While [27] can generate different results driven by the input noise, the learned networks are very likely to be trapped in local optima.

the noise is marginalized out and thus fails to drive the network to generate large visual variations.


As with humans, when our agent visits a new city we would expect it to have to learn a new set of landmarks, but not have to re-learn its visual representation, its behaviours, etc.

we expect the agent to take advantage of existing visual features (convnet) and movement primitives (policy LSTM).


Local w: We opt to use w's that correspond to local m×m blocks of pixels.

we treat x as a h/2 × w/2 × 10 tensor and allow w to operate over any m × m × 1 patch in a single channel of x.


We make our FBN to only depend on text (and not on the image), making its learning task easier.

our FBN performs the following computation: f sent LSTM, dim 256 f phrase linear+mean pool f f bn 3-layer MLP with dropout +3-way softmax Here, w c t and w f t denote the one-hot encoding of words in the sampled caption and feedback sentence, respectively.


If both players -learner and adversary -behave rationally in the sense of minimizing their personal costs, then the Nash equilibrium is the strongest available concept of an optimal choice of a v . A Nash equilibrium is defined as a pair of actions a * = [a * +1 , a * −1 ] such that no player can benefit from changing the strategy unilaterally.

for both players v ∈ {−1, +1}, The Nash equilibrium has several catches.


Each one of these corresponds to a trace-norm regularized learning problem, under a uniform sampling distribution (in the corresponding submatrix) of a noisy low-rank "orthogonal" matrix, and can therefor be learned with Õ(kn A ) and Õ(kn B ) samples respectively.

õ(kn) samples should be enough to learn both inside A and inside B.


Multinomial DBMs: To allow DBMs to express more information and introduce more structured hierarchical priors, we will use a conditional multinomial distribution to model activities of the toplevel units.

we will use M softmax units, each with "1-of-K" encoding (so that each unit contains a set of K weights).


This model is closely related to the IRM (Kemp et al., 2006) and IHRM (Xu et al., 2006), where rows and columns are first clustered using a CRP prior, then each relation R ij is conditionally independent from others given the clusters that row i and column j belong to.


$# ¡ ¡ &% (' ¢) Extension of approximate linear programming to the average-cost setting requires a different algorithm and additional analytical ideas.

our contribution can be summarized as follows: Analysis of the usual formulation of approximate linear programming for averagecost problems.


The present paper attempts to make progress on these challenges.

we make the following contribution: • We propose a class of 'optimistic' approximations to the Gittins index that can be computed with significantly less effort.


We refer to our approach as context selection for exponential family embeddings (CS-EFE).

we introduce a binary hidden vector to indicate which elements the target depends on.


The key idea of SITE is to map the original pre-treatment covariate space X into a latent space Z learned by deep neural networks.

sITE attempts to enforce two special properties on the latent space Z, including the balanced distribution and preserved similarity.


The foundation of our development is the fact [1] that any tree-structured distribution V h£ @ pi e £ ¦ can be factored in terms of its max-marginals.

for each node C 3¥ , the corresponding single node max-marginal is defined as follows: a£ g & ¡ k} t ¡ £¢ ¥¤ §¦ ¨¢ © ¦ © V W£ @ ¦ i e f£ n In words, for each w 5 , a£ n u is the maximum probability over the subset of configurations @ ¦ with element ¦ fixed to .


, the random variables {Y uv } 1≤u<v≤n are independent.

• • The function f , when regarded as an integral operator, has finite spectrum with rank d. , where q k are orthonormal L 2 -integrable basis functions.


Continuing to abuse the notation from section 2, we can say that the LFW database only samples a small subset of G, which is now the set of all transformations that occur in LFW.

for any two images in LFW, x and x , only a small (relative to |G|) subset of their orbits are in LFW.


. . , k f D over K, and mean / covariance functions µ / Σ.

samples over the belief can be drawn using an infinite vector Ω of i.i.d.


In this paper we focus on team Markov games, that are Markov games where each agent receives the same expected payoff (in the presence of noise, different agent may still receive different payoffs at a particular moment.).

there are no conflicts between the agents, but learning the game structure and learning to coordinate are nevertheless highly nontrivial.


. . , β J2 J −1 is the vector of wavelet coefficients, and the rows of W contain the corresponding wavelet basis functions evaluated at x i = i/n.

w is an orthogonal matrix with W li ≈ √ nψ jk (i/n), or W li ≈ √ nφ jk (i/n), for some l; the √ n factor is due to convention in the literature and software implementation.


2 that our goal is to resolve the ambiguity between the true colors x(n) and the illuminant m only up to scale.

we need only estimate the illuminant chromaticity m and true chromaticities x(n) from the observed image v(n), which we can relate from (2) as A key property of natural illuminant chromaticities is that they are known to take a fairly restricted set of values, close to a one-dimensional locus predicted by Planck's radiation law [16].


The Dirichlet distribution draws are constituted via Sethuraman's construction [14]; this allows us to place gamma priors on α V and α T , while retaining conjugacy, permitting analytic Gibbs' sampling (we therefore get a full posterior distribution for all model parameters, while most LDA implementations employ a point estimate for the document-dependent probabilities of topics).

the following hierarchical construction is used for draws from Dir(α V /V, .


7), we can modify Ω as long as exp tr A Ω is changed by the same factor for any A.

we can subtract c from an entire column in Ω; this will affect the result of exp tr A Ω by a factor of exp c, which is independent of A and hence will be subsumed by the normalizer Z.


While this may be appropriate for some settings, the assumption of a temporal ordering restricts the contexts in which such models can be applied.

it is much more restrictive than the assumption of independent sampling that underlies most statistical language models.


If the edge weights are determined by a single similarity measure, then it hard if not impossible to capture all relevant aspects of similarity when using a single similarity measure.

in the case of visual object matching it is beneficial to integrate different similarity measures that focus on different visual representations.


Our goal then is to infer a low-dimensional latent state space model in which optimal control can be performed.

we seek to learn a function m, mapping from high-dimensional images x t to low-dimensional vectors z t ∈ R nz with n z n x , such that the control problem can be solved using z t instead of x t : where ω accounts for system noise; or equivalently z t ∼ N (m(x t ), Σ ω ).


The main contribution of this paper is in providing general convergence guarantees for kernel-based quadrature rules in misspecified settings.

we make the following contributions: • In Section 4, we prove that consistency can be guaranteed even when the integrand f does not belong to the assumed RKHS.


We first conducted experiments by constructing 15 transfer tasks for the 15 subjects.

for each subject, we used the first image matrix as the target matrix and used each of the remaining 10 image matrices as the source matrix each time.


In most cases, functional data are simply treated as high-dimensional vectors for which the standard methods apply.


This paper investigates and pushes forward various ways to enforce orthogonality regularizations on training deep CNNs.

we introduce three novel regularization forms for orthogonality, ranging from the double-sided variant of standard Frobenius norm-based regularizer, to utilizing Mutual Coherence (MC) and Restricted Isometry Property (RIP) tools [17][18][19].


Particularly, X consists of a set of points H l,m in a grid-like manner, and the sets Z i corresponding to S i .


Most algorithms that are being used in practice minimize a convex surrogate loss.

instead of minimizing the number of mistakes on the training set, the algorithms minimize , where ℓ : R → R is a convex function that upper bounds the 0 − 1 loss.


On the other hand, batch algorithms enjoy faster convergence rates, though their per-iteration cost may be prohibitive.

second order methods attain quadratic rate, but constructing the Hessian matrix requires excessive computation.


Hence, the DAE cannot effectively learn mean-shift vectors for locations that are closer than σ to the natural image manifold.

our DAE does not produce meaningful results for input images that do not exhibit noise close to the DAE training σ.


We use the shorthand A −1 I,J = (A −1 ) I,J .


In this model, an algorithm to minimize a convex function can make queries to a first-order "oracle," and the complexity is defined as the smallest error achievable using some specified minimum number of queries needed.

the oracle is queried with an input point x 2 C from a convex domain C, and returns an unbiased estimate of a subgradient vector to the function f at x.


The first goal is achieved by fixing the distances between words with similar frequency vectors to their original values.

if p j and p k lie within some small neighborhood of each other, then the corresponding element D jk in the distance matrix is fixed to the value | p j − p k | 2 .


In contrast, since 1− •C(s) ≥ 1−(1−e −10 )1 = e −10 > 0, MWU successfully converges to equilibrium.

for the same learning rates, MWU e exhibits chaotic behavior whereas MWU converges to Nash equilibrium.


In this work, we propose a unified framework for learning to detect objects in images from data with heterogeneous labels.

we consider the case of image collections for which we would like to predict bounding box localizations, but that (for a significant proportion of the training data) only image level binary annotations are provided indicating the presence or absence of an object, or that weak indications of object location are given without a precise bounding box annotation.


Call this set Ω : For technical reasons, we also need to look at realizations that are good up to time n−1.

for each n, define Crucially, this is F n−1 -measurable.


Consider a canonical Gaussian process where n = 2 and T {t ∈ R 2 : t 2 = 1}.

the process {X t } t∈T can be reparameterized according to the phase of each point t ∈ T : the random variable X t can also be denoted as X φ , where φ ∈ [0, 2π) is the phase of t. φ is the unique number in [0, 2π) such that t = (sin φ, cos φ).


Then Q * satisfies Definition 1, optimal local privacy, and moreover, Before continuing, we give a more intuitive understanding of Proposition 1.

concavity implies that for a, b > 0, log(a we have for any distribution P on X ∈ [−1, 1] d that (in natural logarithms) We now consider the case when X Here the arguments are slightly more complicated, as the coordinates of the random variables are no longer independent, but Theorem 3 still allows us to explicitly characterize the saddle point of the mutual information.


We tested if the model can incorporate the knowledge of a new domain with generative replay.

we sequentially trained our model on classifying MNIST and Street View House Number (SVHN) dataset [25], and vice versa.


The margin vector coefficients change value during each incremental step to keep all elements in D in equilibrium, i.e., keep their KT conditions satisfied.

the KT conditions are expressed differentially as : jES where a e is the coefficient being incremented, initially zero, of a "candidate" vector outside D. Since 9i == 0 for the margin vector working set S = {Sl, .


Exponential Families One of the advantages of our setting is that it applies to any convex loss function, as long as there is an efficient way of computing the gradient.

we can use it for cases where we are interested in modeling g(w|x) is the conditional log-partition function.


Chain graph models are defined over chain graphs that contain a mixed set of directed and undirected edges but no partially directed cycles.

we study the problem of learning the structure of Gaussian chain graph models in a high-dimensional setting.


For these types of studies, we tolerant some false positives but not false negatives.

it is acceptable that an edge presents but the two genes corresponding to this edge do not really interact with each other.


An equivalent representation of the prior is: where u j and v j are latent variables introduced for each dimension j.


In contrast, we address this issue by showing how to analytically differentiate through the fixed point of a nonlinear MPC solver.

we compute the derivatives of an iLQR solver with a single LQR step in the backward pass.


To address this, we employ the alternating direction method of multipliers (ADMM) [18; 19; 20] for parameter agreements.

we encourage θ i• and θ •i to agree with their mean u i• , by introducing an additional term to the Lagrangian L where u i• is updated as an independent parameter.


This notion of identifying bottlenecks is similar to the well-known max-flow, min-cut theorem.

for a graph whose edge weights represent maximum flow capacities between pairs of vertices, instead of the current conditional transition probabilities, the bottleneck edges can be identified as precisely those edges across which the maximum flow is equal to their maximum capacity.


Thus, the screening performance of SIS strongly relies on the structure of Σ.

the (asymptotically) necessary and sufficient condition for SIS being strong screening consistent is Σ satisfying the RDD condition.


It parallels the intractable Algorithm 1, but replaces the selection of optimal subsets by a greedy algorithm.

due to the approximate nature of the greedy algorithm, we allow the algorithms to pick sets slightly larger than k. GREEDI is a two-round algorithm that takes the ground set V , the number of partitions m, and the cardinality constraints l (final solution) and κ (intermediate outputs).


Given an arbitrary probability distribution p(x), in order to obtain a good approximation p(x; θ) in the t-exponential family, we minimize the relative t-relative entropy ( 19) Here q(x) = 1 Z p(x) t denotes the escort of the original distribution p(x).

since p(x; θ) = exp t (�Φ(x), θ� − g t (θ)), (25) using the fact that ∇ θ g t (θ) = E q[Φ(x)], one can take the derivative of ( 24) with respect to θ: the approximate distribution can be obtained by matching the escort expectation of Φ(x) between the two distributions.


We define a notion of partial order on such hierarchies, and identify those events whose probability as computed using different levels of the hierarchy does not agree.

we are interested in those events that receive high probability at more general levels (for example, the system is certain that the new example is a dog), but low probability at more specific levels (in the same example, the system is certain that the new example is not any known dog breed).


The technical conditions required for submodularity of ( 6) are due to averaging of the kernel similarity scores -as the average requires a division by the cardinality |S|.

the analogue of ( 6) which replaces all the averages by sums (i.e.


Also, as discussed in [21], the condition requires that signalto-noise ratio (SNR) goes to infinity.

if we define the SNR as then under Assumption 1(e), it can be easily checked that Since x has k nonzero components, x 2 ≥ kx 2 min , and therefore condition (10) requires that SNR → ∞.


Given the past K fixations, the next fixation is chosen to minimize the model entropy of the edgelet orientations.

, where the entropy of a distribution P(x) is defined as .


For all i ∈ A and n ∈ N, we define T n,i n−1 =0 1{I = i} to be the number of samples of arm i before time n. We will show that under TTEI with parameter β, lim n→∞ T n,1 /n = β.

the algorithm asymptotically allocates β proportion of the samples to true best arm.


Let A k = UΛV be the SVD of where max is elementwise.

the shrinkage operator shifts the singular values down, and truncates any negative values to zero.


However, with more (30%) poisoned data, the attacker can subvert the outlier removal to obtain stronger attacks, increasing test error on MNIST-1-7 to 40%-much higher than the upper bound of 7% for the oracle defense.

defenses that rely on the (potentially poisoned) data can be much weaker than their data-independent counterparts, underscoring the need for outlier removal mechanisms that are themselves robust to attack.


Before we describe our efficient algorithm in detail, we require the definition of a unimodal function.

a unimodal discrete function is monotonically non-decreasing in the interval [1, k] and monotonically non-increasing in the interval [k, p].


As a consequence, at each step t the agent might either behave greedily (exploitation) with probability 1f, by selecting the best action a; with respect to the Q-value functions, a; = argm axaQt (St , a) , or resort to uniform random action selection (exploration) with probability equal to f. The update of the eligibility trace depends on whether the robot selects an exploratory or an exploiting action.

the vector et changes according to (we start with eo = 0) where 0 ::; >.. ::; 1 is a trace-decay parameter [19], and f( Sf) is the CA3-CA 1 vector activity.


To transfer a learned QMDP-net policy, we simply expand its planning module by adding more recurrent layers.

we trained a policy in randomly generated 30 ⇥ 30 grid worlds with K = 90.


Next we construct a tracking model starting a neurally plausible architecture to compute saliency, and show that it can be implemented with widely accepted models of cortical computation.

the model is based on a feature selection mechanism akin to the well known phenomenon of feature-based attention in MT.


Let M denote the set of strategies such that an element µ ∈ M is a function µ : N × S → + .

we define the value function V (x, s|µ , µ) to be the expected net present value for an agent at state x when its competitors' state is s, given that its competitors each follows a common strategy µ ∈ M, and the agent itself follows strategy µ ∈ M. where i is taken to be the index of an agent at state x at time t, and the subscripts of the expectation indicate the strategy followed by agent i and the strategy followed by its competitors.


[28,29] 3 Learning Architecture In this section, we describe the basic modules and the architecture of the learning algorithm.

our learning modules use a combination of temporal slowness and a non-degeneracy principle similar to orthogonality [30,31].


Suppose we would like to perform a sequence of related experiments.

in Section 6.1 we consider the case of vaccine design.


. . , z n+p ) = λβ for some λ ∈ Z * whp as p → +∞.

using that ẑ is guaranteed to satisfy ẑ 2 ≤ 2 2n+p 2 z 2 for all non-zero z in the lattice, it can be derived that ẑ has a "small enough" L 2 norm and therefore indeed satisfies the desired property whp as p → +∞.


In addition to offering a novel theory for place cell remapping, our model also provides an explanation for what determines place field sizes.

a given place cell will be active in the regions of space that are consistent with both with the grid representation (structure) received by that place cell and the sensory experience coded for by that place cell.


the average of the difference between each pair correlations of aligned functional activities across subjects (γ m for last three iterations).

dHA will be finished if the error rates in the last three iterations are going to be worst.


Moreover, these distributions afford the added flexibility of modeling under-and over-dispersed spike counts, a biologically significant feature of neural spiking data [4].

while the Poisson has unit dispersion (its mean is equal to its variance), the binomial distribution is always under-dispersed, since its mean always exceeds its variance, and the negative binomial is always over-dispersed, with variance greater than its mean.


we consider the case for which there is a document D (t) j of words associated with the jth column at time t; in our example below, this will correspond to the jth piece of legislation in year t. It is possible that we may have documents associated with the matrix rows as well (e.g., speeches for the ith legislature), but in our model development (and in our examples), documents are only assumed present for the columns.


Motivated in part by this observation, in recent years boosting algorithms that work by constructing branching programs over the weak classifiers (note that this is in contrast with AdaBoost, which constructs a single weighted sum of weak classifiers) have been developed and shown to enjoy some provable noise tolerance.

the algorithms of [KS05,LS05] have been shown to boost to optimally high accuracy in the presence of random classification noise when run with a random classification noise tolerant weak learner.


For example for degree k = 2 we obtain: where w has been rewritten as w 1 ∈ R D×D , w 2 ∈ R D×D and w 3 ∈ R D×D .

the ranking order of documents d given a fixed query q is independent of w 1 and the value of the term with w 3 is independent of the query, so in the following we will consider models containing only terms with both q and d. we will consider the following degree k = 2 model: where W ∈ R D×D , and the degree k = 3 model: Note that if W is an identity matrix in equation (1), we obtain the cosine similarity with tf-idf weighting.


PASCAL-VOC) and can better validate the necessity of global symbolic reasoning.


Hence, it is not easy to specify sum-product networks by hand.

fully connected networks typically violate those conditions.


Furthermore, since Equation ( 1) is fully differentiable, the Euclidean gradient can easily be derived using standard calculus.

the Euclidean gradient ∇ E = ∂L(θ) ∂d(θ,x) ∂d(θ,x) ∂θ depends on the gradient of L, which we assume is known, and the partial derivatives of the Poincaré distance, which can be computed as follows: Let α = 1 − θ 2 , β = 1 − x 2 and let γ = 1 + 2 αβ θ − x 2 .


In this paper we investigate a simple approach to determining approximately .optimal policies based on a simple direct linear programming approach.

the idea is to approximate the optimal value function by formulating a single linear program and exploiting structure in the MDP and the value function approximation to solve this linear program efficiently.


In this setup, any effect of ŷ(t) i on x i ] is feedback.

a simple way to constrain this relationship is using a linear model x we assume that x is perturbed by an amount that scales linearly with ŷ(t) i .


This is motivated by a recent work that implicitly generates sparsifiers for multistep random walks [4].

we sample the rows of X, the KRP of A and B, using products of quantities computed on the corresponding rows in A and B, which provides a rank-1 approximation to the optimal importance: the statistical leverage scores.


We address this issue by showing that the solutions x (k) , k = 0, • • • , stay close to the segment between x gt and x gt + (1 − p) a+b 2 1.

for points on this segment, we can leverage the independence of t ij to derive the following concentration bound for one step of TranSync: Lemma 3.1 Consider a fixed observation graph G. Let r = (a+b)p (a+b)p+2(1−p)δ and d min be the minimum degree of G. Suppose d min = Ω(log 2 (n)), and p + r(1 − p) = Ω(log 2 (n)/d min ) .


Now consider a protocol in which users send randomized responses of their corresponding bits.


( 1), the marginal likelihood becomes To obtain simple and stable solutions when estimating m and K from the data, we consider point estimates of the parameters m and K, based on a penalized likelihood approach with conjugate priors.

2 The conjugate prior for mean m and covariance K of a multivariate Gaussian is the so-called Normal-Wishart distribution [6], which decomposes into the product of an inverse Wishart distribution for K and a Normal distribution for m, the prior for the Gram matrix K is given by an inverse Wishart distribution with scalar parameter α > 1/2(N − 1) and U being a symmetric positive-definite matrix.


We emphasize that even when restricted to a simpler HMM model, learning on the full data via SVIHMM attains similar results to that of [27] with significant gains in efficiency.

our SVIHMM runs require only under an hour for a fixed 100 iterations, the maximum iteration limit specified in the DBN-EM approach.


This problem can be addressed by introducing a discriminator function and constructing a mini-max loss function.

multiplying (9) with a function f (s 0 ) and averaging under s 0 ⇠ d ⇡0 gives Following Theorem 1, we have w / w ⇡/⇡0 if and only if L(w, f ) = 0 for any function f . This motivates us to estimate w ⇡/⇡0 with a mini-max problem: where F is a set of discriminator functions and z w := E s⇠d⇡ 0 [w(s)] normalizes w to avoid the trivial solution w ⌘ 0.


Second, in unfavorable settings in which the true positive rate is low, it can be shown that the true positives are distributed in such a way that precludes graph connectivity issues (see Section D.3 for more details).

in the random-sample model, for each subspace S t , there is a matrix A sub defined below, whose support is contained within the true positive support of the output of CSC(β) for β ∈ (β L , β U,ρ ).


Furthermore, to obtain the desired bounds we need to extend the usual tools for analyzing spectral methods to the current setting.

these tools need to be adapted to the agnostic settings where there is no underlying true WFA.


These questions fall in the broader inquiry of understanding the joint convergence of no-regret online learning algorithms, an inquiry that lies at the heart of game-theoretical learning for well-grounded reasons: Specifically, studying whether the process converges at all provides an answer as to the stability of the joint learning dynamics, while studying whether it converges to a Nash equilibrium provides an answer as to whether the joint learning dynamics lead to the emergence of rationality.

for the latter point, if, when agents adopt an online learning algorithm the joint action converges to a point that is not a Nash equilibrium, each agent would be able to do better by unilaterally deviating from that algorithm.


The training involves using a second neural network, the critic, to discriminate between real samples and the generator output.

[2,7] introduces a critic built around the Wasserstein distance between the distribution of true images and generated images.


Since all computational strategies rely on kernel evaluation, we prefer the dual approach.

(4) allows an unconstrained optimization and the nonsmooth regions in 1 or 2 can be easily approximated by smooth surrogates [33].


"Gaussian scale mixtures" (GSM) have been proposed to model even larger neighborhoods of wavelet coefficients.

very good denoising results have been obtained by including within subband neighborhoods of size 3 × 3 in addition to the parent of a wavelet coefficient [3].


For any µ ≥ 0, if w ≤ µ then the following set is robustly positively invariant (x(k) ∈ X , ∀k ≥ 0): • If the activation is ReLU, then the network is Globally Input-Output practically Stable.

∀k ≥ 0 we have: The constant ζ ≥ 0 is the norm ball radius for x(0) − x.


Besides, we also use the label precision in each mini-batch, i.e., label Precision = (# of clean labels) / (# of all selected labels).

we sample R(T ) of  small-loss instances in each mini-batch, and then calculate the ratio of clean labels in the small-loss instances.


To deal with high-dimensional feature spaces we used hashing [20].

for the TREC dataset we used 2 18 feature bins and for the proprietary dataset we used 2 24 bins.


Compared with the raw word space which is normally extremely high-dimensional and sparse, it is more reasonable to characterise the clustering structure in the latent feature space-the empirical latent topic distributions as in the MedLDA [16].

we summarise the topic distribution of document i by x i ∈ R T , whose t-th element is 1 Ni Ni l=1 I(z il = t).


The idea has been used in a number of recent developments of spectral methods for latent variable models [1,2,10].

we first whiten the second-order tensor M 2 by finding a matrix W ∈ R V ×k such that W M 2 W = I k . This whitening procedure is possible whenever the topic distribuction vectors {µ i } k i=1 are linearly independent (and hence M 2 has rank k).


Crucially, the only operation on M performed in each iteration of CG is a matrix-vector multiplication (MVM) with M . Since M = K + σ 2 I, speeding up MVM with M is critically dependent on our ability to perform fast MVM with the kernel matrix K . We can apply the algorithm from Section 3 to perform fast MVM.

observe that the i-th row of K is given by k i = [K(x i , x 1 ), .


The neural network takes as input the global embedding vector ρ v , and it outputs the group-specific embedding vectors ρ (s) v . The mathematical expression for ρ (s) v for a feed-forward neural network and a residual network is respectively given by where we have considered the hyperbolic tangent nonlinearity.

the main difference between both network architectures is that the residual network focuses on modeling how the group-specific embedding vectors ρ v . if all weights were set to 0, the feed-forward network would output 0, while the residual network would output the global vector ρ (0) v for all groups.


We repeat iterations only for each word within a document.

we update parameters from an arriving document and discard the document after doing l iterations.


This bound is tight for Lasso, in the sense that the right hand side cannot be improved except for the constant.

the factor O( √ kλ) cannot be removed using Lasso -this can be easily verified with an orthogonal design matrix.


The proof of this result relies on convex analysis and the geometry of exponential families [15,16].

in any minimal exponential family, there is a one-to-one correspondence between exponential parameters (θ α in our notation) and mean parameters (µ α = E[φ α (X)]).


We use similar idea for solving (3) and refer the corresponding algorithm as SymANLS.

at the k-th iteration, SymANLS first updates U k by V k is then updated in a similar way.


To alleviate this problem, in this paper, we consider a semiparametric model for pairwise comparison which incorporates a broad class of distributions of the pairwise measurements, and contains various popular parametric models, such as the Bradley-Terry-Luce model and the paired cardinal model (Gopalan et al., 2014), as special cases.

we assume that the pairwise measurements follow a natural exponential family distribution that is parametrized by a underlying score matrix.


But MW updates do converge to the boundary, so it is not clear how to derive a tight lower bound on this sum.

we cannot rule out the possibility that MW updates converge to a subgame NE (a subgame is obtained from the original game by removing at least one strategy of some player), and if this happens, V i (t) converges to zero.


An immediate conclusion of this result provides a proof of a probabilistic version of properties (i) and (ii) stated above for energies drawn from this distribution.

the global minima of E(X) over DS coincide with those over Π n with high probability.


But this reduction to the matrix setting allows for speedups using a variety of tools.

sketching [6,25,27] or iterative sampling [22,9] lead to routines that run in input sparsity time: O(nnz) plus the cost of solving an O(r log n) sized least squares problem.


Imagine that the procedure (2) of LOC-ALGO is done with an iteration parameter t ∈ Z + .

then for each vertex v ∈ V, we assign the largest iteration number t such that the chosen ball R at the iteration t contains w. Clearly, this is well defined algorithm is run till each node is chosen as the 'ball center' at least once.


More interestingly, when we multiply two PSDD factors, the parameters of the inputs may not persist in the output PSDD.

the PSDD only maintains enough parameters to represent the resulting distribution, which further explains the size differences.


However, only little prior work targets MP variants with non-negativity constraint [5,38,52].

the least-squares objective was addressed and no rigorous convergence analysis was carried out.


Finally, in Section 4 we present some insight obtained from our asymptotic analysis.

in the high-dimensional limit, the original coupled dynamics associated with the algorithm will be asymptotically "decoupled", with each coordinate independently solving a 1-D effective minimization problem via stochastic gradient descent.


It is based on the idea of learning to discriminate between noisy and ground truth images.

we consider approaches where the regularization functional is given by a neural network Ψ Θ with network parameters Θ.


The first stochasticity arises from a stochastic estimate of the objective by random mini-batches.

the batch learning needs to scan the full dataset to compute subgradients, which is often too expensive to deal with large-scale datasets.


In general, the goal of latent variable models is to discover statistical structures (factors) latent in the data, often with dimensionality reduction, and thus to generalize well to unseen examples.

our choice of Gamma-Poisson (GaP) is theoretically as well as empirically motivated, as we elaborate in Section 2.2.


We confirm the very fact that averaging does not stand Byzantine attacks, while Krum does.

we report on attacks by omniscient adversaries -aware of a good estimate of the gradient -that send the opposite vector multiplied by a large factor, as well as attacks by adversaries that send random vectors drawn from a Gaussian distribution (the larger the variance of the distribution, the stronger the attack).


We evaluate our algorithms on several synthetic and real data sets.

we use our approach for density estimation for acceleration data, motivated by an application in earthquake detection using mobile phones.


However, the bound has also a significantly smaller log determinant ln |M T | on the resulting matrix M T , due to the construction of φ t,k via A −1/2 ⊗ .

when the graph is very dense, the log determinant in GOB.Lin is a factor n smaller than the corresponding term for the n independent bandit case (see, e.g., [8], Section 4.2 therein).


This effect is exploited by the simple transcriptional network shown in Figure 5B, in which the output from each DNA switch activates the same DNA switch itself, and mutual inhibition is achieved by competition for RNAP.

we have switch templates D ii with fixed thresholds set by I i , and D ii produces A i as its output RNA.


Once the SSDAs are trained, we construct a new training set that pairs features extracted from the hidden layers of the SSDAs with optimal column weights.

for each image, a vector φ = [f 1 ; ...; f C ] is built from the features extracted from the hidden layers of each SSDA, where C is the number of columns.


The basic principle behind collapsed sampling is that we can reduce the variance of an estimator by making part of the inference exact.

that is, suppose we partition our variables into two sets: X p , and X d . In collapsed importance sampling, the distribution of variables in X p will be estimated via importance sampling, while those in X d will be estimated by computing exactly P (X d |x p ) for each sample x p . suppose we have some function f (x) where x is a complete instantiation of X p [ X d , and a proposal distribution Q over X p . Then we estimate the expectation of f by (1) on samples {x p [m]} M m=1 drawn from a proposal distribution Q.


One important property of this algorithm is that just like MCMC-MLE, it can be shown to asymptotically converge to the maximum likelihood estimator θ * .

3 for fully visible discrete MRF's, if one uses a Gibbs transition operator and the learning rate is set to is the largest magnitude of the gradient, and C 1 is the maximum variation of φ when one changes the values of a single component only: The proof of convergence relies on the following simple decomposition.


In fact, our distribution measures framework encompasses optimization objectives for the expected, robust, and percentile MDP problems as well as for VaR and CVaR.

if µ is the Lebesgue measure (i.e., a uniform density over the unit interval), all percentiles are equally weighted and the µ-robust policy will optimize the expected cumulative reward over the distribution P(M).


Based on the semantic concepts such as right-next-to, holding, etc., it generates the positions for arranging latent distributions of two children nodes.

we first sample a tuple l = (x, y) from p(l|z s ), indicating the offsets between two children's bounding boxes in latent space, where z s ∼ p(z s ) and p(z s ) is generated by our concept mapping operator τ concept taking the relation word of layout node.


The purpose of our work is to incorporate both nonnegativity and sparseness into PCA, maintaining the maximal variance property of PCA.

the goal is to find a collection of sparse nonnegative principal vectors spanning a low-dimensional space that preserves as much as possible the variance of the data.


In this section we show that in the mortal multi-armed bandit setting, the regret per time step of any algorithm can never go to zero, unlike in the standard MAB setting.

we develop an upper bound on the mean reward per step of any such algorithm for the state-aware, budgeted death case.


the Möbius addition of x and y in D n c is defined as when c = 0, one recovers the Euclidean addition of two vectors in R n . Note that without loss of generality, the case c > 0 can be reduced to c = 1.


We will run no-regret dynamics on this zero-sum game, where at every iteration t = 1, . . . , T , the adversary will pick a distribution over functions and subsequently the learner picks a solution x t . For simpler notation we will denote with w t the probability density function on [m] associated with the distribution of the adversary.

w t [i] is the probability of picking function L i 2 L. The adversary picks a distribution w t based on some arbitrary no-regret learning algorithm on the m actions in L. For concreteness consider the case where the adversary picks a distribution based on the multiplicative weight updates algorithm, i.e., Subsequently the learner picks a solution x t that is the output of the ↵-approximate stochastic oracle on the distribution selected by the adversary at time-step t. Write ✏(T ) = q 2 log(m) .


In section 5, we discuss how we evaluated our methods on widely used audio CAP TCHAs and we give our results.

we show that the audio CAP TCHAs used by sites such as Google and Digg are susceptible to machine learning attacks.


Moreover, Krauthgamer, Nadler and Vilenchik [45] and Berthet and Rigollet [41] have shown that for semi-definite programs (SDP) this bound is tight.

the natural SDP cannot even solve the detection problem: to distinguish the data from covariance Σ = λvv T + I from the null hypothesis in which no sparse vector is planted, namely the n samples are drawn from the Gaussian distribution with covariance matrix I.


Although we cannot enforce nonadaptivity, and it is sometimes unreasonable to expect that analysts are entirely non-adaptive, we intend the mechanism to be used for validation.

analysts should do their discovery, training, tuning, development, and adaptive data analysis on unrestricted "training" or "discovery" datasets, and only use the protected database when they wish to receive a stamp of approval on their model, predictor, or discovery.


Our first goal will be to learn which parameters (α, β) are most consistent with our observations-our unlabeled training set-using maximum likelihood estimation.

to do this for a particular training set S ⊂ X, we will solve the problem we are maximizing the probability that the observed labels produced on our training examples occur under the generative model in (1).


However, incorporating the input noise may significantly change the results.

for the cases of multiple neurons, our current results predict that there is no overlap between the active regions of the response functions for ON and OFF neurons.


In fact, we show Theorem 1 in a more general proximal setting.

to minimize F (x) := f (x)+ψ(x) where ψ(x) is proper convex function that can be non-smooth.


Moreover, gcd(L S ) = 1 is a weak restriction, but a restriction nonetheless.

we instead use a functional stopping rule for u (Theorem 3.2) that holds for all G: as u increases, if we find a graph that we previously encountered, then there cannot be any new graphs as u → ∞.


Besides being used as a standalone analysis tool, this approach can inform model-based fMRI analyses to investigate whether the model correctly tracks the brain's internal mechanism.

if a brain region is found to be important in the current analysis, but not using the model-based fMRI analysis, this could mean that the model used to extract neural information is not representing all of the relevant neural signals involved in decision-making and requires further modification.


We include these restrictions by converting inferred covariance(s) into correlation(s) in step 2) of the algorithm in Section 4.3.

the restrictions are enforced by a projection step.


The results and analysis remain qualitatively similar.

given a new data point x 0 , our task is to estimate f (x 0 ), where we restrict the estimator f (x 0 ) to be of the form f ( the estimator is a weighted average of the given noisy labels.


A capsule projection is performed by projecting input feature vectors fed from a backbone network onto the capsule subspace.

an input feature vector is orthogonally decomposed into the capsule component as the projection onto a capsule subspace and the complement component perpendicular to the subspace.


Along the way, for sparse graphs, we provide more general consistency results than were previously known, regardless of privacy.

to the best of our knowledge, no prior results give a consistent estimator for W that works for sparse graphs without any additional assumptions besides boundedness.


However, there might be a simpler way to grow estimate-based dependency trees, one that does not involve complex red-edge rules.

we can just run the original algorithm on a small sample of the data, and use the generated tree.


In these policies, D is the partial realization that we have observed so far, and X D {x ∈ X | (x, y) ∈ D for some y ∈ Y} is the domain of D (i.e., the set of selected items in D).

for any item x, we write δ(x | D) to denote the worst-case utility gain if x is selected after we observe D. In this definition, note that we have extended the utility function f to take a partial realization as the second parameter (instead of a full realization).


the position of the window i in the grid is a latent variable given which we can write the probability of the bag as Relaxing the terminology, E and W are referred to as, respectively, the counting grid and the window size.


Our experimental results show that our model is comparable to other state-of-the-art approaches in action recognition.

our experimental results demonstrate that combining large-scale global features and local patch features performs significantly better than directly applying hCRF on local patches alone.


PGAS uses Gibbs-like steps for the state trajectory x 0:T and the hyper-parameters θ, respectively.

we sample first x 0:T given θ, then θ given x 0:T , etc.


Using this same hash function for each core, a hash value is computed for each representative x C for cluster C in the current clustering C. The communication round ends by redistributing the clusters randomly using these hash values.

the value h π, (x c ) determines which core receives C. The current clustering C is thus repartitioned among cores.


That is due to the nature of the code R . (iii) As discussed in Appendix E, Theorem 4 can also be used to establish an approximate rule of thumb for pricing.

considering two query schemes k 1 IC and k 2 IC, the query price π is to be set as π(k1) π(k2) ≈ k1 k2 .


We model the interacting neuronal populations with a system of stochastic integro-differential equations.

the intrinsic dynamic of each population is modeled using a linear differential operator while the effective connectivity between populations is modeled using causal integral operators.


In particular, the minimizers {Z * (t) j : j ∈ V w \B(t)} are determined as the solution of the (T w , B w )-problem.

b(t) changes at most q times with t. Lemma 3.9.


We provide the first non-trivial upper and lower bounds on the communication complexity of this basic estimation task in various settings of interest.


Another interesting and important question is whether the recovery is stable in the presence of small dense noise.

suppose we observe D = A 0 + E 0 + Z, where Z is a noise vector of small 2 -norm (e.g., Gaussian noise).


Beyond [9], several works have inspired this paper.

accelerated SDCA [26] is an instance of an inexact accelerated proximal point algorithm, even though this was not explicitly stated in [26].


An even more interesting finding is that the solutions found by RM were sparse while achieving lower test misclassification errors than standard deep learning methods.

in the fully connected case, the final solution produced by RM zeroed out 32%, 26% and 63% of the parameter matrices (from the input to the output layer) respectively.


We now show how the key result in Lemma 1 can be used for group-sparse inference.

we show how to compute a p-value for the null hypothesis H 0 : µ ⊥ L, or equivalently, H 0 : P L µ 2 = 0.


Our main idea is to synthesize policies for the unseen pairs of environments and tasks.

our agent learns two sets of embeddings: one for the environments and the other for the tasks.


Though Bregman divergences satisfy this non-negativity property, like metrics, the similarities to metrics end there.

a Bregman divergence need not satisfy the triangle inequality or be symmetric.


Next, we hunt for the stable members of the FGSD that are robust against the perturbation or noise in the datasets.

we will look at the stability of R and F based on FGSD from f (λ) perspective by first analyzing its influence on a single edge perturbation (or in other words analyzing rank one modification of Laplacian matrix).


To do so, we learn one model of LCA for each class of the data.

we use LCA to project each example x i into a lower dimensional space where we hope for two properties:  We label the example x by the class c that maximizes the probability in eq.


However, the original decoupled basis [5] turns out to have optimization difficulties [11].

the non-convexity of the optimization problem means that a suboptimal solution may be found, leading to performance that is potentially worse than the standard coupled case.


All the results for uniform γ and weights α t = 1 hold in this case as well.

by the arguments leading to (5), if we want to minimize m i=1 k l=2 e {ft(i,l)−ft(i,1)} , where f t is as defined in (1), then the following strategy is optimal: in round t output the cost matrix This will ensure that the expression m i=1 k l=2 e {ft(i,l)−ft(i,1)} changes by a factor of at most κ(γ t , α t ) in each round.


These assumptions hold for many target applications, including the ones explored in Section 5.

we note that assumption (A0) can be often satisfied with a careful choice of C. It has been studied that (projected) SGD has a sublinear convergence rate for a smooth strongly-convex objective if the variance of gradient estimates is uniformly bounded [23,21].


More formally, the criterion under study is whether the following implication holds for every sequence of scoring functions (f k ) k≥0 and every data distribution D: where the infima are taken over all scoring functions.

we show that if a surrogate loss is convex in the sense that (y, .)


As with a standard VAE, we make use of variational inference for our model.

we consider the following variational distribution over the latent variables which approximates the true posterior on Z.


if we estimate the label that would be predicted by IR-Int (see 4.1) and the discussion below it), and we plug that in with any weight c ≥ 0, then the isotonic regression function will still have that same label estimate as its value.


We confirmed that dIRM outperformed the other models in all datasets for the all measures.

dIRM showed good results in the Synth2 and Enron datasets, where the changes in relationships are highly dynamic and unstable.


In this paper, we propose a novel approach for learning parametric sparse models for image superresolution.

mapping functions between the LR patch and the sparse codes of the desirable HR patches are learned from a training set.


Now we describe several details about how the above ideas are implemented.

learning root filter η: Given a set of training images (x t , y t ), we firstly learn the root filter η by solving the following optimization problem: η * is learned by only considering the feature vector ω(•).


Consequently, the pdf in (3) can be treated as a probability mass function (pmf) over s ∈ κ(σ (i) (1 : t − 1)) and sampling from it becomes equivalent to sampling Hence, we can interpret a particle x (i) σ(1:t−1) as a sequence of indices of selected graph vertices σ(1:t−1) is a vector of ones assigned to RVs with indices in σ (i) (1 : t − 1).

it holds ind(x , where ind : {0, 1} n → 2 S is a function that assigns to x a set of indices of coordinates of x that are equal to one.


( 22)) has the basic properties of STDP.

a presynaptic spike occurring immediately before a post-synaptic spike will potentiate a synapse, because membrane potentials are positive immediately before a postsynaptic spike.


The construction of F is essentially a reduction of smooth ISSC to a simpler submodular set cover problem, and generalizes the reduction approach in [11].

we first lift the analysis of [11] to deal with multiple thresholds (Section 4.2).


[8], rigid-motion synchronization initializes PGO by computing an estimate of the vertex labels M i (absolute poses) given enough measurements of the ratios M ij .

it tries to find the absolute poses that best fit the relative pairwise measurements.


Our main result has a striking corollary in the context of tree-structured graphs: for any discrete graphical model, the inverse of a generalized covariance matrix is always (block) graph-structured.

for binary variables, the inverse of the usual covariance matrix corresponds exactly to the edge structure of the tree.


To compute subsequent semi-supervised eigenvectors, i.e., at steps t = 2, . . . , k if one ultimately wants a total of k semi-supervised eigenvectors, then one lets Q be the n × (t − 1) matrix with first column equal to 1 and with j th column, for i = 2, . . . , t − 1, equal to x * j−1 (where we emphasize that x * j−1 is a vector not an element of a vector).

, where x * i are successive semi-supervised eigenvectors, and the projection matrix due to the degree-weighted inner norm.


We conduct an ablation study to analyze several design choices of our method.


While several nonparametric divergence estimators exist, relatively few have known convergence properties.

even for those estimators whose MSE convergence rates are known, the asymptotic distributions are unknown.


The improvement of AdaNCG + over [4]'s algorithm is brought by reducing the number of Hessian-vector products for performing each iteration of AdaNCG.

the number of Hessian-vector products of each NCD step in [4] is O(1/ √ 2 ), which becomes O(1/ max( 2 , ∇f (x j ) 2/3 )) for each AdaNCD step in AdaNCG + .


4 L ∞ -approximation of smooth functions: small and efficiently computable coefficients In this section we prove Theorem 3.2.

for every f ∈ C K B the corresponding g f can be approximated by a low degree trigonometric polynomial in L ∞ -norm.


The planning of eye movements is thought to be influenced by bottom-up (scene dependent) saliency as well as top-down (goal relevant) factors [21].

one major view is that two types of maps, the saliency map and the priority map, encode measures used to determine the target of attention [39].


Despite the overall trends in our results, we see some interesting exceptions.

batch-ENS with batch size 5 is significantly better than that with batch size 1, with a p-value of 0.02 under a one-sided paired t-test.


For K = 2, our statistical rate of convergence in Frobenius norm recovers the result of [5].

our theory confirms that the desired local optimum studied by [5] not only exists, but is also attainable by an efficient algorithm.


We note that in the setting of strong convexity, existing stability based approaches impose a much stronger condition of strong convexity on R n (✓).

they require the loss function `(✓, x) to be strongly convex in ✓ at each x 2 X .


Notice that this expression does not depend on q T -in fact, this cost precisely corresponds to the scenario where the firm had not set up a CLM and instead used the initial estimate q 0 to encode.

for α = 1, the firm is entirely neutral to the quality of the estimate q T ; even if the CLM provided an estimate q T which performed worse than q 0 , the cost increase due to the bad choice of q is recouped from payments of the ill-informed participants.


In this section, we extend this result and show how the Martingale type of a pair (W , X ) are related to existence of certain uniformly convex functions.

the following theorem shows that the notion of Martingale type of pair (W , X ) is equivalent to the existence of a non-negative function that is uniformly convex w.r.t.


A useful method for solving this problem is by using linear discriminant functions .

we seek a weight vector wand a threshold () such that w T x < () if its label y = -1 and w T x ~ () if its label y = 1.


We can then obtain estimated values of the parameter priors under the null hypothesis by matching the moments of each Gamma distribution to their historical values.

we set the expectation and variance of the Gamma distribution Ga(α all , β all ) to the sample expectation and variance of C all B all observed in past data: , where C out (S) and B out (S) are respectively the total count ∑ G−S c i and total baseline ∑ G−S b i outside the region.


When the data were generated by LIM, AIC or BIC fared much more poorly whereas MDL recovered the correct model (LIM) across both sample sizes.

under AIC or BIC, FLMP was selected over LIM half of the time for N = 20 (51 % vs. 49%), though such errors were reduced for N = 150 (17 % vs 83 %).


We refer the reader to the supplementary for more details.

consider the case where all blocks are updated in parallel as in the thread-greedy coordinate descent algorithm of Scherrer et al.


We achieve this by leveraging temporal information to learn a richer, more capable generative model.

we extend AIR into a spatio-temporal state-space model and train it on unlabelled image sequences of dynamic objects.


We evaluate a set of baseline learning strategies on these tasks, and show that a novel model incorporating predictive lookahead is a promising approach for learning from a teacher's response.

a surprising result is that it can learn to answer questions correctly without any reward-based supervision at all.


Furthermore, if the dimension of a mode was smaller than the sampling size, we used all the coordinates.

we sampled min(s, N k ) indices for each mode k ∈ [K].


V is a set {V 1 , V 2 , ..., V n } of observable variables (also called endogenous), that are determined by variables in the model (i.e., determined by variables in U ∪ V ), 3.

f is a set of functions {f 1 , f 2 , ..., f n } such that each f i is a mapping from the respective domains of U i ∪ P A i to V i , where U i ⊆ U and P A i ⊆ V \ V i and the entire set F forms a mapping from U to V . each f i in v i ← f i (pa i , u i ), i = 1, ..., n, assigns a value to V i that depends on the values of the select set of variables (U i ∪ P A i ), and 4.


We propose to use the Inexact Proximal point method for Optimal Transport (IPOT) algorithm to compute the optimal transport matrix T * , which provides a solution to the original optimal transport problem (3) [59].

iPOT iteratively solves the following optimization problem: where Algorithm 1 IPOT algorithm [59] 1: end for 9: T (t+1) = diag(δ)Qdiag(σ) 10: end for Here the Bregman divergence D h serves as a proximity metric and β is the proximity penalty.


However, to render our approximation tractable, we further restrict the set of approximating processes.

we require the existence of some expansion in orders of the where the remainder O(ε) contains the dependency on the parents.


ALPACA offers a simple and natural formulation of the patient staging problem as a probabilistic inference query.

given the measurements corresponding to a particular patient, we perform patient staging by: (1) computing a posterior distribution over the event ordering σ (j) , then (2) computing a posterior distribution over the most recent element of the snapshot set τ (j) .


In this paper, we consider the case s = 2, and the biases are the sums of ±1 randomly drawn R times.

when R is not too small, the bias distribution approaches a Gaussian distribution with mean 0 and variance R. The ratio ρ ≡ R/N is referred to as the diversity.


The present contribution studies to what extent we can omit this brief calibration period.

is it possible to successfully transfer information from prior BCI sessions of the same subject that may have taken place days or even weeks ago?


Remark: Of course, the identity of * depends on + .

the policy * will only be part of a globally optimal plan if the set of choices + is optimal.


We construct a "local-geometric-context" vector for each part, and obtain landmark mixture labels by grouping landmark instances with similar local geometry.

for each landmark i and image n, we construct a K-element vector g in that defines the 2D relative location of a landmark with respect to the other K landmarks in instance n, normalized for the size of that training instance.


one might conjecture that it is easy to use local search algorithms to find all the modes.


lC, solid curve), together with the assumption that some fixed response difference ~r = 1 is required for correct discrimination [2].

for any fixed pedestal contrast c, the discrimination threshold ~c satisfies r(c + ~c) = r(c) + 1.


In particular, the linear conjugate gradient (CG) method works as the inner iterative algorithm for solving the linearized Gauss-Newton normal equation, whereas the outer nonlinear algorithm repeatedly takes so-called "Krylov-dogleg" steps, relying only on matrix-vector multiplication without explicitly forming the Jacobian matrix or the Gauss-Newton model Hessian.

our iterative dogleg algorithm can reduce both operational counts and memory space by a factor of O(n) (the number of parameters) in comparison with a direct linear-equation solver.


On the other hand, the analysis of the natural modification to ACAL can be far more subtle, when there is noise.

because the version space is only guaranteed to contain the best classifier with high confidence, there is still a small probability of making a prediction that disagrees with the best classifier h * on each round that we do not request a label.


We now provide a formal statement of the problem studied in this paper in a very general setting, and then define notation required for our specific results.

formal Problem Statement: Let P ∈ F G be an unknown probability measure on a sample space X , from which we observe n IID samples X 1:n = X 1 , ..., X n IID ∼ P . In this paper, we are interested in using the samples X 1:n to estimate the measure P , with error measured using the adversarial loss d F D . for various choices of spaces F D and F G , we seek to bound the minimax rate of estimating distributions assumed to lie in a class F G , where the infimum is taken over all estimators P (i.e., all (potentially randomized) functions P : X n → F G ).


The study is currently being repeated with North America participants and we are curious to see whether we can replicate our findings.

we hope to further improve the classification accuracy through higher data quality of the new EEG recordings.


N (0, 1) entries, while the signal X has one of the following forms: Case I) X = [0 n×(p−c) ; X * ] where the n × c matrix X * is the nonzero part of X.

x has c nonzero columns and does not necessarily have a low-rank structure.


Intuitively (1) and ( 2) have similar effects since large values of p(x i , y i |θ)p(x i |θ , σ i , x i ) dominate the summation in p(x i , y i |θ)p(x i |θ , σ i , x i )dx i .

both methods prefer a parameter configuration such that the product p(x i , y i |θ)p(x i |θ , σ i , x i ) is large for some x i .


In bipartite ranking, the goal is to pick s in S so that the larger s(X), the greater the probability that Y is equal to 1 ideally.

the objective is to learn s(x) such that the r.v.


We incorporate them into the edge feature function.

we define the (and the only) edge feature function as where R i,j = r(x i , x j ) denotes the parent-child relation: r(x i , x j ) = 1 if document x i is the parent of x j , and r(x i , x j ) = 0 for other cases.


Let π be an arbitrary policy and let δ ∈ [0, 1].

we define the oracle-infused policy π + δ as follows: when choosing an action, π + δ explores the policy space with probability 1 − δ (according to its current model), but with probability δ, we force it to take an oracle action.


Our main contribution is to describe how the UCT algorithm can be modified to efficiently incorporate these techniques in practice.

we provide a general approach that significantly reduces the effort needed recursively apply control variates.


The framework of gyrovector spaces provides an elegant non-associative algebraic formalism for hyperbolic geometry just as vector spaces provide the algebraic setting for Euclidean geometry [2,25,26].

these operations are used in special relativity, allowing to add speed vectors belonging to the Poincaré ball of radius c (the celerity, i.e.


We remark that the conditions in Theorem 1 and Proposition 2 are conceptual but that the constants involved are unknown in practice.

this means that the allowed values of the factor ↵ that controls the growth of the sample size are unknown a priori.


Inspired by recently developed first-order methods for optimizing composite functions [6,7,8], the regularized dual averaging (RDA) method we develop exploits the full regularization structure at each online iteration.

at each iteration, the learning variables are adjusted by solving a simple optimization problem that involves the whole regularization term, not just its subgradients.


In this paper, we propose a generic architecture, called BAI-MCTS, that builds on a Best Arm Identification (BAI) algorithm and on confidence intervals on the node values in order to solve the best action identification problem in a tree of arbitrary depth.

we study two specific instances, UGapE-MCTS and LUCB-MCTS, that rely on confidence-based BAI algorithms [8,13].


However, unlike in traditional gradient descent, the stationary distribution is not necessarily just the optimal point.

the harder parts of understanding this algorithm involve understanding the properties of the stationary distribution.


Thus, we generally want each row of C to be V-shaped.

a simple C with V-shaped rows is the classification cost matrix, with entries C y,k = y = k . 1 The classification cost is widely used in multiclass classification.


We now consider a setup with mixtures of three or four Gaussians.

we consider the following four cases, each using a larger sample size of n = 2000: • Case 1, mixture of three Gaussians on a line: 15.


The L(w, S) term is related to the size of the set {y|m ρ (x i , y * i , y, w) ≤ M i }, which can be controlled by margin m ρ 2 (x i , y i , yi ).

for underestimation, It implies that the larger m ρ 2 (x i , y i , yi ), the lower L(w, S).


(b) We demarcate two regimes, which we call semi-dense and semi-sparse, under which common neighbors can be successfully used for both link prediction and clustering.

(c) in the semi-dense regime, the number of common neighbors between the query node q and another node within its cluster is significantly higher than that with a node outside its cluster.


A more important property is that it can be expressed as a difference of two convex functions.

we can express where Though the objective function ( 1) is nonconvex w.r.t.


after each episode, a single gradient-based update is made to the policy, and the policy optimization does not revisit those trajectory data again.


Together with the independence these ensure the argmax requirement.

if G µ (B) is the optimal value of some perturbed density restricted to B, then the event that the optima over ⌦ is contained in B is equivalent to the event that G )) is a probability measure proportional to µ(B) [12].


To get DDP to converge to good policies we slowly changed the model from a model in which control is trivial to the actual model.

we change the model such that the next state is α times the target state plus 1 − α times the next state according to the true model.


If, for all x 2 X u , |x > w sup | is larger than 1, then w semi 6 = w sup .

we get an improved semi-supervised estimator if all points in X u are outside of the margin.


In this paper, we identify a fundamental problem with Q-learning (and other forms of dynamic programming) with function approximation, distinct from those previously discussed in the literature.

we show that approximate Q-learning suffers from delusional bias, in which updates are based on mutually inconsistent values.


In this paper we take just a few steps toward more general predictions.

we consider variations of the problem of prediction by a fixed interval.


. . , h K ] is performed using the backward recurrence: ) which is started with I K = h K and the reconstructed image being I = I o .

starting at the coarsest level, we repeatedly upsample and add the difference image h at the next finer level until we get back to the full resolution image.


When the condition number is roughly equal to the sample size (κ = Θ(n)), then Quartz does better than both ASDCA and SPDC as long as n/τ + ω ≤ n/ √ τ .

this is the case when the data is sparse: ω ≤ n/ √ τ .


In this section, we describe the embedding generation, or forward propagation algorithm (Algorithm 1), which assumes that the model has already been trained and that the parameters are fixed.

we assume that we have learned the parameters of K aggregator functions (denoted AGGREGATE k , ∀k ∈ {1, ..., K}), which aggregate information from node neighbors, as well as a set of weight matrices W k , ∀k ∈ {1, ..., K}, which are used to propagate information between different layers of the model or "search depths".


The plots in Figure 2 show the theoretic limits.

the minimal Ncut value in the kNN case is obtained at a different position than the minimal value in the r-neighborhood case.


In the first step, we keep D fixed and minimize over W . This consists in solving the problem This minimization can be carried out independently for each task since the regularizer decouples when D is fixed.

introducing new variables for (F (D)) 1 2 w t yields a standard L 2 regularization problem for each task with the same kernel K(x, z) = x (F (D)) −1 z, x, z ∈ IR d . In other words, we simply learn the parameters w t -the columns of matrix W -independently by a regularization method, for example by an SVM or ridge regression method, for which there are well developed tool boxes.


We show that, one unfortunately cannot do anything beyond this trivial decomposition, that is, treating each dimension independently, and solving d different estimations problems individually.

the communication cost 2 must be at least d times the cost for one dimensional problem.


In this paper, we propose an optimization-based ABC algorithm that can successfully address the deficiencies of both sampling-and regression-based algorithms.

we show that ABC can be formulated under a unified optimization framework: finding the saddle point of a minimax optimization problem, which allows us to leverage powerful gradient-based optimization algorithms to solve ABC.


However, in contrast to the Gaussian model, non-linear classifiers can achieve a significantly improved robustness.

consider the following thresholding operation T : R d → R d which is defined element-wise as It is easy to see that for ε < 1, the thresholding operator undoes the action of any ∞ -bounded adversary, i.e., we have T (B ε ∞ (x)) = {x} for any x ∈ {±1} d . Hence we can combine the thresholding operator with the classifier learned from a single sample to get the following upper bound.


Figure 2(c) shows one way of doing this, by using a simple cross-validation for the tartuffe data set.

half of the data was held out as a validation set, and for a range of different choices for m, the model was trained and its cross-entropy on the validation set was calculated.


Each vertex has a relatively close selected center.

for every u ∈ V , there is a center opened at distance at most (1 + ) v d(u, v) p x uv 1/p .


In each time period, the algorithm samples an action from the posterior distribution p t (a) = P t (A * = a) of the optimal action.

thompson sampling uses a policy π t = p t . It is easy to see that this is equivalent to sampling a model index θt from the posterior distribution of models and then selecting an action A t = arg max a∈A R θt (a) that optimizes the sampled model.


In this experiment, we evaluate the accuracy of estimating pairwise angular similarity on several datasets.

we test the effect to the estimation accuracy when the Super-Bit depth N varies and the code length K is fixed, and vice versa.


For each object v 2 V, we have a pool of predefined d features, each of which is a real number in [ 1,1].

we can represent each v 2 V as a vector of features x 2 [ 1,1] d . We note that the mapping from v to x can be non-linear and that d can be very large.


Approximate Learning: Exact maximum likelihood learning in this model is intractable, but efficient approximate learning of DBMs can be carried out by using a mean-field inference to estimate data-dependent expectations, and an MCMC based stochastic approximation procedure to approximate the model's expected sufficient statistics [7].

consider approximating the true posterior P (h|v; θ) with a fully factorized approximating distribution over the three sets of hidden units: k |v), where µ = {µ (1) , µ (2) , µ (3) } are the mean-field parameters with q(h i for l = 1, 2, 3.


In this paper we study memory patterns embedded in a balanced network of spiking neurons.

we focus on the maximal capacity of Hebbian cell assemblies.


We are interested in measuring its scalability in addition to the effect of having several rounds on the quality of the solution.

we compare the quality of the solution (i.e., the maximum radius from the k-center objective) produced by the parallel implementation to that of the sequential one-machine implementation of the farthest seed heuristic.


In this section we employ the CS decision trees for the problem of object detection, following a solution setting similar to [9].

we adopt a patch-based abstraction of an image and the aim of the tree-based predictor is to jointly predict, for each patch, the foreground/background class it belongs to and a displacement vector pointing to the object's center.


Suppose G = (V, E) is an undirected graph over p nodes corresponding to the p response variables.

given the underlying graph G, and the set of cliques (fully-connected sub-graphs) C of the graph G, the corresponding conditional random field (CRF) is a set of distributions over the response conditioned on the covariates that satisfy Markov independence assumptions with respect to the graph G. letting { c (Y c , X)} c2C denote a set of clique-wise sufficient statistics, any strictly positive distribution of Y conditioned on X within the conditional random field family takes the form: With a pair-wise conditional random field distribution, the set of cliques consists of the set of nodes V and the set of edges E, so that A key model specification question is how to select the class of sufficient statistics, .


This model of the RDM stimulus is simple but captures the most important properties of the stimulus.

a high coherence RDM stimulus has a large proportion of motion in the correct direction with very low variability of measurements whereas a low coherence RDM stimulus tends to have lower proportions of motion in the correct direction, with high variability (cf.


. . , v J can be separated across different machines.

the "sum" or "coupling" variable s is maintained on a master thread which runs Algorithm 2, and each v j is maintained on a worker thread running Algorithm 3.


We put forth the hypothesis that the efficacy of the mechanoreceptor-CN synapses is regulated according to spike-timing-dependent plasticity (STDP, [1,15]).

we employ a STDP rule specifically developed for the SRM [20].


This explains why practical designing tricks can stabilize the backward process.

by analyzing the local Hessian of residual block, we can answer the questions about skip connection in [Orhan and Pitkow, 2018] via local Hessian.


The regularized GAN also manages to perfectly classify the unregularized GAN's samples as fake (cross-testing FP 0.0) whereas the unregularized GAN classifies the samples of the regularized GAN as real (cross-testing FP 1.0).

the regularized model is able to fool the unregularized one, whereas the regularized variant cannot be fooled.


A gamma process is a completely random measure over X with gamma marginals.

it is a random atomic measure of the form (3), such that for each measurable subset A, the (random) mass G(A) is gamma distributed.


Allowing letters to emit sets of columns lets us accomodate letter templates of variable width.

this means that we can unify segmenting ink into letters and recognizing blocks of ink; figure 3 shows an example of how useful this is.


One of the special properties of RELU activation function is non-negative homogeneity.

for any scalar c ≥ 0 and any x ∈ R, we have σ RELU (c • x) = c • σ RELU (x).


2 Lemma 3.1 implies that the class H K is VC-type.

it is uniform Donsker, as well as uniform Glivenko-Cantelli.


These are analogous to the standard radial basis function kernels exp[−(x − x ) 2 /(2σ 2 )], but we find that they have surprising properties on large graphs.

while loops in large random graphs are long and can be neglected for many purposes, by approximating the graph structure as locally tree-like, here this leads to a non-trivial limiting form of the kernel for σ → ∞ that is not constant.


Viewed from this perspective, serial reproduction defines a stochastic process: a sequence of random variables evolving over time.

it is a Markov chain, since the reconstruction produced on the current trial depends only on the value produced on the preceding trial (e.g.


Therefore, ADO is best used as a tool for confirmatory rather than exploratory analyses.

it is best suited for situations in which the field of potential models can be narrowed to a few of the strongest competitors.


Using the DRMM, the solution I ⇤ c (L) of the activity maximization for class c (L) can be derived as the sum of individual activity-maximizing patches I ⇤ Pi , each of which is a function of the learned DRMM parameters (see Appendix E).

i ⇤ c (L) ⌘ This implies that I ⇤ c (L) contains multiple appearances of the same object but in various poses.


Moreover, we also propose a local method for multi-task regression problems.

we extend the kernel regression method to multi-task setting in a similar way to the classification case.


The algorithm successively narrows down the gap between upper and lower bounds.

ᾱj (t + 1) ≤ ᾱj (t) and α j (t + 1) ≥ α j (t).


Also, the optimality con- ditions enables us to predict important properties of the optimal solution before we actually solve it.

we can explicitly specify the fraction of noisy pixels by means of the ν-trick originally developed for SVMs [8] which was later applied to Boosting [7].


For algorithms minimizing a surrogate loss, the question of consistency reduces to the question of calibration of the surrogate loss with respect to the target loss of interest [5][6][7]16]; in general, one is interested in convex surrogates that can be minimized efficiently.

the existence (and lack thereof) of convex calibrated surrogates for various subset ranking problems, with target losses including for example the discounted cumulative gain (DCG), mean average precision (MAP), mean reciprocal rank (MRR), and pairwise disagreement (PD), has received significant attention recently [9,[11][12][13][15][16][17].


( with initialization w 0,j = 1, for all j = 1, . . . , K. Lemma 4.

[7] For any 0 < ǫ ≤ 1 2 , the MWU algorithm provides the following regret bound for the online learning problem described above: for ǫ = log(d+1) T , we have R(T ) ≤ log(d + 1)T For the rest of the paper, we refer to the MWU algorithm with ǫ = log(d+1) T as the online learning (OL) algorithm, and the update in (5) as the OL update at time t + 1.


Generally, however, the commitment by P 1 creates new dependencies that render the problem inherently non-Markovian with respect to the state s t and thus harder to solve.

even though the dynamics of the environment are Markovian with respect to the state s t , the MVDP only becomes Markov in the Stackelberg setting with respect to the hyper-state η t = (s t , π t:T,1 ) where π t:T,1 is the commitment by P 1 for steps t, .


This result lower-bounds the likelihood at ηδ by explicitly constructing a scaled version of η that satisfies the self-normalization constraint.

if η is chosen so that normalizers are penalized for distance from log µ(Y) (e.g.


This is followed by a summary of the numerical experiments carried for verifying the major claims of the paper.

the empirical findings are a) the new MKL formulation is well-suited for object categorization tasks b) the MD based algorithm scales better than state-of-the-art gradient descent methods (e.g.


Prior to this work, some relevant ideas have been explored [21,16,1].

[21,16] proposed an introspective learning (IL) approach that learns the target model by comparing its outputs on (I, c) and (I / , c).


Inverting the correspondence between multipliers and message parameters yields the converse While local message passing algorithms are convenient for many applications, their convergence is not guaranteed in general.

lBP often fails to converge for networks with tight loops [1] such as the 3×3 lattice of Figure 1(a).


This method builds a representation which is a frequency count of all possible contiguous subsequences of length p. This produces a mapping with features φ and 0 otherwise.

to define a feature space for outputs we count the number of contiguous subsequences of length p on the input that, if starting in position q, have the same element of the alphabet at position q + (p − 1)/2 in the output, for odd values of p. φ We can then learn pre-images using a window also of size p as described in Section 3.2, e.g.


In Section 2.3, we will discuss the sparse case where the column T :,j,k may be entirely zero.

when that case does not arise, entries of P can be interpreted as the transition probabilities of a second-order Markov chain (Z t ) t∈N : If the last two states were j and k, then the next state is i with probability P i,j,k .


We also observe that the discriminator needs enough capacity to provide faithful gradients towards the equilibrium.

instantiating the discriminator with feedforward neural network w./w.o.


In an ambient space Ω, we say that a set C separates the sets A and B if any continuous path in Ω with endpoints in A and B intersects C. Theorem 3.3.

under Assumptions 3.2, let (µ t ) t≥0 be a Wasserstein gradient flow of F such that, for some 0 < r a < r b , the support of µ 0 is contained in B(0, r b ) and separates the spheres if (u m (t)) m∈N,t≥0 is a sequence of classical gradient flows initialized in B(0, r b ) such that µ m,0 converges weakly to µ 0 then (limits can be interchanged) A proof and stronger statements are presented in Appendix C. There, we give a criterion for Wasserstein gradient flows to escape neighborhoods of non-optimal measures-also valid in the finite-particle setting-and then show that it is always satisfied by the flow defined above.


We performed several preprocessing and data formatting steps, which contributed to disguising the origin of the datasets.

we introduced a number of features called probes.


By our theory, ResNet with one-neuron hidden layers is already a universal approximator.

a ResNet with multiple units per layer is in some sense an over-parametrization of the model, and over-parametrization has been observed to benefit optimization [36,4,1].


Suppose problem (1) satisfies the HEB condition with θ ∈ (0, 1/2] and F (x 0 )−F * ≤ 0 , where x 0 is an initial solution.

let η = 1/(36L), and by running Algorithm 1 with R = log 2 0 , we have E[F (x (R) ) − F * ] ≤ , and the computational complexity for achieving an -optimal solution in expectation is O(n log( 0 / ) + Lc 2 max{ 1 1−2θ , log( 0 / )}).


Among countless applications, the SVD is used for optimal low-rank approximation and principal component analysis (PCA).

for k < r, a partial SVD can be used to construct a rank k approximation A k such that both A − A k F and A − A k 2 are as small as possible.


We obtain the following probability model: q c = e f * c . This formulation is closely related to the maximum-likelihood estimate with conditional model q c = e fc / K k=1 e f k (logistic regression).

if we choose a function class such that the normalization condition K k=1 e f k = 1 holds, then the two formulations are identical.


This analytic computation yielded several insights into which combinations of these ingredients enable nonlinear deep networks to achieve dynamical isometry.

deep linear Gaussian networks cannot; the maximum Jacobian singular value grows linearly with depth even if the second moment remains 1.


We note that most research on VSTM is concerned with characterizing how subjects encode a single item in VSTM (for instance, the precision with which a single item can be encoded [1,2]) and, thus, does not consider the joint encoding of multiple items.

we are not aware of any previous work attempting to experimentally probe and characterize exactly how the encoding of an item influences the encoding of other items, i.e.


Note that the precise form of the mapping from unexpected uncertainty to NE spikes is rather arbitrary.

there may be a strong non-linearity, such as a thresholded response profile.


While the performance of EBC is clearly datadependent, SMA (purple lines in Figure 4) exhibits dramatic performance improvement compared to alternatives in all instances (see the Supplement, Section 3, for extensive additional experiments, including comparisons to other classifiers, and varying adversary's budget constraints).

figure 5 (left) looks deeper at the nature of SMA solution vectors w. we consider how the adversary's strength, as measured by the query budget, affects the sparsity of solutions as measured by w 0 .


To obtain a more detailed analysis, we focused on a subset of datasets but extended the configuration budget for optimizing all methods from one hour to one day and to two days for AUTO-SKLEARN.

we clustered our 140 datasets with g-means [27] based on the dataset meta-features and used one dataset from each of the resulting 13 clusters (see Table 6 in the supplementary material for the list of datasets).


Convergence Rate Throughout the paper, we discuss the convergence rate of training algorithms in terms of the average of the `2 norm of the gradients.

we say an algorithm gives an ✏-approximation solution if This metric is conventionally used in the analysis for non-convex optimization [17,32,31].


Since our algorithm involves a sorting step, this shows that the overhead added by the maximum error rule is only about 7× compared to sorting.

this implies that no algorithm that relies on sorted samples can outperform our algorithm by a large margin.


The RCV1-v2 and BioASQ datasets provide label relationships as a graph.

labels in RCV1-v2 are structured in a tree.


Based on this derivation we believe that the natural way of relating R H and C H would be to replace the p 0 in Herbster's potential formulation by q 0 such that 1/p 0 +1/q 0 = 1.

one would have to consider C H q 0 and then define b R H p 0 := 1/C H q 0 .


We study the problem of constructing a hypothesis test for an unknown data-generating function h : R p → R, when h is estimated with a black-box algorithm (specifically, Gaussian Process regression) from n observations {y i , x i } n i=1 .

we are interested in testing for the hypothesis: where H 0 , H a are the function spaces for h under the null and the alternative hypothesis.


It should be noted that our lower bounds remain valid under any source of randomness which may be introduced into the optimization process (by the oracle or by the optimization algorithm).

our bounds hold in cases where the variance of the iterates produced by the algorithm converges to zero, a highly desirable property of optimization algorithms in this setting.


The main goal of this paper is to provide finite-sample analysis of these estimators, via unified analysis of the estimator after bias correction.

we show conditions under which, for β-Hölder continuous (β ∈ (0, 2]) densities on D dimensional space, the bias of fixed-k estimators decays as O n −β/D and the variance decays as O n −1 , giving a mean squared error of O n −2β/D + n −1 .


For the sake of interpretability, we now specialize to the case where the region graph has only two layers, with the first layer corresponding to vertices and the second layer corresponding to hyperedges.

for r, s 2 R, we have r ✓ s only if |r| = 1, and R = V [ F , where F is the set of hyperedges and V denotes the set of singleton vertices.


Our main result applies to a more general context, with a generalized notion of detection that applies to arbitrary SBMs.

• we show that an approximate belief propagation (ABP) algorithm that mitigates cycles achieves the KS threshold universally.


We do so by proving a slightly simpler form of our result, where we assume that k is not too small.


However, in the case of the Mallows mixture model, we show that such a degenerate case provides a lot of useful information about the problem.

it must hold that φ 1 φ 2 , and π 1 and π 2 are fairly close -one is almost a cyclic shift of the other.


[12] suggest that speech signals in a multiple-speaker condition obey to a large extent disjoint orthogonality in time and frequency.

at most one source has a nonzero energy at a specific time and frequency.


It is related to our model in a sense that both models estimate hidden features.

hLinUCB extends linear contextual bandit by inclusion of hidden features and operates under a UCB-like strategy.


Now, define the estimate of prediction loss Note that y t can be readily computed from `t(•), which is available to the algorithm (equivalently, we may assume that the algorithm observed y t ).

due to the linearity of expectation, we have ˜t(w) provides an unbiased estimate of the loss `t(w) for any w 2 W. Hence, by adding a feature cost term we get ˜t(w) + h s(w), c i as an estimate of the loss that the learner would have suffered at round t had he chosen the weight vector w. Algorithm 1 The LQDEXP3 Algorithm Parameters: Real numbers 0  ⌘, 0 <  1, W 0 ⇢ W finite set, a distribution µ over W 0 , horizon T > 0.


The experimental results on real-world datasets confirm that CPLST can reduce the number of learning tasks without loss of prediction performance.

cPLST is usually better than PLST and other related LSDR approaches.


† More details of the equation are provided in Appendix B.

under mild conditions on F , we can expand the operator e hL up to the mth-order (m ≥ 1) such that the remainder terms are bounded by O(h m+1 ).


In this paper, we also use GBDT as a platform to examine the efficiency and effectiveness of decision tree parallelization.

we used PV-Tree or other baseline algorithms to parallelize the decision tree construction process in each iteration of GBDT, and compare their performance.


The Agnostic-degree-profiling algorithm (see Section 3.2) solves exact recovery in any SBM(n, p, ln(n)Q/n) for which exact recovery is solvable, using no input except the graph in question, and runs in o(n 1+ ) time for all > 0.

exact recovery is efficiently and universally solvable whenever it is information-theoretically solvable.


This sparsity requires that the support must be sparse within each prespecified block.

disjoint blocks of data points are given in advance, and an only small subset of atoms can be used in each block.


These results well reflect the properties of methods that generate captions sequentially and compositionally.

while SPICE focuses on semantical analysis, metrics including CIDEr, BLEU-4, ROUGE and METEOR are known to favor frequent training n-grams [11], which are more likely to appear when following a sequential generation procedure.


given an input to N , we define for every j = 1, . . . , d Our main theoretical results are Theorems 1 and 3.


Here we propose to use the Fisher score to construct an attribute contribution matrix A ∈ R M ×L , where an entry A d,l represents the discrimination capability of attribute a d for differentiating the class pair (i, j) indexed by u l (i, j).

given the attribute a d and class pair (i, j), let µ d k and σ d k be the mean and standard deviation of k-th class and µ d be the mean of samples from both classes i and j corresponding to d-th attribute.


( 7) for both positively (y = 1) and negatively (y = 0) labeled pairs 1 of examples.

for the posterior means, we obtain: where the coefficient ν in eq.


We ran the modified algorithm to register one puppet mesh to the remaining 6 meshes in the dataset, obtaining the correct registration in all cases.


[1] present several principles to tune parameters in kernel functions.

when the learned kernel is restricted to be a linear combination of multiple basis kernels, the problem of learning the combination coefficients as well as an SVM classifier is usually called multiple kernel learning (MKL).


This is a so-called minimax lower bound, and gives an indication of the best achievable performance of any passive learning algorithm.

no passive algorithm can learn more rapidly.


Instead we use an approximation that considers only the diagonal of the covariance matrix.

we propose the following, approximate, Mahalanobis distance metric: where the sum runs over the tensor coordinates.


In this section, we prove the correctness of Algorithm disPCA as pre-processing for these applications.

we show that by setting t 1 , t 2 sufficiently large, the objective value of any solution merely changes when the original data P is replaced the projected data P = PV (t2) (V (t2) ) .


As we observe, The best performance among the four algorithms belongs to Ada Newton.

ada Newton is able to achieve the accuracy of R N (w) R ⇤ N < 1/N by 2.4 passes over the dataset which is very close to theoretical result in Theorem 1 that guarantees accuracy of order O(1/N ) after ↵/(↵ 1) = 2 passes over the dataset.


In particular, we prove the impossibility of bounding performance of k-nearest neighbors only in terms of the diameter and the cut size.

we give a parameterized family of graphs for which the number of mistakes of the perceptron is upper bounded by a fixed constant independent of the graph size while k-nearest neighbors provably incurs mistakes linearly in the graph size.


This similarity measure is a generalization of Equation ( 2).

f LBGM is equivalent to the Boosted SSC similarity measure in the restricted case of a diagonal A.


Algorithmically, if initialized at a point with small enough objective value, then a descent algorithm always decreases the objective value and hence stays in this region.

for a generic kernel 3 a 0 ∈ S k−1 , if the sparsity rate 4 θ k −2/3 and the number of measurement m poly(k), initializing at some preconditioned k consecutive entries of y, and applying any descent method that converges to a local minimizer under a strict saddle hypothesis [17,18], produces a near shift-truncation of the ground truth.


During training, each suggested region is answered automatically according to the ground truth matte.

we only provide two possible answers, i.e., foreground or background.


For the fully-connected stage, we use multi-task learning to improve the training of the feature extractors [2,[25][26][27].

the main regression task predicts the crowd density value, while an auxiliary classification task predicts the number of people in the image patch.


With this uniform distribution over trees, we have Ï¡ for each edge, and 9 41 2 @ e f£ n A ¡ y e , so that £ E i p C £ y e . With the set-up of the previous section, the basic form of the upper bounds follows by applying Jensen's inequality [6].

for any pair £ E i t C £ y e s , we have the upper bound £ y e ¡ 9 41 2 4@ £ ge f£ n BA .


In particular, while LDP across all population of users is resilient to any attack based on auxiliary information, LDP across a sub population may be vulnerable to such attacks and additional levels of protection may need to be applied.

if data collector observes that user's response has changed; data collector knows with certainty that user's true counter value had changed.


Thus, multiple metrics have been proposed to evaluate topic models based on the perceived dependencies between words within a topic [7,8,9,10].

[11] showed that the Multinomial assumption was often violated in real world data.


The breakthrough of gradient cost for finding second-order stationary points were achieved in 2016/2017, when the two recent lines of literatures, namely FastCubic [1] and CDHS [8] as well as their stochastic versions [2,35], achieve a gradient cost of Õ(min(n −1.5 +n 3/4 −1.75 , −3.5 )) which serves as the best-known gradient cost for finding an ( , O( 0.5 ))-approximate second-order stationary point before the initial submission of this paper.


To ensure online learnability, it is sufficient to consider an assumption on the dependence of N ∞ (α, F) on α.

an obvious example of such a class is a VC-type class with N ∞ (α, F) ≤ (c/α) d for some c which can depend on n. Assuming that F ⊂ [0, 1] X , the value of the game is upper bounded by 2D T (F) ≤ 4 √ dT log c. for binary prediction, using the Sauer-Shelah lemma ensures that the value of the game is at most 4 dT log(eT ), matching the result of [15] up to a constant 2.


For example, if p( x) is completely monotonic, then, ) is convex, but the converse does not necessarily hold.

concavity of g does not follow from convexity of p( √ x), as the latter only requires that g ≤ g 2 .


A distance between smooth characteristic functions evaluated at a single random frequency is almost surely a distance (Definition 1 formalizes this concept) between these two distributions.

there is no need to calculate the whole infinite dimensional representation -it is almost surely sufficient to evaluate it at a single random frequency (although checking more frequencies will generally result in more powerful tests).


In this section, we present some applications of our theorems and algorithms in machine learning.

we consider the regularized problems with a smooth loss: where (a i , b i ), i = 1, . . . , n denote a set of training examples, R(x) could be the 1 norm x 1 , the ∞ norm x ∞ , or a huber norm [28], or the 1,p norm K k=1 x k p , where k is the k-th component vector of x.


1), which reduces the problem of CI testing to Binary Classification.

the key steps of our procedure are as follows:  . .


Then, we have: We propose to retain r and d, but treat them as constants for the purposes of gradient computation.

we augment a network, which contains batch normalization layers, with a perdimension affine transformation applied to the normalized activations.


For the rest of the paper, we consider the problem of learning an embedding for a vocabulary of size n, which is canonically defined as V = {1, 2, • • • , n}.

we want to learn a vector representation v i ∈ R d for each token i.


It is worth noting that for k-sparse matrices with k constant, this distance does not scale with p or q.

if the absolute value of the elements of Θ (1) and Θ (2) are bounded by Θ max then d(Θ (1) , Θ (2) ) ≤ 2 Having the estimator Θ (i) the algorithm constructs the confidence set for episode i as where > 0 is an input parameter to the algorithm.


Of course, the connection between AdaGrad and dropout is not perfect.

adaGrad allows for a more aggressive learning rate by using But, at a high level, AdaGrad and dropout appear to both be aiming for the same goal: scaling the features by the Fisher information to make the level-curves of the objective more circular.


We provide detailed empirical results that demonstrate the benefits of our approach on a large variety of OpenML datasets.

on many of these datasets, our algorithm either surpasses the accuracy of the state-of-the-art baselines, or provides more compact models while being competent in terms of accuracy.


Our first main contribution is to characterize a rich class of structured sparse regularizers that allow efficient computation of their polar operator.

motivated by [6], we consider a family of structured sparse regularizers induced by a cost function on variable subsets.


We propose a framework for learning representations that relies on auto-encoding variational Bayes, in which the search space is constrained via kernel-based measures of independence.

our method employs the d-variable Hilbert-Schmidt Independence Criterion (dHSIC) to enforce independence between the latent representations and arbitrary nuisance factors.


Case 1: Suppose that d p < d q , which implies that d A − 1 < d q .

we may write for some choice of b i s. This means that we can write (5) as if we knew the set of complex poles Z p = {z 1 , .