. . , T k } ∈ O k (S) is an optimal k-clustering, let r be the number of clusters of T that "do not respect" the partition {S 1 , S 2 }.

r is the number of clusters of T that intersect both S 1 and to have the smallest r. If r = 0, then T is a refinement of S and there is nothing to show.


In fact, many stochastic algorithms assume that S t is a uniform subset of [n], because in this case the sub-sampled Hessian is an unbiased estimator of the full Hessian.

8✓ 2 C, E [H St (✓)] = H [n] (✓), where the expectation is over the randomness in S t . We next show that for any scaling matrix Q t that is formed by the sub-samples S t , iterations of the form Eq.


Contrastive loss [3,7] takes pairs of examples as input and trains a network to predict whether two inputs are from the same class or not.

the loss is written as follows: where m is a margin parameter imposing the distance between examples from different classes to be larger than m. Triplet loss [27,2,19] shares a similar spirit to contrastive loss, but is composed of triplets, each consisting of a query, a positive example (to the query), and a negative example: Compared to contrastive loss, triplet loss only requires the difference of (dis-)similarities between positive and negative examples to the query point to be larger than a margin m. Despite their wide use, both loss functions are known to suffer from slow convergence and they often require expensive data sampling method to provide nontrivial pairs or triplets to accelerate the training [2,19,17,4].


For instance, it admits iterative algorithms that minimize each F r separately and combine the results in a straightforward manner (for example, dual decomposition).

it has been shown that the minimization of decomposable functions can be rephrased as a best-approximation problem, the problem of finding the closest points in two convex sets [25].


Given a Gaussian policy π(a|s, θ) ∼ N θ T φ(s), σ 2 , under the assumption of uniformly bounded rewards and basis functions (Assumption 4.1), we have the following upper bound to the variance of the i-th component of the PGT gradient estimate ∇ θi J P GT µ (θ): As expected, since the variance of the gradient estimate obtained with PGT is smaller than the one with REINFORCE, also the upper bound of the PGT variance is smaller than REINFORCE one.

while the variance with REINFORCE grows linearly with the time horizon, using PGT the dependence on the time horizon is significantly smaller.


Suppose we are given a dataset of labeled sequential data points Z = [(X 1 , y 1 ), .

. . , (X n , y n )] where each sequential data point the t-th time-step data point of the i-th sequential data point is x i,t .


Three experiments simulating various levels of uncorrelated input/output activity were conducted.

excitatory Poisson inputs at 2, 20 and 40 Hz were balanced by inhibitory Poisson inputs at 6.3, 63 and 124 Hz to generate output rates of approximately 2, 20 and 40 Hz, respectively.


As shown in Figure 2, training with = 2 • 10 4 converged to a fixed solution, which provided an estimate f close the original function f . In contrast, training with = 3 • 10 4 converged to an oscillation and not to a fixed point.

after sufficient training, the estimate kept switching between fodd and feven at each iteration of the gradient descent algorithm.


( 3) holds because the discrete matching M * should not change for infinitesimal change of θ, as long as the transformations T X , T Y , and the distance d(•, •) are continuous.

the gradient does not propagate through the matching.


Consider a discrete-time stochastic process X = {X t : t ≥ 0} on the state space N = {1, 2, . . . , n}, where the initial state X 0 is given, say X 0 = i, the next state X 1 is determined by the transition probability P(X 1 = j|X 0 = i) = p ij , and the subsequent states are determined by the transition probabilities where t ≥ 0.

note that the process X is time homogeneous, i.e., the transition probabilities in (1) are independent of t. if the previous and current states are the same, the process will remain in the current state forever.


For any k = 1, . . . , K, the statistical error of the sample-based minimization function defined in (3.3) satisfies that, for any fixed where 2) and (3.3), and m = Theorem 3.4 establishes the statistical error associated with c M k (⌦ [K] k ) for arbitrary ⌦ j 2 B(⌦ ⇤ j ) with j 6 = k. In comparison, previous work on the existence of a local solution with desired statistical property only establishes theorems similar to Theorem 3.4 for ⌦ j = ⌦ ⇤ j with j 6 = k. The extension to an arbitrary ⌦ j 2 B(⌦ ⇤ j ) involves non-trivial technical barriers.

we first establish the rate of convergence of the difference between a sample-based quadratic form with its expectation (Lemma B.1) via concentration of Lipschitz functions of Gaussian random variables [24].


We will show how to translate each instance of X3C, (U, S), to an instance of k-means clustering in the Euclidean plane, X.

x has a grid-like structure consisting of l rows (one for each S i ) and roughly 6m columns (corresponding to U ) which are embedded in the Euclidean plane.


To make this precise, we introduce a time after which for each arm, the empirical proportion allocated to it is accurate.

given β ∈ (0, 1) and > 0, we define It is clear that P(M β < ∞) = 1 for all > 0 if and only if T n,i /n → w β i with probability 1 for each arm i ∈ A.


2.2 becomes prohibitively expensive as the length of the chain T becomes large.

the forward-backward algorithm in the local step takes O(K 2 T ) time.


For any f , U (f ; micro , P) ⌘ U(f ; instance , P).

f We further restrict our study to metrics selected from the linear-fractional metric family, recently studied in the context of binary classification [9].


In turn, computing the overlap between the model and neural RDMs provides evidence about how well a particular model explains the response patterns in a particular brain region.

we performed a region of interest analysis as well as a searchlight analysis by first constructing the RDMs of STG (target RDM) and the model layers (candidate RDM).


Similar to several other analyses of AMP algorithms such as [14][15][16][17][18], the theorem provides a scalar equivalent model for the componentwise behavior of the adaptive GAMP method.

asymptotically the components of the sets θ t x and θ t z in (10) are distributed identically to simple scalar random variables.


The autoregressive property is enforced by multiplying the weight matrices of MADE with suitably constructed binary masks.

we use MADE with Gaussian conditionals as the building layer of our flow.


We are interested in situations where ||f || ν is much smaller than ||f || ∞ .


In Section 3 we analyzed generalized linear models and robust regression using both the (1, µ)-GD property and the (2, µ)-GD property.

the (1, µ)-GD property was critical to obtain dimension-independent norm-based capacity control.


In the presence of outlier data points, there is an inherent unidentifiability between AB and E which can prevent the detection of outliers and hence reduce the performance of subspace clustering.

an outlier y i can be included in the sparse component as e i = y i or included in the dictionary D with its own subspace, which leads to (AB) ii ≈ 1.


In the proposed RankGAN, we relax the training of the discriminator to a learning-to-rank optimization problem.

the proposed new adversarial network consists of two neural network models, a generator and a ranker.


y is the sum of relatively few shifted copies of waveforms, so the coefficients of most dictionary elements will be zero), CBP draws on the basis pursuit denoising, which has been effective in the compressive sensing setting and elsewhere [10], [1].

cBP (with a Taylor basis) recovers coefficients using: Here we denote by F the matrix with columns {f n,j (t)} and F the matrix with columns {f n,j (t)}.


Compared to EM, moment-based algorithms provide a stronger theoretical foundation for learning latent variable models.

it is known that in the realizable setting the output of a MoM algorithm will converge to the parameters of the true model as the amount of training data increases.


For the rest of this section, we build the connection between the proposed robust portfolio optimization and its moment-based counterpart.

we show that they are consistent under the elliptical model.


Based on this observation, we develop a coordinate descent procedure to obtain scalability of the bundle method over large data sets.

we put an outer loop above the bundle method.


We propose the first fast oblivious subspace embeddings for spaces induced by a non-linear kernel without explicitly mapping the data to the high-dimensional space.

we propose an OSE for mappings induced by the polynomial kernel.


Though based on the same goals, however, our methods are quite different.

we are able to cast our optimization as an instance of semidefinite programming.


The bad news is that, in general, condition Q s,q (κ), as well as RE and Compatibility conditions, cannot be verified.

given a sensing matrix A and a representation structure B, it seems to be difficult even to verify that a pair (H, • ) satisfies condition Q s,q (κ) associated with A, B, let alone to synthesize such H which satisfies this condition and results in the best possible error bounds (4), ( 5) for the regular and the penalized 1 -recoveries.


Finally, we provide a lower bound for the problem of detecting clusters in evolving stochastic block model.

it implies that in the case when p = 1 and q = 0 the algorithm of Theorem 5 is optimal (up to constant factor).


Such approaches tend to learn shared structures from a set of relevant tasks and generalize to novel tasks.

our approach is inspired by early work on parameter prediction that modifies the weights of one network using another [32,33,34,35,36,37,38,26,39].


Partitioning-based regression methods are often invariant to monotone transformations of observations.

when f 0 is a step function, standardizing the distance between the observations, and thereby the split points, has no effect on the nature of the estimation problem.


As a consequence of the gradient uniform convergence bounds, we show how to use any algorithm that finds approximate stationary points for smooth functions in a black-box fashion to obtain optimal sample complexity for these models-both in high-and low-dimensional regimes.

standard algorithms including gradient descent [33], SGD [10], Non-convex SVRG [38,3], and SCSG [26] enjoy optimal sample complexity, even when allowed to take multiple passes over the dataset.


Most importantly, this function U (S, c) is identical to V(S, 1 2 (T c)), the Black-Scholes value of the option in (4) when the GBM volatility parameter is in the Black-Scholes analysis.

analogous to to (3), U (S, c) satisfies a differential equation: The following is our main result of this paper.


This paper demonstrates a more principled approach to address this problem, by constructing new bases that emerge from investigating the links between basis construction methods and different power series expansions of value functions.

instead of using the eigenvectors of the transition matrix, the proposed approach uses the average-reward or gain as the first basis vector, and dilates the reward function by powers of the average-adjusted transition matrix.


Let Y s,a,π be a random variable defined as Y s,a,π = (X s,a,π − R(s, a))/γ.

we can see that the pdf of Y s,a,π is a convex combination of the pdfs of X s ,π for each s ∈ S. we have f Ys,a,π (y) = s ∈S T (s |s, a)f X s ,π (y).


A key property of the ANOVA kernel is multi-linearity [4]: where p ¬j denotes the (d − 1)-dimensional vector with p j removed and similarly for x ¬j .

everything else kept fixed, A m (p, x) is an affine function of p j ∀j ∈ [d].


Some recent work on active learning has focused on improved label complexity under certain noise conditions [17,8,18,6,7].

it is assumed that there exists constants κ > 0 and 0 for all h ∈ H. This is related to Tsybakov's low noise condition [16].


In this work, we compare several different techniques for neural program induction, with a particular focus on how the relative accuracy of these techniques differs as a function of the available training data.

if technique A is better than technique B when only five I/O examples are available, does this mean A will also be better than B when 50 I/O examples are available?


In contrast, the cost (4) will heavily penalize the classifier if it does not predict the output for which p LM (•) is positive, and will penalize less if it predicts outputs for which p LM (•) is zero.

this cost will encourage p ✓ (y|x) to cover as much of p LM (•) as possible, a behavior called "coverage-seeking" in [25].


In the rest of Section 4, we show how InfoGAIL can infer the latent structure of human decisionmaking in a driving domain.

our agent only relies on visual inputs to sense the environment.


However, these algorithms are designed for the centralized setting, and are not applicable in the setting of large-scale datasets that are maintained remotely by different sites.

collecting the information from all the remote sites and performing a centralized clustering algorithm is infeasible due to high communication costs, and new distributed clustering algorithms with low communication cost need to be developed.


This restriction rules out a variety of renewal densities that are commonly employed to model neural data [13,14,15].

the log-normal and inverse-Gaussian densities both have increasing regimes on a subset of [0, ∞), as does the gamma density q(u) ∝ u κ−1 e −uκ when κ > 1.


The MLDS explicitly incorporates the dynamics, noise, and tensor structure of the data by juxtaposing concepts in probabilistic graphical models and multilinear algebra.

the MLDS generalizes the states of the linear dynamical system (LDS) to tensors via a probabilistic variant of the Tucker decomposition.


For a given pattern µ, the inhibitory neurons will be free to adjust their steady state firing rates v µ inh to whatever pattern proves to be optimal to maintain stability.

we think of the activity of inhibitory neurons as "auxiliary" variables.


Having set up the framework for infinite CCA, we now describe its applicability for the problem of multitask learning.

we consider the setting when each example is associated with multiple labels.


As a result, we can perform stochastic approximation of (7) while monitoring the KL divergence between the posterior approximates due to the change of m, S, and X across iterations.

we use stochastic mirror ascent [17] in the space of probability densities in RKHS, which was recently proven to be as efficient as stochastic natural gradient ascent [19].


their gradients are L i -Lipschitz), and x is a minimizer of (almost) all f i (x) (i.e.

p i (∇f i (x ) = 0) = 1), then E x k − x goes to zero exponentially, rather then polynomially, in k. reaching a desired accuracy of E x k − x 2 ≤ ε requires a number of steps that scales only logarithmically in 1/ε.


We close this section by noting that the summability condition (4.6) imposes a lower bound on the step-size exponent β that is different from the lower bound in Theorem 3.

if β = 1/2, (4.6) cannot hold for any vanishing sequence of exploration factors ε n ↓ 0.


Thus if 1 − µ < 2 n−t , it is preferable not to explore new arms and to play the best known arm, with mean reward µ, until time n. A fortiori, the best known arm is played until time n whenever its parameter is larger than 1 − 2 n . We denote by A n the first arm whose parameter is larger than 1 − 2 n . We have K n ≤ A n (the optimal policy cannot explore more than A n arms) and The parameter θ An of arm For all k = 1, 2, . .

., let L 1 (k) be the length of the first run of arm k. We have: using the fact that: and To conclude, we write: Observe that, on the event {L 1 (1) + .


In contrast to previous uses of graph-based priors, we define f as the real function f (defined over the entire observation space) evaluated at the graph nodes.

f is defined as a linear function of x, and at the graph node i, Xw, and p(f ) induces a Gaussian prior on w, with precision matrix A = X T ∆X, ) Notice that since ∆ is singular, A may also be singular, and the corresponding prior may therefore be improper.


The most significant weakness of tree models is boundary artifacts.

leaf nodes that are adjacent in the original process may be widely separated in the tree structure (see Figure 3(b)).


Once we could ensure that a pair of corresponding images are mapped to a same latent code and a same latent code is decoded to a pair of corresponding images, (x 1 , G 2 (z 1 ⇠ q 1 (z 1 |x 1 ))) would form a pair of corresponding images.

the composition of E 1 and G 2 functions approximates F ⇤ 1!2 for unsupervised image-to-image translation discussed in Section 2, and the composition of E 2 and G 1 function approximates F ⇤ 2!1 .


We will also consider a vector restricted on a support set.

for a d-dimensional vector x and a support set T ⊂ {1, 2, . . . , d}, depending on the context, x T can either be a |T |-dimensional vector by extracting the elements belonging to T or a d-dimensional vector by setting the elements outside T to zero.


We explore the tradeoff between communication and statistical efficiency for a number of fundamental nonparametric density estimation problems.

we insist that our algorithms are sample-efficient and our goal is to design distributed protocols using a minimum amount of communication.


Further define the matrices Then the marginal probability of T in DVS is We prove Theorem 2 via a perturbation argument that connects DVS to volume sampling.

observe that for ✏ > 0 and |S| n it holds that Carefully letting ✏ !


In this section, we empirically study the proposed dual GAN algorithms.

we show the stable and monotonic training for linear discriminators and study its properties.


We believe a good generative model should be able to synthesize 3D objects that are both highly varied and realistic.

for 3D objects to have variations, a generative model should be able to go beyond memorizing and recombining parts or pieces from a pre-defined repository to produce novel shapes; and for objects to be realistic, there need to be fine details in the generated examples.


Further details of this can be found in Appendix D. As explained earlier our primary interest is to explore the various bounds strengths and weaknesses.

we are interested in their tightness, as well as their ability to support model selection.


Hence, all of the bounds discussed above can be extended to the problem of maximizing a log-supermodular function.

the inequality in Theorem 4.1 shows that the maximum value of the objective function on any graph cover is achieved by a lift of a maximizing assignment on the base graph.


Their main disadvantage is that they do not control the structural complexity of the resulting functions.

if one of the components of a (say, a weight in a neural network) happens to be irrelevant, a Gaussian prior will not set it exactly to zero, thus The goal of supervised learning is to infer a functional relation ¢¡ ¤£ ¦¥ ¨ § © , based on a set of (maybe noisy) training examples ¡ ¥ ¨ § ¦ © "!


. . , F p (X p )), which is specified by the marginal CDFs {F s (X s )} s∈V and the copula density F COP;θ corresponding to the exponential family MRF density.

the non-parametric extension in (11) of the exponential family MRF densities is precisely an exponential family MRF copula density.


The goal is to establish correspondences between the input-data layers A 0 and A 0 .

every cell A 0 (x) in the reference object o ∈ Φ has a certain shift d ∈ D in the searched object o ∈ Φ, and we want to estimate d. Here comes the cornerstone idea of our method: we establish the matching of A 0 (x) with A 0 (x − d) for a shift d if there is a pair of "parallel" paths (we call this pair a siamese path), originating at those nodes and ending at the last layers A L , A L , which match.


The payment function of A has the same form as the VCG payment rule, with the parameters factored in to ensure that the auction remains strategy-proof.

for all j ∈ [n], the We assume that It is typical to assume an upper bound (here, H v ) on the bidders' valuation for any bundle.


We focus on learning under a cluster assumption that is formalized in the next section, and establish that there exist nonparametric classes of distributions, denoted P XY , for which the decision sets (over which the target function is smooth) are discernable from unlabeled data.

moreover, we show that there exist clairvoyant supervised learners that, given perfect knowledge of the decision sets denoted by D, can significantly outperform any generic supervised learner f n in these  Based on this, we establish that there also exist semi-supervised learners, denoted f m,n , that use m unlabeled examples in addition to the n labeled examples in order to estimate the decision sets, which perform as well as f D,n , provided that m grows appropriately relative to n. if the error bound for f D,n decays polynomially (exponentially) in n, then the number of unlabeled data m needs to grow polynomially (exponentially) with the number of labeled data n. We provide general results for a broad range of learning problems using finite sample error bounds.


In this section, we discuss adaptive methods and analyze their convergence behavior in the nonconvex setting.

we focus on two algorithms: ADAM (3.1) and the proposed method, YOGI (3.2).


The population prediction loss γ( Ŷ , n), and group-specific losses γ 0 ( Ŷ , n), γ 1 ( Ŷ , n), for a fixed learning algorithm Ŷ , behave asymptotically as inverse power-law curves with parameters (α, β, δ).

∃M, M 0 , M 1 such that for n ≥ M, n a ≥ M a , Intercepts, δ, δ a in (3) represent the asymptotic bias B( ŶD∞ ) and the Bayes error N , with the former vanishing for consistent estimators.


Another useful property of NormalHedge, which Hedge does not possess, is that it assigns zero weight to any action whose cumulative loss is larger than the cumulative loss of the algorithm itself.

non-zero weights are assigned only to actions which perform better than the algorithm.


Our robustness analysis provides in turn insights onto desirable properties of generative models capturing real-world distributions.

the intriguing generality of our analysis implies that when the data distribution is modeled through a smooth and generative model with high-dimensional latent space, there exist small-norm perturbations of images that fool humans for any discriminative task defined on the data distribution.


It is important to mention that we have not used the whole range of layers in our experiments.

we usually started from layer 2 and finished at layer 8.


In this paper, we formally analyze and empirically evaluate how information is changed by serial reproduction and how this process relates to memory biases.

we provide a rational analysis of serial reproduction (in the spirit of [5]), considering how information should change when passed along a chain of rational agents.


The preferred treatment criterion represents a relaxation of treatment parity.

every decision making system that achieves treatment parity also satisfies the preferred treatment condition, which implies (in theory) that the optimal decision accuracy that can be achieved under the preferred treatment condition is at least as high as the one achieved under treatment parity.


This presents a challenge in applying any method.

it was found that the performance of any scalar-gain algorithm was extremely poor, even with projection of parameter estimates.


The minimax rate is achieved by kernel estimators and local polynomial estimators.

for kernel estimators if we use a product kernel with common bandwidth h n for each variable, choosing h n ∼ n −1/(4+D) yields an estimator with the minimax rate.


While this assumption works well for a hypothetical Mondrian World of flat colors, it does not always hold for natural images.

weiss [2001] found that this model of reflectance and lighting is rarely true for outdoor scenes.


Some researchers have exploited the internal structure of the dataset to speed up first-order methods.

they use ∇f i (x) to approximate ∇f j (x) when the two data samples i and j are sufficiently close.


Consequently, a better understanding of connectivity patterns during MI of different limbs as measured by EEG is required to guide the design of new feature extraction algorithms for BCIs.

it is important to properly address possible volume conduction effects, not confine the analysis to a small subset of electrodes, and consider a broad range of frequency bands.


We explore the use of non-parametric models for partially ranked data and derive efficient procedures for their use for large n. The derivations are largely possible through combinatorial and algebraic manipulations based on the lattice of partial rankings.

we demonstrate for the first time a non-parametric coherent and consistent model capable of efficiently aggregating partially ranked data of different types.


Given the heuristic nature of existing algorithms, it would be most helpful to be able to call an iterative improvement procedure on their output.

we seek an analogue of k-means for hierarchical clustering.


A copy of this MRF is instantiated each time an instance (equivalently, a token, as this relates to the well-known type-token distinction) of X is parsed as v, in the same way that each instance of a symbol in a grammar uses the same categorical distribution to select productions.

an instance of a symbol has an associated region R ⊆ I and the MRF instantiated for that instance is simply the subset of the full-image MRF containing all of the nodes in R and all of the edges between the nodes in R. The energy of this instance is .


we compute ρ i , and update ν (i−1) to ν i upon the arrival of the i-th sample x i .


We start by noting that do-calculus [Pearl, 1995] provides rules to evaluate invariances in the interventional space.

we focus here on the Rule 3, which ascertains the condition such that a set of interventions does not have an effect on the outcome variable, i.e., P (y|do(x, z), w) = P (y|do(x), w).


Proposition 2.1 [4] Let G be a DAG over {Xl , ... , X n }, and let P F G be a distribution.

the multi-information is the sum of local mutual information terms between each variable and its parents (denoted I G).


In order to remove the equality constraint, which will simplify the problem, let's change variables by defining the n×(n−ν) matrix F as {x : Q T D G x = 0} = {x : x = F y}.

f is a span for the null space of Q T ; and we will take F to be an orthogonal matrix.


In this paper, we propose a two-step representation learning method to bridge the feature spaces of different languages by exploiting a set of parallel bilingual documents.

we first formulate a matrix completion problem to produce a complete parallel document-term matrix for all documents in two languages, and then induce a low dimensional cross-lingual document representation by applying latent semantic indexing on the obtained matrix.


All such submodular functions f are called representatives of F . The conversion between an approximation guarantee for F and an approximation guarantee for a representative f of F holds for any choice of the representative.

2 [22] shows that it possible to obtain a (1 − 1/e) approximation ratio for a cardinality constraint.


All operations applied on p are linear with respect to the inputs.

p could be formalized as a linear combination of the inputs although its coefficients are unknown.


Since we want to minimize classification error on future test items, we define the teaching loss function to be the generalization error: The first expectation is with respect to the test distribution p(x, y).

we still assume that test items are drawn iid from the test distribution.


However, the bounds for the ensemble Nyström are tighter than those for any Nyström expert based on a single sample of size m even for a uniform weighting.

for µ = 1/p, the last term of the ensemble bound for norm-2 is smaller by a factor larger than µ max p In this section, we present experimental results that illustrate the performance of the ensemble Nyström method.


The objective of the present paper is to design training methods that scale to larger and harder control problems, even if most of the components were already known.

we combine supervised learning with trajectory optimization, namely Contact-Invariant Optimization (CIO) [12], which has given rise to some of the most elaborate motor behaviors synthesized automatically.


We can substantially enrich the linear model while keeping its overall structure if we permit the coefficients themselves to depend on the input x.

we define (offset function omitted) f (x) = ✓(x) > x, and choose ✓ from a complex model class ⇥, realized for example via deep neural networks.


We demonstrate the effectiveness of our approach in the task of 3D layout prediction from single images, and show that good models are learned when labeling only a handful of random variables.

the same performance as using the full training set can be obtained while only labeling ∼10% of the random variables.


On the lower bound side, the lower bound result in Blum et al.

[3] is only for the special case when k = d. We extend their result to every k and d. we show that the sample complexity for collaborative learning has to be Ω(max{d ln k, k ln d}/ ) for constant δ (Theorem 6).


We generally employ a further approximation when ranking and selecting edges.

we evaluate an edge only in terms of the two variables associated directly with the edge.


For a more realistic evaluation we next compare the same models after consensus re-ranking [10,23].

for a given test image, we first find its nearest neighbors in the training set in the cross-modal embedding space learned by a two-branch network proposed in [34].


Having observed these performances, the task of predicting the performance of any of them on a new dataset can be cast as a matrix factorization problem.

we are seeking a low rank decomposition such that Y ≈ XW, where X ∈ R N ×Q and W ∈ R Q×D , where Q is the dimensionality of the latent space.


(The true coordinate update requires iterating over all of the data.)

sVI iteratively updates an estimate of λ as follows, where λt is a random variable whose expectation is λ VB t and ρ t is the learning rate.


The result, in brief, is that E[Ψ n ] = O(1/n), under conditions that are similar to those above, but stronger.

we require that γ n be proportional to 1/n and that X n be bounded.


Following the evaluation protocol established in [7], we use a retrieval setting to evaluate the responses at each round in the dialog.

every question in VisDial is coupled with a list of 100 candidate answer options, which the models are asked to sort for evaluation purposes.


In this case, we assume that x d = Y d in the model in Figure 1 and consider the standard approach when dealing with real-valued observations, which consist of assuming a Gaussian likelihood function.

as in the standard linear-Gaussian IBP [8], we assume that each observation x d n is distributed as Positive Real-valued Data.


We can see that test accuracies are statistically the same across different algorithms, which validates our analysis on the objective function.

we observe that SMM is consistently more efficient than other methods, especially for a large number of training samples.


The following local alternative (A1), and balancing assumption (A2), are similar to those used in previous works [6][7][8][9] on problem (1).

condition (A1) means that the KL-divergence between N (µ 1 , Σ) and N (µ 2 , Σ) tends to 0 as n → ∞.


Proposition 2 The linear machine in ( 7) is equivalent to the following approximate kernel machine Although both random Fourier features and the Nyström method can be viewed as variants of the unified framework, they differ significantly in the construction of the approximate functional space H a .

the basis functions used by random Fourier features are sampled from a Gaussian distribution that is independent from the training examples.


While C-DAC does a good job of matching human behavior, at least based on the behavioral metrics considered here, we note that this does not necessarily imply that the brain implements C-DAC exactly.

solving C-DAC exactly using dynamic programming requires a representational complexity that scales exponentially with the dimensionality of the search problem (i.e.


With refetching, it can outperform BestFixed.

when all required objects are refetched instantly, this policy has a 13-20% lower miss rate than BestFixed, and almost the same performance as BestShifting( ) for modest .


Random initialisation (Rnd) has also been used for networks with the same number of hidden units.

for networks with 50 hidden units, we have evaluated the performance of 6 different random initialisations.


Because the dimension of V is reduced by one on each iteration of the loop, and because the initial dimension of V is simply the number of selected eigenvectors in step one, the size of Y is distributed as the number of successes in N Bernoulli trials where trial k succeeds with probability λ k λ k +1 .

|Y | cannot be larger than rank(L), and To get a feel for the sampling algorithm, it is useful to visualize the distributions used to select y i at each time step, and to see how they are influenced by previously chosen items.


Alternatively, we use the importance sampling to estimate the conditional likelihoods [24]: Although the SGVB learning framework has shown to be effective in training deep generative models [16,24], the conditional auto-encoding of output variables at training may not be optimal to make a prediction at testing in deep CGMs.

the CVAE uses the recognition network q φ (z|x, y) at training, but it uses the prior network p θ (z|x) at testing to draw samples z's and make an output prediction.


Moreover, our results are also subject to improvements.

we did not use GPUs yet, which has limited our ability to exhaustively explore model hyper-parameters and evaluate the accuracy of large networks.


To explain, we will consider a simple "toy" data distribution where the behavior of these algorithms is more clear.

we will generate three heavy-tailed random variables X, Y, Z according to: Here, σ 1 , σ 2 are scale parameters sampled independently from a Laplace distribution, and e 1 , e 2 , e 3 are sampled independently from a unit Gaussian.


Moreover, these methods do not suffer from "over-shrinking" because λ → 0 as t → ∞.

if the chosen t happens to be too large, the worst we can get is the standard empirical estimator.


In this paper, we investigate the use of Markov chain Monte Carlo sampling to perform approximate inference in general log-submodular and log-supermodular models.

we consider a simple Gibbs sampling procedure, and establish two sufficient conditions, the first guaranteeing polynomial-time, and the second fast (O(n log n)) mixing.


But is superpixel segmentation necessary?

can we simply train with randomly sampled points?


In addition to the distillation loss in KD and the adversarial losses in NaGAN mentioned above, we define a distillation loss from the classifier to the teacher and an adversarial loss between the teacher and the discriminator.

the classifier and the teacher, serving as generators, aim to fool the discriminator by generating pseudo labels that resemble the true labels.


It was shown in [20]  satisfies the following property: the (pseudo)marginals computed by performing tree-reweighted variational inference with the parameters ✓ := B ⇤ trw ( b ) yield the sufficient statistics b .

the approximate backward map B ⇤ trw computes an element in the pre-image of the approximate forward map given by tree-reweighted variational inference.


The new solution, denoted by p i+1 , is computed as the projection of p i onto the level set L i , which is equivalent to solving the following optimization problem: Although the projection is regarded as a quadratic programming problem, it can often be solved efficiently because its solution is likely to be the projection onto one of the hyperplanes of polyhedron L i .

only very few linear constraints of L are active; most of them are inactive.


Aiviv~, where Vi denotes the transpose of v and Vi is the i-th eigenvector with eigenvalue Ai.

the more peaked the spectrum the more aligned (specific) the kernel can be.


It is sufficient that the objective be polynomial-time evaluable, along with its first and second derivatives, 2 We are obviously assuming X is equipped with an appropriate σ-algebra, and R with the standard Borel σ-algebra, such that the joint distribution P over X × R is well defined and κ(•, •) is measurable.

3 let Mn(θ) , and equip the parameter space Θ with the uniform metric • Θ.


To update the basis matrix L, we adopt the block-coordinate descent with warm restarts [2].

each column of the basis L is updated individually while fixing the other columns.


The key step of the Markov decision model is to decide the best action at certain state.

it is to find the optimal decision policy.


To make in-and out-of-distribution samples more separable, we consider adding a small controlled noise to a test sample.

for each test sample x, we calculate the pre-processed sample b x by adding the small perturbations as follows: where " is a magnitude of noise and b c is the index of the closest class.


The Glauber dynamics associated with Ising models in high temperature are fast mixing and offer a natural way to define a martingale sequence.

consider the Doob martingale corresponding to any linear function f for which we wish to show concentration, defined on the state of the dynamics at some time step t  . Also, crucially, t * need not be too large since the dynamics are fast mixing.


These methods are based on using parameterized nonlinear functions to activate each unit in a neural network, with the unit-dependent parameters incorporated into the data-driven training algorithms.

[6] considered the adaptive piecewise linear (APL) unit defined by a mixture of hinge-shaped functions, and [7] used nonparametric Fourier basis expansion to construct the activation function of each unit.


The FO d-DNNF node ∧ r (n x , n r , D, D , V) represents a decomposable conjunction between the d-DNNF nodes n x , n r and a d-DNNF node isomorphic to the ∧ r node itself.

the isomorphic operand is identical to the node itself, except for the size of the domain of the variables in V, which becomes one smaller, going from D to D in the isomorphic operand.


As baseline we used a fairly straightforward means of document matching via its length.

longer documents in one language will be most probably translated into longer documents in the other language.


A second approach, suggested in [13, p. 34], is to approximate the null distribution as a two-parameter Gamma distribution [12, p. 343, p. 359]: this is one of the more straightforward approximations of an infinite sum of χ 2 variables (see [12,Chapter 18.8] for further ways to approximate such distributions; in particular, we wish to avoid using moments of order greater than two, since these can become expensive to compute).

we make the approximation An illustration of the cumulative distribution function (CDF) obtained via the Gamma approximation is given in Figure 1, along with an empirical CDF obtained by repeated draws of HSIC b .


Then By Corollary 8, for each F r , there exists a partition of V into disjoint sets A r1 , .

. . , A rMr such that we can write a↵ 0 (B y ) as the nullspace of either of the matrices . . . . . . , where 1 A is the indicator vector of A ✓ V . For T 0 , this follows directly from Equation ( 6).


We call an optimization method rescaling invariant if the updates of rescaling equivalent networks are rescaling equivalent.

if we start at either one of the two rescaling equivalent weight vectors w(0) ∼ w (0) , after applying t update steps separately on w(0) and w (0) , they will remain rescaling equivalent and we have w(t) ∼ w (t) .


The ex ante coordinated strategy profile for the team is implicitly represented by the pair ( λ, ωT ,σ ).

that pair encodes a coordination device that operates as follows: • At the beginning of the game, a pure normal-form plan σ ∈ Σ is sampled according to the discrete probability distribution encoded by λ.


Furthermore, the performance prediction loss acts as a regularizer that forces the encoder not optimized into a trivial state to simply copy tokens in the decoder side, which is typically eschewed by adding noise in encoding x by previous works [1,24].

when both the encoder and decoder are optimized to convergence, the inference process for better architectures is performed in the continuous space E. starting from an architecture x with satisfactory performance, we obtain a better continuous representation e x by moving e x = {h 1 , • • • , h T }, i.e., the embedding of x, along the gradient direction induced by f : where η is the step size.


A popular family of learning algorithms called temporal difference (TD) methods [1] are designed for this situation.

least-squares TD (LSTD) algorithms [2,3,4] exploit the linearity of the value function to estimate its parameters from sampled trajectories, i.e., from sequences of feature vectors of visited states, by solving a set of linear equations.


Throughout this paper we will take θ t = ατ t with α is fixed.

the sequence {τ t } t≥0 is given by the recursion τ 2 t+1 = F(τ 2 t , ατ t ).


Figure 2 reports numerical results obtained.

leftmost plots report results associated with models trained on sequences from the Highway split and tested on the New York-like City and the Old European Town splits (top-left and bottom-left, respectively); rightmost plots report results associated with models trained on sequences from the New York-like City split and tested on the Highway and the Old European Town splits (top-right and bottom-right, respectively).


At the beginning of every epoch k, for every s ∈ S, a ∈ A such that N s,a ≥ η, it generates multiple samples for P s,a from a 'boosted' posterior.

it generates ψ = O(S log(SA/ρ)) independent sample probability vectors , where M t s,a denotes the vector [M t s,a (i)] i=1,...,S , with M t s,a (i) := 1 κ (N t s,a (i) + ω), for i = 1, . . . , S. (3) Here, κ = O(log(T /ρ)), ω = O(log(T /ρ)), η = T S A + 12ωS 2 , and ρ ∈ (0, 1) is a parameter of the algorithm.


For each of the underlying clustering algorithms, we vary the input parameter of desired number of clusters as 4 : 1 : 10.

we have a total of 7 × 4 = 28 input clusterings.


We do this by using our framework to derive a notational variant of an existing Flanker model [5], and using parameter values from this previous model to simultaneously generate qualitatively accurate predictions in both the flanker and AX-CPT paradigms.

our theory generates plausible behavior in qualitatively different tasks, using the same parameters.


As shown in Table 2, for the random distractor, the 'EM Model' substantially outperforms the 'NaIve Model' in all cases.

for the 'wine' and 'iris' datasets, by introducing an additional class label to every training instance, there is only one class label left out of the class label candidates and yet the performance of the 'EM Model' is still close to the case when there are no additional class labels.


we consider a sequence of problem instances {x R n → R is proper convex, and, m := m (n) with m = δn, δ ∈ (0, ∞).


The increased availability of data in recent years has led several authors to ask whether it is possible to use data as a computational resource.

if more data is available, beyond the sample complexity limit, is it possible to use the extra examples to speed up the computation time required to perform the learning task?


Given a set of points S ∈ (R d ) n consider the task of identifying a subset C ⊆ Y of size k with the lowest possible cost.

instead of searching for k centers in R d , we are searching for k centers in Y , and our runtime is allowed to depend polynomially on |Y |.


Thus we propose to automatically fill the missing entries of M 0 based on the feature co-occurrence information expressed in the observed data, by conducting matrix completion to recover a low-rank and sparse matrix.

we formulate the matrix completion as the following optimization problem where • 1 denotes a ℓ 1 norm and is used to enforce sparsity.


The weights q themselves are chosen by the algorithm based on the data.

in this section we set g(q) to be the Kullback-Leibler divergence of q from ν, g(q) = D(q ν) ; D(q ν) = j q j log(q j /ν j ).


In this section, we develop techniques for applying subgame solving to calculate responses to opponent off-tree actions, thereby obviating the need for action translation.

rather than simply treat a bid of $101 as $100, we calculate in real time a unique response to the bid of $101.


Then, for any γ > 0, the sample complexity of dense grid features can be bounded by .

as was the case with polynomially-exact features, for a fixed d, D( Unfortunately, this scheme suffers heavily from the curse of dimensionality, since the sample complexity is doubly-exponential in d. This means that, even though they are easy to compute, the dense grid method does not represent a useful solution to the issue posed in Section 3.2.


In particular, TranSync is significantly better than coordinate descent on dense graphs in terms of accuracy.

on dense but irregular graphs, coordinate descent did not converge at all when p = 0.8.


We show theoretically that we can iteratively increase the sample size while applying single Newton iterations without line search and staying within the statistical accuracy of the regularized empirical risk.

we can double the size of the training set in each iteration when the number of samples is sufficiently large.


One of the main applications of subspace learning is in reducing the dimensionality of the input.

one may find nested subspaces of dimension 1 ≤ k ≤ n that minimize the distances from the original to the projected samples.


This set contains 675,000 galaxies observed in D = 3750 spectral bins, preprocessed by first moving them to a common rest-frame wavelength and filling-in missing data following [18] but using the more sophisticated weighted PCA algorithm of [5], before computing a sparse neighborhood graph and pairwise distances between neighbors in this graph.

a log-log plot of the average number neighbors m(r) vs. neighborhood radius r (shown in the Supplement), indicates that the intrinsic dimension of these data varies with the scale r. in order to support m = O(d) neighbors, the radius must be above 60, in which case d ≤ 3.


Reward To support multiset classification, we propose a multiset-based reward for the controller's classification action.

consider an image X with m labels Y = {y 1 , ..., y m }.


The main problem we address in this paper is that of finding alternative representations for a given MDP.

we look for finite, linearly independent subsets Φ of F S . These are bases for subspaces that will be used to speed up the search for V π , by limiting it to span(Φ).


We note that other values of κ > 1 can be used, but the choice κ = 2 given above results in simpler semantics.

if a compensation happens to satisfy c-map(X i = x) = c-map(X j = x) = c-map(X i = x, X j = x) for some state x, we have that θ(X i = x) + θ(X j = x) = map(X i = x, X j = x) (i.e., the parameters alone can recover an original MAP value).


Although Algorithm 1 may not find a globally optimal solution, the solution has good performance.

we will theoretically show that the solution obtained by Algorithm 1 improves the performance of the parameter estimation error bound during the multi-stage iteration.


* y) + where ∼ N (0, σ 2 1) , z ∼ IBP(α) and y i ∼ p(y) Again, the goal is to infer A, the probabilities π i and then to associate likely values of Z ij and Y ij with the data.

[24] make a number of alternative assumptions on p(y), namely either that it is iid Gaussian or that it is iid Laplacian.


Thus, the Gaussian width of the error sets of regularized and constrained problems are closely related.

for θ * 2 = 1, with ρ = 1, β = 2, we have w(A r ) ≤ 3w(A c ).


In this paper, we address this problem of diverse feature selection using an approach that falls between that of greedy methods and convex-regularization methods.

we construct regularizers that capture a notion of diversity -unlike regularizers such as Lasso, our regularizers are set functions as opposed to functions of the regression coefficient vector.


This provides a strong indication that an empirical estimate of β(f ) would make a good regularization term in a hierarchical learning scheme.

we should deter the learning algorithm from choosing a classifier that predicts very rare labels.


Second, and more importantly, a key idea in Bayesian RL is the "lifting" of exploration to the space of task parameters.

instead of exploration through action selection, Bayesian RL can perform exploration by sampling task parameters.


In our case of a binary sample space, this equates to selecting one random axis h ∈ {1, . . . , H} and toggling its bit value (thereby changing the binary state in that dimension), leaving the remaining axes unchanged.

the posterior probability computed for each candidate sample is expressed by: where we have introduced a parameter β that allows for smoothing of the posterior distribution.


Finally, as there are r = 20✏ 1 log 1+✏/4 (n) iterations of BLITS, the adaptivity is O log 2 n . Our goal in this section is to show that beyond its provable guarantees, BLITS performs well in practice across a variety of application domains.

we are interested in showing that despite the fact that the parallel running time of our algorithm is smaller by several orders of magnitude than that of any known algorithm for maximizing non-monotone submodular functions under a cardinality constraint, the quality of its solutions are consistently competitive with or superior to those of state-of-the-art algorithms for this problem.


A wide variety of research disciplines, including computer science, economic, biology and social science, involve causality analysis of a network of interacting random processes.

many of those tasks are closely related to subset selection.


This can be done by assigning a prior of zero to any ✓ yielding ln Now, using Equation (7) in the optimal posterior (Equation 4) simplifies to where the normalization constant Z X,Y corresponds to the Bayesian marginal likelihood: This shows that the optimal PAC-Bayes posterior given by the generalization bound of Theorem 1 coincides with the Bayesian posterior, when one chooses `nll as loss function and := b a (as in Equation 2).

moreover, using the posterior of Equation ( 8) inside Equation ( 3), we obtain = n minimizing the PAC-Bayes bound is equivalent to maximizing the marginal likelihood.


Our analysis shows that EMGD is able to find an ǫ-optimal solution by computing O(log 1 ǫ ) full gradients and O(κ 2 log 1 ǫ ) stochastic gradients.

with the help of stochastic gradients, the number of full gradients required is reduced from O( √ κ log 1 ǫ ) to O(log 1 ǫ ), independent from the condition number.


now, using the definition of the g (t) functions, and by the regret bound for the EG algorithm, we have that for any fixed distribution α ∈ ∆ N , Since ∇f (t) ≤ 1, Summing (4) from t = 1 to T , and adding to (3), we get that for any distribution α over Φ, by concentrating α on any given φ i , the above inequality implies that T t=1 f (t) (φ i (x (t) )) − T t=1 f (t) (x (t) ) ≤ O( log(N )T ), and thus we have a no Φ-regret algorithm.


We assume X is specified in a way so that the algorithm can efficiently construct the smallest Euclidian ball containing X .

furthermore, we assume the algorithm has noisy black-box access to f . the algorithm is allowed to query the value of f at any x ∈ X , and it observes y = f (x)+ε where ε is an independent σ-subgaussian random variable with mean zero: E[exp(λε)] ≤ exp(λ 2 σ 2 /2) for all λ ∈ R. The goal of the algorithm is to minimize its regret: after making T queries x 1 , .


The natural extension of this solution to having an unknown and potentially infinite number of tests is called alphaspending.

we choose any sequence of constants {↵ i } i2N such that P i ↵ i  ↵, and on receiving P i , our decision is simply 1 {P i  ↵ i }.


Due to the wide applicability of this class of M -estimators in (1), there has been a line of work on developing efficient optimization methods for solving special instances of this class of M -estimators [14,26], in addition to the papers listed above.

due to the superposition-structure in (1) and the high-dimensionality of the problem, this class seems naturally amenable to a proximal gradient descent approach or the ADMM method [2,17]; note that these are first-order methods and are thus very scalable.


Our main interest is in the local discrepancy function disr S : R d → R parametrised by a given set of samples S = {X 1 , ..., X |S| } and defined as follows: where:  where λ ∈ P(R 1 ), and a set of samples S = {X 1 , ..., X |S| } that is used in a given (Q)MC estimator to approximate I f . The star discrepancy function D * λ with respect to a distribution λ is defined on S as: ..,|S| and F λ stands for the cdf function for λ.

to measure the discrepancy between arbitrary distribution λ ∈ P(R 1 ) and a set of samples S, the set of samples is transformed to the interval [0, 1] via the cdf F λ and the discrepancy between the uniform distribution on [0, 1] and the transformed sequence F λ (S) is calculated.


However, empirically we did not observe convergence to a point other than the ground truth.

gradient descent seems to always find the global minimum, see Figure 4.


We show that for an arbitrary bipartite graph with n vertices on each side and average degree d its multitasking .

when the average degree is concerned, the multitasking capacity of a graph tends to zero, provided that the average degree of a graph is ω(log n).


For any x ∈ R + , the orbits in ( 5) correspond to a particular mechanical word 0, 1 or 0w1 depending on the value of x (Figure 1).

for any word u, let y u be the fixed point of the mapping φ u on R + so that φ u (y u ) = y u and y u ∈ R + .


[10], we assume the first-order statistics-the mean and variance-of a convolutional feature map encodes a certain style attribute.

the information carried by each feature map can be divided into two components: a style (the mean and variance of activations) and shape (the spatial configuration of activations).


Note also that choosing larger values ↵ leads to jumps in the solution sets sizes (c.f., 6.1).

varying the required utility Q, even by a small amount, may not be possible to achieve by the current solution's size (↵ j ) and would require moving to a set larger by at least an ↵ factor (↵ j+1 ).


In this paper, we investigate the automatic generation of tags with properties similar to those generated by social taggers.

we introduce a machine learning algorithm that takes as input acoustic features and predicts social tags mined from the web (in our case, Last.fm).


On the kidney biopsy data set, PX-VB converges in 507 iterations, while VB converges in 7518 iterations.

pX-VB requires 15 times fewer iterations than VB.


The decomposition into c-components forms a partition of the observable vertices, which gives very useful structural information about the causal model.

a fact that is key to our whole analysis is that if N is a semi-Markovian Bayesian network on observable and unobservable variables V ∪ U with c-components C 1 , .


While inheriting the nonlinear modeling power, our model disposes of the Kronecker-product structure in the full covariance and can therefore select an arbitrary subset of tensor entries for training.

given a tensor M ∈ R d1×...×d K , for each tensor entry m i (i = (i 1 , .


The key feature of our algorithm is scalability.

running time of our algorithm with a given parameter value ∆ scales linearly in n, while keeping the relative error bound exactly the same.


5 We evaluate this learning method on relatively simple instances on which commonly used software such as Alchemy can be a few orders of magnitude off from the optimal likelihood of the training data.

table 3 compares the likelihood of the training data under the weights learned by FocusedFlatSAT and by Generative Weight Learning [7], as implemented in Alchemy, for four types of Markov Logic theories.


ELIRL inherits the theoretical guarantees showed by Ruvolo and Eaton [34].

the optimization is guaranteed to converge to a local optimum of the approximate cost function in Equation 4 as the number of tasks grows large.


It can be interpreted as the fraction of all pairs of subjects whose predicted survival times are correctly ordered among all subjects that can actually be ordered.

it is the probability of concordance between the predicted and the observed survival.


In the last section, we have shown that the symmetric NMF (2) can be transfered to problem (3), the latter admitting splitting property which enables us to design alternating-type algorithms to solve symmetric NMF.

we exploit the splitting property by adopting the main idea in ANLS and HALS for nonsymmetric NMF to design fast algorithms for (3).


Most modeling studies, however, neglect the fact that in a natural situation, neural activity results from multiple objects or even complex sensory scenes.

attention experiments require the presentation of at least one distractor along with the attended stimulus.


In Section 3, we showed that the discrepancy is crucial for forecasting non-stationary time series.

if we could select a distribution q over the sample Z T 1 that would minimize the discrepancy and use it to weight training points, then we would have a better learning guarantee for an algorithm trained on this weighted sample.


The set of payoffs reachable in Γ coincides with the set of payoffs reachable in Γ * .

any strategy {λ σ } σ∈Σ1 , {ω σ } σ∈Σ1 over Γ * is payoff-equivalent to the realization-form strategy ω = σ∈Σ1 λ σ ω σ in Γ.


As the exact choice of features, their encoding and comparison metric is not crucial to our technique, we chose a fairly simple representation that was general enough to use in a wide variety of settings, but informative enough to capture the details of objects (given the subtle variation that can distinguish two different cars, features such as [6] were found not to be precise enough for this task).

we apply a first derivative Gaussian odd-symmetric filter to the patch at four orientations (horizontal, vertical, and two diagonal), giving four signed numbers per pixel.


The proposed embeddings should reflect the clinical relationships between procedures and diseases, such that the procedures are assigned to the diseases with short distance.

for the m-th admission, we may recommend a list of procedures with length L, denoted as E m , based on its diseases and evaluate recommendation results based on the ground truth list of procedures, denoted as T m . given {E m , T m }, we calculate the top-L precision, recall and F1-score as follows: Pm+Rm .


Rather than solving this decentralized problem directly, we train each agent to replicate what it would have done with full information in the centralized case.

the vector of state variables x is compressed, and the i th agent must decompress x i to compute some estimate ûi ≈ u * i .


The intuition is that these low-order moments roughly characterize the empirical data distribution, and can be used to form the control variate to correct the noisy gradient to a better direction.

the variance of the augmented "stochastic gradient" becomes smaller as it is derived with more information about the data.


For RegPCP, we construct the graph following the approach suggested in .

we have w ij = exp(−d 2 ij /(2σ 2 )) if x i is within 20NN of x j or vice versa, and w ij = 0 otherwise.


In contrast, our method provides a novel deep model for learning graph representations of 2D visual data Furthermore, our method is related to learning deep models on graph structure [28,11,29].

graph convolutional networks [11] are used to propagate information on our learned graph.


Note that when the original graph G is a tree, the graph is already triangulated and the set T in Theorem 1 is equal to the edge set E. Hence, Theorem 1 implies that the inverse Γ of the augmented covariance matrix with sufficient statistics for all vertices and edges is graph-structured, and blocks of nonzeros in Γ correspond to edges in the graph.

the (m − 1)p × (m − 1)p submatrix Γ V,V corresponding to sufficient statistics of vertices is block graph-structured; in the case when m = 2, the submatrix Γ V,V is simply the p × p block corresponding to the vector (X 1 , .


The remarkable performance came from the BSCI approach, which incorporates the convex LS formulation with the sparse RIR model.

the BSCI approach yielded higher than 90% normalized correlation in source estimates when SNR was better than 20 dB, and it yielded higher than 99% normalized correlation in the low noise limit.


More generally, given a matrix W and a pair of norms k that is, we apply the p-norm on the columns of W and the r-norm on the resulting d-dimensional vector.

the 0 1 loss of a multiclass predictor h W on an example (x, y) is defined as the 0 1 loss equals 1 if h W (x) 6 = y and 0 otherwise.


We can now write the KKT conditions of problems (LRP λ ) and (LRD λ ) as In view of Eq.

( 3), we can see that if | θ * λ , xj < mλ, then the KKT conditions imply that the coefficient of xj in the solution β * λ is 0 and thus the j th feature can be safely removed from the optimization of (LRP λ ).


If λ satisfies the following condition, the optimal solution to problem 6 can not be zero The proof idea is to show when λ is large enough, the trivial solution c = 0 can not be optimal.

if c = 0, the corresponding value in the primal problem is λ γl ∞ .


The k-NN classifier is defined by We analyze the performance of g n,k by comparing it with g(x) = 1(η(x) ≥ 1/2), the omniscient Bayes-optimal classifier.

we obtain bounds on Pr X (g n,k (X) = g(X)) that hold with high probability over the choice of data S, for any n. It is worth noting that convergence results for nearest neighbor have traditionally studied the excess risk R n,k − R * , where R n,k = Pr(Y = g n,k (X)).


We chose these two particular algorithms because they exhibit two extreme behaviors when converted into batch learning algorithms.

if we were to present the classic Perceptron with an example-sequence S drawn i.i.d.


Moreover, instead of fitting a single model with many variables as in a flat single layer approach, our proposal of multi-layer coding requires fitting many small models separately, each with a small number of parameters.

fitting the small models can be done in parallel, e.g.


[29] identified six molecular subtypes of ovarian cancer in this data, labeled C1 through C6.

the C1 subtype was found to have much shorter survival times, and was characterized by differential expression of genes associated with stromal and immune cell types.


We then possibly threshold β I1 with t 1 = 4λ n |I 1 | (to be specified), to obtain I 2 , repeat step 2 with I = I 2 to obtain β I and set all other coordinates to zero; return β.

our algorithm is constructive in that it does not rely on the unknown parameters s, β min := min j∈S |β j | or those that characterize the incoherence conditions on X; instead, our choice of λ n and thresholding parameters only depends on σ, n, and p. In our experiments, we apply only the first two steps, which we refer to as a two-step procedure; the Gauss-Dantzig selector is a two-step procedure with the Dantzig selector as β init [5].


Let V = {1, 2, . . . , n} denote the index set of the data points x i and partition V into c disjoint subsets V j ; i.e., V i ∩ V j = ∅ for i = j and ∪ c j=1 V j = V , where the cardinality of V j is n j so that c j=1 n j = n. We also make use of a matrix representation for the problem in question.


Theorem 2 states that a conditional probability on a countably-dimensional product space is uniquely defined (up to a.e.-equivalence) by a conditionally projective family of marginals.

if we can define a parametric model on each finite-dimensional space Ω I x for I ∈ F(E) such that these models are conditionally projective, the models determine an infinite-dimensional parametric model (a "nonparametric" model) on the overall space Ω E x . Theorem 2 (Extension of Conditional Probabilities).


Therefore, to retrieve the attention for a specific word, we convolve the matrix along the visual dimension using a 1 × 1 dimensional kernel.

similarly, we obtain θ A,V and θ A,Q , which we omit due to space limitations.


Using a mixed loss function, which is a weighted combination of the cross-entropy (XE) loss L XE and the reinforcement learning (RL) loss L RL , helps improve readability and fluency since the cross-entropy loss is based on a conditioned language model, with a goal of producing fluent captions.

the mixed loss function is given by where γ ∈ [0, 1].


(2) We replace ReLU with hard-threshold activations.

we utilize the so-called HELU σ function introduced in [23], which is a continuous and piecewise linear approximation of the scalar hard-threshold operator.


To test whether our method memorizes the training set images, we retrieved the closest images from the training set for each of sample synthetic images.

for each synthetic image for an output neuron Y (e.g.


Both of these are one-shot learning tasks.

a single view of a target object is encoded and a simple classifier (nearest neighbors) must rank test images depicting the same object as being more similar to the encoded target than to images of any other objects.


Each edge e i,k is represented by the description of the next node D(s k ).

deciding which action to take is equivalent to taking a peek at each neighboring node and seeing whether that node is likely to lead ultimately to a target node.


The property above is similar to the discriminatory kernel condition in Cybenko [13].

our results apply to radial basis functions, single layer neural networks, and multilayer neural networks in which the final layer is scaled with n. the statements we make require a "wide" final layer but are still applicable to networks with multiple layers.


For all S ⊂ S let q S and r S be the vectors of the probabilities of (a, b) and (c, d) respectively given that S is the set of parents that are active.

then: this means that if there is only one parent shared between (a, b) and (c, d), M is the sum of two rank 1 matrices, and thus is at most rank 2.


The notion of extracting features of the data which are maximally dependent on the original data is far more general than what we described in this paper.

one may show that feature selection [7] and clustering [8] can also be seen as special cases of this framework.


Thus, for the above result to be non-trivial, we need 3.54λ ≤ 2, namely λ ≤ 0.56.

in the noiseless case, the RoLR is able to estimate the LR parameter with a non-trivial error bound (also known as a "breakdown point") with up to 0.56/1.56 × 100% = 36% of the samples being outliers.


To do this, [11] uses a form of the wavelet operator in the Fourier domain, and generalizes it to graphs.

sGWT takes the Fourier transform of the graph by using the properties of the Laplacian L (since the eigenvectors of L are analogous to the Fourier basis elements).


For a node w, father (w) denotes its parent, T(w) denotes the subtree tree rooted at the node, Ivs(w) denotes the number of leaves in the subtree and path( w) := w is the path from the root to the node.

we use the path w from root to node as the label of the node w. We denote by words(S(x )) the set of all strings w such that wu E nodes(S(x )) for some (possibly empty) string u, which means that words(S(x)) is the set of all possible substrings of x.


We rearrange this summation by the size of wavelet coefficients.

we order the coefficients as the following then the sparsity condition imposed on the density functions is that the decay of the wavelet coefficients follows a power law, where C is a constant.


An important property of such an expectation is that for any quantity X(i): where recall that the expectations on the r.h.s.

are with respect to i ∼ D. when In fact, we will consider only weights s.t.


When data are corrupted by noise of small magnitude such as Gaussian noise, a straightforward extension of SSC is the Lasso type method called "Lasso-SSC" [18,13]  Note that while Formulation (2) has the same form as Lasso, it is used to solve the subspace clustering task.

the support recovery analysis of Lasso does not extend to this case, as X −i typically does not satisfy the RIP condition [18].


The distance measure we use for this is the minimax distance.

while the upper bound requires that we would like to be as high as possible.


(When the query elements are not distinct, the false positive rate may vary significantly, depending on on the distribution of the number of appearances of elements and which ones yield false positives; we focus on the distinct item setting here.)

the false positive rate is a priori the same for any possible query set Q.


This algorithm is not only highly efficient in practice, but also enjoys strong computational and statistical guarantees in theory.

by leveraging a sophisticated characterization of local restricted strong convexity and Hessian smoothness, we prove that within each stage of convex relaxation, our proposed algorithm maintains the solution sparsity, and achieves a (local) quadratic convergence, which is a significant improvement over (local) linear convergence of proximal gradient algorithm in [9] (See more details in later sections).


Furthermore, the functions F z , G d , and H can be maximized independently, since the corresponding non-negativity and normalization constraints are decoupled.

the |Z| + |D| + 1 optimization problems can be solved in parallel.


Instead, we can hope to design algorithms that rapidly learn W . In this section, we propose principled methods for achieving this goal.

in this section, we focus on the consequences of minimizing the function φ i is either the original piecewise function φ i or one of its subfunctions φ k i .


We conclude this section by noting that the solvable case (29) is generic: its key results extend to the general situation defined above (21).

we checked this fact numerically for several values of L. the minimization of F ∞ nullifies as many trial parameters as necessary to express the remaining parameters via independent effective parameters t 0 , t 1 , . . ..


The goal of our paper is to understand the limit points of GDA and OGDA dynamics (points that last iterate might converge to) for general functions f (x, y).

we answer the following questions: • are the stable limit points of GDA and OGDA locally min-max solutions?


The MSG approach above is computationally expensive since it runs local PointNet at large scale neighborhoods for every centroid point.

since the number of centroid points is usually quite large at the lowest level, the time cost is significant.


In this paper we describe how MAP inference can be used to sample efficiently from Gibbs distributions.

we provide means for drawing either approximate or unbiased samples from Gibbs' distributions by introducing low dimensional perturbations and solving the corresponding MAP assignments.


This will happen with arbitrarily high probability for large enough n. Define u = X T y and observe that Hence, the AdaGrad solution w ada / sign(u).

w ada has all of its components equal to ±⌧ for some positive constant ⌧ .


The resolution is to simply note that there is so little probability mass in those far reaches that the cost there is irrelevant.

consider a single center p (and assume x → x − p 2 2 is integrable); the dominated convergence theorem grants where a ball B i may be chosen so that B c i x − p 2 2 dρ(x) ≤ 1/1024.


For convenience, we define a Assume binary data (before normalization, i.e., sum to one).

the chi-square similarity ρ χ 2 becomes ρ χ  9) and χ 2 (2) based on (10).


We found that substantial gains can be made this way.

alice maintains an RBF network that is trained on-line, based on the list of labeled prototypes.


Condition (A) d T k Hd k ≤ 0 (at Step 3) may hold when the local model is not strictly convex (or H is not positive definite).

d k is a direction of zero or negative curvature; a typical exploitation of non-positive curvature is to set ∆" equal to the "step to the trust-region boundary along that curvature segment (in Step 3)" as a model minimizer in the trust region.


We introduce the notation Q(Z, α) to denote the quantile of order 1 − α for the distribution of a random variable Z conditioned on the event Y = −1.

the following quantile will be of interest: where we have used here the notion of generalized inverse F −1 of a càdlàg function F : We now turn to the definition of the ROC curve.


We investigated how model performance varied as a function of training data, and found that LN models were more susceptible to overfitting than CNNs, despite having fewer parameters (Figure 4A).

a CNN model trained using just 25 minutes of data had better held out performance than an LN model fit using the full 60 minute recording.


Then, there exists a sufficiently large k 0 ∈ N such that for all k ≥ k 0 the sequence {f (x k )} k generated by CR satisfies From Theorem 2, it can be seen that the function value sequence generated by CR has diverse asymptotic convergence rates in different regimes of the KŁ parameter θ.

a larger θ implies a sharper local geometry that further facilitates the convergence.


The vast majority of supervised learning techniques assume that source (training) data and target (testing) data are drawn from the same distribution over pairs of example inputs and labels, P (x, y), from which the conditional label distribution, P (y|x), is estimated as P (y|x).

data is assumed to be independent and identically distributed (IID).


An alternative is to make weaker assumptions regarding p (x n ) that are of particular relevance to classification.

the cluster assumption asserts that the data density should be reduced in the vicinity of a decision boundary (e.g., [2]).


Table 2 shows that linear policies can also achieve high rewards on all the MuJoCo locomotion tasks.

for Humanoid-v1 and Walker2d-v1, ARS found policies that achieve significantly higher rewards than any other results we encountered in the literature.


Conditioned on F ∩ G ∩ H, the truncated and untruncated scores are equal.


Fix 2 (0, 1) and L 2 N. A learner strategy, , is ( , L)-private if, for any adversary estimator, b X a , where the probability is measured with respect to the randomness in the target, X ⇤ , and any randomness employed by the learner strategy and the adversary estimator.

2 if a learner employs a ( , L)-private strategy, then no adversary estimator can be close to the target within an absolute error of /2 with a probability great than 1/L.


we choose y such that (x, y) is in some sense similar to the training examples.


To tackle this limitation of conventional attention mechanisms, we propose to allow the attention model to output uncertainty on each feature (or input) and further leverage them when making final predictions.

we model the attention weights as Gaussian distribution with inputdependent noise, such that the model generates attentions with small variance when it is confident about the contribution of the given features, and allocates noisy attentions with large variance to uncertain features, for each input.


We observe that the reconstruction penalty tends to works better than orthogonality constraints.

on the Hollywood2 dataset ISA achieves a mean AP of 54.6% when the reconstruction penalty is used.


Our solution to obtain more robust policies from few demonstrations, which are also capable of diverse behaviors, is to build on GAIL.

to enable GAIL to produce diverse solutions, we condition the discriminator on the embeddings generated by the VAE encoder and integrate out the GAIL objective with respect to the variational posterior q (z|x 1:T ).


The stronger notion, node privacy, corresponds more closely to what was achieved in the case of traditional data sets, and to what one would want to protect an individual's data: it ensures that no matter what an analyst observing the released information knows ahead of time, she learns the same things about an individual Alice regardless of whether Alice's data are used or not.

no assumptions are needed on the way the individuals' data are generated (they need not even be independent).


At the bottom level, we have a set of distinct urns which are used to define the initial and transition probabilities of the HMDP model (and are therefore referred as HMM-urns).

one of HMM urns, u 0 , is set aside to hold colored balls to be drawn at the onset of the HMM state-transition sequence.


observations {(y i , x i )} n i=1 of the pair (Y, X).

we assume the link function f is unknown.


Instead, we chose to treat the task as a classification one.

for each tag we try to predict if a particular artist has "none", "some" or "a lot" of a particular tag relative to other tags.


Finally, it receives the edge semi-bandit feedback and uses it to update M t and B t . It is worth emphasizing that IMLinUCB is computationally efficient as long as ORACLE is computationally efficient.

at each round t, the computational complexities of both Step 1 and 3 of IMLinUCB are O |E|d 2 .


The observation model is defined to be outlier-prone of order n, if p(f |y 1 , ..., y n+1 ) → p(f |y 1 , ..., y n ) as y n+1 → ∞ [3,4].

the effect of a single conflicting observation to the posterior becomes asymptotically negligible as the observation approaches infinity.


It would be considerably better if we updated the upper bound after every stochastic gradient descent step.

this variant, however, is identical to stochastic gradient descent on the original objective function due to the following: in order to compute the gradient of the upper bound we need not compute the upper bound itself.


When Is = Ω(log n/n), the minimum sample complexity is a decreasing function in Is.

as the quality of social network increases, the minimum sample complexity decreases.


For this problem (provided a minimizer exists) every critical point is a global minimizer.

moreover, for any given (not necessarily optimal) A ∈ L 2 , where λ min (Σ 0 ) is the smallest eigenvalue of Σ 0 .


Many inference problems in graphical models can be cast as determining the expected value of a random variable of interest, ¢ , given observations drawn according to a target distribution £ .

we are interested in computing ¤ ¦¥ ¨ § © ¢ .


We use our algorithm to verify its performance in simulation scenario.

we apply our algorithm to two popular setting: (a) a grid graph based pairwise MRF with varying node and edge interaction strengths, and (b) a grid graph based MRF on the weighted independent set (or hardcore) model.


In mean-prediction, we consider the next timestep xt conditioned on xt−1 from the Gaussian prediction [8]: where k X (x) is a vector containing k X (x, x i ) in the i-th entry and x i is the i th training vector.

we set the latent position at each time-step to be the most-likely (mean) point given the previous step: xt = µ X (x t−1 ).


The ball with radius R is denoted by Statement of the Problem In this work, we generalize online stochastic convex optimization to the case of multiple objectives.

at each iteration, the learner is asked to present a solution w t , which will be evaluated by multiple loss functions f 0 t (w), f 1 t (w), .


From the table, we observe that DBA achieves significantly better performance than all the other methods.

for Accuracy@1 scores, DBA can achieve a gain of about 30% relative to two MNB methods, and about 10% relative to two SVM methods; for macro-average F-measures, DBA can achieve a gain of about 50% over MNB methods, and about 25% over SVM methods.


[12] recently proved that the adversarial multi-armed bandit with switching costs is a hard online learning problem, which is a problem where the best possible regret guarantee is Θ(T 2/3 ).

for any learning algorithm, there exists an oblivious setting of the loss values that forces a regret of Ω(T 2/3 ).


Naturally, if f is a neural network approximating f and g approximates g, then we can compose these approximations to get an approximation of the composition.

if g has a Lipschitz constant, then the composition approximation will depend on this Lipschitz constant and the accuracies of the individual approximations.


Even when the estimation does not achieve the best performance, the metric always reduces the MSE from the original NW estimation.

in the Slice, Pol, CPU, NavalC, and NavalT datasets, GGL performs poorly showing the non-Gaussianity of data, while the metric using the same information effectively reduces the MSE Figure 3: Regression with real-world datasets.


A Markov chain (row stochastic matrix) M is irreducible, if any state can be reached from any other state.


Finally, our method is able to project the learned graph representation back into 2D grids, and thus is fully compatible with existing networks.

our method consists of Graph Projection, Graph Convolution and Graph Re-projection.


In order to produce an equivalent QP, however, we must also ensure that the dual variables µ x (y i , y j ), µ x (y i ) are the marginals resulting from a legal density α(y); that is, that they belong to the marginal polytope [4].

we must enforce consistency between the pairwise and singleton marginals (and hence between overlapping pairwise marginals): yi µ x (y i , y j ) = µ x (y j ), ∀y j , ∀(i, j) ∈ E, ∀x.


In a line of recent work [6,26,14] it has been shown that relaxation based on the nonlinear graph p-Laplacian lead to similar runtime performance while providing much better cuts.

for p = 1 one obtains a tight relaxation of the Cheeger cut, see [8,26,14].


Along with the other kernel parameters, we infer the parameters of K t using slice sampling.

we represent K t by its Cholesky factor and sample in that space.


The first manifestation of this impact is improvement in the Lipschitzness 2 of the loss function.

the loss changes at a smaller rate and the magnitudes of the gradients are smaller too.


The last is a classification task on images for which we use CNNs.

experimental Set up: Each method is executed in an asynchronously parallel set up of 2-4 GPUs, it can evaluate multiple models in parallel, with each model on a single GPU.


Despite the difference in illumination and cell density from the training data, and despite the segmentation artifacts caused by the compression of the image sequence, our method shows a high generalization capability and obtains a total loss of 0.78%.

we extract 93.2% of 384 mitosis events which is a significant improvement over the mitosis detection rate reported in [12] (81.5%, 294 events).


One fruitful avenue of analysis has been the use of so-called transportation, or Talagrand inequalities T p , which can be used to prove concentration inequalities on W p [20].

we say that ρ satisfies a is the relative entropy [20].


We generate a synthetic dataset by following the protocols described in [41,25].

the sample size N is set to 1000, and the number of covariates d is set to 100.


We will assume throughout that all potential function values, all message values and all the required products computed by the algorithm can be represented as n-bit natural numbers and thus fall in Z N = {0, . . . , N − 1} where N = 2 n . As expressed by Equation (2), marginal probabilities are obtained by taking products of such n-bit numbers and then normalizing to obtain finite-precision real-valued numbers in the range [0, 1].

it will be convenient to think of values in Z N as elements of the cyclic group of order N with addition and subtraction modulo N . we will make frequent use of the following simple fact: for any fixed x ∈ Z N , if r ∈ Z N is chosen randomly among all n-bit numbers, then x + r mod N is also distributed randomly among all n-bit numbers.


We numerically estimate the growth of regret in L, the exponent of L, in the log-log space of L and regret.

since log(f (L)) = p log(L) + log(c) for any f (L) = cL p and c > 0, both p and log(c) can be estimated by linear regression in the new space.


To capture this intuition, we add an additional term that aims to maximize the information gained about the answer from looking at the image.

we introduce another adversarial regularizer corresponding to the difference in entropies between the base model prediction given the image and the question-only model which we write as We note that this regularizer resembles the conditional mutual information (CMI) between the answer and image given the question I(A; I|Q); however, f Q (q) is not constrained to be the marginal of f (v, q) such that estimating the CMI in this way is ill-defined.


On the other hand, DirtyIMC performs the best among all prediction methods.

it performs slightly better than MF-ALS in terms of accuracy, and much better in terms of AUC.


As a result, the performance of model is no longer limited by the sample size but the amount of available computational resources.

the data size can easily go beyond the size of physical memory of available machines.


In order to address the second drawback, we note that our aim at an iteration t of the algorithm is to reduce the potential function Φ(y).

given the current distribution parameterized by θ Mt we would like to add a new tree θ Tt to the mixture that solves the following problem: θ T ∈ T . Here, T is the set of all tree distributions defined over n random variables.


In such a model, an exogenous Markov process X t taking values in a set X influences rewards.

the expected reward at time t is given by f θ (a, X t ).


In many of these real-world contexts, we also do not have the luxury of collecting large amounts of data and then retrospectively determining when (if ever) the structure changed.

we cannot learn in "batch mode," but must instead learn the novel structure in an online manner, processing the data as it arrives.


In fact, in Section 3 we see how a rate of 1/n on the objective corresponds to a rate of 1/ √ n on the loss.

we perform an oracle analysis of the optimum of the SVM objective (rather than of empirical minimization subject to a norm constraint, as in other oracle analyses of regularized linear learning), based on the existence of some (unknown) low-norm, low-error predictor w. Strong convexity is a concept that depends on a choice of norm.


In this paper, we show ways of sampling intermediate steps of alternating minimization algorithms for computing low rank tensor CP decompositions, leading to the sparse alternating least squares (SPALS) method.

we sample the Khatri-Rao product, which arises as an intermediate object during the iterations of alternating least squares.


Instead, we formulate the estimation equations in terms of the components of the SVDs of the training data matrices.

we take the SVDs of the zero-mean input and output matrices: From these two decompositions, we can write the two correlation matrices as Now we can rewrite the equation for A x to remove the need for the squaring operation and similarly for Ay  Using these identities, we compute A x and Ay using the following steps: 1) Find the SVDs of the data matrices using the expressions in equation 10.


If the subspaces are disjoint or independent 1 , the solution X in both formulations is shown to be such that X ik = 0 only if data points y i and y k belong to the same subspace [7,14,15,25].

the sparsest/lowest rank solution is obtained when each point y i is represented as a linear combination of points in its own subspace.


3), which lead to an effective solution to the challenging SSL task, justified both in theory and practice.

theoretically, instead of competing as stated in the first problem, a good classifier will result in a good generator and vice versa in Triple-GAN (See Sec.


We used a perturbation size large enough to be noticeable by humans on the computer screen but small with respect to the image intensity scale (✏ = 32; see Figure 1a middle).

we chose ✏ to be large (to improve the chances of adversarial examples transfer to time-limited human) but kept it small enough that the perturbations are class-preserving (as judged by a no-limit human).


To do this, we parameterize the set of bias fields using the sine/cosine basis images shown on the right of Figure 1: We optimize Equation ( 8) by simultaneously updating the bias field estimates (taking a step along the numerical gradient) for each image to reduce the overall entropy.

at time step t, the coefficients α j for each bias field are updated using the latent image estimates and entropy estimates from time step t − 1.


Another advantage of the proposed method is the simplicity of the PG algorithm, which doesn't require truncation or the introduction of auxiliary variables, also making the algorithm easily adaptable to challenging inference tasks.

the PG sampler can be directly applied to the sticky HDP-HMM with DP emission model considered in Fox et al.


In Figures 4a and 4b we compare our various strategies of optimization over the same grid on this expensive problem.

the algorithms were restricted to only the exact parameter settings as evaluated by the grid search.


Thus, the variables S i , L i and V i form a Markov chain.

this means that whatever information S i carries about V i is expressed (if the model is well-fit) in the cortical representation L i , making this structure appropriate for value prediction.


Example 1 Suppose that H is the class of singletons.

for h i ∈ H, where h i : {0, 1} n → {−, +}, we have h i (x) = + if x is the binary representation of i, and h i (x) = − otherwise.


In contrast, M3 is consistently better than the flat baselines on all datasets except NEWS20.

m3 performs significantly better on the largest datasets, especially in Macro-F 1 , showing that even very large class hierarchies can convey very useful information, and highlighting the importance of having a scalable, parallelizable hierarchical classification algorithm.


We henceforth refer to this case as the state aggregation case.

we assume that for any t = 0, 1  Note that dim E [Q] = K. We define the distance between Q ⇤ and the hypothesis class Q as The following result establishes that with Q and ⇢ defined above, the performance loss of OCP is larger than 2⇢H(H + 1) in at most K episodes.


We will consider solving the classical convex optimization problem min x f (x), for smooth functions f , by instead solving an associated saddle-point problem which we call the Fenchel Game.

we consider that the payoff function g of the game to be g(x, y) = x, y − f * (y).


We empirically evaluate our loss function on deep networks across different applications, datasets and model architectures.

we study the performance of these models on tasks of adversarial learning, generalization from limited training data, and learning from data with noisy labels.


It follows that the asymptotic complexity of the GMM algorithms is better than for the classical MM algorithm.

the GMM with adjacent breaking and top-k breaking for constant k's are the fastest.


In a 3-clique, there is only one way to do this, so M is unique if G is triangulated.

there exists a surjection from perfect matchings in the expanded dual of G to cuts in G. Furthermore, since we have given edges interior to the cliques of the expanded dual zero energy, every perfect matching M complementary to a cut C of our Ising model (2) obeys the relation This means that instead of a minimum-weight cut in a graph we can look for a maximum-weight perfect matching in its expanded dual.


Notice that the mixed 1 / p -norm in the objective function of ( 10) and ( 12) promotes selecting a few nonzero coefficient blocks c ij α ij .

we find a representation of each incomplete data point using a few other incomplete data points, while, at the same time, find missing entries of the selected data points.


It turns out that this will lead to the well-known exponential weights algorithm [14,15].

we pick T (s) to be exp ( ⌘(s + R)) which exactly upper bounds 1{s  R}.


A different label is fed into DPN to switch among response policies by linking the shared convolution layer with the corresponding control layer, achieving convenient fast policy switch.

the response policies against opponent's n different policies are trained separately and distinguished by different labels.


Node Initialization We associate a state h with each node v in a set of initially unconnected nodes.

z v is drawn from the d-dimensional standard normal N (0, I), and h where τ v is an interpretable one-hot vector indicating the node type.


From a neuroscience perspective, learning high-dimensional functional connectivity probabilistic models opens the door to new studies of brain architecture.

the models estimated with our strategy are well suited to exploring the graph-community structure resulting from the functional integration, specialization, and segregation of distributed networks.


Then Hardt and Roth [30,Theorem 4.2 and 4.7] results in an additive error O(( , where c is the maximum entry in their projection matrix.

we improve Hardt and Roth [30] by an O(c √ k) factor.


Goal In this section, we define the goal of safe exploration.

we ask what the best that any algorithm may hope to achieve is.


It turns out the dependency of F on X * can be essentially decoupled by using degenerated kernels corresponding to a finite number of feature maps.

we consider kernels of form where we assume the number m of features is no larger than the particle size n. Then, the fixed point of SVGD reduces to where the test function f (x) := P x φ (x) no longer depends on the fixed point X * .


This paper will focus on discrete dynamical systems.

time procedes in discrete steps t = 1, 2, 3, . . .. At every step t, the agent selects an action a t from a finite set A and the environment (stochastically) emits an observation o t , taken from a finite set O.


When there is a symmetry between the items, the observation probability should be the same.

if the true configuration is llm, then observing lml and mll should be equally likely as item1 and item2 belong to the same cluster and so on.


3, we empirically confirm that the recall value of geometrically related predicates are significantly increased, such as using, carrying, riding.

predicting predicates which have clear subject-object relative geometry, was helped by the module.


Without the manifold assumption M, the semi-supervised smoothness assumption SSS as usually defined is too weak, and current methods don't lead to improved inferences.

methods that use regularization based on graph Laplacians do not achieve faster rates of convergence.


source assumption for feature-based prediction.

we present generalization error bounds on prediction using low-rank models.


Given these definitions, the training in [4] can be interpreted as guided policy search for the MDP described in the previous paragraph.

the variational distribution q provides a guide policy q(τ |x * ) over trajectories τ {z 0 , ..., z T , x * }: The primary policy p generates trajectories distributed according to: which does not depend on x * .


Let I ′ be the set of non-zero columns of C ′ , and let H ′ be the column-normalized version of C ′ .

column i 2 for all i ∈ I ′ , and H ′ i = 0 for all i / ∈ I ′ .


Models are also evaluated using the Hinge-loss.

let u i ∈ R d be some vector associated with task i.


Our encoder first uses the current question to attend to the exchanges in the history, and then use the question and attended history to attend to the image, so as to obtain the final encoding.

we use the spatial image features V ∈ R d×k from a convolution layer of a CNN.


Let G ✓ K m,n be the completion graph of a partially known (m ⇥ n) matrix A.

then the set of uniquely reconstructible entries of A is exactly the set A e , with e in the transitive closure of G. all of A is reconstructible if and only if G is connected.


the performance drops significantly at θ=18°, showing that it is sensitive to the distortion.


Theorem 3.1 displays an interesting interplay between the desired accuracy ✏, and the penalty we thus pay as a result of performing approximate projections .

as ✏ is made small, k becomes large, and thus so does .


Therefore The equality in (4) holds if and only if a k = b k for all k, which means that AP = PB.

this is true for the permutation P o , which completes the proof of all the statements.


Following [19], the formulation we consider is close to the setting of functional regression, and covers the reproducing kernel Hilbert space (RKHS) setting as a special case.

it reduces to standard linear regression for finite dimensions.


The algorithm to compute this sparse convex approximation is again a simple and fast greedy algorithm, which roughly speaking is a simplification of the Frank-Wolfe algorithm for this particular problem.

to find the projection of q onto Convex(P ), start with any point t 0 ∈ Convex(P ).


When the number of classes k = 2, the dual functional is equivalent to v-SVM.

data are separated by dividing the space into equally ranked regions by the decision rule all input vectors x satisfying br -1 < w . x < br are assigned the rank r (using the convention that bk = (0).


If two zero mean stochastic variables are scaled both with the same factors their magnitude and variance will increase together.

as the magnitudes of one variable increase so will the magnitude and the variance of the other variable.


The joint distributions over factors that we test with are summarized in Figure 5a, which includes varying degrees of sampling bias.

configuration A uses uniform and independent factors; B uses factors with non-uniform marginals but are uncorrelated and independent; C uses uncorrelated but dependent factors; and D uses correlated and dependent factors.


Complementary experiments show abrupt transitions between the different regimes in the borderline regions.

an asynchronous aperiodic regime (region B) is prone to evolve into an asynchronous (region A) or periodic synchronous (region C) regimes (Figure 6).