500 of these input pairs and targets were used as training data, and presented one per step to the LSNN during the episode, where each step lasted 20 ms. LSNN parameters were updated using BPTT to minimize the mean squared error between the LSNN output and the target in the training set, using gradients computed over batches of 10 such episodes, which formed one iteration of the outer loop.

each weight update included gradients calculated on the input/target pairs from 10 different TNs.


Firstly, current adversarial attacks [Nguyen et al., 2015;Huang et al., 2017] are all based on calculating the derivative of the final loss with respect to the input.

regardless of the training procedure, one can always attack the system as long as chain rule is applicable.


This is easily overcome by uniformization (in fact, just by thinning, since we're dealing with a Poisson process).

let G be the set of observed events and Gprev the previous set of thinned events.


For every even r ≤ m/4 and everyunit vector While it might seem at a glance that bounding the high-order moments of X(x) is merely a technical issue, known tools and techniques could not be used to prove Lemmas 3, 4.

earlier work by Kane and Nelson [KN14, CJN18] and Freksen and Larsen [FL17] used high-order moments bounds as a step in proving probability tail bounds of random variables.


Based on the link between stability and generalization, we predict that on realistic data, all risk-minimizing models asymptotically become stable, but the rates of convergence to this ultimate stability differ.

an appropriate scaling of the stability measures will make them independent of the actual sample size used.


Our goal throughout is to minimize a function f : Here e denotes a small subset of {1, . . . , n} and x e denotes the values of the vector x on the coordinates indexed by e. The key observation that underlies our lock-free approach is that the natural cost functions associated with many machine learning problems of interest are sparse in the sense that |E| and n are both very large but each individual f e acts only on a very small number of components of x.

each subvector x e contains just a few components of x.


let v 1 be the principal eigenvector of B T B, where this theorem shows that it is possible to estimate v solely from the covariates using, in fact, a spectral method if, and only if µ > √ γ.


However, the parameter importance (i.e., the connection importance) in a certain network is extremely difficult to measure because of the mutual influences and mutual activations among interconnected neurons.

a network connection may be redundant due to the existence of some others, but it will soon become crucial once the others are removed.


Given a joint distribution p, consider the "(X × X ) T −1 → P(X × X )"valued probability tree ρ = (ρ 1 , .

. . , ρ T ) defined by the values of the mappings ρ t (ǫ) are products of conditional distributions, where conditioning is done with respect to a sequence made from x s and x ′ s depending on the sign of ǫ s . We note that the difficulty in intermixing the x and x ′ sequences does not arise in i.i.d.


A useful method for solving this problem is through linear discriminant functions, which consist of linear combinations of the components of the input variable.

we seek a weight vector W and a threshold () such that w T x < () if its label y = -1 and w T x 2: () if its label y = 1.


There exists constant C depending on f and K such that for n sufficiently large, with probability at least 1 − 1/n, and |M| = | M|.

taking h ≈ n −1/(4+d) optimizes the above rate to d(M, M) = Õ(n −1/(4+d) ).


We study this predictive power of Wn in the standard batch learning framework.

we assume that the training data (xi, yi) for i = 1, ... n are independently drawn from the same underlying data distribution D which is unknown.


In addition, we analyze the effects of the object localization net.

we assess the occurring performance changes of two configurations: (i) by only adding the bounding box regression loss into the objective function ('Train'), i.e., both the segmentation net and the object localization net are trained but only the segmentation net is deployed; (ii) by training and applying the object localization net ('Apply').


For continuously differentiable f , Robbins and Monro (1951) propose a simple iterative stochastic search algorithm for solving (1.1).

at the k-th iteration, we obtain z k sampled from D and take (1.2) where ⌘ is the step-size parameter (also known as the learning rate in machine learning literature), r u f (u k , z k ) is an unbiased stochastic gradient for approximating r u E Z⇠D f (u k , Z), i.e., and ⇧ U is a projection operator onto the feasible set U .


We found both the expert pretraining and RL objective important for generating high quality molecules, thus both of them are kept throughout training.

we use PPO algorithm to train the RL objective with the default hyperparameters as we do not observe too much performance gain from tuning these hyperparameters, and the learning rate is set as 0.001.


In this section, we use three numerical experiments to demonstrate the effectiveness of our method to handle irrelevant/corrupted features.

we test the performance of our method and effect of number of irrelevant features and dimension subspaces d with respect to different λ.


The engineering implications of ACS merit further exploration.

our compression results with overcomplete ACS indicate that the reconstruction quality was significantly higher than with standard CS.


However, their stochastic algorithm is not designed for online learning where online performance is important.

their algorithm alternates between two stages with one stage updating the models using received examples and another stage updating the so-called challenge level using received examples.


For learning the score s(x|X), we take recourse to large-margin classification with structured loss functions [37] to obtain the relative loss objective l(x, x |X) = max(0, s(x |X)−s(x|X)+∆(x, x )).

we want to ensure that s(x|X) ≥ s(x |X) + ∆(x, x ) whenever x should be added and x should not be added to X.


and "Is there a natural hedging strategy (S, c) that (roughly) achieves this value?"

what is the minimax value of the option, as well as the optimal hedge, when we fix the variance budget c and the asset's current price S, but let the number of rounds tend to 1?


We define ρ := β * 2 /σ to be the SNR and ω := r/ β * 2 to be the relative contractivity radius.


At decoder step i, we utilize a context-based attention mechanism with glimpse, similar to Vinyals et al.

[35], which extracts the relevant information from the inputs using a variable-length alignment vector a t . a t specifies how much every input data point might be relevant in the next decoding step t. Let xi t = (s i , di t ) be the embedded input i, and h t ∈ R D be the memory state of the RNN cell at decoding step t. The alignment vector a t is then computed as ) Here ";" means the concatenation of two vectors.


In this paper, we focus on scalable parallelization of Monte Carlo simulation, a problem motivated by the increasingly large inference problems occurring in a variety of fields in science and engineering.

we assume that we are given a large scale inference problem involving an intractable posterior expectation, for example a Bayes estimator, and that Monte Carlo simulation is to be used to approximate the targeted expectation.


The learning problem is that of selecting a function f 2 F with small risk or expected loss  Generalization bounds based on the notion of Rademacher complexity are standard [7].

for the empirical risk minimization (ERM) hypothesis b g n , for any > 0, the following bound holds with probability at least 1 : R n (G) is in the order of O(1/ p n) for various classes used in practice, including when F is a kernel class with bounded trace and when the loss l is Lipschitz.


Next, we upper bound the expected reward of the best (t) arms found in τ (t) fresh probes.

for any the probability of picking an arm with expected reward greater or equal to µ(h) is ( (t)/τ (t))(1 − h).


We resolve the problems described above by assuming that speakers and listeners deviate from normative behavior by assuming a conventional lexicon.

our final convention-based agents assume: (a) There is some single, specific literal lexicon which everyone should be using, (b) and everyone else knows this lexicon, and believes that I know it as well, (c) but in fact I don't.


The cost-based dueling bandit relates to function optimization with noisy comparisons (Jamieson et al., 2012;Matsui et al., 2016) because in both frameworks an oracle compare two actions and the feedback from the oracle is represented by binary information.

the same algorithm can be applied to both frameworks.


For the negative-non-matching-pairs, we employ a progressive mining strategy to obtain the most informative patches possible.

we sort the negatives for each sample by loss in decreasing order and sample randomly over the top M , where M = max(5, 64e 0.6k 1000 ), where k is the current iteration, i.e., we start with a pool of the 64 hardest samples and reduce it as the networks converge, up to a minimum of 5.


Moreover, we derived a general form of the asymptotic free energy, which can be applied to different asymptotic limits.

the standard asymptotic theory requires a large number N of words per document, compared to the number M of documents and the vocabulary size L. This may be reasonable in some collaborative filtering data such as the Last.FM data used in our experiments in Section 4.


By replacing this parameter set with the parameters of a Dirichlet density, we represent the predictions of the learner as a distribution over possible softmax outputs, rather than the point estimate of a softmax output.

this density can intuitively be understood as a factory of these point estimates.


As our learning algorithm approximates the underlying distribution up to small error in total variation norm, all crucial properties of the underlying distribution are preserved.

the analyst is free to compose our learning algorithm with an arbitrary non-private analysis.


( 21) becomes more amenable to both computations and statistical analysis and indeed most previous MTL literature has been focused on this setting, either by designing ad-hoc output metrics [33], linear output encodings [34] or regularizers [5].

in this latter case the problem is cast as that of minimizing the functional A ts f t , f s H (22) where the psd matrix A = (A ts ) T s,t=1 encourages linear relations between the tasks.


When learning with decomposition, each sub-network is allowed to converge independently, which can contribute significantly to the obtained savings.

with enough observed variables, we have realized that the vast majority of sub-networks converge very quickly, sometimes in one iteration (when the projected dataset is complete).


In this paper, we focus on parallel SGD methods, which have received considerable attention recently due to their high scalability [6,8,32,13].

we consider a setting where a large dataset is partitioned among K processors, which collectively minimize a function f . Each processor maintains a local copy of the parameter vector x t ; in each iteration, it obtains a new stochastic gradient update (corresponding to its local data).


In the ideal setting (say k → ∞), we will have P ij ≡ Pij = (1/d max )w j /(w i + w j ).

the random walk will move from state i to state j with probability equal to the chance that item j is preferred to item i.


Note that there are other bootstrapping techniques applicable to sequential, dependent data with lower coverage error, such as the double bootstrap [13], block-block bootstrap [1] and Markov or Sieve bootstrap [28].

the Markov bootstrap has been shown to have a lower coverage error for Markov data than the block bootstrap under certain restricted conditions [10].


That is, (MMS) performs as well as the best strategy that sees the covariate sequence in advance.

knowledge of Σ, not T , is necessary for the learner.


Our method finds a local minima of exactly (13), while [12] finds a global minima of an approximation to (13).

[12] requires approximating the search domain by hyperplanes and an additional branch-and-bound algorithm while ours does not.


In its simplest form, an MLN is a set of weighted first-order logic formulas, and can be viewed as a template for generating a Markov network.

given a set of constants that model objects in the domain, it represents a ground Markov network that has one (propositional) feature for each grounding of each (first-order) formula with constants in the domain.


Then, during play, Libratus would solve endgames that are reached using a significantly finer-grained abstraction.

those endgames have no card abstraction, and they have a fine-grained betting abstraction.


Since our algorithm optimizes the metrics at every node, we first examine the resulting per-node decisions.

how accurately can we predict the correct subcategory at any given node?


The details of this tracking framework are described in [7] and are briefly summarized here.

the posterior distribution of the unknown motion parameters is represented using a discrete set of samples and is propagated over time using particle filtering [3 , 7].


Although it is possible to optimize with these criteria using Monte-Carlo Markov sampling or other approximations, we instead develop a more efficient deterministic algorithm that takes advantage of the hGMRF structure in the FoGSM model.

we compute and take x as the optimal denoised subband.


That is, there exist constants a m,p 0, such that, for every f 2 W m p (⌦), we have (c) E continuously maps B ↵ p,q (⌦) into B ↵ p,q R d for all p 2 (1, 1), q 2 (0, 1] and all ↵ > 0.

there exist constants a ↵,p,q 0, such that, for every f 2 B ↵ p,q (⌦), we have For detailed conditions on ⌦ ensuring the existence of E, we refer to [8, p. 181] and [9, p. 83].


Our test evaluation measurement is the Mean Reciprocal Rank (MRR), which is the average of the reciprocal of the rank of the correct answer.

for each English word e, we calculated the discriminative score of respective methods when e is paired with each Hebrew word in the test set, and then found the rank of the correct word (1 for the highest).


Second, this collection of data sets presents a very different test bed from the usual image and speech data sets found in the neural network literature.

these data sets span a wide variety of domains, including agriculture, credit scoring, health outcomes, ecology, and engineering applications, to name a few.


In order to ensure spatial coherence and limit computation load, we employ the PBIl algorithm not on single time courses, but on averages of spatial clusters of voxel time courses.

we first spatially cluster voxel locations as shown in Figure 1, then compute average time course for each cluster and then explore the averages via PBIL for model building.


Note that the variance parameter of q(w) converges to 1 in one iteration, though underestimates the true posterior variance 1 + D. Intuitively, the convergence speed of w and q(w) suffers from strong coupling between the updates of w and z.

the update information has to go through a feedback loop w → z → w • • • .


This principle was independently hinted at by Tong and Koller (Tong and Koller, 2000) with a useful generative interpretation.

they proved that SVM are a limiting case of their Restricted Bayesian Classifiers.


Differences in scores may be caused by small differences in our implementation.

the difference in our score on Enduro and that reported by [25] might be due to us not using a play-time limit during evaluation.


Before stating our convergence results for Algorithm 1, we will need a notion of smoothness relative to our subspaces.

we say F is L i -Π i -smooth if for all i there are constants L i > 0 such that for all y ∈ im(Π i ), Finally, let F min := min y∈span{im(Πi)}i F (x 0 + y) be the global minimizer of F subject to the constraints from the Π i .


Our hypothesis is that we have higher confidence in the correct predictions, that is, σ + will typically be smaller than σ − .

we hypothesize that the relative difference between the average deviations, ∆(σ) = 2 σ−−σ+ σ++σ− , is larger than 0.


[6], the number of genes are reduced by utilizing clinical information that is provided together with the microarray expression dataset.

survival analysis via univariate Cox regression with patient survival times is used to select a subset of genes closely associated with breast cancer.


If a block had a large eigenvalue, near 1 or −1 (times its λ max ), then that block could itself be broken into two distinct clusters.

the clustering C would not accurately capture the cluster structure of the matrix P . Hence, condition (3) amounts to requiring that no other cluster structure is present, in other words that within each block, the Markov chain induced by P mixes well.


In the control phase, we update parameters selectively so as not to overwhelm the model with steady-state data.

we estimate the noise standard deviation σ s for each variable s as the root mean squared error on the training data, and update the model with an example only if its (current) prediction error exceeds 2σ s . 3  Given our model and an initial condition (the T past states, controls, and disturbances for the M AHUs), we optimize the cost of a length-L trajectory with respect to control inputs at every step.


While this result may seem paradoxical, note that the basic phenomenon appears in a classical (non-private) prediction market with a transaction fee, although to our knowledge this observation has not yet appeared in the literature.

a classical prediction market with budget bound B 1 , trades of size 1, and a small transaction fee α, will still have an α-incentive to participate, and the worst case loss will still be Θ(B 1 ); this loss, however, can be extracted by as few as Θ(1) participants.


The function τ was originally introduced by Alexander [1,2] in the context of exponential inequalities for empirical processes indexed by VC classes of functions, and Giné and Koltchinskii [15] generalized Alexander's results.

they proved (see [15, p. 1213]) that, for a VC-class of binary-valued functions with VC-dim(F) = d, the ERM solution f n = arg min f ∈F with probability at least 1 − Ks −1 e −s/K for some constants C, K and any s > 0.


HMMs have been extensively studied and used both theoretically and in numerous application areas.

financial modeling with HMMs has been considered since their introduction by Baum et al.


Although the formula looks similar to the Gaussian case, it is important to note that the conditional embedding operator allows us to compute the conditional expectation of any f ∈ F, regardless of the distribution of the random variable in feature space (aside from the condition that h(•) is in the RKHS on X s , as noted by Song et al.).

we do not need to assume the random variables have a Gaussian distribution in feature space.


At the heart of our approach is the idea to translate this concept to basic symbolic theorem provers, and hence combine their advantages (multi-hop reasoning, interpretability, easy integration of domain knowledge) with the ability to reason with vector representations of predicates and constants.

we keep variable binding symbolic but compare symbols using their subsymbolic vector representations.


The integrated architecture in this paper combines feed forward control and error feedback adaptive control using neural networks.

for our proof of convergence, we require the softassign algorithm to return a doubly stochastic matrix as *sinkhorn theorem guarantees that it will instead of a matrix which is merely close to being doubly stochastic based on some reasonable metric.


Usually in real applications, although there are a large number of features, the feature space is quite sparse, which provides us a possibility of designing a nearly lossless approach to reduce the number of effective features.

in a sparse feature space, many features are (almost) exclusive, i.e., they rarely take nonzero values simultaneously.


Motivated by applications to large graphs, our second contribution deals with the computational complexity of graph reconstruction.

we investigate to which extent the learning rates can be preserved when replacing the empirical reconstruction risk by a computationally cheaper Monte-Carlo version, obtained by sampling with replacement B n 2 pairs of nodes.


We will now construct the proposed algorithm.

we will first extend the continuous-time Markov process proposed in [36] and develop a process whose marginal stationary distribution has a density proportional to exp(−βU (x)) for any given β > 0.


However, for reasonable choices of W, a straightforward application of these techniques can lead to algorithms with trivial bounds.

in the case of W consisting of m × n matrices with trace-norm at most r, standard online regret bounds would scale like O r √ T . Since for this norm one typically has r = O √ mn , we get a per-round regret guarantee of O( mn/T ).


Theorem 2 can be viewed as a bid distribution-dependent analogue of the revenue covering framework [4] and of the smooth mechanism framework [13].

the quantity µ(D) is the datadepenent analogue of the worst-case µ quantity used in the definition of µ-revenue covering in [4] and is roughly related to the µ quantity used in the definition of a (λ, µ)-smooth mechanism in [13].


With each node i in the taxonomy, we associate a unit-norm weight vector u i ∈ R d . Then, we define the conditional probabilities for a nonroot node i with parent j by We now describe a classifier, called H-BAYES, that is the Bayes-optimal classifier for the H-loss.

h-BAYES classifies any instance x with the multilabel When no ambiguity arises, we write p i instead of p i (x).


These networks trained much faster than the MAP networks, since they were initialized with previously trained networks.

we found that using VCL led to ∼3x, ∼2x, and ∼4x convergence speedups for HCP to NKI, HCP to Buckner and HCP to WU120, respectively.


Vector field estimators have to take into account that the numerical representation of a vector depends on the coordinate system it is measured in.

the estimate should be invariant with respect to a rotation of the coordinate system.


The output of each neuron g l,i can be formulated by g l,i : where b l ∈ R n l is the bias vector.

a dendritic layer output f D l is the composition of branches and can be formulated by when d = 1, the dendritic layer is the same as an affine fully connected layer with linear activation functions.


However, for most large real-world graphs such as the World Wide Web ( [1]), the degree distribution deviates significantly from the Poisson distribution and has high variance.

most large matrix-completion datasets such as the much publicized Netflix prize dataset and the Yahoo Music dataset exhibit power-law distributed degrees, i.e., the number of vertices of degree d is proportional to d −β for a constant β (Figure 1).


Finally, we remark that even though we are mostly interested in the case where `= O(1), our dependencies on `seem to be better than the results of [3] even in the setting of Gaussian perturbations.

our running time (as well as our tolerance for error) grows polynomially with n `, whereas the running time of [3] grows with n 3 `.


Further support of the normalized model comes form the fitted parameter values: for the model with no normalization, the response level of the highest contrast stimulus r 4 was not well constrained 2 (r 1 =6.

the fit prior slope parameter is in good agreement with values from a previous study [16].


We modify our meta algorithm to work in this model by using the fact that we only store noisy linear sketches of the private matrix during the updates and low-rank factorization is computed through post-processing on only the noisy sketches.

we can use the generic transformation [23] to maintain the sketch of the updates.


We extend the randomized-feature approach to the task of learning a kernel (via its associated random features).

we present an efficient optimization problem that learns a kernel in a supervised manner.


However, our objective (1) is not strongly convex when the dimension is very large or there are redundant features.

the Hessian matrix H(w) of the smooth function (w) is not positive-definite.


(5) and P = D-1 S, then (1 -A), x are solutions of (2).

the NCut algorithm and the matrix P have the same eigenvectors; the eigenvalues of P are identical to 1 minues the generalized eigenvalues in (2).


P is indistinguishable from Q on the RHKS F. This population definition can be approximated using sample expectations.

given n samples from P as X = {x i ⇠ P, i 2 [n]}, and m samples from Q as Z = {z i ⇠ Q, i 2 [m]}, the following is a finite sample approximation: and the witness function is approximated as: k(x, z j ).


Our insight, conversely, is to sample extra features from continuous latent distributions rather than certain modes from enumerations, by learning a GAN model (see Figure 2).

we address two challenges that impede good translation under imbalanced scenarios: (1) through which base-novel class pairs we can translate; and more fundamentally, ( 2) through what objectives for GAN training we can estimate the latent distribution of novel classes with limited observations.


We fit the dataset to both a standard 110-topic LDA model and a DiscLDA model with restricted forms of the transformation matrices {T y } y=20 y=1 .

the transformation matrix T y for class label c is fixed and given by the following blocked matrix This matrix has (C + 1) rows and two columns of block matrices.


At this point, we use the 'kernel' trick: instead of explicitly specifying H or the mapping Φ, we take the approach of specifying inner products.

given any positive definite kernel K : S × S → R, it is well known (Mercer's theorem) that there exists a Hilbert space H and Φ : S → H such that K(x, y) = Φ(x), Φ(y).


Nonetheless, thanks to the generality of our formulation, the learned ranking functions and metric functions from most of these works can be naturally admitted into our optimization objective function.

our formulation brings the possibility to take advantage of these existing research results in the diversified ranking setting.


Finally, for a matrix A, let A + denote its (uniquely defined) Moore-Penrose pseudoinverse.

for two vectors x, y ∈ R n , and the degree matrix D G for a graph G, we define the degree-weighted inner product as if a vector x has unit norm, then x T D G x = 1.


Spatial interactions in natural images are often well-described by Markov models [13] and first-order stationary Markov processes are known to have sinusoidal KLT bases [29].

for the LUV color channels, our filters are similar to the discrete cosine transform (DCT) bases that are often used to approximate the KLT.


We note that while the algorithm is simple, its analysis is not straightforward and contains some counterintuitive ideas.

we show that the true (unknown) variation can be replaced with a crude proxy called the observed variation (to be later defined), while still maintaining mathematical tractability of the problem.


Fortunately, V ↵ has a closed form, and W ↵ has been studied before [2].

if ↵ > 1 2 , then V ↵ (q) = c ↵ q ↵ , where c ↵ is a constant with a closed form given by Lemma B.15.


The matching algorithm places the model into the image using maximum-likelihood.

it finds the configuration L of parts in model Θ in image I maximizing Figure 1: This graphic illustrates density map quality at various resolutions.


In our experiments, we treat the template actions as tags.

on each video and for each template action, we use the set of Action Bank action detection scores collected at different spatiotemporal scales and correlation volumes.


Diversity: This term is used to enforce spatial relationship constraints on the locations of selected templates.

their locations should not be too close to each other, because we want the learned templates to be diverse, so that they can cover a large range of image shape patterns.


Recently, [14,12] showed that the Bayesian approach has good theoretical properties for the nonparametric problem.

it achieves the minimax optimality under weak assumptions.


To track the potentially changing generating structure, the datapoints must potentially be differentially weighted.

datapoints should be weighted more heavily after a change occurs.


In this paper, we propose a direct approach (called DSPCA in what follows) that improves the sparsity of the principal components by directly incorporating a sparsity criterion in the PCA problem formulation and then relaxing the resulting optimization problem, yielding a convex optimization problem.

we obtain a convex semidefinite programming (SDP) formulation.


We assess the predictive performance of the collected samples from the various algorithms by computing a Monte Carlo estimate of the predictive log-likelihood of a held-out data point under the model.

for a held out point y * we have   m is the probability of choosing atom m for sample t. We did not use the Rao-Blackwellized estimator to compute Eq.


An additional interesting problem for future research is how to translate our results to the regret minimization setting.

it would be nice to see a conversion of algorithms like UCB [5] to a distributed setting.


Also, let X c (t) denote the set of all nodes that are in an infected state at time t in cascade c. We know the infection of each node was the result of an unknown, previously infected node to which it is connected, so the component of the likelihood function for each infection will be dependent on all previously infected nodes.

the likelihood function for a fixed given A is The likelihood function is composed of two terms.


In a related work Neu and Rosasco [2018] consider the problem of linear least squares regression and show that the iterates produced by GD on least squares objective are related to the solutions of ridge regression.

for any given regularization parameter of ridge regression, Neu and Rosasco [2018] show that there exists a weighing scheme for GD iterates that is exactly equal to the ridge solution.


The focus of this work is to apply this idea of life-long learning to the matrix completion problem.

given columns of a matrix that arrive online over time with missing entries, how to approximately/exactly recover the underlying matrix by exploiting the low-rank commonality across each column.


Weighted majority votes are a theoretically well understood and widely used in practice type of ensemble predictor.


Hedging the bets extension (HB): we extend the hedging idea [5] to handle sets of query images.

we find the subtree in the hierarchy that maximizes the information gain while maintaining an overall accuracy above a threshold over the set of example images.


This error bound achieves the squared error rate σ 2 rd(log d)/n as in Theorem 3 when the noise level σ is of no smaller order than Θ ∞ , but not of smaller order.

(17) does not imply exact recovery when σ = 0.


In this section, we begin with a precise formulation of the problem, and then develop some key properties of the regularizer and loss function.

we define a notion of decomposability for regularizing functions r, and then prove that when it is satisfied, the error ∆ = θ − θ * of the regularized M -estimator must satisfy certain constraints We use these constraints to define a notion of restricted strong convexity that the loss function must satisfy.


Suppose Algorithm 1 is run with 1/η > max t max x∈X |∇ T t x t |.

then, for every H 0 the regret is bounded as where r = rank( T t=1 ∇ t ∇ T t ) ≤ d and we can use the theorem to show that as long as we bound |∇ f (x t ) T x t | by a constant-a significantly weaker requirement than assuming bounds on the diameter of X and on the norms of the gradients-one can find a norm • H for which the quantities D H and G H are properly bounded.


The most important of these measurements is the "QT interval".

drug-induced prolongation of the QT interval (so called Long QT Syndrome) can result in a very fast, abnormal heart rhythm known as torsade de pointes, which is often followed by sudden cardiac death 2 .


In comparison, EigenPro iteration using top-k eigen-directions costs O(md + kd).

applying preconditioner P in EigenPro requires left multiplication by a matrix of rank k. This involves k vector-vector dot products resulting in k • d additional operations per iteration.


When the function is smooth enough, we show that acceleration can be achieved by a stable discretization of this ODE using standard Runge-Kutta integrators.

we prove that under Lipschitz-gradient, convexity and order-(s + 2) differentiability assumptions, the sequence of iterates generated by discretizing the proposed second-order ODE converges to the optimal solution at a rate of O(N −2 s s+1 ), where s is the order of the Runge-Kutta numerical integrator.


This is the hierarchical Baysian model that assumes multiple tasks sharing the same GP prior [18].

the negative logarithm of p {y i,j }, {f j }|Σ is where l(y i,j , f j (i)) = − log p(y i,j |f j (i)).


Interestingly, this estimator, f = max(ID(K))−1 , appears to be the estimator for a hypergeometric random sample of size D s = max(ID(K)) − 1.

suppose we randomly pick D s balls (without replacement) from a pool of D balls and we observe that k balls are red; then a natural (and unbiased) estimator for the total number of red balls would be D D s k ; here k = k − 1.


We prove some stronger results in more general settings.

we consider a regressor which, by combining aspects of both tree-based regression and kernel regression, adapts to intrinsic dimension, operates on general metrics, yields a smooth function, and evaluates in time O(log n).


Hence, by controlling the behavior of the IS link around the origin, it is possible to control the margin of the optimal classifier.

the margin is a decreasing function of the curvature of the loss at the origin, φ (2) (0).


That is, where  The parameters for FastGRNN: ) are trained jointly using projected batch stochastic gradient descent (b-SGD) (or other stochastic optimization methods) with typical batch sizes ranging from 64 − 128.

the optimization problem is given by: where L denotes the appropriate loss function (typically softmax cross-entropy).


It would be quite interesting to understand what kind of adaptive-feedback patterns give rise to easy problems, for which a regret of O( √ T ) is attainable.

is there a problem with oblivious losses and adaptive feedback whose minimax regret is Θ(T 2/3 ), as is the case with adaptive losses?


ConQ[sa] is a restriction of Q[sa] obtained by intersecting each cell in its domain, dom(Q[sa]), with [s → a].

conQ[sa] is a partition function defined on some partition of [s → a] (rather than all of Θ), and represents Q-values of cells that are consistent with [s → a].


Because this stage can be viewed as a standard phase retrieval through lifting, we can simply use accuracy guarantees that are already established in the literature (e.g., [3,6,5]).

we use [5, Theorem 2] which established an error bound that holds uniformly for all valid B .


However, existing convergence analyses do not apply to the currently used greedy algorithms.

all existing MP variants for the conic hull case [5,38,52] are not guaranteed to converge and may get stuck far away from the optimum (this can be observed in the experiments in Section 6).


This means that by using the kernel (6) in the computation of (4), one can make an assumption that the integrand f has smoothness of degree α in each variable.

one can incorporate one's knowledge or belief on f into the construction of weighted points {(w i , X i )} via the choice of α.


Using the plain robust optimization formulation in Algorithm 2, even the optimal policy π is not considered safe in this example.

the robust return of π is min ξ ρ(π , ξ) = −9, while the optimistic return of π B is max ξ ρ(π B , ξ) = +10.


The main property of such feature formation process, desirable for our applications, is the fact that topics combine additively.

the same bin may have contributions from multiple topics, and this models the fact that the bin height is the count of edges in a neighborhood which may include parts generated by different components.


In this section we will present two views to illustrate how such noise suppression can be achieved.

in Section 4.1) we show that under certain setup, linear Fredholm kernel suppresses principal components with small variance.


People like to retweet an interesting post soon after it is created and retweeted by others, but may gradually lose interest, so the intervals between retweets become longer over time.

the stream begins in a self-exciting state, in which previous retweets increase the intensities of future retweets, but eventually interest dies down and events are less able to excite one another.


The technique we propose uses subsamples of the training set containing n  N component functions that we grow geometrically.

we start by a small number of samples and minimize the corresponding empirical risk added by a regularization term of order V n up to its statistical accuracy.


As can be observed, models trained with our method mostly better generalize to unknown data distributions.

our method always outperforms the baseline by a statistically significant margin when the training images are from Night scenarios.


Thus, it is the Laplace exponent of a subordinator.


The magnitude of Γ st is a scalar multiple of the correlation of X s and X t conditioned on X \{s,t} , and encodes the strength of the edge (s, t).

the sparsity pattern of Γ st reflects the edge structure of the graph: Γ st = 0 if and only if X s ⊥ ⊥ X t | X \{s,t} .


Uniformity testing is arguably the most extensively studied problem in distribution property testing [GR00, Pan08, VV14, DKN15b, Gol16, DGPP16, DGPP17] and its sample complexity is well understood.

it is known [Pan08, CDVV14, VV14, DKN15b] that Θ(n 1/2 / 2 ) samples are necessary and sufficient for this task.


Wilson and Ghahramani [20] join machine learning and econometrics efforts by proposing a model for both mean and covariance regression in multivariate time series, improving previous work of Bru [21] on Wishart Processes in terms of computational tractability and scalability, allowing more complex structure of dependence between Σ(t) and Σ(t + ∆).

they propose a continuous time Generalised Wishart Process (GWP), which defines a collection of positive semi-definite random matrices Σ(t) with Wishart marginals.


Although promising directions within the sampling community such as Elliptical Slice Sampling (ESS, [5]) have been proposed, they have been shown to be particularly slow compared to variational methods.

[6] showed that their automated variational inference (AVI) method can provide posterior distributions that are practically indistinguishable from those obtained by ESS, while running orders of magnitude faster.


The main objective of our work is the active allocation of the sampling resources for anomaly detection, in the setting where anomalies are defined as extreme values.

we consider a variation of the common setting of minimal feedback also known as the bandit setting [14]: the learner searches for the most extreme value that the sources output by probing the sources sequentially.


Each class is associated with a stem distribution and a suffix distribution.

we assume that stems and suffixes are independent given the class, so we have where c k , t k , and f k are the class, stem, and suffix associated with ℓ k , and t.f indicates the concatenation of t and f . we generate a label by first drawing a class, then drawing a stem and a suffix conditioned on the class.


L "observed features" j ∈ {1, . . . , R} "noise features" j ∈ {R + 1, . . . , 40} However, we can do much better by reorganizing the computations in such a way that the Jacobian matrix J d is never explicitly required.

substituting ( 6) into (1), ) we observe that it suffices to solve the single linear system, By organizing the computations this way, the number of least squares problems that must be solved is substantially reduced from k to only one.


We apply this relabeling operation iteratively to obtain context-dependent atom vectors where h V are shared across layers.

the final atom representations arise from mimicking the set comparison function in the WL isomorphism test, yielding The set comparison here is realized by matching each rank-1 edge tensor h to a set of reference edges also cast as rank-1 tensors W (0) [k] ⌦ W (1) [k] ⌦ W (2) [k], where W[k] is the k-th row of matrix W. Eq.


Our inequalities are tight under natural assumptions and can greatly simplify the analysis of common sequential decision problems.

we apply them to sequential hypothesis testing, best arm identification, and sorting.


Specifically, if evolving x t with a numerical integrator using discrete time increment h induces an error bounded by O(h K ), the integrator is called a Kth order integrator, e.g., the popular Euler method used in SGLD [3] is a 1st-order integrator.

[17] proved the bounds stated in Lemma 1.


( 22) corresponds to the maximum element chosen within the channel c of Z t . the normalization in Eq.


In this paper, we examine a large scale energy disaggregation task, and apply a novel extension of sparse coding to this problem.

we develop a method, based upon structured prediction, for discriminatively training sparse coding algorithms specifically to maximize disaggregation performance.


However, we can reduce this time to O(log n) by using the JL lemma, which informs us that pairwise distances can be approximately captured by compressive projection onto d = O(log n) dimensions.

for d > (4 + 2β) log n ǫ 2 /2 − ǫ 3 /3 , let Ω denote an n × d matrix with i.i.d.


Both of these steps are computationally expensive operations, and future work could address improving the efficiency of our approach.

it seems likely that similar spectral features can be approximated without computing all the pairwise distances, which would result in significant speedup.


The appealing additive property of the objective function J makes it trackable to analyze each split when the search tree grows.

we split the k-th leaf node into two child nodes k(1) and k(2) if and only if it increases the overall objective function J k(1) + J k(2) > J k . Moreover, we optimize each split by choosing the one that maximizes J k(1) + J k (2) .


In the proposed approach, the random variables (RVs) linked by a tree substructure across different levels compete for their label assignments, while in the existing approaches the goal is to cooperate in the label assigns, which is usually achieved by averaging.

usually the label assignment of a parent node is enforced to be compatible with the label assignment of its children by averaging.


(See the supplementary material for the derivations.)

while F(γ GA ) = F(γ M A ), they achieve this value for different reasons: γ GA follows an optimal schedule along a bad path, while γ M A follows a bad schedule along a near-optimal path.


Additionally, airlines have an incentive to provide sufficient margin such that passengers can make connections and they don't have to re-ticket them.

they tend to set up routes such that, even in a worse than average scenario, the original route will tend to succeed.


Nonparametric models have also been proposed for capturing nonlinear relations [35,24,22].

[24] extended the linear tensor learning to the nonparametric learning problem using a kernel method and proposed a convex regularization method and an alternating minimization method.


DNNs can be trained greedily, in a layer-by-layer manner, as in Restricted Boltzmann Machines [12] and (sparse) auto-encoders [3,17], or by learning all layers simultaneously, which is relatively efficient if the layers are convolutional [18].

the advent of massively-parallel GPUs has recently made it possible to train deep convolutional networks on a large scale with excellent performance [7,14].


Note that the training sets and testing sets for a classifier are disjoint.

lR and DRF takes D L i as the training set and D U i and D S i for testing sets, while SSDRF takes D L i and D U i for training and D U i and D S i for testing.


This means that any regularizer that depends on scale (such as 1 and 2 norms) is effectively meaningless since the minimization in (1) will drive Ω(θ) to 0 (as this doesn't affect the loss).

in such discrete setting we need a scale-invariant regularizer, such as the 0 pseudo-norm.


In our implementation, we found that selecting β aggressively helped speed up recovery of our algorithm.

we selected β = 1/ √ n. Complexity: The complexity of each iteration within a single stage is O(kmn), since it involves calculating the rank-k approximation 3 of an m×n matrix (done e.g.


The 4th column of figure 1 shows the result of doing this for the experimental kernels.

it shows that the clear spectral peaks suggested by the Wiener filter (in the 3rd column) may be artefactualthey can arise from a form of whitening.


From both perspectives, redundancy reducing algorithms such as Principal Component Analysis (PCA), Independent Component Analysis (ICA), Independent Subspace Analysis (ISA) and Radial Factorization [11; 21] have received considerable interest since they yield image representations that are favorable for compression and image processing and at the same time resemble properties of the early visual system.

iCA and ISA yield localized, oriented bandpass filters which are reminiscent of receptive fields of simple and complex cells in primary visual cortex [4; 16; 10].


2, D-TS and D-TS + have much smaller regret STD than that of ECW-RMED in the non-Condorcet dataset, where certain preference probabilities (for different arms) are close to 1/2.

the STD of regret for ECW-RMED is almost 200% of its mean value, while it is only 13.16% for D-TS + .


Note that the planted group assignment of SBM is invariant under a global permutation of the group labels.

as long as the set of vertices in the same group share the same label, the label itself can be anything.


It is relatively straightforward to include additional model sophistication into (10), one example of which we will consider in the context of the image-processing example.

in many applications it is inappropriate to assume a Gaussian model for the noise or residual n . In Section 4.3 we consider the following augmented noise model: with gamma priors placed on α λ and α 2 , and with p = 1, . . . , M . The term λ n • m n accounts for "spiky" noise, with potentially large amplitude, and πn represents the probability of spiky noise in data sample n. This type of noise model was considered in [18], with which we compare.


In Section 3, we analyze the algorithmic convergence.

we extend recent results [5] on efficient margin maximizing to show that the algorithm can converge to the maximum achievable margin within a preset precision in a finite number of steps.


However, video frames are more than items in a set.

in DPP, the ground set is a bag -items are randomly permutable such that the most diverse subset remains unchanged.


One line of related research studies the communication complexity of convex optimization.

[40] studied two-processor convex minimization in the same model, provided a lower bound of Ω(n(log n + log(1/ ))) bits on the communication cost of n-dimensional convex problems, and proposed a non-stochastic algorithm for strongly convex problems, whose communication cost is within a log factor of the lower bound.


The learned features improve accuracy by a significant 4% to 5% across four still image recognition datasets.

we show best classification results to date 61% on the STL-10 [14] dataset.


This observation, combined with Theorem 2, leads to the following interesting consequence.

we can numerically estimate Λ in order to give a lower bound on ρ = log(1/p1) log(1/p2) for any hash family H in which every function induces at most T parts of measure at most 1/2.


Since the Poincaré Ball has a Riemannian manifold structure, we can optimize Equation (2) via stochastic Riemannian optimization methods such as RSGD [6] or RSVRG [39].

let T θ B denote the tangent space of a point θ ∈ B d . Furthermore, let ∇ R ∈ T θ B denote the Riemannian gradient of L(θ) and let ∇ E denote the Euclidean gradient of L(θ).


The Hessians r 2 f are M -Lipschitz continuous over the set C, i.e., for any x,  In this section, we introduce a generic framework to reach an (✏, )-SOSP of the non-convex function f over the convex set C, when C has a specific structure as we describe below.

we focus on the case when we can solve a quadratic program (QP) of the form up to a constant factor ⇢  1 in a finite number of arithmetic operations.


The maximal clustering over the elements of D(v), C ?

(D(v)), is defined as: the clustering with maximal energy over the set of elements, D(v) can be written as the energy of any cluster, C, in that clustering multiplied by a clustering with maximal energy over the elements D(v)\C.


We also give numerical results which demonstrate good performance on a benchmark problem.

we show that our algorithm achieves much better performance on this problem than OP-MDP [2] when both algorithms generate the same number of successor states, despite the fact that OP-MDP assumes knowledge of all successor state probabilities in the MDP, whereas our algorithm only samples states from a simulator.


From F = 3 to 25, the proposed algorithm (AnchorFree) gives very promising results: for the three considered metrics, AnchorFree consistently gives better results compared to the baselines.

the ClustAcc's obtained by AnchorFree are at least 30% higher compared to the baselines for all cases.


, PCP yields a unique solution which agrees with the correct (L, S) with high probability if there exists a dual certificate Y obeying where = 1 32 p n log n . The proof of the above proposition adapts the idea in [1,12] for uniform errors to non-uniform errors.

the proof exploits the properties of R ⌦ associated with non-uniform errors, which are presented as Lemma 1 (established in [9]) and Lemma 2 in Appendix A.1.


The aforementioned margin bounds are all dimensionality independent.

the bounds are solely characterized by the margins on the training data and do not depend on the dimension of feature space.


After each full Gibbs sweep, we obtain a sample z (t) (and corresponding counts N (t) j of the number of points assigned to cluster j), which can be used to infer the Bernoulli parameters µ (t) j .

we use T of these samples to estimate the probability that a file x i needs to be changed given files in the starter set S. averaging predictions over the T Gibbs samples, we estimate: x n . (6) A restricted Boltzmann machine (RBM) is a Markov random field (MRF) whose nodes are (typically) binary random variables [17].


We first consider the scenario where the observations are exact.

suppose we are given m observations {T ijk } (ijk)∈Ω , where Ω is sampled from uniformly at random from We propose to recover matrices A, B, C and therefore tensor T = Pair(A, B, C) using the following convex program, minimize where M * denotes the nuclear norm of matrix M, which is the sum of singular values of M, and S A , S B , S C is defined in Proposition 1.


The result also suggests that instead of selecting top few eigenvectors, as is normally done in KPCA, selecting them by our method and then applying L 1 regularization yields better result.

in case of IONOSPHERE and BREAST-CANCER data sets top |N | (5 and 3 respectively) eigenvectors do not contain the representative ones.


In this paper we show that the improved factor is also achieved by InfoBoost, a modification of AdaBoost developed by Aslam [Asl00] based on a different motivation.

consider the problem of finding a consistent hypothesis for m examples labeled by a k literal disjunction.


To showcase also the viewpoint prediction of our detector we insert a CAD model inside each estimated 3D box, matching its orientation in 3D.

for each detection we automatically chose a CAD model out of a collection of 80 models whose 3D bounding box best matches the dimensions of the predicted box.


A small modification to the unconstrained formulation (7) extends our tractable Stein discrepancy computation to any domain defined by coordinate boundary constraints, that is, to 1 for all j.

for each dimension j, we augment the j-th coordinate linear program of (7) with the boundary compatibility constraints max , and k 6 = j.


The "polyhedral lemma", found in [5, Theorem 5.2] and [13, Lemma 2], proves that the distribution of η Y , after conditioning on {Y ∈ A} and on P ⊥ η Y , is given by a truncated normal distribution, with density The interval endpoints a 1 (Y ), a 2 (Y ) depend on Y only through P ⊥ η Y and are defined to include exactly those values of r that are feasible given the event Y ∈ A.

the interval contains all values r such that r • η + P ⊥ η Y ∈ A.


Specifically, we propose methods for finding computationally efficient versions of a given target expression.

finding a new expression which computes an identical result to the target, but has a lower complexity (in time and/or space).


Below, we present such a network that takes both the given conditional variables and an extra set of random variables as inputs, and then passes through a deep neural network with nonlinear transformations to produce the samples of the target variables.

our network is built on the fact that for any distribution P on sample space K and any continuous distribution Q on L that are regular enough, there is a function G : L → K such that G(x) ∼ P, where x ∼ Q [12].


If it is possible to "encode" members of a class of distributions using a carefully chosen subset of the samples, then this yields an upper bound on the sample complexity of distribution learning for that class.

by constructing compression schemes for mixtures of axisaligned Gaussians and general Gaussians, we obtain new upper bounds on the sample complexity of learning with respect to these classes, which we prove to be optimal up to logarithmic factors.


Layer-wise coordination modifies the structure of Transformer from two aspects: First, each layer in the decoder attends to the corresponding layer in the encoder.

the encoder and decoder have the same number of layers and layer i in the decoder can extract information directly from layer i in the encoder, instead of the last layer of the encoder like Transformer.


To mitigate this problem we develop an importance sampling approach, where sequences are drawn from a proposal distribution q(•) that has an increased likelihood of generating extensions in which X e occurs, and then uses a weighted empirical estimate.

we will sample extensions The Poisson Superposition Importance Sampler (PSIS) is an importance sampler whose proposal distribution q is based on Poisson superposition.


In this paper, we propose Contrastive Learning (CL), a new learning method for image captioning, which explicitly encourages distinctiveness, while maintaining the overall quality of the generated captions.


The images were obtained from several sources.

we have re-labeled subsets of the Caltech Motorcycle and Faces database 3 , to obtain the subordinates of sport and cross motorcycles, and male and female faces.


α and β are updated from the entire document.

when we compare this algorithm with VB-LDA, it looks like a hybrid variant of a batch updates for α and β, and incremental updates for γ j , Here, we consider an incremental update for β to be analogous to CVBLDA, in which β is updated for each word.


Following the delayed and perturbed leader More generally, follow-the-perturbedleader optimizes over all information available to the algorithm, plus some additional noise to smoothen the worst-case analysis.

if the cost vectors are delayed, we naturally interpret followthe-perturbed-leader to optimize over all cost vectors c t delivered in time for round t when picking its point x t . the tth leader becomes the best choice with respect to all cost vectors delivered in the first t rounds: (we use the superscript d to emphasize the delayed setting).


This framework accounts for the effect of the measurement noise on the variability of the estimate θ.

it provides a characterization of the distribution p( θ|θ stim ) of the estimate for a given stimulus direction in terms of its expected value and its variance as a function of the measurement noise.


To do so, we build upon tools in amortized variational inference for developing a probabilistic meta-learning approach.

our method builds on model-agnostic meta-learning (MAML) [9], a few shot metalearning algorithm that uses gradient descent to adapt the model at meta-test time to a new few-shot task, and trains the model parameters at meta-training time to enable rapid adaptation, essentially optimizing for a neural network initialization that is well-suited for few shot learning.


Similarly, our notion of preferred treatment ensures envy-free decision making at the level of sensitive attribute groups.

with preferred treatment classification, no sensitive attribute group would prefer the outcomes from the classifier of another group.


Instead of drawing exactly m samples from a distribution p, we first draw m 0 ⇠ Poisson(m), and then draw m 0 samples from p. As a result, the number of times different elements in the support of p occur in the sample become independent, giving much simpler analyses.

the number of times we will observe domain element i will be distributed as Poisson(mp i ), independently for each i.


In Figure 1(c), we show the result of SGLD using about 1000 samples.

we generate 100k samples, discard the first 2k for burnin, and then keep every 100'th sample.


Therefore, once initialized within RC , the iterates produced by a descent algorithm will stay in RC .

at any stationary point q ∈ R 10 , the local optimization landscape can be characterized in terms of the number of spikes (entries with nontrivial magnitude 12 ) in ζ.


Again, the standard large deviation bound implies that there exists a universal constant c > 0 such that for all j ≥ 1: .

with this choice, there exists a constant c > 0 such that There exists a constant c > 0 such that for any L: if we choose s(Z n 1 ) = m −1 e (2k) −1 cn ∆(Z n 1 ) 2 in Figure 1, then ∆R(Epoch-Greedy, H, T ) ≤ 2L + 1 + c k∆ −2 ≤ 2 8k(ln m + ln(T + 1)) c∆ 2 + 1 + c k∆ −2 .


For bounds directly on function values, see [26].

we now consider θn = 1 n n−1 k=0 θ k and, following [4,5], we make extra assumptions regarding the smoothness of each f n and the fourth-order moment of the driving noise: (H6) For each n 1, the function f n is almost surely twice differentiable with Lipschitz-continuous Hessian operator f ′′ n , with Lipschitz constant M . for all θ 1 , θ 2 ∈ H and for all n 1, , where • is the operator norm.


This alignment is accomplished through an adversarial minimization across domain, analogous to the prevalent generative adversarial approaches [15].

a domain discriminator, D(•), is trained to classify whether a particular data point arises from the source or the target domain.


To demonstrate this, we adapt the proposed DirBN structure to the following models: PFA+DirBN Poisson Factor Analysis (PFA) is a popular framework for topic analysis (DPFA , DPFM , GBN (Zhou et al., 2016) can be viewed as a deep extension to PFA).

we use the Bayesian nonparametric version of PFA named BGGPFA (Zhou et al., 2012), where θ d is constructed from a negative binomial process and φ k is drawn from a Dirichlet distribution.


For instance, it follows from Proposition 1 that for standard Gaussian measurements, τ 2 (T) = 0 with high probability once n < δ m /2.

the situation can be much better for random spd measurements ( 14) as exemplified for measurements ∼ N (0, I) in Appendix H. it turns out that τ 2 (T) = Ω(1/r) as long as n = Ω(m • r).


It captures the idea that not all variation is important; clusters can vary in their emission parameters for a particular dimension and that variation still might not be interesting.

an important dimension is one where there is a gap between parameter values.


There is some universal constant c 0 ≥ 1, such that for any c ≥ c 0 and any ∈ (0, 1), if we run algorithm 1, using parameters η, T satisfying then after S = log 4 (9/ ) epochs of the algorithm, wS+1 satisfies E[F ( wS+1 )−min w F (w)] ≤ .

by taking η = Θ(1) and T = Θ(1/λ), the algorithm attains an -accurate solution after O(log(1/ )/λ) stochastic iterations of without-replacement sampling, and O(log(1/ )) sequential passes over the data to compute gradients of F (•).


The standard approach to parameter estimation in statistics is through maximum likelihood, which chooses parameters ✓ by maximising (To keep the notation light, we reserve n to index the data samples.

x n denotes the n-th |V|-dimensional data vector and x mn refers to the n-th observation of node m.) This estimator has played a central role in statistics as it has many desirable properties including consistency, efficiency and asymptotic normality.


These two classes of matrix factorization problems have attracted significant interest in the research community.

convex formulations of noisy matrix factorization have been shown to admit strong theoretical recovery guarantees [1,2,3,20], and a variety of algorithms (e.g., [15,16,23]) have been developed for solving both matrix completion and robust matrix factorization via convex relaxation.


But we can also see the iteration as a noisy attempt to reach the parallel coordinate optimum λ VB t . While λ is an unbiased estimate of this quantity, we will show that Eq 1 uses a biased estimate but with reduced variance.

this new perspective opens the door to other ways of updating λ t based on the noisy estimates of λ VB t . we use a Kalman filter to track the progress of λ t based on the sequence of noisy coordinate updates.


M-step We update the parameters, L i , T i and V i , of π i,l to obtain π � i,l based on the expectation in the E-step.

take log of the likelihood P r(r T = 1, z 0:T i , T ; π i,l ) with π i,l substituted with π � i,l and focus on terms involving the parameters of π � i,l : In order to update, L i , we partially differentiate the Q-function of Eq.


To resolve the curse of dimensionality, in many practical applications, we approximate the value of each state by some linear map of its feature s 2 < m , where d < S to reduce the dimension.

we assume that V ⇡ (s) ⇡ T s w ⇤ for some w ⇤ 2 < m . To compute w ⇤ , we formulate the problem as a Bellman residual minimization problem that w); < 1 is a discount factor, and r s,s 0 is the random reward of transition from state s to state s 0 .


Sadly, through upper and lower bounds we show that for the most biased coin problem this estimate-then-explore approach requires quadratically more flips than our proposed algorithm that adapts to these unknown parameters.

we show that when ✓ 1 ✓ 0 is sufficiently small one cannot use a static estimation step to determine whether ↵ = 0 or ↵ > 0 unless a number of samples quadratic in the optimal sample complexity are taken.


4 The perturbed image x 0 + r * S shown in    In this work, we precisely characterized the robustness of classifiers in a novel semi-random noise regime that generalizes the random noise regime.

our bounds relate the robustness in this regime to the robustness to adversarial perturbations.


Psychophysical data suggest that temporal modulations of stimulus amplitude envelopes play a prominent role in the perceptual segregation of concurrent sounds.

the detection of an unmodulated signal can be significantly improved by adding amplitude modulation to the spectral envelope of a competing masking noise.


In SIFT congealing, smoothness is achieved through soft clustering and the properties of the SIFT descriptor.

to compute the descriptor, the gradient is computed at each pixel location and added to a weighted histogram over a fixed number of angles.


Then, the matrices X and − Ḡ can be simultaneously singular-value-decomposed by Ū and V . Moreover, the set X c ⊂ X , which is defined as is a non-empty convex compact set.

by Proposition 4.2, for every k ≥ 0, the point X k has a unique projection Xk ∈ X c onto X c . Let Since we have Q k F = 1 for all k ≥ 0.


We found that on such trials the model significantly deviated from the subjects' beliefs.

subjects gave probability lower than one that non-example numbers within the range spanned by observed examples were consistent with the true concept.


Rather, we build on Bayesian optimization (BO) [10,12,4], which leverages Bayesian modeling in an attempt to achieve more efficient optimization.

bO maintains a posterior over the unknown function based on previously observed experiments, e.g.


All of our experiments are performed for the task of binary classification with a smooth variant of the hingeloss (see Shalev-Shwartz and Zhang [2013a]).


Then η is also an admissible natural parameter, because we can write p(x) = h(x) exp( t(x) � η − Ã(η)), where t(x) = H � t(x), h(x) = h(x) exp(t(x) � b), and Ã(η) = A(H η + b).

the natural parameter is only unique up to affine transformations.


However, efficiently finding the solution that minimizes (3) is rather difficult.

a simple relaxation method cannot be applied directly here.


Our method is based on a new data augmentation representation, following the recent developments in Bayesian logistic regression [13], which is a direct data augmentation scheme with only one layer of auxiliary variables and does not need to tune in order to get optimal performance.

for the above posterior inference problem, we can show the following lemma.


This geometric picture allows us to make the following important observation about the objective value max z• e of the SDP relaxation.

the largest value that z• e can take depends on the angle θ that x makes with e. as θ decreases, the relaxation becomes tighter and as the vector deviates from e, the relaxation gets looser.


Given the feature representation of all images from the retrieval database, , where x im i = ImgEnc(I i ), we can compute a sampling probability based on the distances between the history representation s t to each image feature, x im i .

the sampling probability is modeled using a softmax distribution over the top-K nearest neighbors of s t : π(j) = e −dj / K k=1 e −d k , j = 1, 2, ..., K, where d k is the L2 distance of s t to its k-th nearest neighbor in {x im i } N i=0 .


Ben-David et al [1,2,3] put forward the goal of establishing a Theory of Clustering.

attempts have been made by [4,2,3] to study and theoretically justify the stability-based approach of evaluating the quality of clustering solutions.


In this work we show how the total derivative rule leads to an intuitive visual framework for creating gradient estimators on graphical models.

previous "policy gradient theorems" are easily derived.


We address the first key challenge in our AltMinPhase algorithm (Algorithm 1) by initializing x as the largest singular vector of the matrix S = 1 m i y 2 i a i a i T . Theorem 4.1 shows that when A is sampled from standard complex normal distribution, this initialization is accurate.

if m ≥ C 1 n log 3 n for large enough C 1 > 0, then whp we have x 0 − x * 2 ≤ 1/100 (or any other constant).


The gradient and Hessian of F (w, v) equal respectively.

at (w, v) = (0, −1/ ), the gradient is 0 and the Hessian equals 0 − − 0 , which is arbitrarily close to 0 if is small enough.


We deliberately make the evaluation more challenging by preparing training and testing sets that are disjoint in terms of class labels.

we use the CUB-200-2011 [32] dataset with 200 bird classes and 11,788  images.


We give a positive answer to the first question, when the base algorithm is the perceptron and the noise is label flip with a constant probability.

we prove that the expected number of iterations required by the resulting algorithm equals (up to a constant factor) to that of the perceptron in the noise-free setting.


This however does not provide a satisfactory solution mainly because it does not take into account how the ranking is performed.

when there are some errors in the ranking, it does not learn how to compensate these errors although the threshold based approach tries to learn the best threshold with respect to these errors.


Because of the harder noise model, they necessarily require multiple noisy copies per clean example and the unbiased estimation schemes also become fairly complicated.

their techniques break down for non-smooth losses such as the hinge loss.


We showed that for smooth problems, we can preserve the performance benefits of parallelization over centralized stochastic optimization even when we relax synchronization requirements.

we presented methods that take advantage of distributed computational resources and are robust to node failures, communication latency, and node slowdowns.


The subspace row(Y ) does not necessarily contain the desired sparse vector e T i X 0 , but it does contain some approximately sparse vectors.

consider following vector in row The vector v is a superposition of a sparse signal x0 and its scaled shifts | is small 7 and x 0 is sparse enough, z can be viewed as small noise.


Furthermore, a variant of our algorithm even runs in sublinear time, if A 0 is large and ψ def = OPT (1) / A 0 is not too small.

we obtain time o( A 0 ) when OPT (1) ≥ ( −1 log(mn)) 4 and A 0 ≥ n( −1 log(mn)) 4 .


The shift of a word's meaning can be revealed by comparing it against a set of anchor words [18,17], which is a direct result of the distributional hypothesis [10,13,6].

let {1, • • • , l} be the indices of the l anchor words, common to two different corpora.


In addition to their score, we also evaluate the algorithms by the number of samples they use.

we show that CBMPI/DPI use 6 times less samples than CE.


the ICA algorithms can be regarded as an optimization method of such criteria.


We find that our statistic is able to correctly reject the null at a much earlier point than the MPLE alone.

our statistic manages to reject the null for τ ≥ 0.04, while the MPLE requires a parameter which is an order of magnitude larger, at 0.4.


The latter could be called "loss stability" (in terms of "loss sensitivity coefficients") for the sake of informative names.

writing z 1:n ⇡ z 0 1:n when these n-tuples differ at one entry (at most), an equivalent formulation to the above is n = sup z1:n⇡z 0 if two samples S n and S 0 n differ only on one example, then kA(S n ) A(S 0 n )k H  n . Thus our definition implies stability with respect to replacing one example with an independent copy.


Empirical evaluations on the MNIST digit recognition and TIMIT speech recognition tasks illustrate that SafetyNets enables practical, low-cost verifiable outsourcing of deep neural network execution without compromising classification accuracy.

the client's execution time is 8×-80× lower than executing the network locally, the server's overhead in generating proofs is less than 5%, and the client/server exchange less than 8 KBytes of data during the IP protocol.


(29) converges to Now we use the fact that r can be chosen arbitrarily.

let r = q 1/(2+δ/2) , where δ > 0 is the same quantity appearing in condition 3d.


In view of ( 4) and ( 6), a reasonable alternative would be the norm of the residual map R : R m×n → R m×n , which is defined as Intuitively, the residual map measures how much a solution X ∈ R m×n violates the optimality condition (4).

x is an optimal solution to (1) if and only if R(X) = 0.


Unlike the previous example, our analysis shows that here the reconstruction of both the prior and the characteristics of sensory noise is relatively reliable (Figure 3 and Supplementary Material), without major biases, even when we decouple the internal representation of the noise from its objective counterpart (except for underestimation of the noise lower bound s 0 , and of the internal noise σHigh , Figure 3 c).

in all cases the exponent k prior of the prior over speeds can be recovered with good accuracy.


In the reweighted Bethe setting, we also present a useful characterization of the concave region of the Bethe entropy function in terms of the geometry of the graph.

we show that if the region graph consists of only singleton vertices and pairwise edges, then the region of concavity coincides with the convex hull of incidence vectors of single-cycle forest subgraphs of the original graph.


It is not hard to see that the equivalence relation over complete parent configurations, K = ΦΦ , is equal to the componentwise (Hadamard) product of the individual equivalence relations for each parent variable.

unfortunately, the dual problem ( 4) is still expressed in terms of the indicator matrix Y over child variable values, which is still not acceptable.


At each time point t we may have a corpus of documents {y nt t } Nt nt=1 , where y nt t is a P -dimensional observation vector, and N t denotes the number of documents at time t. The model presented in Section 2.1 is readily extended to handle this situation.

at each time point t, for each document n t , the ERL representation for word count p, denoted by y nt p,t , is y nt p,t = g z nt p,t , p = 1, . . . , P, t = 1, . . . , T, n t = 1, . . . , N t , where z nt t ∈ R P and P is the vocabulary size.


The representational framework proposed here suggests that linear decoding may be used for this purpose.

we can record neural responses for a variety of stimuli and reverse-engineer the map between spikes and the relevant latent variables (or, if the assumed generative model is linear as here, the stimuli themselves).


measures optimized by traditional compression algorithms, such as peak signal-to-noise ratio (PSNR), they avoid artifacts such as blur and blocking, producing visually more pleasing results [1][2][3][4][5][6][7][8][9][10].

visual quality can be improved by incorporating generative adversarial networks (GANs) [14] into the learning process [4,15].


When γ is defined over the interval [0, 1], the Riemannian exponential map at p is defined as Exp p (v) = γ(1).

the exponential map takes a position and velocity as input and returns the point at time 1 along the geodesic with these initial conditions.


(ii) Characterization of the optimal tuning curves (encoders) for sensory cells in a more general setting than hitherto considered.

we study the optimal shift of tuning curve centers, providing an explanation for observed experimental phenomena [18].


Our result is an easily computable distribution p which comes within a factor of 3 of " 1 p and, as a result, within a factor of 9 in terms of sample complexity (in practice we expect this to be even smaller, as the factor of 3 comes from consolidating bounds for a number of different worst-case matrices).

to put this in perspective note that the definition of p does not place any restriction either on the access model for A while computing p , or on the amount of time needed to compute p . we are competing against an oracle which in order to determine p has all of A in its purview at once and can spend an unbounded amount of computation to determine it.


2 Given a parametric model M, a maximum likelihood estimator (MLE) is a function f MLE : S → Θ such that for any data P ∈ S, f MLE (P ) is a parameter that maximizes the likelihood of the data.


To simplify the notation we write k T x f for f, k x H , and f T Lg for f, Lg , where f, g ∈ H and L : H → H, even if H is infinite-dimensional.

a Gaussian process GP(m, k) has a dual representation in an RKHS H [12]: there exists µ ∈ H and a positive semi-definite linear operator Σ : H → H such that for any x, x ∈ X , ∃φ x , φ x ∈ H, the mean function has a realization µ in H, which is defined by the reproducing kernel k(x, x ) = ρ 2 φ T x φ x ; the covariance function can be equivalently represented by a linear operator Σ.


We can see that the mass of the posterior distribution becomes concentrated around the sample mean nc i=1 xi nc as β → ∞.

after we determine the assignments of data points to clusters, we update the means as the sample mean of the data points in each cluster.


Although we have not yet developed a complete theory for the general procedure, we are able to obtain a learning bound for the capped-L 1 regularization.

if the target function is sparse, then the performance of the solution after merely twostages of our procedure is superior to that of Lasso.


In this section, we propose a practical private subspace clustering algorithm based on the exponential mechanism [18].

given the dataset X with n data points, we propose to samples parameters where ε > 0 is the privacy parameter.


These experiments also illustrate the benefits of the proposed model for inducing dependencies in the feature selection process.

the dependencies obtained are suitable for the multi-task learning problems considered.


For instance, if G = {g 1 } then it may be impossible to have a non-trivial bound.

if the family of state predictors is not rich enough, then, it may not be possible to handle the non-stationarity of the data.


For the next rounds of training, all the models are trained only on the 9K images that comprise our full evaluation+feedback dataset from Table 2.


(  for the mean and the covariance functions.

let M α and M β be the numbers of basis functions used to model the mean and the covariance functions, respectively.


First, the resulting draws of random distributions will place their mass on only a few terms.


The second algorithm focuses on statistical efficiency.

we use the mistake bounds for traditional perceptron training methods and ask if quantum computation lends any advantages.


. . , X n2 , and by Xc n1:n2 the average of the other random variables.

we sometimes use the notation i / ∈{n1,...,n2} for the second sum when n is implied from the context.


In this case, when F G contains all Borel measurable distributions on X , minimax bounds have been established under very general conditions (essentially, when the sample space X is an arbitrary totally bounded metric space) in terms of covering numbers of X [50,39,23].

in the particular case that X is a bounded subset of R d of full dimension (i.e., having non-empty interior, comparable to the case X = [0, 1] d that we study here), these results imply a minimax rate of Sobolev IPMs: The closest work to the present is [26], which we believe was the first work to analyze how convergence rates jointly depend on (Sobolev) smoothness restrictions on both F D and F G . for Sobolev spaces F D = W s,p and F G = W t,q with p, q ≥ 2 (compare our Example 4), they showed Our main results in Sections 4 and 5 improve on this in two main ways.


The solution ŵ can therefore be interpreted as the superposition of a vector which has small `2 norm, and a vector which has small k-support norm, with the parameter a regulating these two components.

as a tends to zero, in order to prevent the objective from blowing up, û must also tend to zero and we recover k-support norm regularization.


For example, a sub-tree 'a-b-c-d', where 'a', 'b', 'c' and 'd' represent symbols and '-' represents an edge between symbols, covers sub-trees 'd', 'a-c-d' and 'b-d'.

c u (t) = λ if u matches t allowing the node skip, 0 otherwise.


This concept was introduced by InfoGAN [14], where latent codes are utilized to discover the salient semantic features of the data distribution and guide the generating process.

the regularization seeks to maximize the mutual information between latent codes and trajectories, denoted as I(c; ⌧ ),which is hard to maximize directly as it requires access to the posterior P (c|⌧ ).


So in our experiments, we take the results of SVD as the ground truth and see how close our algorithms get to it.

the goal is to determine how large the parameter that determines the size of the sketch needs to be to get close to the exact scores.


Recall that FR i 1 (η ) denotes the random variable that is the regret of playing FPL(η ) in a step phase on block i with respect to the first expert.

this will be negative if expert 2 is the best expert on block i, even though globally expert 1 is better.


There is also an alternative, more direct motivation for diffeomorphic mappings in the context of dimensionality reduction, however.

a diffeomorphic mapping has the property that it does not lose any information.


In this paper we propose a method for automatically recovering a detailed parametric model of non-rigid body shape and pose from monocular imagery.

we represent the body using a parameterized triangulated mesh model that is learned from a database of human range scans.


Crucially, we show that learning an optimal Bayesian network structure with vertex cover number at most k can be done in polynomial time for any fixed k. Moreover, vertex cover number provides an upper bound for tree-width, implying that inference is also tractable; thus, we identify a rare example of a class of Bayesian networks where both learning and inference are tractable.

our main theoretical result shows that an optimal Bayesian network structure with vertex cover number at most k can be found in time 4 k n 2k+O(1) (Theorem 5).


In contrast to model based approaches to OPE [5], importance sampling provides an unbiased estimate of the performance of the evaluation policy.

importance sampling is robust to partial observability, which is often prevalent in real-world domains.


The methods above are typically applied to imaging data offline, after the experiment is complete; however, there is a need for accurate and fast real-time processing to enable closed-loop experiments, a powerful strategy for causal investigation of neural circuitry [10].

observing and feeding back the effects of circuit interventions on physiologically relevant timescales will be valuable for directly testing whether inferred models of dynamics, connectivity, and causation are accurate in vivo, and recent experimental advances [11,12] are now enabling work in this direction.


Interestingly, this migration to object proposals has simply pushed the problem (of searching for a set of bounding boxes) upstream.

a number of object proposal techniques [8,32,41] involve the same enumeration + NMS approach -except they typically use cheaper features to be a fast proposal generation step.


Informally, the doctrine codifies the notion that not only should protected attributes, such as race and gender, not be explicitly used in making decisions, but even after the decisions are made they should not be disproportionately different for applicants in different protected classes.

if an unprotected feature, for example, height, is closely correlated with a protected feature, such as gender, then decisions made based on height may still be unfair, as they can be used to effectively discriminate based on gender.


This is accomplished by employing a grid search over λ and a combined grid/iterative search over σ 2 .

for each fixed grid value of λ we seek a minimizer over σ 2 by evaluating the validation risk at a coarse grid of σ 2 values and then performing a Golden search over the interval defined by the two σ 2 values on either side of the coarse grid minimum.


The decision making of the ant agent thus relies on the competition between its active and passive neurons.

the number of spikes needed for an ant to go foraging or self-grooming depends on the temporal dynamics of the system; it varies from one observation episode to another.


The second set of experiments consists of targetted attacks.

altering the input image to obtain an arbitrary target segmentation map as the network response.


The empirical estimation of R(f ) (also p f (0)) can be obtained by kernel density estimation, which is defined as: Hence, the approximation of f * can be found by learning algorithms associated with R σ z (f ).

in theory, for any f : there holds R(f In this paper, we assume that f * (x) = mode(Y |X = x) = w T * x for some w * ∈ R p . Following the ideas of ridge regression and Lasso [22], we consider the robust linear estimator for learning the conditional mode function.


We start with a low-dimensional mapping of the original data that is guaranteed to preserve the value of a shift-invariant kernel (specifically, the random Fourier features of Rahimi and Recht [8]), and convert this mapping to a binary one with similar guarantees.

we show that the normalized Hamming distance (i.e., Ham-ming distance divided by the number of bits in the code) between any two embedded points sharply concentrates around a well-defined continuous function of the kernel value.


The goal of this paper is to expand the class of models amenable to fast exact inference by developing and exploiting a novel representation for factors with properties similar to the three above.

we investigate models with latent count variables, and we develop techniques to represent and manipulate factors using probability generating functions.


Any f ∈ L 2 (Ω) which can be -approximated with polynomial in 1/ number of steps of gradient descent is infinitely differentiable.

f function must belong to the intersection of all Sobolev spaces on Ω. Gradient descent for periodic functions on R. Let us now consider a simple but important special case, where the reach can be analyzed very explicitly.


To model the responses of thousands of neurons in a free viewing experiment, we take an alternative approach and directly estimate a receptive field shift for all neurons from the tracked pupil position solely based on optimizing the predictive performance of the network.

we feed the pupil location p ∈ R 2 into a network that predicts a shift ∆x = (∆x, ∆y) ∈ R 2 for each time point which is added onto all locations x i = (x i , y i ) of the spatial transformer readout.


In this paper, we study the setting where each individual contributes a sensitive binary value, and we wish to infer the population proportion via hypothesis tests, subject to DP.

we derive uniformly most powerful (UMP) tests for simple and one-sided hypotheses, optimizing finite sample performance.


We justify this idea of noise adding by identifying a certain class of discriminators that have good generalization properties.

we prove that projection discriminator, introduced in [32], has a good generalization property.


It is not hard to see that the set {� pair σ : σ ∈ T } spans a r(r−1) 2 dimensional space, and hence the nullity of the above matrix is at most r(r−1)− � r(r−1) Thus by Theorem 13, we get that CCdim(� pair ) ≥ r(r − 1) − � r(r−1) 2 − 2 .

for r ≥ 5, this gives CCdim(� pair ) > r, and therefore establishes that no convex surrogate ψ operating on a surrogate target space � T ⊆ R r can be classification calibrated with respect to � pair on the full probability simplex Δ r(r−1) .


Our main result is an impossibility, showing that even for learnable submodular functions it is impossible to find a non-trivial approximation to the minimizer with polynomially-many samples, drawn from any distribution.

this implies that minimizing a general submodular function learned from data cannot yield desirable guarantees.


However, a significant amount of research has been carried out on sparse PCA recently, and we thus propose to leverage some of the recently proposed relaxations and heuristics to solve this rank 1 sparse PCA problem (see [8,20] and references therein).

the Truncated Power iteration (TPI) algorithm of [20] can easily be modified to compute Ω * k,q which corresponds to a generalization of the rank 1 sparse PCA in which in general a = b and k = q.


In this way, we develop a set of grid points that is typically a very small subset of the entire possible grid, which is adapted to the parts of the belief space typically inhabited by the agent.

given a grid resolution r and a belief state b we can compute the coordinates (grid points g i ) of the belief simplex that contains b using an efficient method called Freudenthal triangulation [2].


As α, F p , F n and b are all constants for the purpose of this analysis, we may optimize for b 1 in the equivalent expression The derivative with respect to b 1 is This equals 0 when This further yields that the false positive rate is minimized when b 2 = b * 2 , where This result may be somewhat surprising, as here we see that the optimal value b * 2 is a constant, independent of b.

the number of bits used for the backup filter is not a constant fraction of the total budgeted number of bits bm, but a fixed number of bits; if the number of budgeted bits increases, one should simply increase the size of the initial Bloom filter as long as the backup filter is appropriately sized.


Although the generated samples obey the distribution of the original samples, the distance between the generated samples and the original samples may also be large.

the degree of similarity between x and G(z) is not large enough.


A fragmentation-coagulation process (FCP) is a continuous-time Markov process π ≡ (π(t), t ∈ [0, T ]) over a time interval [0, T ] where each π(t) is a random partition in Π S . Since the space of partitions for a finite S is finite, the FCP is a Markov jump process (MJP) [18] : it evolves according to a discrete series of random events (or jumps) at which it changes state and at all other times the state remains unchanged.

the jump events in an FCP are either fragmentations or coagulations.


Comparatively, Gaussian RBF works well for a wider range of k true 's, but only if the hyperparameter σ is selected carefully.

rBF-Median (black dashed line) works generally well, despite being slightly conservative (i.e.


We previously explored an alternative approach for inference, where we directly used the confidences in ŷq as weights in an importance-weighted SVM.

for each query, we trained a model with all M data points, but modulated their influence according to the soft indicator vector ŷq , such that less confident points incurred lower slack penalties.


The unnormalized relative entropy projection step can be performed efficiently by using a procedure similar to the Sinkhorn balancing used for the Birkhoff polytope in [12].

given a nonnegative matrix X ∈ R m×n + , one alternately scales the rows and columns to match the desired row and column sums until some convergence criterion is met.


In the second iteration, t = 2, we now lock x(3) = 1.

we calculate the MMs with the added constraint that x(3) to 1.


Instead, we approximate the s-Delaunay graph as was previously proposed for Euclidean distance case in [16,17,15].

we adopt the approximation proposed in [15] by simply extending Algorithm 2 to inner product similarity function s(x, y) = x, y .


The proportion of the mass that a parent j at level k − 1 transmits to a given child a depends on the weight w ja between the parent and the child (normalized by the sum of the weights of the edges from j to all its children), and the probability mass P k−1 SS (j) that is assigned to node j.

the portion a child event a is able to obtain from the total discounted mass depends on how authoritative its parents are, and how strong the relationship between the child and its parents.


On the last stage, we obtain the generated high-resolution image after passing the last hidden state h m−1 through a convolutional layer with 3 × 3 kernel size.

the drawing bot, denoted as Î = C(S, z), is decomposed as h 0 = F 0 (z, F ca (e)); Here, z is a noise vector usually sampled from a standard normal distribution.


As a result, the nodes in the RNN are more capable of "skipping" across the network, allowing unimpeded information flow across multiple time steps, thus alleviating the problem of learning long term dependencies.

such effect is more prominent in the long run, due to the network's recurrent structure.


In fact, human behavior in 4 l moons U h R is similar to 4 l moons U .

having random highlighting is similar to having no highlighting in how it affects human categorization.


But it is easy to fix this assumption by adding edges between the latent variables as shown in figure 2b.

we use a second order approximation to the motion of the camera: Truth factor analysis Jacobs Structure:  Note that we do not assume that the 2D trajectory of each point is smooth.


Based on the correctness of this procedure, we describe a dynamic program to compute the choice probabilities of a general offer set S. The key idea is to decompose these probabilities to include the position at which a product is chosen.

for i  k and s 2 [k], let ⇡(i, s, k) be the probability that product a i is chosen (i.e., first among products in S) at position s after the k-th step of the RIM.


However, in addition to tp-rate maximization, some performance guarantees in terms of the fprate are usually required.

it is reasonable to require (following the Neyman-Pearson approach) that, in the long term, the average fp-rate of the agent will be below some given threshold 0 < γ < 1.


Note that although the curves for small d are different from those of ORF, when d is large (d > 32 in practice), the kernel estimation is almost unbiased, and the variance ratio converges to that of ORF.

it is clear that SORF can provide almost identical kernel approximation quality as that of ORF.


Similar to Lemma 1, by making additional assumptions concerning the prediction method used we can derive a slightly simpler bound.

for a labeled set L that is an α cover, we assume unlabeled nodes are labeled with the weighted majority vote of neighbors in L. In other words, set ŷi = y i for i ∈ L, and set ŷi = y for i / ∈ L with y such that j∈L:y j =y W i,j ≥ j∈L:y j =y W i,j .


Similar to deep network approaches [11,14], recent works tried to avoid feature engineering and employed sparse coding of image patches to learn features from "scratch", for texture analysis [15] and object recognition [30,3].

orthogonal Matching Pursuit [18] is a greedy algorithm that incrementally finds sparse codes, and K-SVD is also efficient and popular for dictionary learning.


In contrast, in this paper we propose a novel generative model that is able to reason jointly about the 3D scene layout as well as the 3D location and orientation of objects in the scene.

given a video sequence of short duration acquired with a single camera mounted on a moving car, we estimate the scene topology and geometry, as well as the traffic activities and 3D objects present in the scene (see Fig.


According to this analysis, by setting T = Θ(1/λ), then as long as the total number of batches is at least Ω(log(1/ )), and λ = Ω(1/m), then the algorithm will attain an -suboptimal solution in expectation.

without any additional communication, we extend the applicability of distributed SVRG (at least for regularized least squares) from the λ = Ω(1/ √ m) regime to 1/λ = Ω(1/m).


This model constitutes a relatively direct extension of existing techniques for real matrices [15,12].

we have introduced a probit link function and a simple auto-regression construction to impose statistical correlation in the traits of a person/entity at consecutive times.


Showing the guarantees for this algorithm on non-monotone functions requires multiple subtle components.

we require that at every iteration, any element is added to the solution with low probability.


An important point to note is that the complexity of these algorithms depends simply on the size k and the number of nodes n. There is no loss in generality in assuming that the size-k FVS F is fully connected and each of the feedback nodes has edges to every non-feedback node.

after re-ordering the nodes so that the elements of F are the first k nodes (T = V \F is the set of nonfeedback nodes of size n − k), we have that J = J F J T M J M J T 0, where J T 0 corresponds to a tree-structured subgraph among the non-feedback nodes, J F 0 corresponds to a complete graph among the feedback nodes, and all entries of J M may be non-zero as long as ).


Note that different base measure functions f lead to specific distributions of Y , for example, Bernoulli, Gaussian, binomial, and other common distributions.

the parameterization in (2.1) generalizes several existing models of ranking, crowdsourcing, and ordinal embedding.


Although the improvement is on the non-dominating term, multi-stage ORDA is an optimal algorithm for both stochastic and deterministic optimization.

for deterministic strongly convex and smooth f (x) (M = 0), one can use the same algorithm but only replaces the stochastic subgradient G(x, ξ) by the deterministic gradient ∇f (x).


Generally, FOMs use the DGF to choose steps: given a gradient and a scalar stepsize, a FOM moves in the negative gradient direction by finding the point that minimizes the sum of the gradient and of the DGF evaluated at the new point.

the next step can be found by solving a regularized optimization problem, where long gradient steps are discouraged by the DGF.


We refer to an assignment of values to all the random variables as a labeling.

a labeling is a vector x ∈ L n , which specifies the label x a assigned to each random variable X a .


Our result shows that, with a suitable choice of the regularization parameter λ, the rescaled estimation error is of the order r max{d 1 , d 2 }/n + √ rd 1 d 2 /n up to a logarithmic factor.

for n min{d 1 , d 2 }, the leading term of the error is r max{d 1 , d 2 }/n.


Inductively suppose we have solved the first tasks through programs stored below address 7 ¦ A ¢ , and that the most recently found program starting at address 7 ¡ £¢ ¥¤ i a7 9 ¦ A ¢ actually solves all of them, possibly using information conveyed by earlier programs.

to find a program solving the first 8 ¦ tasks, OOPS invokes Try as follows (using set notation for ring W ): we spend roughly equal time on two simultaneous searches.


In this paper, we consider an interesting and not uncommon scenario of multitask learning, where the tasks are heterogeneous and bear a union support.

each task can be either a regression or classification problem, with the inputs lying in a very high-dimensional feature space, but only a small number of the input variables (i.e., predictors) are relevant to each of the output variables (i.e., responses).


These results indicate that our approach generalizes well across problem types and datasets.

it generalizes from synthetic graphs to real ones, from SAT graphs to real-world social networks, and from graphs with roughly 1,000 to graphs with roughly 100,000 nodes and more than 10 million edges.