. . , f n : [0, 1] → R may be nonconvex and the notation [n] denotes the set {1, 2, . . . , n}.

our goal is to develop an algorithm that achieves an objective -close to the global optimal value for any > 0 with a complexity that scales along with the properties of f . we present an algorithm that simultaneously achieves the best known bounds when f i are convex and also for general f i , while scaling much better in practice than the straightforward approach when considering f used in robust estimation such as Huber Loss, Tukey's biweight function, and MCP.


For the latter scenario, we show in Appendix C.1 that local exponential stability holds under a certain smoothness condition on the discriminator.


To do so, we outline a general framework for graph generative models; by contrast to the pioneering work of Caron and Fox [12], models within our framework are stationary across steps of the graph sequence.

our model grows the graph by instantiating more latent atoms of a single random measure as the dataset size increases, rather than adding new atoms to the measure.


The previous lemmas suffice to show that the solutions of the MILP problem can be decoded into valid DAGs of bounded treewidth: Theorem 1.

any solution to the MILP can be decoded into a valid DAG of treewidth less than or equal to w. the decoding of an optimal solution solves (1).


We hope that the connection between SMC and neural network technologies will inspire further research into adaptive SMC methods.

application of the methods developed in this paper to adaptive particle smoothing, high-dimensional latent models and adaptive PMCMC for probabilistic programming are particular exciting avenues.


We make use of graph convolution G conv from [11] to further propagate information on the graph.

for a single graph convolution with its parameter W g ∈ R d× d, the operation is defined as where f can be any nonlinear activation functions.


. . , U m } ∈ C m (S) to be their common refinement.

the clusters of U consist of those elements that are in the same clusters of both S and T . Formally, This result can be thought of as expressing a relationship between seperation and the lattice of partitions of S which will be important to our later robustness extension Lemma 6.


Showing the superiority of our method especially for type II errors.

methods based on HSIC fail to detect weak dependencies in the time series.


The primary objective of our experiments is to demonstrate that PGRD is able to use experience online to improve the reward function parameters, thereby improving the agent's obtained objective return.

we compare the objective return achieved by PGRD to the objective return achieved by PGRD with the reward adaptation turned off.


In the context of the kernel market, this suggests that participants need only interact with the mechanism by pushing updates that lie in the span of their own data.

we only need to consider updates of the form df = αk((x, y), •).


We then go further by showing that SVIHMM still converges using approximate messages within an uncertain parameter setting where θ is learned simultaneously with the state sequence x.

we approximate τ by monitoring the change in belief residuals with a sub-routine GrowBuf, outlined in Alg.


One way to express this notion of cost is to incorporate it directly into the utility function.

we consider an augmented utility function U (a, q; s) that depends on the approximate density.


Their base operation is evaluating the proximal operator of a function, which involves solving a small convex optimization problem that often admits a closed-form solution.

the proximal operator prox f (x 0 ) : R → R of a function is defined as If f is convex, the fixed points of the proximal operator of f are precisely the minimizers of f . In other words, prox f (x * ) = x * iff x * minimizes f . This fix-point property motivates the simplest proximal method called the proximal point algorithm which iterates x (n+1) = prox f (x (n) ).


We use this colouring to define our experiments.

for the ith experiment, select all vertices whose colour is 1 in the ith bit.


In such settings, revenue sharing is a central part of running such a marketplace for the intermediary, and fixedpercentage revenue sharing schemes are often used to split the revenue among the platform and the sellers.

such revenue sharing schemes require the platform to (i) take at most a constant fraction α of the revenue from auctions and (ii) pay the seller at least the seller declared opportunity cost c for each item sold.


Initialization: The α, β and γ weights are initialised in a bottom-up fashion on the decomposition tree -we initialise the weights of a vertex after we have initialised the weights of all its children.

we first do a depth-first search of D starting from r: When we reach an edge (v, w) ∈ E(T ), if neither v or w is a leaf then we set β ab ((v, w)) ← θ (v,w),a,b otherwise assuming w is a leaf we set α a (v) ← 1 (dummy edge).


Finally, we extend in Section 4 some of the known consistency results of the Lasso and multiple kernel learning [3,4], and give a partial answer to the model selection capabilities of our regularization framework by giving necessary and sufficient conditions for model consistency.

we show that our framework is adapted to estimating consistently only the hull of the relevant variables.


By this measure, EGT/AS and EGT perform slightly worse relative to their performance as measured by iteration count.

cFR + takes about 800 gradient computations in order to reach 1mbb in either game, whereas EGT/AS takes about 1800.


Despite the rich promise of sparse coding models, we believe that their development has been hampered by their expensive computational cost.

learning large, highly overcomplete representations has been extremely expensive.


To evaluate the performance of Neurify, we test it on nine models trained over five datasets for different tasks where each type of model includes multiple architectures.

we evaluate on fully connected ACAS Xu models [16], three fully connected Drebin models [5], three fully connected MNIST models [21], one convolutional MNIST model [42], and one convolutional self-driving car model [2].


To overcome such difficulties, we borrow the lower bound analysis for bandit pure exploration problems in [7].

our analysis considers the query distribution of any active query algorithm A " pϕ 1 , .


Another outcome of these results relates to the level at which it is useful to phrase the problem of factoring graphical probabilistic models.

the perspective we are promoting here is that probability distributions defined by graphical models should be viewed as multi-linear functions, and the construction of jointrees should be viewed as a process of constructing arithmetic circuits that compute these functions.


Let R be a ⌫-selfconcordant barrier for K. Then, for ⌘k  12d , the regret of OPTIMISTICBCO can be bounded as follows: (k + 1) 2 .

for ⌘ = T 8/13 d 5/6 , = T 5/26 d 1/3 , k = T 1/13 d 5/3 , the following guarantee holds for the regret of the algorithm:  This work was partly funded by NSF CCF-1535987 and IIS-1618662 and NSF GRFP DGE-1342536.


Suppose we have a probability distribution p b (x) = f b (x)/Z b defined on a space X , where f b (x) can be computed efficiently for a given x ∈ X , and we are interested in estimating the partition function Z b . Annealed importance sampling (AIS) is an algorithm which estimates Z b by gradually changing, or "annealing," a distribution.

one must specify a sequence of K + 1 intermediate distributions p k (x) = f k (x)/Z k for k = 0, . . . K, where p a (x) = p 0 (x) is a tractable initial distribution, and p b (x) = p K (x) is the intractable target distribution.


We also establish efficiency and asymptotic performance guarantees that apply to agnostic reinforcement learning, where Q ⇤ does not necessarily lie in Q.

we consider the case where Q is the span of pre-specified indicator functions over disjoint sets.


Compared with many other sketching algorithms for dimension reductions such as stable random projections, CRS exhibits a significant advantage in that it is "one-sketch-for-all."

we demonstrate the effectiveness of CRS in efficiently computing the Hamming norm, the Hamming distance, the l p distance, and the χ 2 distance.


On the other hand, [12] shows that the rate n /( +p) is often asymptotically optimal in a minmax sense.

the latter is the case for H = W m 2 (X), f 2 W s 2 (X), and s 2 (d/2, m], that is, when using a Sobolev space as the underlying RKHS H, then all target functions contained in a Sobolev of lower smoothness s > d/2 can be learned with the asymptotically optimal rate n 2s 2s+d .


However, by restricting the noise in the bags to a class of "nice" distributions, we can show that the above algorithm converges to the global optima at a linear rate.

within a small number of steps the algorithm is able to learn the correct set of positive points in each bag and hence the optimal classifier w * as well.


In the full version of the paper, for the specific problem of portfolio selection, where f t (x) = − log(r t • x), we give a faster implementation whose per iteration running time is independent of the number of iterations, using the more sophisticated "online Newton method" of [HKKA06].

we have the following result: Theorem 4.


The proposed iFDM in Figure 1a can be particularized to resemble some other models that have been proposed in the literature.

we recover: i) the iFHMM in [23] by choosing the state space X = {0, 1}, x tm = s tm and L = 1, ii) the ICA iFHMM in [23] if we set X = R, L = 1 and assume that p(x tm |s tm = 1, x (t−1)m , s (t−1)m ) = p(x tm |s tm = 1) is a Gaussian distribution, and iii) a BNP counterpart of the LDS [9] with on/off states by assuming L = 1 and X = R, and letting the variables x tm be Gaussian distributed with linear relationships among them.


We build on the proof of this result to show that the class of n-bidder, m-item MBARPs has a pseudo-dimension of O m 3 log n . We note that when we analyze the class of MBARPs, we assume additive reserve prices, rather than bundle reserve prices.

each item has its own reserve price, and the reserve price of a bundle is the sum of its components' reserve prices, as opposed to each bundle having its own reserve price.


The resulting constraints are fixed by the joint distribution of the state x and the optimal actions u * .

they are fully determined by the structure of the optimization problem (1) that we wish to solve.


Just the sequence of actions, when replayed intelligently through a learned controller, leads to compelling path tracing performance.

sequences of actions can robustly be interpreted in context of current image observations without any explicit 3D reconstruction or localization.


With this in mind, we conducted a preliminary experimental evaluation of Algorithm 1, implemented using a popular algorithm for learning decision trees in place of the required ERM oracle.

we use the J48 algorithm from Weka v3.6.2 (with default parameters) to select the hypothesis h k in each round k; to produce the "alternative" hypothesis h ′ k , we just modify the decision tree h k by changing the label of the node used for predicting on x k . Both of these procedures are clearly heuristic, but they are similar in spirit to the required optimizations.


Our empirical results show that in practice the constants inside the Θ-notation are significantly tighter than the theoretical proof might suggest, and in fact feature hashing performs well for a larger scope of vectors.

for a synthetic set of generated bit-vectors, we show that whenever ε 2 δ the constant hidden by the Θ-notation is at least 0.75 (except for very sparse vectors, i.e.


Beyond Ω min 1/γ, n/V queries, , and hence, Theorem 3.2 implies that the misclassification rate of 4 From Answering Queries to Model-agnostic Private Learning In this section, we build on our algorithm and results in Section 3 to achieve a stronger objective.

we bootstrap from our previous algorithm an ( , δ)-differentially private learner that publishes a final classifier.


Bertsimas and de Ruiter [7] show that even the affine approximations of ( 1) and ( 4) (where recourse decisions are restricted to be affine functions of respective uncertainties) are equivalent.

we have the following Lemma which is a restatement of Theorem 2 in [7].


A key aspect of our framework is that the conditional posteriors of all the model parameters are available in closed form using data augmentation strategies that we will describe below.

since we model binary label matrix as thresholded counts, we are also able to leverage some of the inference methods proposed for Bayesian matrix factorization of count-valued data [27] to derive an efficient Gibbs sampler for our model.


Extending the adaptive feed-forward layer to RNNs, we presented an adaptive LSTM that significantly increases model capacity and statistical efficiency while being more robust to hyper-parameters.

we obtain new state of the art results on the Penn Treebank and the WikiText-2 word-modeling tasks, using ~20-30% fewer parameters and converging in less than half as many iterations.


The expected posterior risk contains a term for each variable Xi and each instantiation to its parents.

it contains a term for each of the parent variables U.


Here we automatically create arbitrarily many diverse agents by parameterizing one Go program.

we use different parametrizations of Fuego 1.1 [10].


In either case we denote u i = f (x i ), and in LIN:LIN also v i = g(x i ), and learn by attempting to minimizing the objective in (3), where (•) is again a continuous loss function such as the square root of the logistic.

we learn by optimizing the problem (3) with the additional constraint U = sign(W q X), and possibly also V = sign(W d X) (for LIN:LIN), where We optimize these problems by alternatively updating rows of W q and either rows of W d (for LIN:LIN ) or of V (for LIN:V ).


The survival function S(t) gives us the probability a patient survives up to time t, while the hazard function λ(t) gives us the instant risk of patient at time t. We define a Gaussian process prior over the hazard function λ.

we choose λ(t) = λ 0 (t)σ(l(t)), where λ 0 (t) is a baseline hazard function, l(t) is a centred stationary Gaussian process with covariance function κ, and σ is a positive link function.


The U-Tree algorithm retains the subtree of distinctions i at a leaf Lj if the sum of the KS statistical differences over the fringe nodes F(Lj , i) of the subtree is (1) larger than the sum of the KS differences of all other subtrees, and (2) exceeds some threshold 8.

the tree is extended from leaf L j with a subtree i of new distinctions if for all subtrees m -=I-i: and Whenever the tree is extended, the action values of the previous leaf node are passed on to the new leaf nodes.


Figure 2(b) shows that the performance of all the algorithms is consistent with that seen for both the uniform and beta payoff distributions.

while the mean regret per turn is somewhat higher than that seen for the uniform distribution, it is still lower than when payoffs are from the beta distribution.


. . , ω kN , for k = 1, . . . , K. These expectations are available in closed form and can thus easily be computed.

the expectation of each Pólya-Gamma variable ω kn is very efficient to compute and is available in closed form [20] E The M-step involves a maximization w.r.t.


But we did not consider this to be a good choice.

consider making the helicopter fly in the increasing -direction, so that $ ¤£ ¡ £ $£ ¡ §¦ p£ ¡ £ ( starts off as $ F ¡ ¡ ¡ ( (say), and has its first coordinate £ slowly increased over time.


For example, if X 1 and X 2 are independent given X 3 , X 4 , and We cast the problem of learning the structure of a deep neural network as a problem of learning the structure of a deep (discriminative) probabilistic graphical model, G dis .

a graph of the form where " " represent a sparse connectivity which we learn, and "→" represents full connectivity.


Meanwhile, we also add regularization terms over the missing features, c(M ij , 0), ∀(i, j) ∈ Ω, to avoid overfitting.

we use the least squared loss function c(x, y) = 1 2 x − y 2 .


Describing the contents of images is useful for automated image captioning and conversely, the ability to retrieve images based on natural language queries has immediate image search applications.

in this work we are interested in training a model on a set of images and their associated natural language descriptions such that we can later rank a fixed set of withheld sentences given an image query, and vice versa.


A unique feature of Quick Shift is that it has a segmentation parameter τ which allows practioners to merge clusters corresponding to certain less salient modes of the distribution.

if a local mode is not the maximizer of its τ -radius neighborhood, then its corresponding cluster will become merged to that of another mode.


We can justify this choice by noticing that it is the expectation of the conditional distribution P (h|v), namely where the sigmoid is applied to the vector in an element-wise fashion.

using the conditional expectation as the nonlinearity is a fundamental ingredient in the variational lower bound that justifies the greedy layer-wise procedure [2].


For the Ising model on the grid, the critical edge parameter for high-temperature is .

we are in high-temperature if and only if θ ≤ θ c , and we can reject the null hypothesis if the MPLE estimate θ > θ c . To generate departures from the null hypothesis, we give construction parameterized by τ ∈ [0, 1].


First, we note that there is an inverse relationship between the tail exponent q in (H2) and the decay rate β of the algorithm's step-size sequence γ n ∝ n −β .

higher values of q imply that the noise in the players' observations is smaller (on average and with high probability), so players can be more aggressive in their choice of step-size.


Final Pruning Before outputting event hypotheses, we perform a final round of pruning to remove some duplicate events.

we delete any event for which there is another higher-scoring event within 5 degrees distance and 50 seconds time.


On the blog selection and influence maximization applications, the proposed continuous optimization approach outperforms STOCHASTIC-GREEDY in terms of the running time/utility tradeoff.

for blog selection we can compute a solution with the same utility 26⇥ faster than STOCHASTIC-GREEDY with " = 0.5.


It is interesting to see that several well-known nonparametric estimators for classification and regression can be seen as special cases of this estimator (or a slightly more general form) for different choices of the output space, its metric and the loss function.

the approach shows a certain analogy of a generalization of regression into a continuous space (manifold-valued regression) and regression into a discrete space (structured output learning).


In order to reduce the model's computational cost, we also propose an alternative to computing the candidate alignment-plan matrix at every step.

we propose a model variant that reuses the Algorithm 1: Pseudocode for updating the alignment plan and commitment vector.


These can be uniquely defined as the solutions of the following Bellman equations: Now, consider the trajectory {(x t , a t )} underlying a learning agent, where x 1 is randomly chosen from P 0 , and define and π t (a|x) = P[a t = a|u t−1 , x t = x].

π t denotes the policy followed by the agent at time step t (which is computed based on past information and is therefore random).


In this work, we develop a fully nonparametric causal inference method to automatically discover causal relationships from purely observational data.

our proposed method does not require any a priori assumptions on the functional form of the interaction between the variables or the noise structure.


The method exploits the structure of the data to optimize the model by finding and frequently updating the most predictive decision variables.

for each i 2 [d] at time t, a lower bound r t i is derived (which we call the marginal decrease) on the amount by which the cost function will decrease when only the i th coordinate is updated.


the assumption of bounded output) for sake of presentation, while it is proved and presented in full generality in the Appendix.

the values of all the constants are given explicitly.


Given the network polynomial f , we can answer any query with respect to the belief network.

let e be an instantiation of some network variables, and suppose we want to compute the probability of e. W e can do this by simply evaluating the polynomial f while setting each evidence indicator λ x to 1 if x is consistent with e, and to 0 otherwise.


Because the pitch of natural speech does not change noticeably within a single frame, we model the corresponding normalized signal within a T-F unit by a single sinusoid to obtain the AM repetition rate.


In the introduction we argued that the visit count N n (x) (and consequently, N n (x, a)) is not directly useful in practical settings, since states are rarely revisited.

n n (x) is almost always zero and cannot help answer the question "How novel is this state?"


Moreover, one can prove that the weight vector β tends to zero as ε grows.

for ε ≥ 0.1 we have β ≈ 0, in which case the logloss approximates the deterministic value log(2) = 0.69.


We treat the ambiguity of e as a hidden class z over e and make use of the query log as a data source for mining the relationship among e, w and z.

the query log can be viewed as a multi-class, multi-label and multi-instance corpus {(X n , Y n )} n=1,2,...,N , in which each pattern X corresponds to a named-entity e and is characterized by a set of instances {x m } m=1,2,...,M corresponding to all the contexts {w m } m=1,2,...,M that co-occur with e in queries, and the label Y contains all the ambiguities of e. Our data was based on a snapshot of answers.yahoo.com crawled in early 2008, containing 216563 queries from 101 classes.


( 6), for its ease of implementation and practical effectiveness [35].

at iteration t, let β t c be the current value of β c . We compute f (β c )'s subgradient s t , then perform the following update, where the max( ) function implements the projection operation such that the update does not fall outside of the feasible region β c ≥ 0.


This learnt representation is far more precise than a rate code with independent Poisson spike trains [11].

shuffling the spike trains in response to identical inputs drastically degrades this precision.


In order to allow the time point at which the goals are achieved to be influenced by the optimization, we will re-frame the goal driven part of the problem in a canonical time and in addition to optimizing the controls, also optimize the mapping from canonical to real time.

we introduce into the problem defined by ( 1) & ( 2) the canonical time variable τ with the associated mapping with θ as an additional control.


Information regularization across multiple regions can be performed, for example, by minimizing the maximum information per region, subject to correct classification of the labeled points.

we constrain each region in the cover (Q ∈ C) to carry at most γ units of information.


To optimize (7), we utilize Monte Carlo methods to approximate expectations and stochastic gradient descent (SGD) for parameter optimization.

the gradients can be expressed as in the TSBN model, if we define vmt = σ(w 2m h t + w 4m v t−1 + c m ) and ĥjt = σ(u 1j h t−1 + u 2j v t + u 3j v t−1 + d j ), the gradients for w 2m and u 2j can be calculated as Other update equations, along with the learning details for the TSBN variants in Section 2.3, are provided in the Supplementary Section B.


As a result, the first key element of modeling question generation was to recognize the compositionality of these questions.

there are conceptual building blocks (predicates like size(x) and plus(x,y)) that can be put together to create the meaning of other questions (plus(size(Red), size(Purple))).


In particular, such an algorithm would imply a breakthrough result for the problem of learning k-juntas (functions that depend on only k n input variables; it is not known how to learn ω(1)-juntas in polynomial time).

we present an algorithm and a set of natural conditions such that any sparse polynomial f satisfying these conditions can be learned from random examples in time polynomial in n and 2 s . any f whose coefficients have been subjected to a small perturbation (smoothed analysis setting) satisfies these conditions (for example, if a Gaussian with arbitrarily small variance is added independently to each coefficient, f satisfies these conditions with probability 1).


3, with a metabolically motivated firing rate penalty corresponding to p = 1, using the procedure described in App.

e. for the same total RMS firing rate budget (obtained by increasing in Eq.


In many cases the posterior is ambiguous because the image is (ie, multiple people are present).

it may be surprising that the pair in the bottom-right both are recognized by the region model -this suggests that the the iter-region dissimilarity learned by the color histograms is a much stronger than the foreground similarity.


We established nearly tight finite-sample convergence guarantees showing that NNQL can accurately estimate optimal Q function using nearly optimal number of samples.

our results state that the sample, space and computational complexities of NNQL scale polynomially (sometimes linearly) with the covering number of the state space, which is continuous and has uncountably infinite cardinality.


In many relevant settings, including the high-temperature regime, the dynamics are rapidly mixing and hence offer an efficient way to sample from Ising models.

the mixing time in η-high-temperature is We may couple two executions of the Glauber dynamics using a greedy coupling (also known as a monotone coupling).


Experimental results demonstrate that the proposed scheme can outperform previous spatial-domain counterparts by a large margin.

it can achieve a compression ratio of 8.4× and a theoretical inference speed-up of 9.2× for ResNet-110, while the accuracy is even better than the reference model on CIFAR-10.


Critically, the response scaling parameter, γ, has been shown to bear a relationship to decision threshold placement in EBRW [18].

equation 1 is equivalent to EBRW's mean response (averaged over many trials) with decision threshold bounds placed γ units away for the starting point for evidence accumulation.


The number of projection vectors required by our approach is not only independent of the size of the dataset but is also fixed, depending only on the number of classes.

we show that for any K class labeled dataset with independent subspace structure, only 2K projection vectors are required for structure preservation.


In this way, each sub-model could independently identify their optimal sub-assignments.

for example, we can use the parameters: Since sub-models M i and M j become independent after relaxing the single equivalence constraint X i ≡ X j , computing these parameters is sufficient to reconstruct the MAP solution for the original model M. we have that θ( We propose then that the parameters of a compensation, with MAP values c-map(.


This can be interpreted graphically, as the level-order cost curves lie further away from the origin as cost increases (Figure 2a).

the cost function can always be used as a proxy for either error rate.


But unlike previous works, we use minibatches as a theoretical tool to improve convergence rates of both nonconvex PROXSVRG and PROXSAGA.

by carefully selecting the minibatch size, we can improve the IFO complexity of the algorithms described in the paper from O(n/✏) (similar to PROXGD) to O(n 2/3 /✏) (matching the smooth nonconvex case).


As in [7], we use contrast filters that have spatial characteristics and contrast response properties closely mirroring well-known characteristic receptive-fields of LGN neurons [9].

we use a bank of second-order Gaussian derivative filters that span multiple octaves in spatial scale, that have peak sensitivity approximately inverse to filter size and that have contrast gain properties independent of size.


The idea here is to allow the learner to preemptively "guess" the gradient at the next step and optimize for this in the FTRL update.

if e g t+1 is an estimate of the time t + 1 gradient g t+1 based on information up to time t, then the learner should play: This optimistic FTRL algorithm admits the following guarantee: Lemma 4 (Lemma 1 [Rakhlin and Sridharan, 2013]).


By using the kernel embeddings of distributions [9], we can effectively represent the distributions without density estimation while preserving necessary distribution information.

the latent SMMs map each distribution into a reproducing kernel Hilbert space (RKHS), and find a separating hyperplane that maximizes the margins between distributions from different classes on the RKHS.


This is effectively what is accomplished using SBL as will be demonstrated below.

we will derive necessary conditions required for a non-degenerate basic feasible solution to represent a local minimum to L(γ).


In order to adapt one domain to the others, one approach is to consider a penalty proportional to the distance between each distribution and the weighted mean of the rest.

let P s be the distribution over data points x in domain D s , and P /s = 1 S 1 P S s 0 =1 w ss 0 P s 0 the distribution of data from all other domains D ws /s .


We discuss the interaction between the parameters of the dynamics (learning rate and averaging rates) and the covariation of the noise process.

we show how the asymptotic rate of covariation affects the choice of parameters and, ultimately, the convergence rate.


The mGP provides a flexible framework for characterizing the dependence structure of real data, such as the examined MEG recordings, capturing certain features more accurately than previous models.

the mGP provides a hierarchical functional data analysis framework for modeling (i) strong, locally smooth sharing of information, (ii) global long-range correlations, and (iii) abrupt changes.


All three representations have been shown to be effective on certain object recognition tasks.

recent work [6] has shown that part-based models outperform global templates and bag-of-words on challenging object recognition tasks.


An event E occurs in the history Φ i , written Φ i E, if and only if some component occurs in that history: ∃e ∈ E : Φ i e. Events can therefore be thought of disjunctively.

they specify a set of possible state-action transitions from a Dec-POMDP, local to one of its agents.


In the cases where CMB fails to identify all the PC nodes, global causal discovery methods cannot identify them either.

structures failing to satisfy Lemma 3 can have different orientations on some edges while preserving the skeleton and v-structures, hence leading to Markov equivalent structures.


This will capture the amount by which a agent i can affect the game in the eyes of agent j.

it is the maximal amount by which an agent i can affect the transition distribution of agent j by changing i's action at each state s: Definition 4 (Influence).


We are interested in a theoretical explanation of the good performance of (RF) 2 presented in the previous section.

we derive the theoretical performance bounds of a two-class (RF) 2 , for simplicity.


Non-convergent dynamics: Outside the class of congestion games, there exist several negative results in the literature concerning the non-convergence of MWU and variants thereof.

in [15] it was shown that the multiplicative updates algorithm fails to find the unique Nash equilibrium of the 3 × 3 Shapley game.


( 10) with respect to f is equivalent to minimizing serves as a good critic of f (x) when it makes the cost g(x){µ w (x)f (x) − 2µ z (x)} larger for x at which f makes a larger error |µ w (x)f (x) − 2µ z (x)|.

g maximizes the objective above when g(x) = µ w (x)f (x) − 2µ z (x) for any f , and the maximum coincides with the least-squares objective in Eq.


Since we never know the real property of the (unknown) probability space, it is more likely the requirement on the probability space for the consistency with the top-k true loss can be satisfied than that for the top-l (l > k) true loss.

it is risky to assume the requirement for the permutation-level 0-1 loss to hold.


. . , y N } is a probability measure on 2 Y , the set of all subsets of Y. P is called a determinantal point process (DPP) if there exists a positive semidefinite matrix K indexed by the elements of Y such that if Y ∼ P then for every A ⊆ Y, we have Determinantal Point Process: Here K A = [K ij ] yi,yj ∈A is the restriction of K to the entries indexed by elements of A, and we adopt det(K ∅ ) = 1.

we will refer to K as the marginal kernel, as it contains all the information needed to compute the probability of including any subset A in Y ∼ P. A few simple observations follow from Equation (1): the diagonal of K gives the marginal probabilities of inclusion for individual elements of Y, and the off-diagonal elements determine the (anti-) correlations between pairs of elements: large values of K ij imply that i and j tend not to co-occur.


Assumption A holds with κ = 0, one can easily verify that for n ≥ s there exists a filter See [15, Appendix C] for the proof.

this implies that x can be recovered efficiently from observations (1): if instead of the filtering problem we were interested in the interpolation problem of recovering x t given 2n + 1 observations y t−n , ..., y t+n on the left and on the right of t, Assumption A would imply a kind of simplicity of x.


While the ✓ (c) j⌫ 0 is drawn as in (6), assigning ✓ ji } are executed as they would be in ( 4) and ( 5), without an MH step.

a construction like ( 6) is used to assign elements of {✓ ⇤ i } to spikes, but after this step the update equations for {U (c) ji } are implemented as in the original (conjugate) model.


To address this issue, we propose a hybrid optimization scheme.

for a small constant ǫ > 0, if is not empty, we switch the iteration to the following gradient descent step (GD-step): where α is a step size chosen by the following line search procedure: for constants α 0 > 0, β, γ ∈ (0, 1) and m = 0, 1, • • • , find the smallest integer m with α = α 0 β m such that the following inequality holds: The detailed steps of the algorithm are presented in Algorithm 1.


. . , g), for any g ∈ H. , and b = U 2 2η are constants.

if Essentially the same theorem holds for SILK, but now with a slightly larger constant a = 4 η(1 In addition, denote g * the minimizer of the batch learning cumulative risk t R(x t , y t , g), and f * the minimizer of the minimum expected risk with R(f * ) := min f E (x,y)∼P (x,y) R(x, y, f ).


We reported the emergence of cell models exhibiting strongly nonlinear behaviour.

we demonstrated that the estimated cells were essentially computing products between outputs of two linear filters that had V1 simple cell characteristics.


[2016] and is presented in Appendix D. Our first lemma, proved in Appendix B.1, confirms that if h✓ h0i , ✓ ⇤ i > 0, then h✓ hti , ✓ ⇤ i > 0 for every t and w hti 1 2 (0.5, 1).

the estimates of the Population EM 2 remain in the correct hyperplane, and the weight moves in the right direction, too.


Many previous works focused on the settings when only unlabeled data from the target domain are available [Huang et al., 2006, Sugiyama et al., 2008, Yu and Szepesvári, 2012.

a line of research has been established based on distribution discrepancy, a loss induced metric for the source and target distributions [Mansour et al., 2009, Ben-David et al., 2007, Blitzer et al., 2008, Cortes and Mohri, 2011, Mohri and Medina, 2012.


. . , W d ∈ R n d−1 ×n d , in which n 0 = n and n d = c. Discussions about networks with more advanced architectures like convolutions, pooling and skip connections can be directly generalized [1].

we have in which w k = W d [:, k] and σ is the nonlinear activation function.


[12,13,14,15,16] and references therein) and the phenomenology, that we shall review briefly here, is well known.

for β small enough, the global minimum of the Bethe free energy corresponds to the so-called paramagnetic state As we increase β, above a certain value β R , the model enters a retrieval phase, where the free energy has local minima correlated with the factors X and Y .


However, there exists a simple reparameterization of the spike and slab prior that is more amenable to approximate inference methods.

assume a Gaussian random variable w qm ∼ N ( w qm |0, σ 2 w ) and a Bernoulli random variable s qm ∼ π sqm (1 − π) 1−sqm .


Despite sharing the multi-branch principle, our ONE method is fundamentally different from such existing methods since our objective is to improve the training quality of any target network, but not to propose a new multi-branch building block.

our method is a meta network learning algorithm, independent of the network architecture design.


For our validation score, we choose a function of the form: where g is an L-Lipschitz loss function.

the logistic loss and the hinge loss are 1-Lipschitz, whereas the 0/1 loss is not L-Lipschitz for any L. Other examples of 1-Lipschitz but non-convex losses include the ramp loss: g(w, x, y) = min(1, max(0, 1 − yw x)).


We initially compute R(  We split the computation of the inverse centrality R(i) into two terms, namely R(i) = T (i) + S(i), where T (i) and S(i) are the sum of the resistances of vertex i to each descendant and nondescendant, respectively.

we compute κ(i) and T (i), i = 1, . . . , n with the following leaves-to-root recursions by computing κ(1) then T (1) and caching the intermediate values.


Each video is represented using three modalities, i.e., static appearance, motion, and auditory.

three visual and audio local keypoint features are extracted for each video: scaleinvariant feature transform (SIFT) [18], spatial-temporal interest points (STIP) [19] and melfrequency cepstral coefficients (MFCC) [20].


Using ACNN to incorporate the side information can improve the performance more significantly.

for ACNN 2-conv, the error rate decreases 0.94% (11% relatively) from 8.49% to 7.55%, while the error rate decreases 0.65% (18% relatively) from 3.57% to 2.92% for ACNN 4-conv.


Then , where log h(x) B k and T (x) B k are the statistics that we pre-compute for (8).

if and the statistics log h(x) B k = 0 and T (x) B k = ( x B k , xx T B k ) can be pre-computed.


Second, in terms of classification, the DualSGD-Hinge and DualSGD-Logit outperform other methods for almost all datasets except the poker data.

the DualSGD-based methods achieve the best mistake rates 4.83±0.21, 8.35±0.20, 19.28±0.00 for the cod-rna, ijcnn1 and airlines data, that are, respectively, 32.4%, 11.3%, 8.8% lower than the error rates of the second best modelstwo recent approaches FOGD and NOGD.


First, with the proper rescaling we show that �R N ,w (x), R N ,w (x � )� ≈ κ S (x, x � ).

then, we also show that the functions obtained by composing bounded linear functions with R N ,w are approximately the bounded-norm functions in H S . the functions expressed by N under varying the weights of the final layer are approximately bounded-norm functions in H S . For simplicity, we restrict the analysis to the case k = 1.


The second group of methods, including Gibbs sampling [28] and sequential Monte Carlo (SMC) [11,24], has the benefit of being asymptotically consistent.

under mild assumptions they can often be shown to converge to the correct solution if simply given enough compute time.


In kernel PCA, the encoding linear operator applies to the s-th tensor power x ⌦s of the data.

the encoding and decoding are parameterized by a linear operator A 2 R k⇥d s , H pca k,s = (h A , g A ) : h A (x) = Ax ⌦s , , g A (y) = A † y , where A † denotes the pseudo-inverse of A.


S l are generated with a deterministic pseudo random number generator from a preassigned random seed.

we can reproduce S l just from the original random seed.


In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical performance.

we derive sublinear (O(1/t)) convergence on general smooth and convex objectives, and linear convergence (O(e −t )) on strongly convex objectives, in both cases for general sets of atoms.


: inc(X) ≤ K max{r,log n} n . Lemma 3.3.

we have  a choice of ν 3 = (6inc3(X)) p (2deg 3 (Y )) 1−p , ν 1 = (6inc1(X)) p (2deg 1 (Y )) 1−p for any p ∈ [0, 1] is a valid choice that guarantees exact recovery.


A second synthetic dataset consists of a full matrix of pairwise comparisons derived from a given ranking of n items, with added uncertainty for items which are sufficiently close in the true ranking of items.

given a positive integer m, we let C In Figure 2, we measure the Kendall ⌧ correlation coefficient between the true ranking and the retrieved ranking, when varying either the percentage of corrupted comparisons or the percentage of missing comparisons.


The observation process N is a marked point process [8] defined on [0, ∞) × R m , meaning that each point, representing the firing of a neuron, is identified by its time t ∈ [0, ∞), and a mark θ ∈ R m . In this work the mark is interpreted as a parameter of the firing neuron, which we refer to as the neuron's preferred stimulus.

a neuron with parameter θ is taken to have firing rate in response to state x, where H ∈ R m×n and Σ tc ∈ R m×m , m ≤ n, are fixed matrices, and the notation y 2 M denotes y T M y.


We address this challenge by developing a dual formulation for (5).

given a convex and differentiable function (z), we can rewrite it in its convex conjugate form as (z) = max α∈Ω αz − * (α) , where * (α) is the convex conjugate of (z) and Ω is the domain of dual variable [4].


By the symmetry it is now clear that the pivoting condition of u and w with respect to v implies the pivoting condition of u and v with respect to w and of v and w with respect to u.

the pivoting condition is a property of the triplet {u, v, w}.


The massive number of templates required for high coverage is a serious impediment for the template approach because each template application requires solving a subgraph isomorphism problem.

it takes on average 7 seconds to apply the 5006 templates to a test instance, while our method takes less than 50 ms, about 140 times faster.


Our main contribution is a characterization of a useful class of skills based on a graphical representation of the agent's interaction with its environment.

we use betweenness, a measure of centrality on graphs [1,2], to define a set of skills that allows efficient navigation on the interaction graph.


A subtlety worth noting is that, for the case of known distributions, the complexity may be affected by the choice of the oracle.

some calculations may be "pushed" inside the oracle.


A set X ⊂ M, where is called geodesically convex if any two points of X are joined by a geodesic lying in X .

if x, y ∈ X , then there exists a path γ : [0, 1] → X such that γ(0) = x and γ(1) = y.


The trade-off parameters of OR-PCA are fixed as λ 1 = λ 2 = 1/ √ p. The performance is evaluated by the similarity between the subspace obtained from the algorithms and the groundtruth.

the similarity is measured by the Expressed Variance (E.V.)


However, it is unclear whether a universal kernel can be constructed from persistence landscapes in a way similar to the definition of k U σ .

we are not aware of a proof that the construction of persistence landscapes, considered as functions in L 2 , is continuous with respect to d W,q for some 1 ≤ q ≤ ∞.


We introduce artificial dynamics simply as a constructive strategy for obtaining samples of a sequence of distributions converging to the distribution of interest.

initially we sample from and easy-to-sample distribution.


We treat skip-thoughts in the framework of encoder-decoder models 1 .

an encoder maps words to a sentence vector and a decoder is used to generate the surrounding sentences.


Then we have which is a standard result from the singular value decomposition (SVD) [12].

the condition of ||∆Aw|| 2 ≤ λ||w|| 2 is equivalent to the condition that the maximum eigenvalue of ∆A is upper bounded by λ.


Since r < δ Hp = H p (δ up ) ≤ δ up , by Lemma 3.1 we have, for some M up > 0.

now, since r and r − M up r 3 are both monotonic for 0 ≤ r ≤ δ up , we have (see H(r, δ up ) ≤ r < δ Hp = H p (δ up ), i.e, H(r, δ up ) < H p (δ up ).


This allows us to perform Rao-Blackwellization, conditioning on the samples at the neighbors of x s rather than using x s 's samples directly.

using this trick we can often get much higher quality estimates from the inference for small N . if the variable state spaces are sufficiently small that they can be discretized (for example, in 3 or fewer dimensions the discretized domain size d may be manageable) but the resulting factor domain size, d b , is intractably large, we can evaluate (4) on the discretized grid for only O(dN b−1 ).


For any α+1 , which is monotonic increasing in α in the valid range [0, 1].

1 = Ψ((1, 0); 0) < Ψ((1, 1); 0) = 3 2 and the estimator is biased towards being axis-aligned.


For the HDP-SLDS, we additionally sample R. Sampling z 1:T As shown in [5], the mixing rate of the Gibbs sampler for the HDP-HMM can be dramatically improved by using a truncated approximation to the HDP, such as the weak limit approximation, and jointly sampling the mode sequence using a variant of the forward-backward algorithm.

we compute backward messages m t+1,t (z t ) ∝ p(y t+1:T |z t , y t−r+1:t , π, θ) and then recursively sample each z t conditioned on z t−1 from where ).


This is in large part because notions of individual fairness require making stronger assumptions on the setting under consideration.


To verify that the improved results for NLTop are not the result of an increased number of parameters, we additionally trained another DeepStruct model containing more parameters, which is called DeepStruct++ in Table 2.

we fixed the original DeepStruct potentials and learned two additional 2-layer multilayer perceptrons that further transformed the unary and pairwise potentials.


To make use of the fingerprinting lemma, we consider a variant of Algorithm 2 that does not truncate the quantity a j − p j to the range ±2 when computing the score z j i for each element i.

we consider scores based on the quantities We prove two main statements: first, that these untruncated scores are equal to the truncated ones with high probability as long as the mechanism's answers are accurate.


All these algorithms are locally-optimal search procedures, and thus sensitive to initial values.

if nodes from two different roles are initialized to have the same role, the output is likely to merge all nodes in both roles into a single role.


Let us say that v is a branching node if it has more than one child.

(we partition the nodes of the tree into three disjoint categories: leaves, single-child nodes, and branching nodes.)


In reality many experiments will finish early and we can adaptively exploit such situations.

rather than follow the fixed offline schedule we may choose to use fewer labs and hence improve CPE.


There is no compelling reason for these updates to depend at all on these magnitudes because the weights in the top layer already ensure appropriate scaling.

for each doubling of the magnitudes of the targets, the updates to the lower layers quadruple for no clear reason.


As we shall see (in Section 4), our algorithms converge much faster than other competing methods for a variety of problems.

on several machine learning datasets, our methods are at least twice as fast as Newton's methods in finding a high-precision solution while other methods converge slowly.


Graphical models provide a convenient means of representing conditional independence relations among large numbers of random variables.

each node s in a graph is associated with a random variable x s , while the set of edges E is used to describe the conditional dependency structure of the variables.


Our goal is to estimate the posterior distribution of the latent utility function given the discrete data.


m ∈ α 1 and m ∈ α 2 , and 2. the corresponding variable assignments are different, i.e.

y 1 m = y 2 m . we put an edge between all pairs of nodes that correspond to inconsistent assignments.


Our algorithm is different from most previous work in that we are explicitly modeling the opponent's learning algorithm and not simply his current policy.

we would like to model players from H ∞ × B 0 .


While classical, this empirical approach is unsatisfactory for tree posterior estimation due to the combinatorially exploding size of tree space.

sRF does not support trees beyond observed samples (i.e., simply sets the probabilities of unsampled trees to zero), and is prone to unstable estimates for low-probability trees.


If W has covariance matrix Σ, then (W, W x) has covariance , and V ar(W |W x) = Σ − Σxx T Σ x T Σx whenever x T Σx = 0 [15].

the conditional variance is the Schur complement of the vector variance x T Σx in the full covariance matrix V . By substituting sample covariance matrices for their population counterparts, we arrive at a new deflation technique: Schur complement deflation Schur complement deflation, like projection deflation, preserves positive-semidefiniteness.


In order to ensure the identifiability of the model in KCDC, we need to ensure that we apply the same kernel to the response variable when computing the KCDC measures S X→Y and S Y →X .

we fix a kernel k r with some fixed hyperparameters and apply it both to Y when computing S X→Y and to X when computing S Y →X .


The key idea behind using Lyapunov functions to show stability of the system (1) is similar to that of gradient descent on strictly quasiconvex functions: if one can show that, given a policy π, applying the dynamics f on the state maps it to strictly smaller values on the Lyapunov function ('going downhill'), then the state eventually converges to the equilibrium point at the origin (minimum).

the assumptions in Theorem 1 below imply that v is strictly quasiconvex within the region of attraction if the dynamics are Lipschitz continuous.


We partially labelled some of the images.

for 7 out of 25 images we annotated the car views using bounding boxes (Figure 3).


Weight perturbation operates by explicit exploration of the entire M N -dimensional weight space; only one component of a particular update will be in the direction of the true gradient for a given input, while the other components can be viewed as noise masking that signal.

an update can be written as ∆W = ∆W (the 'learning signal', the actual gradient) + (∆W − ∆W ) (the 'learning noise'), where the average is taken over all values of ψ.


no eigenvalue of R appears in more than one Jordan block.


Let T be the corresponding "true" transition operator, which maps the t-th sample X to the t + 1-th in that chain.

t (X t |X t−1 ) = P(X t | X)C( X|X t−1 )d X. T produces P(X) as asymptotic marginal distribution over X (as we consider more samples from the chain) simply because P(X) is the marginal distribution of the joint P(X)C( X|X) to which the chain converges.


The algorithms in [8,10,11] can also be considered variants of SPA, with different deflation procedures and pre-/post-processing.

the algorithm in [8] avoids normalization -for real-word data, normalization at the factorization stage may amplify noise and damage the good conditioning of the data matrix brought by pre-processing, e.g., the tf-idf procedure [8].


Note, that M ij = x i • e j , where e j is the j-th unit vector.

the j-th target is a linear function that picks the j-th column out of M. It is important to understand that the matrix M, which we call the problem matrix, specifies n learning problems: In the jth problem each of the n instances (rows) are labeled by the jth target (column).


• Comparing the left and middle panels of Figure 2 implies that having more users is more informative than having more items.

if the total dataset size M × N is equal, the performance tends to be better when M > N (left panel of Figure 2) than M < N (middle panel of Figure 2).


The UCCRL algorithm receives (bounds on) the Hölder parameters L as α as inputs.

if these parameters are not known, then one can still obtain sublinear regret bounds albeit with worse dependence on T . we can use the modelselection technique introduced in [17].


In contrast, an email network is more naturally viewed as a network of actors exchanging documents, i.e., actors are associated with nodes while documents are associated with edges.

an email network defines a multinetwork in which there may be multiple edges (one per email) between any pair of actors.


Unfortunately, this exponential mapping would distort the relationship between perturbations in the latent state (whose size is set by the covariance matrix Q) and the resulting fluctuations in firing rates.

the size of firing-rate fluctations would grow exponentially with the mean, an effect that would then add to the usual linear increase in spike-count variance that comes from the Poisson output distribution.


We say that an algorithm A has β-approximate max-information of k if for every distribution S over n-element datasets, I β ∞ (S; A(S)) ≤ k, where S is a dataset chosen randomly according to S. We denote this by An immediate corollary of our definition of approximate max-information is that it controls the probability of "bad events" that can happen as a result of the dependence of A(S) on S. Theorem 4.

let S be a random dataset in X n and A be an algorithm with range Y such that for some β ≥ 0, P[(S, We remark that mutual information between S and A(S) would not suffice for ensuring that bad events happen with tiny probability.


Without loss of generality, we can assume that w (a) is such that i w (a) i = 1 for all a, 1 ≤ a ≤ r. Given this normalization, RANKCENTRALITY estimates w (a) as stationary distribution of an appropriate Markov chain on G. The transition probabilities are 0 for all (i, j) / ∈ E. For (i, j) ∈ E, they are function of Pa .

transition matrix p(a) = [p Finally, p(a) i ] be a stationary distribution of the Markov chain defined by p(a) .


For the vector case, this property represents a sufficient condition such that log(R) can be defined as a real-valued matrix.

under property (P1), there exists a (not necessarily unique 2 ) matrix log(R) ∈ M d (R) whose matrix exponential e log(R) = R; cf., Culver [1966], Higham [2014].


Then, expected error of the coded linear inverse is strictly less than that of uncoded: where J 2 and J 4 are the submatrices of FΛF := J 1 J 2 J 2 J 4 n×n and the matrix Λ is defined in (8).

(J 1 ) k×k is GΛG , (J 2 ) k×(n−k) is GΛH , and Consider an alternative way of doing linear inverse using n > k workers.


In this paper, we present an end-to-end generative model that jointly synthesizes 3D shapes and 2D images via a disentangled object representation.

we decompose our image generation model into three conditionally independent factors: shape, viewpoint, and texture, borrowing ideas from classic graphics rendering engines [Kajiya, 1986].


The estimation of low-rank matrices arises in various contexts, including principal component analysis, spectral clustering, collaborative filtering, and matrix completion.

consider the class of matrices Θ ∈ R k×m that have rank r ≤ min{k, m}.


We also make the surprising observation that for ⇣ = 0, the singular value distribution of f (W X) is the same as the singular value distribution of X.

the eigenvalues of the data covariance matrix are constant in distribution when passing through a single nonlinear layer of the network.


As BHCD runs, several merges can have the same score.

at the first iteration all pairs of vertices connected by an edge have the same score.


Our distributed representation, in which neurons jointly encode the sampling trajectory, provides a potential solution to this problem.

it allows several chains to be embedded in a single network.


Finally, we perform an experiment to test whether the DeepGenMSM is able to generate genuinely new configurations that do exist for Alanine dipeptide but have not been seen in the training data.

can the generator "extrapolate" in a meaningful way?


Apart from its theoretical advantages, the Radon machine also has several practical benefits.

it is a black-box parallelisation scheme in the sense that it is applicable to a wide range of machine learning algorithms and it does not depend on the implementation of these algorithms.


It suffices to set γ t = q(Y )/p(Y ), where p is the distribution of Y t and q is any distribution with mean X * .

the choice Ỹt := γ t (x t )x t x t with γ t (x) = N (x|0, X * )/N (x|0, Σ t ) satisfies this.


Draw two independent samples Z 1 , Z  [Rakhlin et al., 2015] as well as their data-dependent counterparts [Rakhlin et al., 2011].

the final ingredient needed for expressing our learning guarantees is the notion of discrepancy between target distribution and the distribution of the sample: The discrepancy is a natural measure of the non-stationarity of the stochastic process Z with respect to both the loss function L and the hypothesis set H. note that if the process Z is i.i.d., then we simply have = 0 provided that q t s form a probability distribution.


Applying a linear filter to an image amounts to a two-dimensional convolution of the image with the impulse response of the filter.

if I is the input image, H the impulse response, i.e.


Since the optimal solution for the unconstrained problem has 2 norm no more than 1/λ (see the full version for a proof), we will focus on optimizing θ over the constrained set C = {a ∈ R p | a 2 ≤ 1/λ}, which will be useful for bounding the 1 sensitivity of the empirical loss.

before we formally introduce the covariance perturbation algorithm due to [19], observe that the optimal solution θ * can be computed as θ * only depends on the private data through X y and X X.


Thus, each computation unit generates an m-dimensional vector where m is the number of weight vectors (W's rows) or neurons.

a convolution layer embodies an embedding of text regions, which produces an m-dim vector for each text region.


While the motivating optimization problem (Q) is similar to that considered in the COSSO (Lin and Zhang, 2006) for smoothing splines, the SpAM backfitting algorithm decouples smoothing and sparsity, through a combination of soft-thresholding and smoothing.

spAM backfitting can be carried out with any nonparametric smoother; it is not restricted to splines.


The optimal policy π * for an MDP (S, A, T, H, s(0), R) is the policy that maximizes the expected sum of rewards.

the optimal policy is given by π The linear quadratic regulator (LQR) control problem is a special class of MDPs, for which the optimal policy can be computed efficiently.


How to enforce such a division in a decentralized and distributed way, is tackled in this paper, using a spiking neuron network architecture.

a spatio-temporal model called SpikeAnts is shown to enforce the emergence of synchronized activities in an ant colony.


In addition, the hierarchical structure on β (t) kt+1kt is similar to the one in Zhou (2015), which provides an intrinsic shrinkage mechanism on β (t) kt .

each k t is expected to be sparsely connected by a subset of k t+1 .


We note that n 0 is exactly the minimum number of samples required for structured signal recovery from linear measurements when using convex regularizers [3,1].

the optimization problem succeeds at recovering an unknown weight vector w * with high probability from n observations of the form y i = ⟨a i , w * ⟩ if and only if n ≥ n 0 .


While most recent work has focused on studying the properties of simple well-known norms, we take the opposite approach in this paper.

assuming a given prior knowledge, how can we design a norm that will enforce it?


As a first step we wish to find its projection P x (Z) onto the tangent space.

we wish to find the three matrices M , N 1 and N 2 such that P projecting onto A's columns, denoted P A , is exactly equal to AA † We can similarly define P A ⊥ ,P B and P B ⊥ .


The first equation of ( 4) implies that both U k and V k are bounded and the upper bound decays when the λ increases.

as long as λ is not too close to zero, then the RHS in (4) gives a meaningful bound which will be used for the convergence analysis of local search algorithms in next section.


First, we note that it is easy to extend CRH to learn nonlinear hash functions via the kernel trick [26].

according to the generalized representer theorem [27], we can represent the projection vectors w x and w y as where φ x (•) and φ y (•) are kernel-induced feature maps for modalities X and Y, respectively.


We establish a new connection between value and policy based reinforcement learning (RL) based on a relationship between softmax temporal value consistency and policy optimality under entropy regularization.

we show that softmax consistent action values correspond to optimal entropy regularized policy probabilities along any action sequence, regardless of provenance.


Unlike the   In a final experiment we tested the ability of DS-PSRL algorithm in continuous state and action domains.

we implemented the discrete infinite horizon linear quadratic (LQ) problem in Szepesvári [2015, 2011]: x t+1 = A ⇤ x t + B ⇤ u t + w t+1 and c t = x T t Qx t + u T t Ru t , where t = 0, 1, ..., u t 2 R d is the control at time t, x t 2 R n is the state at time t, c t 2 R is the cost at time t, w t+1 is the 'noise', A ⇤ 2 R n⇥n and B ⇤ 2 R n⇥d are unknown matrices while Q 2 R n⇥n and R 2 R d⇥d are known (positive definite) matrices.


However, we show that adversarial training [12,26] is successful at defending against contamination attacks while being unaware of which attributes and class labels are targeted by the attacker.

the attack is mitigated by training a model that is independent of information that is specific to individual parties.


Suppose the weak-learning condition is given by (C, B), If Booster employs the OS algorithm, then the average potential of the states never increases in any round.

loss suffered after T rounds of play is at most (0).


Define then we have Since we only have finite samples, the expectations in Proposition 1 have to be approximated by empirical moments.

they are computed by averaging over indices j = 1, 2, . . . , n. For each permutation (a, b, c) ∈ {(2, 3, 1), (3, 1, 2), (1, 2, 3)}, we compute The statement of Proposition 1 suggests that we can recover the columns of C c and the diagonal entries of W by operating on the moments M 2 and M 3 .


Without manageable efficient updates at each round, learning the task relationship matrix automatically may impose a severe computational burden.

we need to make predictions and update the models in an efficient real time manner.


In this work we investigate compound HD (hierarchical-deep) architectures that integrate these deep models with structured hierarchical Bayesian models.

we show how we can learn a hierarchical Dirichlet process (HDP) prior over the activities of the top-level features in a Deep Boltzmann Machine (DBM), coming to represent both a layered hierarchy of increasingly abstract features, and a tree-structured hierarchy of classes.


Building upon the recently suggested probabilistic interpretation of CCA [3], we propose a nonparametric, fully Bayesian framework that can deal with each of these issues.

the proposed model can automatically select the number of correlation components, and effectively capture the sparsity underlying the projections.


Most work in the area of learning from private data forms an intrinsic analysis.

a complex algorithm is run by the owner of the data, directly on that data, and a single output is produced which appropriately indicates the desired parameters (modulo noise).


More precisely, we set T = u t+1 q ⊗ u q ⊗ u q + =q σ u ⊗ u ⊗ u in (4) and then find optimal u t+1 q by minimizing the least squares objective given by (4).

each inner iteration is a simple least squares problem over the known entries, hence can be implemented efficiently and is also embarrassingly parallel.


In this paper we take a different approach in order to jointly learn the processed fluorescence signal and the inverse covariance estimate.

we convolve the fluorescence time series f i with a kernel α and then pass the convolution through the logistic function σ(x), Note for α 0 = −α 1 (and α k = 0 otherwise) this convolution filter approximates the threshold filter in Equation 3.


In contrast to these studies, however, we propose that model averaging behavior is abandoned once the observer has committed to a particular hypothesis.

subsequent perception is conditioned only on the chosen hypothesis, thus sacrificing optimality in order to achieve self-consistency.


The goal is to learn an influence function F that accurately captures the diffusion process.

accuracy of the learned influence function is measured in terms of the squared error with respect to the true model: the expectation is over the seed set and the randomness in the diffusion process, but not the data loss process.


This turns out to be fundamentally different to our noisy setting.

the results of [8] state that exact recovery is impossible, and even with approximate recovery, a huge number of tests (i.e., higher than polynomial) is needed unless = O q , where q max is the maximum allowed reconstruction error measured by the Hamming distance, and is maximum adversarial noise amplitude.


Second and more importantly, we provide a tighter bound on the maximum correlation of the incorrect vectors.

in each iteration of the OMP algorithm, there are n − k possible incorrect vectors that the algorithm can choose.


Given how well EXP does as an approximate recognition model for DBM, and EXP does not bother to detect change points or modify its learning rate in response to detected change points, there might exist a generative model for which EXP would be an exact Bayesian recognition model.

one might consider a model that assumes the underlying real-valued hidden variable to undergo persistent stochastic changes with constant noisy characteristics, such as a Gaussian process, which then gives rise to noisy binary or categorical observations.


7 and 8, respectively, at each level, into coordinate blocks of equal size is beneficial.

we decompose the mixture model into blocks containing equal numbers of BNs.


In fact, if ω * = 0, and if Assumption A is not fulfilled, then 3 ∃ g ∈ ∆P * such that A T g = 0.

given the matrices A and C, there are two distinct vectors θ 1 and θ 2 in ∂P such that A T θ 1 = A T θ 2 .


The Supplementary Material goes in detail about the preprocessing, including R/MATLAB scripts to generate the data.

the we generated data from "interventions" simulated from a model that was fitted to an actual study on premature infant development [3], where the dose is the number of days that an infant is assigned to follow a development program and the outcome is an IQ test at age 3.


2 The functional relationship between ✓ and its joint counterpart e ✓ = e ✓ i,pa(i) (x) is mediated through a kernel-dependent conditioning operation ⌧  ( e ✓) = ⌧  ( e w, e µ µ µ, e B) = ✓ and can be computed in closedform for a wide range of kernels, including Gaussian, cosine, logistic and other kernels with sigmoid CDF.

for block decompositions e , we obtain y 0 (y 0 e µ (j) See Appendix A.1 for a detailed derivation.


The underconstrained nature of the problem can be remedied by imposing constraints on and .

we could cast the problem within a probabilistic framework and impose specific prior distributions on and with higher probabilities for values that achieve certain desirable properties.


Here, we show that the Z statistic, Z = i , which is the core of our testing algorithm, can accurately distinguish whether two words are very similar based on surprisingly small samples of the contexts in which they occur.

for each pair of words, a, b that we consider, we select m 1 random occurrences of a and m 2 random occurrences of word b from the Google books corpus, using the Google Books Ngram Dataset.


This does not imply independence however.

for i The following is a standard definition giving a measure of the dependence of the random variables Z t within a stationary sequence.


Moreover, the computational time is that of the univariate test, which means that it can be very fast.

a desirable requirement is that the null distribution of the test statistic does not depend on the marginal distributions of X and Y , i.e., that the test is distribution-free.


Despite its promising performance D-PPCA for SfM exhibits some limitations.

we assume the independence of the affine motion matrix parameters in (15).


First, in all the methods mentioned above, the side-information has to be defined in terms of equivalence constraints.

a user needs to define a grouping structure for the input data by informing the model which data-points belong together.


By sampling a recommendation from a joint probability distribution over Σ 1 , Σ 2 , the coordination device introduces a correlation between the strategies of the team members that is otherwise impossible to capture using behavioral strategies.

in general there exists no behavioral strategy for the team player that is realization-equivalent to the normal-form strategy induced by the coordination device, as the following example further illustrates.


[24] bypassed their lower bounds under a stronger assumption known as singular value separation: the difference between k-th singular value and all k -th singular values for k > k is at least ω( √ n).

our result shows that we do not need singular value separation while using significantly less space-O(nk/α 2 ) as compared to O(n 2 )-if we are ready to pay for a small multiplicative error.


We call these experiments because the goal of solving these problems is to disambiguate nondeterministic action models.

the agent tries to reach an informative state where the current model is nondeterministic.


Semi-supervised learning is more powerful than supervised learning by using both labeled and unlabeled data.

the manifold regularization framework, together with kernel methods, leads to the Laplacian SVM (LapSVM) that has demonstrated state-of-the-art performance.


The missing information for each state is The two agents are also tested in a reward task in the unbounded environment for assessing whether the exploration of EB-CRP-PIG leads to efficient reward acquisition.

we assign a reward to each state equal to the Euclidian distances from the starting state.


Note that the upper bound in Theorem 1 matches the lower bound up to logarithmic terms and lower order additive factors.

changing the selection strategy for the second arm, B t , from uniform to anything more sophisticated is not expected to yield significant benefits in the adversarial regime.


After the n-th sample is received, the posterior is updated according to Bayes rule: for n = 1, . . . , N . Note that the MNN is deterministic, so the likelihood (per data point) has the following simple form 3 Therefore, the Bayes update in Eq.

3.3 simply makes sure that P (W|D n ) = 0 in any "illegal" configuration (i.e., any W 0 such that g x (k) , W 0 = y (k) ) for some 1 ≤ k ≤ n. the posterior is equal to the prior, restricted to the "legal" weight domain, and re-normalized appropriately.


Finally, we assume that the target projections are unknown within the injection site.

we only know y j outside the support of x j , which we denote supp x j , and we wish to only evaluate error for the observable voxels.


We here presented a hierarchical model for neural population dynamics in the presence of nonstationarity.

we concentrated on a variant of this model which focuses on nonstationarity in firing rates.


A number of other natural normalization techniques have a similar (and, sometime, even stronger) effect.

they all offer similar improvements in the training performance.


While this approach offers a general solution, it is unclear how individual neurons could achieve this goal.

it requires that the individual neuron can change its nonlinear transfer function arbitrarily, i.e.


In comparison, RECURSIVE-RLS [9] leads to essentially the same accuracy while improving computations.

the sets J h are never larger than d eff ( ).


PPM-α Noise Model For α > 0, a neutral noise model P is called positional pairwise majority preserving (or PPM-α) if for every ground truth θ (and the corresponding true ranking σ * ) and every i < j, the quantity is non-negative for every i < j , and at least α for some i < j .

for i < j , the probability that σ * (i) and σ * (j) go to positions i and j respectively in a vote should be at least as high as the probability of them going to positions j and i respectively (and at least α greater for some i and j ).


Time series of financial returns often exhibit heteroscedasticity, that is the standard deviation or volatility of the returns is time-dependent.

large returns (either positive or negative) are often followed by returns that are also large in size.


The so-generated two datasets are then matched and we use the estimation error to quantify the quality of the match.

assume that y i is associated with the observation x i .


In this paper, by relating risk to robustness, we derive a novel result that further motivates the usage of a CVaR objective in a decision-making context.

we show that the CVaR of a discounted cost in an MDP is equivalent to the expected value of the same discounted cost in presence of worst-case perturbations of the MDP parameters (specifically, transition probabilities), provided that such perturbations are within a certain error budget.


The rise of this error as k increases captures the decay of memory traces in (1).

the central goal of this paper is to obtain a deeper understanding of the memory properties of neural networks for sparse signals by studying the memory curve E(k) and especially its dependence on W. we are interested in classes of network connectivities W and input statistics for which E(k) can remain small even for k N . Such networks can essentially perform compressed sensing of their past inputs.


Define the critical signal level for a shrinker η by where X = xũṽ is an arbitrary rank-1 matrix with singular value x.

x critical (η) is the smallest singular value of the target matrix, for which η still outperforms the trivial zero shrinker η 0 .


In this paper, we answer this question affirmatively.

we investigate the problem of LinBET characterized by finite (1 + ϵ)-th moments, where ϵ ∈ (0, 1].


We can further modify QuadSketch in a manner similar to Product Quantization [7].

we partition the d dimensions into m blocks B 1 . . . B m of size d/m each, and apply QuadSketch separately to each block.


Note that unlike RKHS, this space of functions, K p X H, is density dependent.

this now allows us to formulate the following optimization problem for semi-supervised classification/regression in a way similar to many supervised learning algorithms: The Fredholm learning framework solves the following optimization problem 1 : The final classifier is c(x) = (K pX f * ) (x), where K pX is the operator defined above.


The resulting coreset consists of O(dk/ε 2 ) points that approximates the sum of square distances for any k-piecewise linear function (k segments over time).

we can use this coreset to compute the k-piecewise linear function that minimize the sum of squared distances to the input points, given arbitrary constraints or weights (priors) on the desired segmentation.


In this paper, we extend these ideas, and we show that-from the statistical perspective-we are able to obtain a low-rank approximation that comes with improved statistical guarantees by using a variant of this more traditional notion of statistical leverage.

we improve the recent bounds of Bach [3], which provides the first known statistical convergence result when substituting the kernel matrix by its low-rank approximation.


The sequence of the splitting order strength of connection between brain regions.

the earlier (i.e., smaller ) a region or a cluster regions becomes disconnected from the rest of the brain, the weaker it is connected with the rest of the brain.


We have recently been exploring the use of PG algorithms in an incredibly difficult and exciting control domain -fluid dynamics -and as such applied the shell distribution to a fluid dynamical system.

we applied learning to a system used to sudy the dynamics of flapping flight via a wing submerged in water (see Figure 4 for a description of the system (Vandenberghe et al., 2004)).


we employ an off-the-shelf 3D face model as a simulator to generate profile face images with varying poses.


One of the challenges of using latent variable models for exploratory data analysis, however, is developing models and learning techniques that accurately reflect the intuitions of the modeler.

when analyzing multiple specialized data sets, it is often the case that the most salient statistical structure-that most easily found by fitting latent variable models-is shared across all the data and does not reflect interesting specific local structure.


Before proceeding, we recall a few definitions from convex analysis [17].

a convex function f (x) with a bounded (sub • if there exists a constant β > 0 (often called the modulus of strong convexity) such that, for any α ∈ [0, 1], it holds: In the sequel, we use the standard Euclidean norm to define Lipschitz and strongly convex functions.


We can also analyze the DBE assumption itself.

we show that π E is not H's best response when π R is a best response to π E . Theorem 3.


as pointed out in [8], the gradient of the regularized Sinkhorn approximation can be obtained directly from the dual solution as for any a ∈ R n and b ∈ R m . This characterization is possible because of well-known properties of primal and dual optimization problems [17].


The problem with sampling from the collapsed distribution is that for a proper sampler we need to consider all cluster probabilities including those related to clusters which are highly implausible and unlikely to be chosen for a given instance.

most of the time we discard the very computations that made sampling so expensive.


A more complex assumption, known as missing at random (MAR) [15], states that for every i, p(R i | L) = p(R i | L * ).

every missing value is determined by a biased coin flip that is independent of missing data values conditional on the observed data values.


This distribution now has two modes, one representing the posterior given that a jump occurred, the other representing the posterior given no jump.

importantly, we are more likely to infer a jump, and resample from scratch, if the observation r t+1 is far from that expected under the hypothesis of no jump, x t+1 • w L t . the probability that no jump occurred (and that we therefore resample according to the posterior distribution given drift -effectively, the chance that the sample "survives" as it would have in the no-jump Kalman filter) -is proportional to P (r t+1 | x t+1 , w t = w L t , no jump).


Define C = {c : ∃a ∈ A such that c = C(a)}.

c is the set of all possible configurations.


Further, joint parameter fits to 2AFC and GNG choice data indicated that the principal difference in the two tasks was in the nondecision time and decision threshold; the rate parameter (representing the evidence accumulation process) was similar in both tasks.

they suggested that the nondecision time was shorter, and the decision threshold higher than in the 2AFC task (Figure 4B).


Here we make use of the discriminative pose and shape estimate from Section 4 to give us the initial distribution for the posterior.

given the discriminative model for the shape, p(ν|X), position, p(τ |X), and articulated pose, p(θ|X), of the body, we can let (with slight abuse of notation) y The outlined stochastic optimization framework also requires an image likelihood function, p(I|Y), that measures how well our model under a given state Y matches the image evidence, I, obtained from one or multiple synchronized cameras.


In this work, we adopt a proximal-gradient method to handle the non-smoothness of Ω(W).

we first reformulate the overlapping-group-lasso penalty Ω(W) into a max problem over auxiliary variables using dual norm, and then introduce its smooth lower bound [20,9].


The second limitation is that the quality-diversity decomposition, while cleverly leading to a convex optimization, limits the power of modeling complex dependencies among items.

only the quality factor q i is optimized on the training data.


While these methods can obtain good class posteriors, we note that in order to obtain better accuracy/coverage, it is not sufficient to take naive Bayes' output p(y|x) and find a monotone mapping from that to a set of hopefully better class posteriors (e.g., [16]).

in order to obtain better accuracy/coverage, it is also important to rearrange the confidence orderings that naive Bayes gives to documents (which our method does because of the normalization).


We also systematically evaluated the JS-divergence and the multi-information as a function of dimensionality.

we started with the bivariate marginal distribution of two randomly selected pixels.


These techniques are applicable to an unspecified spike sorting algorithm.

a procedure of online estimating both individual spike and noise spectrum is first applied.


These efficiency results are nontrivial; a crucial step in obtaining them is to exploit special properties of squared Euclidean distance.

we write our cost function in three different, but completely equivalent, ways; and we switch back and forth between these: 1.


Using word vector learning as inspiration, we propose an objective function that abstracts the skip-gram model of [8] to the sentence level.

instead of using a word to predict its surrounding context, we instead encode a sentence to predict the sentences around it.


• Under the canonical Erdös-Rényi sampling, we show that the Schatten k-norm of an incoherent rank-r matrix can be approximated within any constant multiplicative error, with number of samples scaling as O(dr 1−2/k ) (see Theorem 1).

this is strictly smaller than the number of samples necessary to complete the matrix, which scales as O(dr log d).


Despite the centrality of this problem, not much is known on the runtime required to guarantee Equation ( 1) with other values of α.

a natural question is how the runtime changes when enlarging α from 0 to 1 γ .


Instead, we measure the sparsity of solutions produced by PSM and VSM by appealing to complementary slackness.

the (i, j)-th element of the inverse covariance matrix is deemed to be nonzero if and only if |W ij − Σij | = ρ.


The last decade has witnessed an explosion in the development of models, theory and computational algorithms for "big data" analysis.

distributed computing has served as a natural and dominating paradigm for statistical inference.


• Probabilistic interpretation of existing regularization techniques: We show that the standard regularized logistic regression is a special case of our framework.

we show that the regularization coefficient ε in (3) can be interpreted as the size of the ambiguity set underlying our distributionally robust optimization model.


In this paper, we show that Online Mirror Descent is, in a sense, universal.

for any convex online learning problem, of a general form (specified in Section 2), if the problem is online learnable, then it is online learnable, with a nearly optimal regret rate, using Online Mirror Descent, with an appropriate distance generating function.


The progress of AI is raising a lot of concerns 4 .

it is becoming clear that keeping an AI system under control requires more than just an off switch.


We show that convergence in an adversarial divergence implies some standard notion of topological convergence.

we show that provided an objective function is a strict adversarial divergence, convergence to in the objective function implies weak convergence of the output distribution to .


We observe that the branch with the 3 × 3 convolution (rate = 1 × 6) contributes most, whereas the branches with large rates (i.e., longer context) contribute less.

information from image features in closer proximity (i.e.


Hence, it's natural to produce captions following such a hierarchical structure.

we propose a two-stage framework for image captioning, as shown in Figure 2.


Since ( unh , F H,λ ) is SLN-robust, it follows that for hinge : (y, v) → max(0, 1−yv), ( hinge , F H,λ ) is similarly SLN-robust provided λ is sufficiently large.

strong 2 regularisation (and a bounded feature map) endows the hinge loss with SLN-robustness 4 .


PLST can be viewed as the counterpart of PCA in the label space [5] and is feature-unaware.

it considers only the label information during reduction.


We show in Theorems 4.9 and 4.10 that the set of CCEs is a strict subset of policy equilibria.

every CCE can be thought of as a policy regret equilibrium, but no policy regret play might not converge to a CCE.


To tackle the issue, we add a constraint that enforces matching between the generated sentence and the ground-truth text in the missing parts.


3 with adaptive exploration method has a constant regret bound.

[11] analyzes a general adaptive submodular function maximization in bandit setting.


then defining the mth residual of the EBP algorithm r(m) can be bounded in size by whenever ρ converges to 0, the algorithm converges to the global minimum.


In this approximation, we assume that the covariance matrices of X and X|Y are the same.

the covariance of X is independent of Y .


The common feature in all of these methods is that they intend to increase the local computations with the aim of reducing the overall iterations, and hence, lowering the communications.

these methods are designed to perform as much local computation as possible before making any communications across the network.


Thus, we see that our optimization approach generally achieves competitive results with random features at lower computational costs, and it offers the most improvements when either the base kernel is not well-suited to the data or requires a large number of random features (large D) for good performance.

our method reduces the sensitivity of model performance to the user's selection of base kernels.


While previous studies (by Mirrokni and Gomes, and by Niazadeh et al) focused on revenue-sharing schemes in static double auctions, in this paper, we take advantage of the repeated nature of the auctions.

we introduce dynamic revenue sharing schemes where we balance the two constraints over different auctions to achieve higher profit and seller revenue.


Preliminary numerical experiments (presented in the supplementary material) illustrate our theoretical findings.

we provide examples of Lipschitz MDPs, for which the regret under DEL does not seem to scale with S and A, and significantly outperforms algorithms that do not exploit the structure.


That might allow us to reformulate the construction algorithm for G x in a way that allows for a more straightforward approach as compared to the parametric ansatz that we have proposed.

our method relies upon the requirement that the target function can be expressed in terms of an underlying potential function g. This leads to the intriguing and nontrivial question: Is it possible to mathematically guarantee the existence of such a potential?


Second, in many recent models, loss is measured not with L p distances (as is conventional in nonparametric statistics [49,47]), but rather with weaker losses, such as where F D is a discriminator class of bounded, Borel-measurable functions, and P and Q lie in a generator class F G of Borel probability measures on a sample space X .

gANs often use losses of this form because (1) can be approximated by a discriminator neural network.


We are then free to use any value of γ on the LHS and maintain a valid bound.


This makes 5 H the natural scale for measuring $ ¢ 5( and suggests that using far fewer than H kernel evaluations we can get good approximations of $ &¢ ( .

for a chosen (fixed) value of " let us say that 2 $ ¢ 5( is trivial if 2 $ &¢ ( #" ¡ %$ 7 29 '@ H § Having picked some threshold " (for SVM expansions " is related to the classification offset) we want to determine whether $ &¢ ( is non-trivial and, if so, we want to get a good relative error estimate for it.


This potential drift, known as the Bereitschaftspotential (BP, see [8] for an excellent survey), shows variation in distribution over scalp with respect to the body part being moved.

the BP related to movement of left versus right arm shows a strong lateral asymmetry.


We remark that PROXSAGA in Algorithm 2 differs slightly from [4].

it uses minibatches where two sets I t , J t are sampled at each iteration as opposed to one in [4].


We use the binary SVM to define the set of easy samples as X e = {x i , y i w 0 φ i (x) ≥ 1}.

a positive sample is easy if it is assigned a score that is greater than 1 by the binary SVM.


In the appendix, we list several important properties of strongly convex functions.

the Fenchel conjugate of a strongly convex function is differentiable.


Our approach relies on a blocked Gibbs sampling algorithm that alternates between sampling the number of considered chains and the global variables conditioned on the current value of matrices S and X, and sampling matrices S and X conditioned on the current value of the remaining variables.

the algorithm proceeds iteratively as follows: • Step 1: Add M new new inactive chains using an auxiliary slice variable and a slice sampling method.


In our case, the values have been chosen on the basis of simplicity and do not appear to dramatically affect the predictive capacity of the model in the simulation results.

we wished to avoid tuning these parameters to the available data set.


We followed [3,4], where the motivation was to aid the prediction of a variable that is expensive to measure by using abundant measurements of correlated variables which are less expensive to acquire.

we estimated Cadmium and Copper at the validation locations using measurements of related variables at the training and test locations: Nickel and Zinc for Cadmium; and Lead, Nickel and Zinc for Copper.


Our goal is to examine how humans adapt their decision-making to different reward environments.

we focus on whether human participants have veridical prior beliefs about reward rates.


Recent successes of deep neural networks have spanned many domains, from computer vision [1] to speech recognition [2] and many other tasks.

sequence-to-sequence recurrent neural networks (RNNs) with long short-term memory (LSTM) cells [3] have proven especially successful at natural language processing (NLP) tasks, including machine translation [4,5,6].


Then, after drawing a new c t , we increment the relevant counts accordingly.

if c t = K + 1 (i.e., a new ancestor is to be drawn), we update n = n + 1, set n K+1 = 1, m ct = m ct + 1, m ct,K+1 = 1, and set up a new (empty) HMM urn with color K + 1 (i.e.


Our research on network pruning is partly inspired by [9], not only because it can be very effective to compress DNNs, but also because it makes no assumption on the network structure.

this branch of methods can be naturally combined with many other methods introduced above, to further reduce the network complexity.


To mitigate this problem one can leverage several tactics commonly used in the GP literature, including the use of pseudo-inputs (Csató and Opper, 2002;Gal et al., 2014;Hensman et al., 2013;Quiñonero-Candela and Rasmussen, 2005;Titsias, 2009), Kronecker-factorized covariances (Casale et al., 2017;Stegle et al., 2011;Rakitsch et al., 2013), and low rank structures (Casale et al., 2015;Lawrence, 2005).

in the instantiations of GPPVAE considered in this paper, we focus on low-rank factorizations of the covariance matrix.


We now turn to the study of a fundamental quantity associated with the property of classification calibration with respect to a general multiclass loss �.

in the above example, we saw that to develop a classification calibrated surrogate loss w.r.t.


Both in that model, and for our algorithm and analyses, we assume that each value in the database is a real vector with norm at most one.


Suppose that γ(r) ≥ C r 1+κ , ∀ r ≥ 0 for some C > 0 and κ > 0.

then if u t * ≤ M for all t and η t = η t −β , then Assuming h is bounded, optimizing over β yields a rate of R t (x) = O(t κ 1+κ ).


The position of c i is equal to a convex combination of the 5 pentagon corners' (x, y) coordinates, where the coordinates are weighted by the elements of θ i .

circles c i at the pentagon's corners represent single-membership θ i 's, while circles on the lines connecting the corners represent θ i 's with mixedmembership in 2 communities.


In maximum coverage problems, for instance, it is quite reasonable to assume that the covered area is only a function of the chosen items and their states.

the gain can be computed as ḡi (y) = f (dom (y) ∪ {i} , φ) − f (dom (y) , φ), where φ is any state such that φ ∼ y and φ[i] = 1.


We set L = 3 for all the experiments and use the same hyperparameters in both algorithms.

the stick-breaking prior parameter γ is set to 1.0; the symmetric Dirichlet prior parameter for the topics is set to 1.0; the prior for level proportions is skewed to favor high levels (50, 20, 10).


In the personalized setting (as in the department store example), we allow the learning algorithm to return different functions for different players.

our goal is to return classifiers f 1 , .


Our results showed that for the first time an ADP algorithm (CBMPI) performed extremely well in both small 10×10 and large 10×20 boards and achieved performance either better (in the small board) or equal with considerably fewer samples (in the large board) than the state-ofthe-art CE methods.

the best policy learned by CBMPI obtained the performance of 51, 000, 000 lines on average, a new record in the large board of Tetris.


For all practical examples in this paper, Ω is a two-dimensional grid and corresponds to different locations in a two-dimensional image.


We argue that the task of identifying the reward function should amount only to identifying the (behavioral) equivalence class to which θ belongs.

identifying the equivalence class is sufficient to get perfect generalization to new tasks.


In the sequel, we discuss an extension of our approach to handle the approximate case D ≈ T A with T and A as in (1).

we have in mind the case of additive noise i.e.


If additionally is strongly convex over the path of the iterates ↵ t then there exists ⌧ > 0 and ⌫ > 0 such that for all t > ⌧, Illustration of the abstention stumps on a variable X.

this theorem holds for the exponential loss (u) = exp(u) and the logistic loss ) since they are strongly convex over the compact set containing the ↵ t s. We first define a family of base hypotheses, abstention stumps, that can be viewed as extensions of the standard boosting stumps to the setting of classification with abstention.


This result demonstrates, consistently with experiments, that neural encoding must be tailored to the task at hand.

when analyzing sensory neural data, one must pay careful care to the task being performed.


Moreover, even though the infimum above is not attained in general, the problem in W resulting after partial minimization over D admits a minimizer.

since the first term in (2.1) is independent of D, we can first optimize the second term with respect to D. we can compute the infimum In this way we could end up with an optimization problem in W only.


In our experiments, we are also able to predict surprisingly well the relative performances of trained residual networks based only on their initialization hyperparameters, in a variety of settings.

we find that the quality of initialization for tanh resnets is determined by trainability (how much gradient explosion on average) while that for (↵-)ReLU resnets is determined by expressivity (how far can two different input vectors be pulled apart) (see Section 6).


One-point Gradient Estimates [FKM05]: Suppose one has to estimate the gradient of a function f : R p → R at a point w ∈ R p via a single query access to f . [FKM05] showed that one can approximate f (w) by p β f (w + βu)u, where β > 0 is a small real parameter and u is a uniformly random vector from the p-dimensional unit sphere S p−1 = {a ∈ R p : a 2 = 1}.

more precisely, For finite, nonzero values of β, one can view this technique as estimating the gradient of a smoothed version of f . Given β > 0, define f (w) = E v∼B p [f (w + βv)] where B p is the unit ball in R p . f = f * U βB p is the convolution of f with the uniform distribution on the ball βB p of radius β.


Putting it All Together In the next section we show how to combine these three techniques into one algorithm and make them work seamlessly.

we provide and analyze how to combine these techniques into an Algorithm 1 which can be used to at least halve the error of a current policy.


Since the vectors u, v used by hash function h u,v are sampled independently, then for a query hyperplane vector w and a database point vector x, Next, we use the following fact proven in [25], where u is sampled as defined above, and θ a,c denotes the angle between the two vectors a and c. Using (4) and ( 5), we get: Hence, when θ x,w − π We note that unlike traditional LSH functions, ours are asymmetric.

to hash a database point x we use h u,v (x, x), whereas to hash a query hyperplane w, we use h u,v (w, −w).


In the Atari benchmark, the system is able to achieve high reward on the task with extremely low sample complexity.

the model is able to learn the paddle parameters quickly from random trajectories, improving the control precision, and leading to high reward, as shown in Figure 6 Figure 5: Even though the baseline is able to achieve lower MSE over one-step predictions of the dynamics of the Cartpole environment (left), the physics engine-based controller is able to achieve a higher reward very quickly (right).