15,16].

most regularization approaches are currently limited to imposing, or learning, tasks structures expressed by linear relations (see Sec.


In the neural net setting the expansion arriving at the output node is adjusted during training and our techniques for proving hardness results fail in this case.

we conjecture that the features learned from the k training examples cannot help to improve its average performance, provided its training algorithm is based on the Gradient Descent or Weight Decay heuristic.


"), a learning algorithm could of course simulate standard O(n log n) sorting algorithms; this number of queries is necessary and sufficient.

when using equivalence queries, the user must be presented with a complete ordering (i.e., a permutation π of the n elements), and the feedback will be a mistake in the proposed permutation.


Model-free reinforcement learning has achieved state-of-the-art results in many challenging domains.

these methods learn black-box control policies and typically suffer from poor sample complexity and generalization.


A small k will be very sensitive to noise and necessarily lead to many classes receiving a probability of zero, which is undesirable for our application.

if k is too large, samples from far outside the neighborhood of a will influence P k (y|a).


When the equality holds, the variances of PE α and PE α are asymptotically the same.

in general, the use of PE α would be more recommended.


Without finding similar images that match the input scene configuration, we would need to apply an object detector densely across the entire image for all object categories.

our model can constrain which object categories to look for and where.


This can be solved by a first order method like mirror descent algorithm for saddle point problems.

the primal and dual updates in algorithm 1 solve two separate maximization and minimization problems 1 .


Deterministic neural nets have been shown to learn effective predictors on a wide range of machine learning problems.

as the standard approach is to train the network to minimize a prediction loss, the resultant model remains ignorant to its prediction confidence.


As can be seen in Tables 1 and 2, ACV closely follows CV on the MNIST dataset.

the approximation of LOOCV based on influence functions [14] performs poorly in the regime where the model is significantly overfit and hence it cannot be used for effective model selection.


Enforcing multi-view consistency is sufficient to infer a consistent set of 2D keypoint positions (and depths) across different views.

this consistency alone often leads to a degenerate solution where all keypoints collapse to a single location, which is not useful.


Another approximate k-NN idea is in [1], one of the first k-NN approaches to use a priority queue of nodes, in this case achieving a 3-fold speedup with an approximation to the true k-NN.

these approaches are based on the notion that any points falling within a factor of (1 + ε) times the true nearest neighbor distance are acceptable substitutes for the true nearest neighbor.


A probabilistic model that defines a clique of size n has the ability to model any distribution over these random variables.

the task of learning and inference on such a model becomes computationally intractable.


MARL can be simply seen as independent RL, where each learner treats the other agents as part of its environment.

the strategies of other agents are uncertain and changing as training progresses, so the environment becomes unstable from the perspective of any individual agent and thus it is hard for agents to collaborate.


The resulting PDEs involve O(n) spatial variables.

our analysis studies the limit as the dimension n → ∞, with a constant step size.


large number of features and not enough samples.

inference is usually tractable since not all features are relevant to the learning task at hand.


Prior k-means clustering algorithms that determine the number of clusters present in the data have primarily involved a method for iteratively modifying k using various statistical criteria [13,14,18].

this work derives this capability from a Bayesian nonparametric model, similarly to the DP-Means algorithm [15].


Indeed, X Y-adaptive identifies the most important direction after few phases and is able to perform an allocation which mimics that of the oracle.

x Y and G do not adjust to the empirical gaps and consider all directions as equally important.


Importantly, presenting more data with the same statistics did not encourage the fitting of over-complicated model structures.

if data was generated by using more 'combos' (4 'doublets'), model learning converged to a model with a correspondingly higher number of latents (Fig.


BALD always outperforms random sampling and usually outperforms or obtains equivalent performance to MES.

bALD significantly outperforms MES in 9 cases, while MES is better than BALD in only 2 cases.


This is consistent with previous empirical findings in [19].

actually one can attain the optimal trade off by solving the following optimization problem: we should label more query if we want the bound to hold with a larger probability.


Each pair (v i , g i ) ∈ V then generates a number of examples in this extended binary problem equal to the number of arcs of the p-graph g i for a total of N i=1 |A(g i )| examples.

the set Z = {z a i } is linearly separable in the higher dimensional problem if and only if there exists a consistent solution for the original PLM problem.


The parameters are chosen by grid search over a small portion of data, and the parameters of the benchmark algorithms are fine-tuned (see Appendix F for details).

we set the discretization level δ = 0.05, the window size z = 3, the step size η k = (kδ/20 + 100) −1 , and the regularization coefficient ζ i,j ≡ ζ = 10 −8 .


Figure 5 (bottom) shows how LLC can smoothly interpolate between any two digits.

the first row interpolates between left and right slanting digits, the second between fat and thin digits, the third between thick and thin line strokes, and the fourth between having a larger bottom loop and larger top loop.


We have conducted comprehensive performance evaluations by testing our method under different circumstances.

to better understand the effect of augmenting logistic regression with semantic relatedness and use of overlapping-group-lasso penalty to enforce group level feature selection, we study the model adding only augmented logistic regression loss and adding only overlapping-group-lasso penalty separately, and compare with the APPLET method.


In contrast, the simplicity of our generative model allowed us to draw an analytical picture of the problem which in turn suggested a systematic approach for solving it.

we used methods from robust control to discover the fastest neural-like sampler for our generative model, and to study its structure.


We consider the problem of detecting humans and classifying their pose from a single image.

our goal is to devise a statistical model that simultaneously answers two questions: 1) is there a human in the image?


In this paper, we aim to address the missing piece in the literature: to recover a geometrically consistent and physically plausible 3D scene and jointly solve all three tasks in an efficient and cooperative way, only from a single RGB image.


It is worth noting that the best earned per-trial reward rates achievable by DBM and FBM are quite close.

as long as a softmax policy is being used, there is no disadvantage to incorrectly assuming environmental volatility as long as the assumed prior mean parameter is appropriately tuned.


coefficients In this section, we theoretically characterize the performance of affine policies for random instances of (1) for a large class of generative distributions including both bounded and unbounded support distributions.

we consider the two-stage problem where constraint coefficients A and B are i.i.d.


Average performance for different training/testing split proportions is in Figure 2  We next compare the performance on further benchmark datasets 1 by selecting training data via various biased sampling schemes.

for the sampling distribution bias on labels, we use P (s = 1|y) = exp(a + by)/(1 + exp(a + by)) (datasets 1 to 5), or the simple step distribution P (s = 1|y = 1) = a, P (s = 1|y = −1) = b (datasets 6 and 7).


The second condition states that the corresponding entries in the extended state variable y = [x T , u T ] are sufficiently distinguishable from each other.

if the trajectories corresponding to this group of state variables are observed, non of them can be well approximated as a linear combination of the others.


Estimating on-line the membrane potential of the presynaptic cell from its spiking history amounts to computing the posterior probability distribution, p (u t |s 1..t ).

since equations 1 and 3 define a hidden Markov model, the posterior can be written in a recursive form: the posterior at time step t, p(u t |s 1..t ), can be computed by combining information from the current time step with the posterior obtained at the previous time step, p(u t−1 |s 1..t−1 ).


We remark that we do not require the selection procedure to be variable selection consistent.

we do not require the selection procedure to only selected relevant features.


It is natural to ask if this is also true in an average sense for every y.

we know an information retrieval system based on DR maps always has a blindspot, but is this blindspot behaviour a typical case?


We experiment with 4 kernels from this dataset.

we chose the "pixel features", the "Pyramid Histogram Of Gradients", bio-inspired "Sparse Localized Features", and SIFT descriptors as our four views.


A very efficient and simple way to address this flaw is to use them in the context of randomization-based ensemble methods.

the core principle is to introduce random perturbations into the learning procedure in order to produce several different decision trees from a single learning set and to use some aggregation technique to combine the predictions of all these trees.


Influence of channels On the Cifar-10 dataset, we model SACNNs with different numbers of channels c (i.e., 8,16,32) to study its influence on the local structure representations learning.

we observe the following two tendencies from Figure 3 (b).


Denote p(i | A) the probability of choosing item i when faced with alternatives in the set A.

given two items i and j, and any two sets of alternatives A and B containing i and j, the axiom posits that the odds of choosing item i over item j are independent of the rest of the alternatives.


These two different splits differ in how close the seen classes are related to the unseen ones.

in the SCS-split setting, there exists one or more seen classes belonging to the same parent category of each unseen class.


We will refer to the function 1 − RI( Ŝ, S) as the Rand index, although strictly speaking the Rand index is RI( Ŝ, S), the fraction of image pixel pairs on which the two segmentations agree.

the Rand index is a measure of similarity, but we will often apply that term to a measure of dissimilarity.


In addition to specifying the constraint violation distribution π as we did for our previous bound, we will specify (implicitly) a particular choice of the violation budget θ.

we will consider solving the following SALP: It is clear that ( 6) is equivalent to (4) for a specific choice of θ.


However, there are some generative models that are more forgiving of these problems.

consider Yu and Dayan's [13] diffusion-jump model, which replaces Equation 1with with σ j σ d . Here, the weights usually diffuse as before, but occasionally (with probability π) are regenerated anew.


( 5), we develop a subgradient descent approach for solving the optimization problem.

in each iteration of subgradient descent, we compute the gradient L(p, γ) with respect to p and γ as follows where , a SVM solution to the combined kernel K(p).


The voted perceptron has some advantages in ease of implementation, and efficiency.

(the voted perceptron algorithm is more stable numerically, as it does not need to compute a partition function. )


While such estimates are theoretically fundamental, they can be completely useless in the presence of outliers.

direct covariance or inverse covariance estimation and its resulting PCA are very sensitive to outliers.


. . , K L ) of n into the L groups with no two groups differing in size by more than 1.

theorem 4 also applies to non-constant cost functions because the number of groups is fixed.


In the medical diagnosis application, we can define F t (S) to be proportional to the number of candidate diseases which are eliminated after performing the set of tests S on patient t. If we assume that a particular test result always eliminates a fixed set of candidate diseases, then this function is submodular.

this objective is the reduction in the size of the version space [5,6].


The grid size was selected for economy of computation.

the grid lower and upper bounds for were ad-hoc, though they were inspired by the literature, while for the same for C, we enlarged the lower range to focus on the region of the parameter space where the stability-based bounds have a better chance to be effective: the stability-based bounds grow with C in a linear fashion, with a coefficient that was empirically observed to be close to one.


We propose an upper bound for the expected total reward U * (T, B) of the oracle by relaxing the hard constraint to an average constraint and solving the corresponding constrained LP problem.

let p j ∈ [0, 1] be the probability that the agent takes action k * j for context j, and 1 − p j be the probability that the agent skips context j (i.e., taking action A t = 0).


A well-known conditional exponential model is the linear-chain conditional random field (CRF) [6] in which x and y both have sequence structures.

where f i (y j , y j−1 , x) is the score of the i-th feature at position j.


That is, P (y = 1|x i = r) > P (y = 1|x i = r ) if r > r . • The monotonicity is non-trivial in that P (x i ) has positive probability for more than one feature value.

there exists r > r and > 0 such that P (x i = r), P (x i = r ) > .


More recently,  showed that by averaging the smoothed gradient estimates and still using the self-concordant barrier as regularization, one can achieve a regret of e O T 5/8 .

denote by ḡt = 1 k+1 P k i=0 b g t i the average of the past k + 1 incurred gradients, where b g t i = 0 for t i  0.


Then we have inf Now, we claim that the mapping u � → regret � p (pred * � (u)) is continuous at u = u p . To see this, suppose the sequence {u m } converges to u p . Then we have The last equality holds by definition of pred * � .

it is easy to see the term on the right goes to zero as u m converges to u p . Thus regret � p (pred * � (u m )) converges to regret � p (pred * � (u p )) = 0, yielding continuity at u p . this implies ∃δ > 0 such that where the last inequality holds since p � ψ * � (u) is a strictly convex function of u and u p is its unique minimizer.


In terms of query trees, this is the average depth of a leaf chosen according to π.

let T be any tree whose leaves include the support of π.


if E(X i ) are approximated by p i (i = 1, . . . , N ), then E(Y ) can be approximated by f ( N i=1 θ i p i ).


The posterior mean µ t (x, ✓) and variance 2 t (x, ✓) conditioned on the previous observations (x 1 , ✓ 1 , y 1 ), ..., (x t 1 , ✓ t 1 , y t 1 ) are computed analogously to (5) [20].

a robust optimization formulation in this setting is to seek x that solves we seek to find a configuration x that is robust against any possible parameter vector ✓ 2 ⇥.


As computational objects, they explicitly distinguish between the sampler for the random variables from their joint distribution, and the assessor of their joint density.

a CGPM is required to sample/assess the joint distribution of a subset of output variables x [r,Q] conditioned on another subset x [r,E] , and marginalizing over Listing 1 Computational interface for composable generative population models.


For the closed convex cone W(x) := conv cl cone(R − x), and for any x, x ′ ∈ R, In other words, Next, we formulate the test as a convex optimization program, when a finite number of samples are given.

using the samples in X ⊂ R instead of all the points in R, we can define an approximation of W(x) as which is the tangent cone (also known as the descent cone) at x with respect to the data hull conv(X).


This idea has been proposed by [9] and [10].

bubeck and Lee [9] suggested to keep all the balls from past iterations and then compute the minimum enclosing ball of their intersection.


First, we observe that the random initialization C 1 , C 2 is necessarily somewhat biased, in the sense that C 1 and C 2 never divide P 1 exactly into two halves.

||C 1 ∩ P 1 | − |C 2 ∩ P 1 || ≥ N − 1 2 − with high probability.