It may be impractical to compute the minimum vertex cover size k * , but Lemma 1(b) may still be used if we can bound k * from above and below.

given lower and upper bounds l and u on k * , |U (k)| ≤ (k − l + 2)u.


Because, by assumption ¥ is head-velocity independent, it follows that % is inversely proportional to head-velocity (assuming perfect integration), % ¥ 1 .

the anticipation time tends to be smaller for fast head rotations and larger for slow head rotations.


As a first step, we generalize the Kesler's Construction originally defined for single-label classification (see [6]) to the PLM setting, thus showing that the embedding problem can be formulated as a binary classification problem in a higher dimensional space when new variables are appropriately defined.

consider the vector y(a) = (M ωs(a) − M ωe(a) ) ∈ R s defined for every preference arc in a given preference constraint, that is a = (ω s , ω e ) ∈ A(g).


Recurrent neural networks (RNNs) live at the heart of many sequence modeling problems.

the incorporation of gated additive recurrent connections is extremely powerful, leading to the pervasive adoption of models such as Gated Recurrent Units (GRU)  or Long Short-Term Memory (LSTM) [Hochreiter and Schmidhuber, 1997] across many NLP applications Rocktäschel et al., 2015;McCann et al., 2017].


Since designing such metrics by hand is difficult, we explore the problem of learning a metric from examples.

we consider relative and qualitative examples of the form "A is closer to B than A is to C".


Applicability of these results to biologically plausible neuronal models is questionable.

models of spiking neurons are known to have modes of synchronous global oscillations.


However, in this paper we will focus on a useful property of this type of factorization that seems to have passed unnoticed thus far.

we call it the "stochastic-factorization trick": Given a stochastic factorization of a square matrix, P = DK, swapping the factors of the factorization yields another transition matrix P = KD, potentially much smaller than the original, which retains the basic topology and properties of P. The stochasticity of P follows immediately from the same property of D and K. What is perhaps more surprising is the fact that this matrix shares some fundamental characteristics with the original matrix P. it is possible to show that: (i) for each recurrent class in P there is a corresponding class in P with the same period and, given some simple assumptions about the factorization, (ii) P is irreducible if and only if P is irreducible and (iii) P is regular if and only if P is regular (see [9] for details).


The most studied sampling in the literature on stochastic optimization is the serial sampling, which corresponds to the selection of a single block i ∈ [n].


Each of these risks has an empirical counterpart where the expectation with respect to P is replaced by an average over an i.

the following empirical version of R φ,σ will be used The main focus of this paper is the analysis of learning algorithms that minimize the empirical φ-risk regularized by the RKHS norm R φ,σ , and their limit as the number of points tends to infinity and the kernel width σ decreases to 0 at a suitable rate when n tends to ∞, λ being kept fixed.


. . , A N ⊂ V, the random measure H ∼ DP(G, α) has the property This implies for the present construction that renormalized restrictions of the random measure are also Dirichlet process distributed.

if A ⊂ V and H A is the marginal renormalized random measure on A induced by the DP on V with concentration parameter α and base measure G, then H A ∼ DP(G A /G(A), αG(A)), where G A is the base measure restricted to A.


It is worth noting, though, that some steps in our algorithm are computationally inefficient.

our membership oracle needs to determine whether a given coverage probability vector is implementable.


Our proposed algorithm integrates the proximal newton algorithm with multi-stage convex relaxation based on the difference of convex (DC) programming, and enjoys both strong computational and statistical guarantees.

by leveraging a sophisticated characterization of sparse modeling structures (i.e., local restricted strong convexity and Hessian smoothness), we prove that within each stage of convex relaxation, our proposed algorithm achieves (local) quadratic convergence, and eventually obtains a sparse approximate local optimum with optimal statistical properties after only a few convex relaxations.


These papers all deal with the issue of optimizing the data representation for a given query and how an inverted index should be stored and managed for general queries.

[3,14] address the problem of computing the top-k results without scanning over the entire inverted lists.


Second, the bound is polynomial in all constants of interest, such as the number of items L and the number of maximization steps K in each episode.

we would like to stress that it is not linear in the number of contexts Y K at step K, which is exponential in K. Finally, note that our bound captures the shape of the optimized function f . because the function f is adaptive submodular, the upper bound on the gain of the policy π g from step k forward, G k , decreases as k increases.


However, it can be calculated in a time that is no more than a forward-backward step on a single data point.

in a mini-batch setting with batch size B, if the backpropagation on the mini-batch can be done in time BT , the running time of the Path-SGD on the mini-batch will be roughly (B + 1)T -a very moderate runtime increase with typical mini-batch sizes of hundreds or thousands of points.


We propose two computationally efficient algorithms that match the query complexity lower bound within log n factor and are completely parameter free.

our iterative-update method to design Monte-Carlo algorithm provides a general recipe to develop any parameter-free algorithm, which are of extreme practical importance.


They discovered that regularized estimation based on the Schatten 1-norm, which is a popular technique for recovering low-rank matrices via convex optimization, can also be applied to tensor decomposition.

the  (7,8,9) tensor of dimensions 50 × 50 × 20 from partial measurements; see [23] for the details.


We introduce a new local flatness condition for the objective function (Assumption 1), under which Runge-Kutta discretization obtains convergence rates even faster than O(N −2 ), without requiring high-order integrators.

we show that if the objective is locally flat around a minimum, by using only gradient information we can obtain a convergence rate of O(N −p ), where p quantifies the degree of local flatness.


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

we derive a convergence rate of the form n −θb , where 0 < θ ≤ 1 is a constant characterizing the (relative) smoothness of the integrand.


We draw y ab repeatedly by choosing a community assignment (z a→b , z a←b ) for a pair of nodes (a, b), and drawing a binary value from a community distribution.

the conditional probability of a link in MMSB is p(y ab = 1|z a→b,i , z a←b,j , β) = K i=1 K j=1 z a→b,i z a←b,j β ij , where β is the blockmodel matrix of community strength parameters to be estimated.


Assume now that we have some partial spectral information available.

fix k ∈ {0, 1, . . . , n − 1} and assume we know eigenvectors u i and eigenvalues λ i for i = 1, . . . , k. We now define a parametric distribution D(α, β 1 , .


Each of the LRMA sampling methods has strengths we can draw on and weaknesses we can improve upon.

our cosine tree sampling method can be viewed as combining the representativeness of RP sampling with the adaptivity of RLS, which explains its empirically dominant rank efficiency.


Our method has more difficulty matching RL in these challenging environments, but nevertheless it displays substantial learning on most of them and matches or even exceeds RL on some.

figure 2: Results on Atari games as measured on the tasks' true reward.


This paper proposes an end-to-end approach for learning machine learning models that will be used in the loop of a larger process.

we consider training probabilistic models in the context of stochastic programming to directly capture a task-based objective.


These divergences are always nonnegative and only zero when two distributions are equal: p G = p data .

the generator induces a distribution p G that is identical to the data distribution p data , and two discriminators now fail to recognize the real or fake samples since they return the same score of 1 for both samples.


We develop a "walk-sum" formulation for computation of means, variances and correlations that holds in a wide class of Gauss-Markov models which we call walk-summable.

this leads to a new interpretation of BP in trees and of LBP in general.


When the sparsity ratio k/n and SNR are both fixed, m can satisfy (10) while growing only linearly with k. In contrast, Theorem 1 shows that with sparsity ratio and SNR • MAR fixed, m = Ω(k log(n−k)) is necessary for reliable sparsity recovery.

the number of measurements must grow superlinearly in k in the linear sparsity regime with bounded SNR.


Optimization scheme based on Rényi divergences of different orders have very similar structure.

gradient could be evaluated in a similar way: where s|x, spike = ds sδ(x−s•v)P (s|spike)/P (x|spike), and similarly for s|x .


The purpose is to obtain a lower-bound on the performance assuming an Oracle decoder that can perfectly identify the workers' skill levels (here a spammer or a hammer) and perform an optimal decoding.

we consider the following: where minimization is with respect to the choice of a decoder for a given kIC code.


The algorithms have multiplicative updates and regret bounds that depend logarithmically on the dimension of the input space.

they proposed a version of EG where the weights are not normalized, called EGU.


Related research has developed heuristics which reduce the belief space representation.

entropy-based representations for heuristic control [8] and full value-function planning [9] have been tried with some success.


For an image volume V = V (x, y, z, c), we propose to calculate the topographical map T (disaffinity map) as where x, y, z denote the spatial coordinates, c denotes the color coordinate, and G x , G y , G z denote the spatial gradients of V (nearest neighbor differencing).

any edge with significant deviation in any color channel will correspond to a "mountain" in the topographic map.


One instantiation of Mallows model is based on Kendall's distance to measure the number of pairwise permutations between the group order and the individual order.

the probability of any observed order 𝒚, given the group order 𝝎 is: where 𝑑 is the Kendall's distance.


The fundamental question arising in our setting is: What are optimal strategies for the learner?

is the popular low-density rejection strategy optimal?


Here we show how to construct non-independent joint distributions over firing rates using copulas.

this approach can be used to combine arbitrary marginal firing rate distributions.


Graphical models and message-passing algorithms defined on graphs are a growing field of research.

the belief propagation (BP, or sum-product) algorithm has become a popular means of solving inference problems exactly or approximately.


First, we show that well-known concepts from statistical learning theory can be directly applied to reason about learning from data an approximately revenue-maximizing auction.

for a set C of auctions and an arbitrary unknown distribution F over valuations in [1, H] samples from F are enough to learn (up to a 1 ✏ factor) the best auction in C, where d C denotes the pseudo-dimension of the set C (defined in Section 2).


We then turn our attention to identifying the roots of BatchNorm's success.

we demonstrate that BatchNorm impacts network training in a fundamental way: it makes the landscape of the corresponding optimization problem significantly more smooth.


Our experiments focus on the setting of rate 1/3 and information block length of 50 for concreteness 1 .

the encoder maps K = 50 message bits to a codeword of length n = 150.


Moreover, it turns out that if we apply the tapering schedule induced by the construction of our estimator to the population parameter Ω ∈ F α , we may express the tapered population Ω as a sum of block matrices in exactly the same way that our estimator is expressed as a sum of block matrices.

the tapering schedule is presented next.


To address this point, we consider the two variants of Algorithm 1 from Section 3.2 and 3.3 with linear approximators.

we consider the family of linearly parameterized value functions Q w (s, a) = w T φ(s, a) with bounded weights w 2 ≤ w max and uniformly bounded features φ(s, a) 2 ≤ φ max .


In terms of collaborative filtering/matrix completion, there is another line of established research that focuses on robust matrix completion, in which a small portion of elements or rows in the underlying low-rank matrix is assumed to be arbitrarily perturbed [18,19,20,21].

the stability of alternating minimization solutions was analyzed with respect to malicious data manipulations in [22].


A "Biased" scale-free model that combines the preferred attachment model [7] with a mixed-membership model.

we generated M = 60, 000 1-edges as follows: (a) pick a vertex i with probability proportional to its degree; (b) randomly pick a destination community k from θ i ; (c) find the set V k of all vertices v such that θ vk is the largest element of θ v (i.e.


That expected generalization error of W can be much smaller than the right side of (2) (see also [14]).

such bounds are not taking into account the input-output relation of the algorithm, and uniform bounding seems to be too stringent for these applications.


A key idea of the proposed method is that because the outcome of an intervention depends on both the inherent effects of the intervention and the participants' expectations, the effectiveness should be inferred in a like-for-like fashion.

the response observed in, say, the sub-group of participants assigned to the control group whose feedback professes belief in the control group assignment should be compared with the response of only the sub-group of the treatment group who equally professed belief in the control group assignment.


In this paper, we consider consolidation using a model whose complexity brings to the fore consideration of computational issues that are invisible to simpler proposals.

it treats cortex as a hierarchical structure, with hierarchical codes for input patterns acquired through a process of unsupervised learning.


Our framework also indicates that in some cases it might be beneficial to use non-consistent surrogates.

a non-consistent surrogate might allow optimization only up to specific accuracy, but exponentially faster than a consistent one.


It was suggested that a new notion of connectivity called propagation connectivity is a sufficient condition for the solution of random MAX-3LIN problem with a planted solution to be unique [26,Proposition 2].

it is claimed that if the hypergraph corresponding to an instance of MAX-3LIN is propagation connected, then the optimal solution for MAX-3LIN is unique and there is an efficient algorithm that finds it.


[7] show by duality that this is equivalent to the distance proposed in (11).

generalizing distribution matching to apply to transforms other than the logistic leads us directly to our new transduction criterion.


. . , F n (y)) behaves like where h 1 , h 2 : [0, 1] → R + are continuous decreasing functions, and we would like to map D-dimensional real vectors into n-bit binary strings in a locality-sensitive manner, where the notion of locality is induced by the kernel K. We will achieve this goal by drawing F n appropriately at random.


While LRMC+SSC, SSC-CEC and SSC-Lifting perform similarly for low-rank matrices, SSC-Lifting performs best among all methods for high-rank matrices.

when the percentage of missing entries, ρ, is more than 70%, SSC-Lifting performs significantly better than other algorithms.


In general, the Bayesian model is quite well supported by the data.

the data reflect the inverse relationship between relative matching speeds and discrimination thresholds predicted by the slow-speed prior of the model.


Since f (•) = h(A(•)) by ( 2) and h(•) is strictly convex on dom(h) by Assumption 2.1(b), it is easy to verify that the map X → A(X) is invariant over the optimal solution set X .

there exists a z ∈ dom(h) such that for any X * ∈ X , we have A(X * ) = z.


A permutation π is a bijective function π : {1, . . . , n} → {1, . . . , n} associating with each item i ∈ {1, . . . , n} a rank π(i) ∈ {1, . . . , n}.

π(i) denotes the rank given to item i and π −1 (i) denotes the item assigned to rank i.


Methods of the second family are based on a maximum likelihood (ML) estimator with an 1 penalty (Yuan & Lin, 2007;Banerjee et al., 2008;Friedman et al., 2008).

if S denotes the empirical covariance matrix, the solution is the matrix Θ which solves the optimization problem An example of an extension to both models (the regression and ML approaches), and the first to explicitly consider additional classical network properties, is the work by Liu & Ihler (2011), which modifies the 1 penalty to derive a non-convex optimization problem that favors scale-free networks.


( 3) is essentially the maximization of the similarity between the stochastic part of V , S V with the stochastic matrix formed as the product of the K stochastic matrices X 1 , • • • , X K , weighted by D V . Before describing the algorithm solving (3), we first derive the solution to another related problem that we term as the stochastic matrix sandwich (SMS) problem, from which we can construct a solution to (3).

in an SMS problem one minimizes the following objective function with regards to an m × n stochastic matrix X, as where A and B are two known stochastic matrices of dimensions m × m and n × n, respectively, and C is an m × n nonnegative matrix.


We tested our approach against Clique Averaging algorithm (CAVERAGE), since it was the best performing approach in [2] on the same type of experiments.

cAVERAGE outperformed Clique Expansion [10] combined with Normalized cuts, Gibson's Algorithm under sum and product model [9], kHMeTiS [11] and Cascading RANSAC [2].


( 8) via a conditional gradient algorithm [4].

in Algorithm 1 we implement the block-coordinate Frank-Wolfe algorithm proposed by Lacoste-Julien et al.


Our goal is to allow practitioners to specify a mis-clustering rate η * , and by inverting this relationship, to determine the right magnitude of the perturbation σ * allowed.

our work can provide a practical method to determine the tradeoff between data perturbation and the loss of clustering accuracy due to the use of X instead of X.


The proof is based on a potential function argument.

we will choose the confidence function g(x) so that is a potential function, which serves as a repository for guarding our loss (we will chose g(x) to be an odd function, and hence will always have Φ t ≥ 0).


In all cases, the adversary can cause at most γ 0 regret and it is necessary for the adversary to play x T 1 ∈ ABC(Σ, γ 0 ), which implies that (MMS) is optimal.

for x T 1 ∈ X = ABC(Σ, γ 0 ) and y T 1 ∈ Y = L(B t ), we have sup for all T > 0, which implies the result.


Theorem 3.5 tells us that if the set S j has high conductance, then there will be a set S k much larger than S j where the absorption probability a(k) remains large.

a(k) will not drop much if S j is closely connected with the rest of graph.


The key idea is to introduce weights related to β 1 (s) to the loss functions so as to make them tighter bounds of (1−NDCG).

we introduce weights to the pairwise and listwise losses in the following way, It can be proved (see Proposition 1 in [4]) that the above weighted losses are still upper bounds of (1−NDCG) and they are lower bounds of the original pairwise and listwise losses.


We are interested in a theoretical guarantee for these policies: the near-optimality guarantee.

a policy is near-optimal if its worst-case utility is within a constant factor of the optimal worst-case utility.


Roughly, B t (a) represents the greatest mean reward value that is statistically plausible.

b t (a) is typically constructed so that B t (a) → E y∼p * a [R(y)] as data about action a accumulates, but with high probability Like Thompson sampling, many UCB algorithms only select actions that are optimal under some p ∈ P. Consider an algorithm that constructs at each time t a confidence set P t ⊂ P containing the set of distributions that are statistically plausible given observed data.


(2) Only if n → ∞ and m i → ∞ simultaneously does the two-layer generalization bound uniformly converge.

if the number of documents for some query is finite, there will always exist document-layer error no matter how many queries have been used for training; if the number of queries is finite, then there will always exist query-layer error, no matter how many documents per query have been used for training.


Recall that for high dispersal we want every feature to be equally active.

we want the mean squared activation of each feature to be roughly equal.


, and use the peaks for that pair to initialize our source activity array ↵.

we compute the likelihood of each phase observation under a set of von Mises distributions centered at each peak (along with a uniform distribution for the garbage source), then smooth these likelihoods over time and frequency using a Gaussian blur, and initialize ↵ from the logarithm of these smoothed likelihoods.


A majority of the prior work on the choice model is about the MLM and its variants such as the hierarchical MLM [5], the multinomial probit model [6], and, generally, random utility models [17].

the attraction effect cannot be represented by these variants of the MLM [13].


At any given point in our compilation we have some frontier F, which is the set of variables that have some but not all factors included in the current SDD.

then we select the variable in our current SDD that is, on the graph of our model, closest to the "center" induced by F. we use argmin X 2X max F 2F distance(X , F ).


r m (with rows X ⇤ of rank r is identifiable if 2r < 1 [see 20, Theorem 3.2].


In this experiment we consider the regime of clustered eigenvalues described in the introduction and summarized in Table 3.

we construct a synthetic matrix A ∈ R 30×30 with the smallest 15 eigenvalues clustered in the interval (5, 5 + ∆) and the largest 15 eigenvalues clustered in the interval (θ, θ + ∆).


Professional photographers always employ the exposure-blending with luminosity masks to perform image post-processing [20].

they create different layer masks for different objects and adjust each of objects independently.


This linesearch guarantees that the algorithm produces a sufficiently large decrease of the cost function at each iteration, but it may require several function evaluations to find l. This algorithm is guaranteed to converge to a critical point of (8) as long as the step sizes are chosen commensurate with the norm of the Hessian [8].

nesterov has recently shown that if f has a Lipschitz-continuous gradient with Lipschitz constant L, then the algorithm will converge at a rate of 1/k where k is the iteration counter [15].


The representation in (6) constitutes a proper generative construction for {✓ (c) jn } in the presence of spikes that co-occur within the refractory time, but it complicates inference.

recall that G (c) In the original construction, (4) and ( 5), in which refractory-time violations are not account for, the Gibbs update equations for {U (c) ji } are analytic, due to model conjugacy.


In the convex representation, the ξ parameters do not actually represent a probabilistic quantity, but rather arise as parameters in a variational inequality.

we write, The algorithm is then equivalent to that in §4.1.2 except that the expectation is taken of x 2 as the E step, and the diagonal weighting matrix becomes, where σ i = E (x 2 i |y; ξ i ).


Starting from the entire image, the Tree-RL approach sequentially acts on the current search window either to refine the object location prediction or discover new objects by following a learned policy.

the localization agent is trained by deep RL to learn the policy that maximizes a long-term reward for localizing all the objects, providing better global reasoning.


In fact, in such case, we can increase x i and decrease x j to increase f (x).

let and define Since r i (x) > r j (x), we can always select a proper α > 0 to increase f (x).


In the random setting, we use the term almost every (a.e.)

graph G ∼ G(p) satisfies a certain property Q (for example, connectedness) if the property Q holds asymptotically almost surely 2 (a.a.s.)


The central idea behind the TS-ReLU network is to learn a linear classifier directly from the rich, expanded representation Z 1 , rather than collapsing it back to the lower dimensional X 1 .

in a standard ReLU network, the hidden layer activity X 1 is sent through a linear classifier f X (W X X 1 ) trained to minimize some loss function L X (f X ).


Implementing language acquisition systems is one of the most difficult problems, since not only the complexity of the syntactical structure, but also the diversity in the domain of meaning make this problem complicated and intractable.

how linguistic meaning can be represented in the system is crucial, and this problem has been investigated for many years.


To address this, we give the following relaxation of the framework that allows concepts to be learned much more effectively and at the same time preserves the original motivation of the KWIK model-it's better saying "I don't know" rather than making a mistake.

we allow the algorithm to make at most k mistakes.


Counterfactuals under this assumption are written in a way that emphasizes they only depend on treatments assigned within that block.

for any a (j) , Y j (a (j) [ã j ]) ≡ Y j (ã j ).


In our setup, q is given by ( 1) where we let L go to infinity but we tie the parameters of all models after a specific level T (with T ≪ L).

we impose the condition that for all components with index i > T the variational distributions for the stick-lengths q vi (v i ) and the variational distributions for the components q ηi (η i ) are equal to their corresponding priors, i.e., q vi (v i ; In our model we define the free energy F as the limit F = lim L→∞ F L , where F L is the free energy defined by q in (1) and a truncated DP mixture at level L (justified by the almost sure convergence of an L-truncated Dirichlet process to an infinite Dirichlet process when L → ∞ [6]).


Here, we take an alternative approach using an attractive feature of the Parzen estimator -it only requires the view similarity between any two images, not the actual views.

if we have an image descriptor so that the distance between descriptors for two images approximates the similarity of views between the objects, we can calculate the Parzen density without ever computing the views.


In this section, we study the problem of learning the graph structure of an underlying GLM graphical model given iid samples.

we assume that we are given n samples X n 1 = {X (i) } n i=1 , from a GLM graphical model: We have removed node-wise terms for simplicity, noting that our analysis extends to the general case.


While results for kernel algorithms are similar, they all differ considerably from those obtained with the multilayer neural network.

we can observe that they are performing reasonably well with all types of representation while the multilayer neural network performance is highly dependent on the representation fed as input.


To our knowledge, this is one of the most general frameworks for learning a broad class of hash functions.

many previous loss-based techniques [20,24] are not capable of optimizing mappings that involve non-linear projections, e.g., by neural nets.


Recall that N ∈ N denotes the number of possible next states.

for each state x ∈ X and each action u available at x, it holds that P (x, u, x � ) = 0 for all but at most N states x � ∈ X.


Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator.

we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance.


Instead of using full connections between the adjacent rows/columns, we show that certain local connections, corresponding to a sparse row/column-wise transform matrix, can also formulate densely connected affinity.

we introduce the (a) one-way connection and the (b) three-way connection as two different ways to implement Eq.


The dataset consists of the propagation of short textual phrases, referred to as Memes, via the publication of blog posts and main-stream media news articles between March 2011 and February 2012.

the dataset contains seven groups of cascades corresponding to the propagation of Memes with certain keywords, namely "apple and jobs", "tsunami earthquake", "william kate marriage"', "occupy wall-street", "airstrikes", "egypt" and "elections."


Both take into account the last sample of the immediate reward, r t . They differ in how they account for the reward from the "next state": either, for Q(1), as q t (the last reward received from the state visited after the last visit to S t ) or, for model-based RL, as p t (the last sample of the reward at the true successor state).

while TD(1) will incorporate the reward observed following R t , regardless of the state, a model-based system will instead consider the expected successor state [21].


Arguing about the ∞-norm margin is convenient when dealing with sparse hyperplanes, i.e., with hyperplanes having only a small number of relevant features (e.g., [14]).

we say that a training set S = ((x 1 , y 1 ), ..., (x m , y m )) is linearly separable with margin γ > 0 when there exists a hyperplane w with ||w|| q ≤ 1 such that if the underlying margin is an ∞-norm margin (and α is not close to 1) then w tends to share the sparsity properties of w. See also Section 3.


First, we compare the perceptron with the graph kernel K 1 0 to the conceptually simpler k-nearest neighbors algorithm with either the graph geodesic distance or the resistance distance.

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


In this paper, we show that IHT schemes with approximate projections for the group sparsity problem yield much stronger guarantees.

our result holds for arbitrarily large , and arbitrary group structures.


In these cases, it is crucial that classifiers exhibit good robustness properties.

a sufficiently small perturbation of a datapoint should ideally not result in altering the estimated label of a classifier.


For example, (1) has the following CCD approximation However, CCD also introduces strong bias which makes it insufficient to capture the complexity of inferred posterior distributions on real data (see Figure 5).

certain clades may depend on their sisters.


We define constraints on uncertainties about a decision process based on its interactions with a reference policy.

state-action trajectories through the decision process are available that were produced from a policy that may be quite different from the optimal one.


In what follows we propose a non-parametric Bayesian framework for modeling intensity functions for event data over time.

we describe a Dirichlet process framework for learning the unknown rate functions, and learn a set of such functions corresponding to different categories.


As a result, their model can be trained by optimizing a convex objective function through a variant of Blahut-Arimoto alternating minimization algorithm, whereas our model is more complex and the objective function becomes non-convex.

training a simple chain structured CRF model [19] in our framework turns out to be intractable even if using Blahut-Arimoto's type of alternating minimization algorithm.


Interestingly, the response time pattern across image groups (Figure Supp.2a)) was inversely correlated to the perceptual bias pattern (Figure 2a)) (Pearson correlation = 1, p < 0.01; two-tailed Pearson correlation test).

subjects made quicker decisions for the hazard group, then pets group, and then vegetables group.


Then, we seek to approximate this matrix by another whose entries satisfy the triangle inequalities.

m ik ≤ m ij + m jk for every triple (i, j, k).


Assumption 1 does not hold for the hinge loss (see Appendix A for details), but it does hold for a modified hinge loss, where we apply the hinge separately to each view: Example 2 (Modified Hinge Loss).

suppose that L(θ; x, y) L is the sum of 3 hinge losses, one for each view.


By taking = ˜ q✏ 2 p 7/2 /2 in Proposition 4.1 we get a ˜ -approximation to µ(0) with computation bounded by a polynomial in p, 1/˜ .

the existence of an FPRAS for the mapping µ 7 !


Yet, roughly speaking, we notice that the value of § for which the above claims (approximately) hold is related to the "amount of uniformity" in @ &% 7! .

a crucial step in the above proof assumed that each @ % 7!


On the contrary, the penalty formulation leads to a non-smooth and non-convex optimization problem, which is difficult to solve with standard methods for sparse learning.

the softthresholding-based coordinate descent method for LASSO does not apply due to the Lyapunov regularization term.


Abbeel and Ng [1] provide several refinements of the analysis in Theorem 1.

suppose that each sample trajectory has length H ≥ (1/(1 − γ)) ln(1/( H (1 − γ))), and that an P -optimal policy is found in each iteration of the projection algorithm (see Step 1 above).


Thus we focus on (13) the local Hessian with respect to z.

we analyze the first part of ( 13), the Gauss-Newton matrix, which is a good positive semidefinite approximation to the Hessian [Martens, 2016, Chen, 2011.


It arises frequently in applications such as machine learning and signal processing; see a recent survey [7].

each smooth functions {g i } N i=1 can represent: 1) a mini-batch of loss functions modeling data fidelity, such as the 2 loss, the logistic loss, etc; 2) nonconvex activation functions for neural networks, such as the logit or the tanh functions; 3) nonconvex utility functions used in signal processing and resource allocation, see [4].


Using this estimate, we can construct a confidence interval with asymptotically nominal coverage.

in the next section, we outline the proof of Theorem 1.


Here, we formulate a Coupled Topic Model by extending LDA [2] to model multiple groups of documents.

it associates the t-th group with a mixture of topics, characterized by a DP sample D t . With this given, the words in a document are generated independently, each from a topic drawn from D t . To exploit the statistical dependency between groups, we further introduce a set of latent DPs to link between these mixtures, as described above.


To remedy this, we explore several alternatives to storing the full hidden state in memory.

we consider performing attention over: 1) the embeddings e (t) , which capture the semantics of individual words; 2) slices of  Since the embeddings are computed directly from the input tokens, they don't need to be stored.


We translate the high-level Problem 1 into the following question: Given r, c , what is the probability that the entry H(i, j) of the hidden dataset takes on value 1?

for each entry H(i, j) we are interested in computing the quantity: Here, Pr(H ) encodes the prior probability distribution over all hidden matrices in X .


The key biologically relevant follow-up question is then: how should we distinguish between the elements in the shortlist?

what reliable signatures of the differences between these architectures could be extracted from data obtainable from experiments that use today's neurophysiological tools?


The classification error is given by: Note that only the relative strength of f c compared with the alternatives is important.

the decision rule given in (2) does not change when we add the same numerical quantity to each component of f (x).


Therefore, our analysis focuses on E[log w (i) ].

assuming perfect transitions, the expected log weights are given by: each log w (i) can be seen as a biased estimator of log Z b , where the bias δ = log Z b − E[log w (i) ] is given by the sum of KL divergences ).


. . , x n ] with n data points in R p and the corresponding response vector y ∈ R n , the goal of RLSR is to learn a ŵ such that, ( ŵ, Ŝ) = arg min we wish to simultaneously determine the set of corruption free points Ŝ and also estimate the best model parameters over the set of clean points.


As shown in the table, we provide matching upper and lower bounds (up to a log factor) for all function and algorithm classes.

our bounds establish the optimality (up to log factors) of accelerated SDCA, SVRG, and SAG for randomized finite-sum optimization, and also the optimality of our deterministic smoothing algorithms for non-smooth composite optimization.


Due to subsampling, the patch size in the continuous and discrete models are related by a multiplicative factor.

a patch of size e k with discretization corresponds to a patch S k of diameter e k s k−1 s k−2 .


Optimal foraging theory [16] explores the ideal strategies for solving this problem.

the marginal value theorem shows that a forager's overall rate of return is optimized if it leaves a patch when the instantaneous rate (the marginal value) of finding food within the patch falls below the long-term average rate of finding food over the entire environment [3].


While the role of multiple passes is well understood if the goal is empirical risk minimization [3], its effect with respect to generalization is less clear and a few recent works have recently started to tackle this question.

results in this direction have been derived in [10] and [11].


These two operations are repeated for different sequences of image patches.

in Equation 2, the term "preserved information in s about x" expresses how weJl x may be reconstructed on the basis of s. it is defined as the negative of the square of the reconstruction error.


It has been shown that the MMV model provably improves the standard CS recovery by exploiting the block-sparse structure [10,21].

in the MMV model we consider the reconstruction of the signal represented by a matrix W ∈ IR d×n , which is given by a dictionary (or measurement matrix) A ∈ IR m×d and multiple measurement vector B ∈ IR m×n such that Each column of A is associated with an atom, and a set of atom is called a dictionary.


Therefore, geometry in diffusion space is meaningful and can be interpreted in terms of the Markov chain.

as shown in [18], quantizing this diffusion space is equivalent to lumping the random walk.


The RPFB is an alternative approach that we describe in Section 3.

in the fixed-window history-based approach (or window-based, for short), we only look at a fixed window of the immediate past values of X 1:t . we use samples in the form of Z t X t−H+1:t with a finite integer H that determines the length of the window.


In [8], our setting, generalized to pre-partitions of any number of subsets, was addressed by a heuristic extension of the probabilistic IB algorithm, yielding improved empirical results.

the algorithm in [8] was based on a modification of the IB stable-point equation, which amplified the impact of features characterizing a formed cluster across all, or most, subsets.


Direct measurements of the sounds received at the ears for sources at different locations in space show that disparate directions are associated with very similar localization cues.

there is a similarity between ILD and ITD cues for directions near the center of gaze and directions with eccentric elevations on the vertical plane.


As additional input, the network is provided with a task-specific embedding vector.

we have f (x, t; θ µ ) = h [x; ψ t ] T , θ h , where h(•), again, denotes the output of the neural network (here with parameters θ h ) and ψ t is the t-th row of an embedding matrix ψ ∈ R K×L (we choose L = 5 for our experiments).


Then, a reasonable rule is to follow expert e in stage s with a probability that is proportional to some monotone function of M e (s − 1).

when this probability is proportional to exp{η s M e (s−1)}, for a certain choice of η s , this algorithm is known to minimize regret [1,2].


In this work, we are interested in non-linear state space models.

we consider the case where a Gaussian process (GP) (Rasmussen and Williams, 2006) is responsible for modelling the underlying dynamics.


Otherwise, for instance, if , it is impossible to distinguish between the case when x * 3 = 1 and the case when x * 2 = λ 1 and x * 1 = λ 2 .

we do not restrict the feature matrix to be non-negative, which is more general than the traditional NMF and is potentially useful for many applications.


This means that the unfavorable cases, i.e., the Brownian realizations for which M − M ε > ε, are not concentrated on some subsets of Brownian realizations matching some evaluations in t 1 , ..., t Nε .

the PAC guarantee also holds when restricted to the Brownian realizations matching the evaluations in t 1 , ..., t Nε only.


By Lemma 3.7, E[I j ] ≥ 2n ε / √ k and thus by Hoeffding's inequality, .

with probability 5/6, at least n ε √ k players found an ε-best arm.


Computing a basis for the nullspace of the coefficient matrix in formulation (5) of the HMM filter recovers, according to Lemmas 1 and 2, the direction of one column of B.

the direction of the column corresponding to observation y k , i.e., b y k . From such basis vectors, we can construct a matrix C ∈ R X×Y where the yth column is aligned with b y .


As shown in Proposition 4.2, the first term in (4.4) is the quadratic form associated with the population Hessian matrix evaluated at Θ.

such a quadratic form is universally lower bounded by the second term, which does not depend on Θ, and is further lower bounded by β/(d under the conditions of Proposition 4.2, the population loss function defined in (4.3) is strongly convex.


We define the set of all weights that satisfy the -soft-margin condition with respect to a distribution P via: Of particular interest is W( , Dn ), the set of all the weights that satisfy the -soft-margin condition with respect to the empirical data distribution.

any w ∈ W( , Dn ) predicts with at least a margin on all but a ( ) fraction of the data.


This method allows us to substantially expand the population-sizes for which population dynamics can be characterized-beyond the number of simultaneously imaged neurons.

we demonstrate using recordings in mouse somatosensory cortex that this method makes it possible to predict noise correlations between non-simultaneously recorded neuron pairs.


. . , C b−1 ), we can calculate the posterior after the bth minibatch: we treat the posterior after b − 1 minibatches as the new prior for the incoming data points.


Foremost is the analysis of the distribution over the move size δ, which we show to be a discrete univariate distribution that generalizes both the binomial and hypergeometric distributions.

we prove that it is always log-concave [3], so it can be sampled in constant expected running time.


As a second main contribution, we provide a novel sound approach to the challenging problem of learning with Sinkhorn loss, recently considered in [6].

we leverage the smoothness of the Sinkhorn approximation to study the generalization properties of a structured prediction estimator adapted from [15] to this setting, proving consistency and finite sample bounds.


In addition to the transition probabilities, we now assume that the probability distribution of cost has uncertainty.

for each pair of s ∈ S n and a ∈ A(s), the true probability mass function 2 , f (•|s, a), for the random cost, C(s, a), is in the following uncertainty set that is characterized with the nominal probability mass function, f 0 (•|s, a): where X (s, a) is the support of C(s, a).


We present a general joint task learning framework, in which each task (either object localization or object segmentation) is tackled via a multi-layer convolutional neural network, and the two networks work collaboratively to boost performance.

we propose to incorporate latent variables bridging the two networks in a joint optimization manner.


We show that some off-the-shelf optimization methods can be used to obtain a solution ĥ close to a scaled and circularly shifted version of the ground truth.

ĥ satisfies C f R ĥ ≈ ±e j for some j ∈ [n], i.e., R ĥ is approximately a signed and shifted version of the inverse of f . Given solution ĥ to (P1), one can recover f and x i (i = 1, 2, . . . , N ) as follows: Here, we use x −1 to denote the entrywise inverse of x.


When using the best fixed hyper-parameter setting across all datasets (bottom row in Table 1), all active learning algorithms achieve less improvement compared with PASSIVE (7% improvement with the best fixed learning rate).


However, since D is not a simple signal, but a more involved data structure, it is not clear how to run existing algorithms on D. In the supplementary material we show how to apply such algorithms on our coresets.

we can run naive dynamic programming [4] on the coreset and get a (1 + ε) approximate solution in an efficient manner, as we summarize as follows.


Often, the input coordinates are not linearly related, but instead the input vector is the result of a nonlinear generative process.

data often can be accurately described as the product of unknown random variables.


Since the samples in the exploitation steps are biased (toward the arm suggested by the learning algorithm using previous exploration samples), we do not use them to learn the hypothesis for the future steps.

in methods we consider, exploitation does not help us to improve learning accuracy for the future.


In this paper, we present theoretical analysis of SON -a convex optimization procedure for clustering using a sum-of-norms (SON) regularization recently proposed in [8,10,11,17].

we show if the samples are drawn from two cubes, each being one cluster, then SON can provably identify the cluster membership provided that the distance between the two cubes is larger than a threshold which (linearly) depends on the size of the cube and the ratio of numbers of samples in each cluster.


The zooming dimension essentially coincides with the covering dimension of (X, D) 6 for the worst-case payoff function µ, but can be (much) smaller when µ is "benign".

zooming dimension would "ignore" a subtree with high covering dimension but significantly sub-optimal payoffs.


We define where the neighborhood of user u is given by and u consists of the revealed ratings of user u restricted to items that have been jointly explored.

the neighborhoods are defined precisely by cosine similarity with respect to jointed explored items.


Note that while they are based on the operations of the underlying Poisson processes, due to theorems 3, 5, and 7, we operate directly on the DPs, without the need of explicitly instantiating the associated Poisson processes or Gamma processes.

let µ t be the base measure of D t , then if the acceptance probability q is a constant, then α t = qα t−1 + α ν .


We now have another instance of the iterative algorithm proposed in Section 2, but for undirected graphical models.

we have just derived an EDML algorithm for such models.


We can see that the result using language features is better than that using audio features.

compared with "speaking turns", "topic turns" improves the performance of our model to learn the influence of participants in meetings.


Consequently there exists a solution to the equation (6).

we aim to find the solution closest to a given initial point x 0 2 X : Using the pseudoinverse and Lemma 22 item vi, the solution to ( 7) is given by where A † and A ⇤ denote the pseudoinverse and the adjoint of A, respectively.


The max-norm is defined as: Similarly to the rank, but unlike the trace-norm, generalization and learning guarantees based on the max-norm hold also under an arbitrary, non-uniform, sampling distribution.

defining mc(X) = X 2 max (no normalization is necessary here), Õ(mc(X)(n + m)) samples are enough for generalization w.r.t.


The key contribution of this paper is to show that, for a certain class of PTL problems, the optimal common representation shared across infinitely many parameterized tasks can be obtainable.

we develop an alternating minimization algorithm à la [1,2] for finding the entire continuum of solutions and the common feature subspace (or the common set of variables) among infinitely many parameterized tasks.


We now demonstrate that this matters in practice.

we show that in some sense, AUPR and AUPRG are as different from each other as AUPR and AUROC.


Input: The base generative model p θ (x) The (set of) constraints f φ (x) 1: Initialize generative parameter θ and constraint parameter φ 2: repeat Connections to adversarial learning For implicit generative models, the two objectives w.r.t φ and θ (Eq.8 and Eq.12) are conceptually similar to the adversarial learning in GANs [15] and the variants such as energy-based GANs [26,55,54,50].

the constraint f φ (x) can be seen as being optimized to assign lower energy (with the energy-based distribution q(x)) to real examples from p d (x), and higher energy to fake samples from q(x) which is the regularized model of the generator p θ (x).


We learn our CCM in a feed-forward manner.

we begin from the top level, training the independent (f k,0 ) classifiers first, in order to maximize the classification performance on the training data.


This produces a model very similar to previous machine learning work on "model-free" approximate POMDP solvers [5,6], which attempt to form good solutions without explicit knowledge of the environment's structure or dynamics.

we model working memory as a discrete memory system (a collection of recent observations) rather than a continuous "belief state" (an inferred probability distribution over hidden states).


We randomly select m = 4sµ s = 160 data to form D m , out of with |Σ| = 2mr 2 = 5120 entries of the 160 × 160 sub-matrix are fed into the algorithm.

roughly 0.5% entries out of the whole pairwise label matrix Z ∈ {0, 1} 1000×1000 are observed.


Hence we make the simplifying assumption that Pr(x, y) and Pr ′ (x, y) only differ via Pr(x, y) = Pr(y|x) Pr(x) and Pr(y|x) Pr ′ (x).

the conditional probabilities of y|x remain unchanged (this particular case of sample selection bias has been termed covariate shift [12]).


It may be computationally expensive to calculate the 3rd order derivatives needed for working out the rate of change of the metric tensor, and so an obvious approximation is to assume these derivatives are zero for each step.

for each step we can assume that the metric is locally constant.


. . , S (r) , where S (r) has the property that min i,i L (r) i,i ≥ L (r) .

we stop SWAP when perturbing S (r) by one variable increases or does not change the resulting loss.


Our final experiment is to apply CGMMN to distill knowledge from Bayesian models by learning a conditional distribution model for efficient prediction.


We note, however, that a weaker result, that still suffices for answering Question 1 in the affirmative, can be proven using a naive improper learning algorithm.

we show below how to learn H n,3 efficiently with a sample of Ω n 3 2 examples.


Namely we consider four different scenarios where the outliers are generated from models with different graphical structures.

each sample is generated from the following mixture distribution: We also consider the scenario where the outliers are not symmetric about the mean and simulate data from the following model For each simulation run, Θ is a randomly generated precision matrix corresponding to a network with 9 hub nodes simulated as follows.


We apply this RL based interactive retrieval framework to the task of image retrieval, which we call dialog-based interactive image retrieval to emphasize its capability in aggregating history information compared to existing single turn approaches [13,14,15,16].

a novel end-to-end dialog manager architecture is proposed, which takes natural language responses as user input, and delivers retrieved images as output.


The main idea is similar to the approach taken by Boosting and SVM.

we construct a non-linear predictor by first mapping the original features into a higher dimensional space and then learning a linear predictor in that space, which corresponds to a non-linear predictor over the original feature space.


In this paper, we integrate both feature extraction and metric learning into one joint optimization framework and propose a new bilevel distance metric learning model.

the lower level characterizes the intrinsic data structure using graph regularized sparse coefficients, while the upper level forces the data samples from the same class to be close to each other and pushes those from different classes far away.


Since h ∈ H can either win a point (x, y, p) or lose it, the model resembles a binary classification task, where the "label" of (x, y, p) is the identity of the winner.

a triplet (x, y, p) would be labeled 1 if the agent produced a better prediction than the opponent, and zero otherwise.


• Locked SVRG: This is the locked version of the asynchronous variant of Algorithm 1 using SVRG schedule.

we use a concurrent read exclusive write locking model, where all threads can read the parameters but only one threads can update the parameters at a given time.


Each strategy variable D j has domain A ∪ j , and given its parent θ j , its CPD chooses an action from A ∪ j according to the mixed strategy σ θi→ai j .

if j = i then Pr(D j = a j |θ j ) is equal to σ j (a j |θ j ) for all a j ∈ A j,θj and 0 for all a j ∈ A ∪ j \ A j,θj ; and if j = i we have Pr(D j = a i |θ j ) = 1.


On the other hand, if M t grows too rapidly, the problem of redundancy arises.

if centroid updates obtained with a small fraction of M t are similar to the updates obtained with M t , then it is waste of computation using M t in its entirety.


For compactness, we discuss the practicability of leveraging kernel-level sparsity for acceleration.

we point out that the winograd convolution [18] is ideal to accelerate kernel-level sparse models.


This minimax lower bound shows that we cannot hope for a better bound than (3) (up to log factors) if we only know the value of L * T . Second-order bounds Another type of improved regret bound was derived by Hazan and Kale [2011b] and involves a second-order quantity called the quadratic variation: where µ T = 1 T T t=1 t is the mean of all loss vectors.

(Q T /T is the sum of the empirical variances of all the K arms).


With q ij := q(y i = j) we define µ ij (q) = q ij for all i > m train and µ ij (q) = 1 if y i = 1 and 0 otherwise for all i ≤ m train .

we are taking the expectation in µ over all unobserved labels Y test with respect to the distribution q(Y test ).


In this paper, we propose a new approach to covariance estimation, which is based on constrained maximum likelihood (ML) estimation of the covariance [7].

the covariance is constrained to have an eigen decomposition which can be represented as a sparse matrix transform (SMT) [8,9].


3: For i = 2, 3, . . . , n perform: rithm, Λ i (s) represents the ordered intersection of the redundant-sets A j Cj up to the controller C i (i.e., 1 ≤ j ≤ i) constrained by the order of priority.

each set Λ i (s) contains a set of actions in state s that are all i -ascending with respect to the superior controllers C 1 , C 2 , .


The KGV has an interesting relationship to the mutual information among the original variables, DB CB DB 1 .

as shown in [6], in the case of two discrete variables, the KGV is equal to the mutual information up to second order, when expanding around the manifold of distributions that factorize in the trivial graphical model (i.e.


Moreover, we also conduct asynchronous parallel experiments on the shared memory architecture.

our experiments show that the proposed sparsification technique significantly reduces the conflicts among multiple threads and dramatically improves the performance.


Compared with classical factorization Z V , such tri-factorization supports the asymmetric transitivity property of directed links.

the existence of two directed links (i, j) (z i W z j ), (j, k) (z j W z k ) increase the likelihood of (i, k) (z i W z k ) via representation propagation z i → z j → z k , but not the case for (k, i) due to asymmetric W . Then we have a lemma as follows: Lemma 3.2.


It gives both positive and negative results on the existence of bad local minimum under different conditions.

it studies many common types of neuron activation functions and shows that for a class of neurons there is no bad local minimum, and for other neurons there is.


However, the better approximation comes at a cost of increased computational complexity for both structure [4] and parameter learning [1] as compared to generative models.

unlike Bayesian networks or junction trees [3], (a) the likelihood of a CRF structure does not decompose into a combination of small subcomponent scores, making many existing approaches to structure learning inapplicable, and, (b) instead of computing optimal parameters in closed form, with CRFs one has to resort to gradient-based methods.


( 1) is the baseline model with a 2-layer stacked architecture, while the other three models add extra skip connections in different ways.

note that these extra skip connections all cross the same time length k. ( 2) and (3) share quite similar architectures.


To define the inverse of each of these operators, a reference must be made to the dimensionality of the original tensor.

given X ∈ R I and A ∈ R IJ , where I, J ∈ N M , M ∈ N, we have imjm , where A (m) ∈ R Im×Jm for all m. The factorization exponentially reduces the number of parameters needed to express A from (1) , where ⊗ is the Kronecker matrix product [1].


Define a function, where We consider the following discriminative model; we model the log-odds ratio of the posterior class probability to be the sum of a bilinear function with respect to the EEG signal amplitude and linear with respect to the second order statistics of the EEG signal: The first term of the equation ( 1) can be interpreted as a spatio-temporal projection of the signal, under the bilinear model, and captures the first order statistics of the signal.

the columns u r of U represent R linear projections in space (rows of X).


We empirically evaluate amortized variational filtering using multiple deep dynamical latent Gaussian model architectures on a variety of sequence data sets.

we use AVF to train VRNN [7], SRNN [12], and SVG [9] on speech [14], music [5], and video [39] data.


Thus, the beta part of the corresponding approximate factors can be removed and the multivariate Gaussian distributions simplify to univariate Gaussians.

the approximate factors ψik with i = 1, . . . , n and k = y i are: where sik , pik , μik , νik , μyi ik and νyi ik are free parameters to be estimated by EP.


Hence, we here explore whether our model could still predict our data if attention might only modulate neuronal gain, but do so non-uniformly across neurons and tasks.

we investigate whether modulating the gain of only the neurons that are loudest, best-tuned, or most informative about the stimulus, or of all neurons equally but in a task-dependent manner, may account for the data.


For simplicity, we only consider binary classification problem in this section.

ȳP i ∈ {−1, 1} T . Also, let number of positives (i.e.


The ith marginal of the final layer of the variational DGP posterior (12) depends only on the ith marginals of all the other layers.

the consequence of this property is that taking a sample from q(f L i ) is straightforward, and furthermore we can perform the sampling using only univariate unit Gaussians using the 're-parameterization trick' (Rezende et al., 2014;Kingma et al., 2015).


These constraints are obtained by matching local maps acquired at different points in time (e.g., when closing a large cycle).

if the k-th map overlaps with some map j acquired at an earlier point in time, our approach localizes the robot relative to this map using once again local scan matching.


But any update to θ that tries to fit this value (i.e., makes )) forces θ1 > 0, which is inconsistent with the assumption, θ1 < 0, needed to generate the high bootstrapped value.

any update that moves (s 2 , a 2 ) higher undercuts the justification for it to be higher.


since what we care about is the generalization performance of the algorithm, a better method is to pick θ to maximize the log-likelihood of data that wasn't used to calculate the "features" b, because when we see a test example, we will not have had the luxury of incorporating information from the test example into the b's (cf.


Visualizations have the potential to help us answer several important questions about why neural networks work.

why are we able to minimize highly non-convex neural loss functions?


However, one dimensional algorithms cannot be easily extended to the multivariate case.

the ideas of root bracketing and function transversal are not well defined [2]; given a particular bracket of a continuous surface, there will be an infinite number of solutions to the equation f ( x) − t = 0, since the solution in multiple dimensions is a set of surfaces, rather than a set of points.


Regardless of the experimental apparati used (e.g., electrodes or calcium imaging), real-time decoding of individual neuron responses requires identifying and labeling individual spikes from recordings from large populations.

real-time decoding requires real-time spike sorting.


Let U be any vector in R d . A single Byzantine worker can make F always select U .

a single Byzantine worker can prevent convergence.


In this work, we propose a novel hybrid approach for tackling problems such as (3), that combines the ideas of the CHW and PST frameworks.

we suggest using a SGD-like low-regret update for the variable w, while updating the variables ξ and b via a PST-like step; the dual update of our method is similar to that of CHW.


The IB-CNN consists of three stacked convolutional layers with activation functions, two average pooling layers, an FC layer, and the proposed IB layer to predict the AU label.

the first two convolutional layers have 32 filters with a size of 5 × 5 and a stride of 1.


Therefore, we discard ∆(i) from the beginning of the permutation, when we calculate loss(T c ) in Algorithm 2.

we eliminate the corresponding components of vectors G and ∆ when calculating the cosine similarity between them.


Our model reproduced this behavior and provided a straightforward explanation: latent-to-observed weights (w ij ) in the MAP model were so strong that whenever a latent was switched on it could almost only produce triplets, therefore doublets were created by spontaneous independent activation of observeds which thus produced embedded and mixture doublets with equal chance.

doublets were seen as mere noise under the MAP model.


This implies that the guarantees associated with this class of algorithms extend to our procedure.


Contributions: (i) we propose a new algorithm to compute C(U ) exactly in O(q 2 p) steps, the first non-iterative algorithm to compute C(U ); (ii) we compare its performance against state-of-the-art iterative algorithms, and observe a much faster convergence.

our algorithm scales much faster than O(q 2 p) in practice; (iii) we implement our algorithm on a GPU, and show that it computes the cost of all (more than 2 billion) trees with ≤ 11 nodes, in ≤ 2.5 hours.


In this paper, we borrow techniques from NLP and apply them to symbolic computation.

we make use of an n-gram model over mathematical operations, inspired by n-gram language models.


Our analysis uses recent results from matrix sketching.

we use the fact that we can approximately solve `p-regression problem using random matrix sketches [29].


Had we ignored the minimality requirement in definition 2 and chosen to factorize Y < X < Z < R y using the chain rule, we would have obtained: |Z does not hold in the graph.

existence of one admissible sequence based on an order O of variables does not guarantee that every factorization based on O is admissible; it is for this reason that we need to impose the condition of minimality in definition 2.


We assume that P workers write updates at regular time intervals (referred to as "clocks").

let u  where S p,c ✓ W p,c = ([1, P ] \ {p}) ⇥ [c s, c + s 1] is some subset of the updates u written in the width-2s "window" W p,c , which ranges from clock c s to c + s 1 and does not include updates from worker p. the noisy state xp,c consists of three parts: 1.


As its name suggests, D-TS selects both the first and the second candidates according to Thompson Sampling.

d-TS maintains a posterior distribution for the preference matrix, and chooses the pair of arms for comparison according to two sets of samples independently drawn from the posterior distribution.


A Bayesian action-graph game (BAGG) is a tuple (N, Θ, P, {A i,θi } i∈N,θi∈Θi , G, {u α } α∈A ) where N is the set of agents; Θ = i Θ i is the set of type profiles; P is the type distribution, represented as a Bayesian network; A i,θi ⊆ A is the type-action set of i given θ i ; G = (A, E) is the action graph; and for each α ∈ A, the utility function is u α : C (α) → R. Intuitively, this representation captures two types of structure in utility functions: firstly, shared actions capture the game's anonymity structure: if two action choices from different type-action sets share an action node α, it means that these two actions are interchangeable as far as the other players' utilities are concerned.

their utilities may depend on the number of players that chose the action node α, but not the identities of those players.


However, with active oracle accesses, SVRG is also possible and the upper bound becomes: Our main contributions in this paper are: (1) presenting a precise formal oracle framework for studying parallel stochastic optimization; (2) establishing tight oracle lower bounds in this framework that can then be easily applied to particular instances of parallel optimization; and (3) using the framework to study specific settings, obtaining optimality guarantees, understanding where additional assumptions would be needed to break barriers, and, perhaps most importantly, identifying gaps in our understanding that highlight possibilities for algorithmic improvement.

• For non-smooth objectives and a stochastic prox oracle, smoothing and acceleration can improve performance in the layer graph setting.


While LCD/BLCD algorithms aim to identify a subset of causal edges via special structures among all variables, we focus on finding all the causal edges adjacent to one target variable.

we want to find the causal identities of each node, in terms of direct causes and effects, with respect to one target node.


The empirical error on a finite sample S is er S (h) = 1 |S| (x,y)∈S I[h(x) = y], where I is the indicator function.

we use h * denote the best classifier in H. h * = arg min h∈H er D (h).


Our goal is to construct a graph from a set of pixels.

we want to construct a graph grounded in the space of these pixels.


In modeling multivariate time series, it is important to allow time-varying smoothness in the mean and covariance process.

there may be certain time intervals exhibiting rapid changes and others in which changes are slow.


The use of soft attention over both types of inputs made strong generalization possible.

on a family of block stacking tasks, our neural network policy was able to perform well on novel block configurations which were not present in any training data.


the number of customers sampling dishes from the " Indian Buffet", let m k be the number of customers who have sampled dish k, let K + be the total number of dishes sampled, and denote by K h the number of dishes with a particular selection history h ∈ {0; 1} n . K h > 1 only if there are two or more dishes that have been selected by exactly the same set of customers.


Finally, we quantify the diversity of a set of captions by averaging their pair-wise editing distances, which leads to two additional metrics.

when only a single caption is generated for each image, we report the average distance over captions of different images, which is defined as the diversity at the dataset level.


However, the higher order dependencies can not be filtered out by these linear transforms.

the absolute values of neighboring wavelet coefficients (but not their signs) are mutually dependent.


. . , DJ are orthonormal, the distribution of missing features is invariant to rotations in the J-dimensional subspace from which they are drawn.

all directions in that space are equally likely.


Small-error linear filters Linear filter ϕ o with a small 1 norm in the spectral domain and small recovery error exists, essentially, whenever there exists a linear filter with small recovery error [8,4].

indeed, let us say that x ∈ C(Z n 0 ) is simple [4] with parameters m ∈ Z + and ρ ≥ 1 if there exists x is (m, ρ)-simple if there exists a hypothetical filter φ o of the length at most m + 1 which recovers x τ with squared risk uniformly bounded by As proved in [15, Appendix C], we have and, for a moderate absolute constant c, with probability 1−α.


Then the symbol P Ω denotes the orthogonal projection onto Ω, namely, Similarly, the symbol P ⊥ Ω denotes the orthogonal projection onto the complement space of Ω.

p Ω + P ⊥ Ω = I, where I is the identity operator.


1 To complete the reduced model, we need the conditional distributions P(G N j f N( j 1) ) and P( f N j f N( j 1) G N j ).

how many "ones" do we expect in an interval, given the event status at the beginning of the interval, and what is the probability that an event is occurring at the beginning of the next interval, given also the number of ones in the current interval?


Note that d KT (V, σ * ) counts the number of times alternative a is approved while alternative b is not for all a, b ∈ A with b σ * a.

let n Next, we show that L(V, a) is a monotonically increasing function of SC APP (a).


The information from this unique match can then be propagated to the neighbouring uncertain matches.

the certain unique matches will support the close-by uncertain ambiguous matches in the image.


We wish to remedy this violation by making an 2 -minimal adjustment of e ij , e jk , and e ki .

the vector e is projected orthogonally onto the constraint set {e : e ij − e jk − e ki ≤ b ijk }.


Further define the JS divergence as: ∆(f g) = 1 2 (D(f g) + D(g f )).

the KL and JS divergences between two Bernoulli random variable with parameters p and q are denoted with D(p q) and ∆(p q) respectively.


The approximate inference algorithm aims to compute good approximations to the real posteriors by iteratively optimizing the above described variational bound.

given the approximations q t (y i ) ∼ N (µ t yi , Σ t yi ) (similar forms for z i and w i ) and q t (α j i ) ∼ Γ(a t ij , b t ij ) from the t th iteration the update rules are as follows: Update for q t+1 (y i ): Alternating between the above described updates can be considered as message passing between the low-dimensional regression outputs and higher dimensional output labels, which in turn are constrained to be sparse.


We additionally study the total cost sufficient for learning, for an abstract notion of the cost of requesting the labels of a given number of examples at once.

we find that for sublinear cost functions, it is often desirable to request labels in large batches (i.e., buying in bulk); although this may increase the total number of labels requested, it reduces the total cost required for learning.


Our main result, Theorem 8, states that as the minipool size goes to infinity, E[∇ (z (t) , θ)] tends in the direction of the gradient of the population zero-one loss Z(θ), where the expectation is with respect to the randomness of the minipool X M and label y.

let z (t) be chosen via uncertainty sampling with the parameters θ: ).


Datasets: Following [6], we conduct experiments on two fine-grained bird datasets, CUB [22] and NABirds [21].

cUB dataset contains 200 categories of bird species with a total of 11,788 images.


We improve the theoretical analysis and empirical performance of algorithms for the stochastic multi-armed bandit problem and the linear stochastic multi-armed bandit problem.

we show that a simple modification of Auer's UCB algorithm  achieves with high probability constant regret.


We study inference on factor graphs using Gibbs sampling, the de facto Markov Chain Monte Carlo (MCMC) method [8, p. 505].

our goal is to compute the marginal distribution of some query variables using Gibbs sampling, given evidence about some other variables and a set of factor weights.


We investigate how would the identity information contained in the identification supervisory signal influence the learned features.

we experiment with an exponentially increasing number of identities used for identification during training from 32 to 8192, while the verification signal is generated from all the 8192 training identities all the time.


The algorithms are derived in a common framework.

we prove the exact formulas for the cost function in each case that close the gap between previously known lower bounds related to Shannon and Rényi entropy.


We adopt the same idea of data-preprocessing as in [14].

we firstly remove the batch effect by applying the surrogate variable analysis proposed by [13].


This approach is based on uniform convergence theory.

an upper bound for any class of distributions can be achieved by bounding the VCdimension of an associated set system, called the Yatracos class (see [7, page 58] for the definition).


Here we use the Syntax-Directed decoder proposed in [28] to guarantee both the syntax and semantics validity.

• Syntax constraints: The AST generation follows the grammar of loop invariants described in Eq 4.


The plot shows a tight upper bound, indicating that it is safe to use our model's derived filtering slack δ.

the achieved eigen-error always remains below the requested tolerable error specified as input, and the slack chosen given the tolerable error is close to being optimal.


We generalize the BMF framework by imposing that H is low-rank.

we impose the rank-1 expansion H = Kc k=1 u :k v T :k , where u :k and v :k are column vectors (thus their outer product is a rank-1 matrix), each of them is modeled here by a Gaussian distribution: and K c is the number of such rank-1 matrices such that K c < min(K l , K r ), i.e., H is low-rank.


The estimator E[L 1 |no-fail] is no longer an unbiased estimator of R(f ), but the bias is small.

since we are after a classifier f with a small risk, we prove an upper-bound on R(f ).


Finally, we like to mention that, if the conditional distributions are not equal modulo translations, then the situation may become more involved.

if we are in a situation with p(f * P (x)|x) > 0 and p(f (10) are in general impossible, and weaker self-calibration inequalities require additional assumptions on P. We refer to [8] where the case = 0 is considered.


To investigate the sequential strategy of subjects, we progressively updated a GP using a subject's clicks on each trial, and exploited this GP to evaluate the next click of the same subject.

we attempted to know to what degree a subject follows a GP.


The fact that SafetyNets operates only on elements in a finite field F p during inference imposes a practical challenge.

how do we convert floating point inputs and weights from training into field elements, and how do we select the size of the field p?


The observant reader may note that the learned rewards do not match the true rewards specified by the grid world.


According to the above selection of hyperparameters on the synthetic datasets, we chose a wide range of hyperparameters to test KCDC on.

both on the response and input we used either a log kernel k(x, x ) = − log( x − x q + 1) with q in [2, 3,4] or an RBF kernel with bandwidth [1, 1.5, 2] times the median heuristic.


Can we tighten the dependence on rank for these problems?

can we bring the dependence on r down from r 3/2 to linear?


, where i j ∼ N (0,  Then the compressed lasso is sparsistent: Persistence (Greenshtein and Ritov [10]) is a weaker condition than sparsistency.

the assumption that E(Y |X ) = β T X is dropped.


umbrella, teddy bear), benefiting from the joint reasoning with frequent concepts over the concept hierarchy graph.

applying the techniques of incorporating high-level semantic constraints designed for classification task into pixel-wise recognition is not trivial since associating prior knowledge with dense pixels itself is difficult.


. . , x p follow an iid uniform distribution on [2 for some > 0 then there exists a polynomial-in-p time algorithm which solves the subset-sum problem whp as p → +∞.

even though the problem is NP-hard in the worst-case, assuming a quadratic-in-p number of bits for the coordinates of x, the algorithmic complexity of the typical such problem is polynomial in p. The successful efficient algorithm is based on an elegant application of a seminal algorithm in the computational study of lattices called the Lenstra-Lenstra-Lovasz (LLL) algorithm, introduced in Lenstra et al.


If K t = K for all t, then we say that we using a (nonadaptive) stationary controller K. In this setting, it is straightforward to check if our system is stable.

it is BIBO stable if and only if the magnitude of all the eigenvalues of D = D t = A+BK t are strictly less than 1.


In this section, we propose Influence Maximization Linear UCB (IMLinUCB), detailed in Algorithm 1.

notice that IMLinUCB represents its past observations as a positive-definite matrix (Gram matrix) M t ∈ d×d and a vector B t ∈ d . let X t be a matrix whose rows are the feature vectors of all observed edges in t steps and Y t be a binary column vector encoding the realizations of all observed edges in t steps.


The expression in Theorem 3.1 can be simplified by examining its Taylor expansion (see Appendix A).

for the special case of ✓ = 1 where the adversary establishes only one connection per server, line (5) simplifies to This suggests that the first-timestamp estimator has a probability of detection that decays to zero asymptotically as log(d)/d.


All emission densities belong to the Hölder class, H 1 ( , L).

they satisfy, for all Here b c is the largest integer strictly less than .


To give more quantitative insight into these trends, we consider each of the six different experimental settings (i.e., (3 datasets) × (unsupervised vs. supervised)) as trials and consider what performance trends are likely to generalize.

we use the non-parametric Wilcoxon Signed-Rank Test [33] to quantify the differences between the different aggregators across trials, reporting the T -statistic and p-value where applicable.


The explosion of data and an increase in model size has led to great interest in training deep neural networks on distributed systems.

distributed stochastic gradient descent has been extensively studied in both deep learning and high-performance computing communities, with the goal of accelerating large-scale learning tasks [13,10,9,47,3,14,30,31,6,44,38].


Furthermore, every village j borrows only from those MFIs i ∈ B j that offer the lowest interest rate.

i∈Bj ,r * i =r * m j x * j,i (1 + r * i − e j ) = d j for any MFI m j ∈ argmin i∈Bj r * i , and x * j,k = 0 for any MFI k such that r * k > r * mj .


We allow h to be a polynomial in x ∈ R m and X to be a basic closed semialgebraic set.

even if ξ is fixed, ( 1) is a hard problem in general.


We used production data regarding Tweet engagement to perform an offline evaluation of our techniques.

the training data was generated as follows: for a particular day d, we collect data for all Tweet impressions (i.e., Tweets shown to a user) generated during that day.


This leads to the cut functional cut CE , Note that in contrast to the hypergraph cut (1), the value of cut CE depends on the way each hyperedge is cut since the term |e ∩ C| |e ∩ C| makes the weights dependent on the partition.

the smallest weight is attained if only a single vertex is split off, whereas the largest weight is attained if the partition of the hyperedge is most balanced.


Since depth estimates in the monocular case are much noisier than in the stereo case, we employ a more constrained model than the one utilized in [7].


Section 3 investigates how to extend the ERM principle in this situation.

probability bounds proving the generalization ability of minimizers of a non-asymptotic approximant of the risk previously introduced are established.


Regularization has emerged as a dominant theme in machine learning and statistics, providing an intuitive and principled tool for learning from high-dimensional data.

regularization by squared Euclidean norms or squared Hilbert norms has been thoroughly studied in various settings, leading to efficient practical algorithms based on linear algebra, and to very good theoretical understanding (see, e.g., [1,2]).


We focus in this paper on convergence of strategies as this implies the other three forms of convergence as well.

we will say an algorithm converges against a particular opponent if and only if lim t→∞ x t = x * .


We show that the answer is "yes" by developing efficient sketched second order methods with regret guarantees.

the three main contributions of this work are: 1.


For now, in our initial simulations, we show that our derived rule can learn synaptic weights which can unmix, or demultiplex, mixed spike trains.

it can recover independent point processes embedded in distributed correlated input spike trains, using an adaptive single-layer feedforward spiking network.


The proof of the above theorem constructs a sequence for which no reasonable play can attain sublinear external regret.

the only way the learner can have sublinear external regret is if they choose to have very small utility.


Compared to using one large RNN that has a lot of freedom of modeling the input-output mapping, ThalNet imposes local structure to how the input-output mapping can be implemented.

it encourages the model to decompose into several modules that have stronger intra-connectivity than extra-connectivity.


As it can be noticed, SMC-learning outperforms both the compared algorithms.

the generalization over the action space performed by tile coding negatively affects the learning performance because of the non-linearity of the dynamics of the system.


This paper introduces SafetyNets, a new (and to the best of our knowledge, the first) approach for verifiable execution of deep neural networks on untrusted clouds.

safetyNets composes a new, specialized IP protocol for the neural network's activation layers with Thaler's IP protocol for matrix multiplication to achieve end-to-end verifiability, dramatically reducing the bandwidth costs versus a naive solution that verifies the execution of each layer of the neural network separately.


(Of course, the endpoint of one interval must be the same as the initial point of the next interval.)

define J i (s) to be the likelihood of the most likely trajectory covering the time interval [τ 1 , τ i ], accounting for the first i observations, and ending at state s. The we can compute J as follows.


In this paper, we take a more fundamental approach to understanding the robustness of supervised classification algorithms by extending well-understood results for supervised batch learning in statistical learning theory.

we seek to understand the sample complexity of Probably Approximately Correct (PAC)-learning in the presence of adversaries.


An important class of high order interactions occurs when the simultaneous effect of two variables on a third may not be additive.

it may be possible that X ⊥ ⊥ Z and Y ⊥ ⊥ Z, whereas ¬ ((X, Y ) ⊥ ⊥ Z) (for example, neither adding sugar to coffee nor stirring the coffee individually have an effect on its sweetness but the joint presence of the two does).


It is known that if the models are mis-specified, model averaging is superior to model selection.

let n be the sample size, then the worst case regret of the former decays at the rate of O(1/n) while the worst case regret of the latter decays at the rate of O(1/ √ n).


While in theory the SSVM framework can be employed in conjunction with any loss function, in practice its feasibility depends on the computational efficiency of the corresponding lossaugmented inference.

given the current estimate of the parameters, it is important to be able to efficiently maximize the sum of the score and the loss function over all possible outputs.


Given p agents of different radii {r i } p i=1 with given desired initial and final positions, {x i (0)} p i=1 and {x i (T )} p i=1 , along with a cost functional over trajectories, compute collision-free trajectories for all agents that minimize the cost functional.

find a set of intermediate points {x i (t)} p i=1 , t 2 (0, T ), that satisfies the "hard" collision-free constraints that kx i (t) x j (t)k > r i + r j , for all i, j and t, and that insofar as possible, minimizes the cost functional.


An attractive property of compression is that it enjoys two closure properties.

if a base class admits compression, then the class of mixtures of that base class, as well as the class of products of the base class, are compressible (Lemmas 3.6 and 3.7).


Surprisingly, even when analytic forms of the entropy or KL divergence are available, sometimes it is better to use (1) because it will have lower variance.

this occurs when q φ (z|x) = p(z|x), i.e.


The market maintains a market state q t ∈ R d at time t = 0, . . . , T , with q 0 = 0.

each trader t = 1, . . . , T arrives sequentially and purchases a contract dq t ∈ R d , and the market state is updated to q t = q t−1 + dq t . q t = t s=1 dq s , the sum of all contracts purchased up to time t. The price paid by each participant is determined by a convex cost function C : R d → R. Intuitively, C maps q t to the total price paid by all agents so far, C(q t ).


Note that c/2 is the expected degree of the G ER (p, c/p) ensemble.

when the Erdős-Rényi random graphs have a bounded average degree (c = O(1)), we can obtain a minimum sample complexity of n = Ω(log p) for structure estimation of Ising models.


We propose a hierarchical prior of the following form: where g : R → R is a suitable injective transformation.

we impose a Gaussian process [11] as a prior on the parameters γ i .


Therefore, if k is not characteristic, which means γ k (P, Q) = 0 for some F k , the maximum risk is zero).

if k is characteristic, then the maximum risk is obtained only when P = Q.


It basically states that the Kantorovich metric is a solution to the Monge-Kantorovich problem, when its cost function is equal to the base metric for the Kantorovich metric.

for two measures µ and ν, and a cost function f ∈ [S × S → R], the Monge-Kantorovich problem computes: The set of measures ξ with marginals µ and ν is also known as the set of couplings of µ and ν.


[35], which has been successfully applied to the problem of item set recommendation.

we have a set of items V = {1, 2, . . . , n} that we want to recommend to the user.


In order to compare with methods such as DECO which is concerned with the Lagrangian formulation of lasso, we modify our algorithm accordingly.

in step 5 of Algorithm 1, we solve Herein, in our experiments, the regularization parameter is set to be λ = 2 X T ∞ .


As the locality principle commonly exists in object-to-object relations in the real world, the tailor module enables the inference on the dynamics of objects focusing on the relations with neighbour objects.

it crops a "horizon" window of size w from the object masks centered on the position of the dynamic object whose motion is being predicted, where w indicates the maximum effective range of object-to-object relations.


To the authors' best knowledge, this is the first reinforcement learning algorithm for which consistency has been demonstrated in a continuous space framework.

the recently advocated "direct" policy search or perturbation methods can by construction at most be optimal in a local sense [SMSMOO , VRKOOj.


We develop a framework which provides convergence conditions and error bounds along with simple proofs and connections to methods in numerical linear algebra.

we show that if the Gaussian precision matrix is generalized diagonally dominant, then any Hogwild Gibbs sampler, with any update schedule or allocation of variables to processors, yields a stable sampling process with the correct sample mean.


To measure prediction performance, we use standard 5-fold cross-validation strategy by computing the root mean square error (RMSE) between the predicted and actual values of the cognitive scores on the testing data only.

the whole set of subjects are equally and randomly partitioned into five subsets, and each time the subjects within one subset are selected as the testing samples and all other subjects in the remaining four subsets are used for training the regression models.


These methods are based on extending the Lasso formulation [8] by modifying the l 1 penalty to account for the group structure.

yuan & Lin [12] proposed the Group Lasso, which solves arg min β , where X G 1 , .


For a phone sequence, we produce N -grams by the standard transformation of the stream; e.g., for bigrams (2-grams) the sequence of phones, t 1 , t 2 , ..., t n , is transformed to the sequence of bigrams of phones t 1 t 2 , t 2 t 3 , ..., t n−1 t n . We then find probabilities of Ngrams with N fixed.

suppose we are considering unigrams and bigrams of phones, and the unique unigrams and bigrams are designated d 1 , ..., d M and d 1 d 1 , ... d M d M , respectively; then we calculate probabilities and joint probabilities where #(t k = d i ) indicates the number of phones in the conversation side equal to d i , and an analogous definition is used for bigrams.


Also of interest is the comparison with bounds for low-rank matrices, for which X Σ ≤ √ rank X X Fro .

for n × m rank-k X with entries bounded by B, X Σ ≤ √ knmB, and the second term in the right-hand side of (7) becomes: Although this is the best (up to log factors) that can be expected from scale-sensitive bounds 4 , taking a combinatorial approach, the dependence on the magnitude of the entries in X (and the margin) can be avoided [9].


Instead of computing λ k+1 exactly according to (13), we propose to approximate λ k+1 with the help of Lemma 3.1 and set µ k+1 to this approximation.

let t k be the total number of consensus steps done before k-th iteration of PDA, and let q k ≥ 1 be the number of consensus steps within iteration k. to approximate P C (x) in (13).


The motivation underlying these methods fundamentally differs from ours, leading to distinct differences in the sampling algorithm.

[4] supports innovation and deletion of particles, but does not support variation of locations.


We conjecture that far more powerful results can be obtained by adapting the convergence theorems in the literature [6,7,9].

younes [7] gives convergence results when the gradient of the energy ∂E(x; ω)/∂ω is bounded by a term that is linear in ω (and hence unbounded).


Unfortunately, while various such multiplicative weights have been designed for voting with subtrees [3,4,5], they are not known to yield statistically consistent prediction.

in fact, the best known result to date [5] presents a weighting scheme which can provably achieve an excess risk 3 (over the Bayes classifier) of the form o P (1) + C • min T R(h T ), where R(h T ) denotes the misclassification rate of a classifier h T based on subtree T . the excess risk might never go to 0 as sample size increases, which in contrast is a basic property of the pruning alternative.


The generality of this outcome allows one to consider using a wide range of alternative importance sampling schemes, while employing appropriate G -weights to cancel any bias.

we will determine blocks on-line by following deterministic greedy search paths.


Likewise we may perform the linearization in (6) as follows: In other words, we use the rescaled estimate ỹ to provide an upper bound on the concave part of the loss function.

this leads to the following instantiation of standard convex-concave procedure: instead of the structured estimation loss it uses the loss bound l(x, y, ỹ, f ) In the case of Γ(η) = 1 this can be simplified significantly: the terms in f (x, y) cancel and l becomes we replace the correct label y by the rescaled estimate ỹ.


Another line of work is that of so-called direct mode estimators which estimate the mode from practical statistics of the data [17,18].

[18] shows that the simple and practical estimator arg max x∈X [n] f n (x), where f n is a kernel-density estimator, is a consistent estimator of the mode.


Then for any f ∈ C s 0 ∩ W s 2 with s ≤ r, we have Consequences similar to those of Theorems 1 and 2 can be drawn for Theorem 3.

the rate in (16) coincides with that of (15) with c = 1/2.


We evaluate deep metric learning algorithms on fine-grained object recognition and verification tasks.

we consider car and flower recognition problems on the following database: • Car-333 [29] dataset is composed of 164, 863 images of cars from 333 model categories collected from the internet.


On the other hand, the supervised FSDR approaches include supervised principal component analysis [9], sliced inverse regression [10], and kernel dimension reduction [11].

for multi-label classification, a leading supervised FSDR approach is canonical correlation analysis [CCA; 6,12] which is based on linear projections in both the feature space and the label space.


We also demonstrate empirically that our algorithms outperform existing methods, such as those of [5,18,14], for ARMP and the matrix completion problem by an order of magnitude and are also more robust to noise and sampling schemes.

results show that our SVP-Newton method is significantly robust to noise and performs impressively on a more realistic power-law sampling scheme for the matrix completion problem.


V ar(â vw,s ) = (s − 1) Interestingly, the variance (15) says we do need s = 1, otherwise the additional term (s − 1) D i=1 u 2 1,i u 2 2,i will not vanish even as the sample size k → ∞.

the choice of random distribution in VW is essentially the only option if we want to remove the bias by premultiplying the data vectors (element-wise) with a vector of random variables.


6 Properties of Simulated Co-evolution, Networks and Cascades * In this section, we perform an empirical investigation of the properties of the networks and information cascades generated by our model.

we show that our model can generate coevolutionary retweet and link dynamics and a wide spectrum of static and temporal network patterns and information cascades.


In addition it stores the structure of the tree T , encoded using the Eulerian Tour Technique 5 .

starting at the root, we traverse T in the Depth First Search (DFS) order.


• In Theorem 2, we derive a bound on the expected function values.

we can prove that in the vanishing noise regime, acceleration (with appropriate averaging) achieves a faster rate, see Corollary 2 and the discussion in Remark 3.


We consider the ERM estimator fz := arg min f ∈F P n (•, f ).

given an n-sample z, ERM selects any fz ∈ F minimizing the empirical risk P n (•, f ).


Connections to other DP-based topic models PFA is a nonnegative matrix factorization model with Poisson link that is closely related to other DP-based models.

[32] showed that by making p(h kn = 1) = 1 and letting θ kn have a Dirichlet, instead of a Gamma distribution as in (1), we can recover LDA by using the equivalence between Poisson and multinomial distributions.


The notation (R x y) means that x and y are related by R. In miniKanren queries, data flow is not directionally biased: any input to a relation can be unknown.

for example, a query (R X y) where y is known and X is an unknown, called a logic variable, finds values X where X and y are related by R. given R and f defined as before, the query finds inputs X to f such that (f X) = y.


In our offline version of UCSG, in each phase k we let both players each select their own optimistic policy.

after Player 1 has optimistically selected π 1 k , Player 2 then optimistically selects his policy π 2 k based on the known π 1 k . the model-policy pair M 2 k , π 2 k is obtained by another extended value iteration on the extended MDP under fixed π 1 k , where Player 2's action set is extended.


This is meaningful because in most applications, the time complexities for evaluating gradients at different points are of the same magnitude.

the worse-case time complexities of first-order methods are usually proportional to a fixed parameter times T . In large-scale settings, however, if we have already spent time computing the (full) gradient at x, perhaps we can use such information to reduce the time complexity to compute full gradients at other points near x.


The second term indicates the regularization effects due to the max-margin posterior constraints, which is consistent with our intuition.

for those data with non-zero Lagrange multipliers (i.e., the data are around the decision boundary or misclassified), the second term will bias the model towards a new posterior distribution that favors more discriminative representations on these "hard" data points.


The ability of CNNs to learn local stationary structures and compose them to form multi-scale hierarchical patterns has led to breakthroughs in image, video, and sound recognition tasks [18].

cNNs extract the local stationarity property of the input data or signals by revealing local features that are shared across the data domain.


( 5) is satisfied whenever α ≥ β min(α 0 , (1 − γ)/L).

both line search criteria at any iteration k are satisfied in a finite number of trials.