It is common to resort to well-studied approximations [57], unless exact techniques such as dynamic programming or submodular optimization [41,30,50,21] are applicable.

the complexity of those approximations increases with the size of the largest variable index subset r. many of the models considered to date do not exceed pairwise interactions.


Furthermore sharpness defined above depends on the scale of w and can be artificially increased or decreased by changing the scale of the parameters.

sharpness alone is not sufficient to control the capacity of the network.


[1] have shown that the error rate on unseen test samples can be upper bounded by the disagreement between the classification-decisions obtained from independent characterizations (i.e., views) of the data.

in the web-page example, misclassification rate can be indirectly minimized by reducing the rate of disagreement between hyperlink-based and content-based classifiers, provided these characterizations are independent conditional on the class.


In the independent cascade model, a node a influences each of its neighbors b with some probability q ab , independently.

given a seed set of nodes S, the set of nodes influenced is the number of nodes connected to some node in S in the random subgraph of the network which contains every edge ab independently with probability q ab .We define f (S) to be the expected number of nodes influenced by S according to the independent cascade model over some weighted social network.


2 implies that free energy upper bounds the negative log likelihood.


Put another way, a rank value y induces the vector (Yl, ... , Yk-d = (+1, ... , +1 , -1 , ... , -1) where the maximal index r for which Yr = +1 is y-1.

the prediction of a ranking rule is correct if Yr(w• x -br ) > 0 for all r. If the algorithm makes a mistake by ranking x as fj instead of Y then there is at least one threshold, indexed r, for which the value of W• x is on the wrong side of br , i.e.


Let Y = U T XV denote the random variable in the tensor subspace and suppose the data points have a zero mean.

the weighted variance can be estimated as follows: , so we also have: Finally, we get the following optimization problems: The above two minimization problems (4) and ( 5) depends on each other, and hence can not be solved independently.


An equivalence class [Z] consists of ces that they are generated from the distribution in (3) with equal probabilities and can be reduced to the same left-ordered form.

the probability of [Z] is We assume that the first K + features are represented i.e.


In the analysis of neural responses, once we fix our time resolution the response becomes a set of discrete "words," and estimates of the information content in the response are de-termined by the probability distribution on this discrete space.

what all of these examples have in common is that we often need to draw some conclusions with data sets that are not in the asymptotic limit N K. while we might use a large corpus to sample the distribution of words in English by brute force (reaching N K with K the size of the vocabulary), we can hardly do the same for three or four word phrases.


Unfortunately, the MI kernel approximation based on these terms cannot be shown to be positive definite anymore l l !

in the moment we use a less elegant and, we feel , less accurate approximation (details can be found in [5]) based on first-order Taylor expansions.


process is completely characterized by P x n = p, hence this problem coincides with the previous one.

for Markov chains, we seek to estimate the transition matrix M . instead of producing a probability distribution Px n , the estimator M maps every sequence For two distributions p and q over [k], let L(p, q) be the loss when p is approximated by q.


Step 2, although the sampling process terminates when a solved node n is reached, we associate every configuration x >n of X >n with probability g(x >n |x ≤n )/v n which is appropriate in lieu of (1).

we can show that this two-step sampling scheme induces a proposal distribution, denoted q S (x), which can be expressed as: where AND(T x ∩ S) is the set of all AND nodes of the partial solution tree T x ∩ S. By applying (3), and noticing that the upper bound is the initial heuristic for any node in OPEN(S) and is exact at any solved node, we re-write q S (x) as q S (x) actually provides bounded importance weights that can use the refined upper bound U S : Proposition 2.


In this subsection, we consider the setting where all the elements are observed (with noise) and the goal is to recover the underlying low-rank tensor without noise.

since all the elements are observed only once, X is simply a vectorization (M = N ), and the lefthand side of inequality (10) gives the quantity of interest ∥X(∆)∥ 2 2 = Ŵ − W * F . the remaining task is to bound X * (ϵ) mean as in the following lemma.


The gradient is Note that, for graph matching problem, we have W T = W. Thus, the second derivative is Therefore, Φ(x, x) is a concave function in x and has a unique global maximum.

it can be obtained by setting the first derivative to zero ( ∂Φ(x,x) ∂x kl = 0), which gives we obtain the update rule in Eq.


In each iteration, it is allowed to gather information by asking for a single pairwise comparison between two items-or, using the dueling bandits jargon, to pull two arms.

it selects two items i and j, and then observes either preference i j or j i; the former occurs with probability p i,j as defined in (1), the latter with probability p j,i = 1 p i,j .


As the temporal structure of the auditory receptive fields was less dominant than their spectral structure (Figure 2C), we focused solely on the spectral domain and did not attempt to address temporal information.

the inputs for the new model (n = 100,000) were short-time frequency spectra of 128 pixels each (24 pixels = 1 octave).


f Y |X is a highly nonlinear function of x, and even η, and x is high-dimensional.


By definition, the true subspace satisfies Π * = U U .

the estimator U can be computed from Π using eigenvalue decomposition.


Comparing the distributionally robust (b) and robust (c) optimal solution for the optimal PRV setting problem, we observe, that the objective value of the distributionally robust counterpart is substantially smaller than the robust one.

the distributionally robust solution is less conservative than the robust solution.


In the proposed framework, modal decomposition is first obtained in terms of learned observables g; thus, the values of g must be back-projected to the space of the original measurements y to obtain a physically meaningful representation of the dynamic modes.

we modify the loss function by employing an additional term such that the original measurements y can be reconstructed from the values of g by a reconstructor h, i.e., y ≈ h(g( x)).


Collecting all elements of positive score is a natural criterion, but it is costly to perform exactly if the output set is very large.

it would be desirable to develop sparsity enforcing variations of Equation (2), for example by adopting ideas from compressed sensing [25].


Learning of bottom-up connections (i.e., feedforward weights) is driven by such error signals through local synaptic plasticity.

in contrast to previous approaches (Marblestone et al., 2016), in our framework a given neuron is used simultaneously for activity propagation (at the somatic level), error encoding (at distal dendrites) and error propagation to the soma without the need for separate phases.


A better importance distribution, which more accurately approximates the posterior, results in importance weights with lower variance.

we estimate the variance of the (log) importance weights as a measure of inference network quality (see Table 3).


These attributes pertain to various physical and chemical properties of the cell.

a flow cytometry data set is a random sample from a patient-specific distribution.


Specifically we define this bump function in the transformed space as where r is the radius from the origin, r e is the maximum radius of any point generated in the initial scaling or subsequent evaluations, and r ∞ is a parameter set to 1.5r e by default.

the acquisition function also decays and new points are never suggested arbitrarily far away.


Because Υ(c, m) is entirely dependent on the two matrices Υ d and Υ e as well as our kernel matrix, we can calculate an estimate for any set of nodes in the graph by explicitly calculating only those rows of Υ which correspond to those nodes and then simply multiplying that sub-matrix by the vector y.

if we have an efficient way to compute arbitrary entries of the kernel matrix K 1 , we can estimate functions anywhere in the graph.


Additionally, we would like to quantify the possible loss (if any) in predictive accuracy for gaining interpretability.

we also use two black-boxes, random forest and XGBoost to benchmark the performance without accounting for interpretability.


During learning, we adopt the policy gradient technique [31] to overcome the non-differentiable problem.

by viewing a set of data samples collectively and evaluating their quality through relative ranking, the discriminator is able to make better assessment of the quality of the samples, which in turn helps the generator to learn better.


Let (Ω, d Ω ) be a compact metric space, B(Ω) be its Borel σ-algebra, and X := M 1 (Ω) be the set of all Borel probability measures on Ω.

then the topology describing weak convergence of probability measures can be metrized, e.g., by the Prohorov metric d X (P, P ) := inf ε > 0 : P(A) ≤ P (A ε ) + ε for all A ∈ B(Ω) , P, P ∈ X , where  the map ρ : M 1 (Ω) → H Ω satisfies the assumptions of Theorem 2.2, and hence the Gaussian-type RBF kernel k σ (P, P ) : is universal and obviously bounded.


More generally, functions of the form f (λ) = λ −α , α ≥ 0, allow for combining shared features and task-specific features to some degree tuned by the exponent α.

moreover, the regularizer (2.2) ensures that the optimal representation (optimal D) is a function of the tasks' covariance W W . we propose to solve the minimization problem for functions f belonging to an appropriate class.


Using the Chebyshev interpolant of log(1 + αx), we approximate log | K| by For stochastic estimation of tr(T j (B)), we only need to compute z T T j (B)z for each given probe vector z.

we compute vectors w j = T j (B)z and ∂w j /∂θ i via the coupled recurrences This gives the estimators each derivative of the approximation costs two extra MVMs per term.


Based on our ICA model, each component is a putative genomic expression program of an independent biological process.

our hypothesis is that genes showing relatively high or low expression level within the component are the most important for the process.


Therefore, by scanning the solution regions from left to right according to the sorted s values, and maintaining some intermediate computed variables, we are able to find the optimal solution, as summarized by the following lemma: Lemma 1.

the subproblem (5) is convex with respect to d and can be solved exactly in O( P 0 log( P 0 )), where P 0 is the number of nonzero elements in tensor P. we can efficiently update d since P is a very sparse tensor with only a small number of nonzero elements.


We assume this occurs by the interneurons having a very large membrane resistance (or correspondingly a very small conductance) and moderate capacitance.

the leakage term, -n, which is the only term in the second line of Eq.


Furthermore it has a slightly bigger area under the LTP part as compared to the LTD part.


This is the reason that p match in the other terms.

if η is small, the repulsive effect from matching behavior due to the diffusion effect is expected to weaken.


Thus, misclassifying the boundary words of a correct phrase blocks the generation of the candidate and produces a missed phrase.

we move the start or end prediction vectors toward the misclassified boundary words of a missed phrase.


Theorem 2 implies that γ m in ( 12) satisfy the monotonicity condition: γ 1 ≥ γ 2 ≥ .

. . ≥ γ M . equations ( 12) constitute the parametric submodular minimization problem as defined above, which reduces to the monotonic parametric max-flow problem for pairwise E. Let • denote the largest integer not exceeding an argument of the operation.


In Section 2, we defined that an instance is an anomaly when its different views are generated from different latent vectors.

for an anomaly score, we use the probability that the instance uses more than one latent vector.


Due to the cardinality constraint, Problem (1) is not only non-convex, but also NP-hard in general (Natarajan, 1995).

it is desirable to develop efficient computational procedures to approximately solve this problem.


Figures 2 (c) and (d) show that the responses of two groups of participants who answered the arithmetic problem in (a) and (b) differed as predicted (χ 2 (1) = 5.208, p < 0.05, using Yates' chi-square correction).

asking participants to solve arithmetic problems is an effective method for testing reference frame inference and perceived orientations can influence higher level cognition.


While the distribution is symmetric about zero (and underestimates are often compensated by overestimates) it indicates that these noisy measurements are a major bottleneck in achieving better error rates.

it would be very interesting to consider more sophisticated risk estimation methods.


Note that EXPLORE-k ensures an absolute-error guarantee, which is weaker than the individually relative-error guarantee of k-AS.

the same EXPLORE-k lower bound also applies to k-AS.


By applying Taylor's expansion to ℳ, we have where e is the -th column of I.

let Δ be the -th element of Δ and be the largest eigenvalue of J such that we can conclude that ≡ is the -th componentwise rate of convergence.


Here's the basic argument for uniform π: we show that if the optimal tree T * requires Q * queries in expectation, then some query must (again in expectation) "cut off" a chunk of H of π-mass Ω(1/Q * ).

the root query of the greedy tree T G is at least this good (cf.


Let T unf olded ∈ R m×n 2 be the unfolded tensor T such that T unf olded (:, (j 1 − 1)n + j 2 ) = T (j 1 , j 2 , :) for 1 ≤ j 1 , j 2 ≤ n. Without noise, the right singular vector of this matrix is u 1 ⊗ w 1 which corresponds to the singular value σ 1 .

selecting k 2 indices of this singular vector with largest magnitudes would recover the index set J 1 × J 2 .


The cross-entropy loss assumes normalized hypotheses.

here, we assume that the source functions are normalized for every x: For any ⌘ > 0 and z ∈ , we define a normalized weighted combination h . We will first assume the conditional probability distributions D k (⋅x) do not depend on k. Theorem 5.


Since f is unknown, R ⇤ could be absorbed into the link function.

we assume that B ⇤ is orthonormal for identifiability.


A good query plan aims to minimize the total size of intermediate results.

it is highly desirable to have a mechanism which can estimate join sizes very efficiently, especially for the lower-order (2-way and 3-way) intersections, which could potentially result in much larger intermediate results than higher-order intersections.


the weights {w i , w α i } are marginal and conditional entropies defined on the beliefs {µ i , µ α }, respectively, the optimal weights should satisfy the following KKT condition where Hi = w i H(x i ; µ i ) + α w α i H(x i |x i+1:c ; µ α ) is the (weighted) average entropy on node i.


This mutual information is equivalent to the notion of rate R in the classical rate distortion theorem [Cover and Thomas, 2012].

it is well-known that the distortion rate function D(R) is convex and monotone decreasing in R. by maximizing mutual information R we are guaranteed to minimize distortion D(R), and hence D * .


In a linear factor model, a user's rating of an item is modeled as a linear combination of these factors, with user-specific coefficients and item-specific factor values.

given a N × M preference matrix for N users and M items, a K-factor model fits it with a N × K coefficient matrix U and a M × K factor matrix V as U V . Various computational methods have been successfully developed to implement such an idea, including probabilistic matrix factorization (PMF) [13,12] and deterministic reconstruction/approximation error minimization, e.g., max-margin matrix factorization (M 3 F) with hinge loss [14,11,16].


The update (7) depends on the variational parameters φ − j of all other words.

unlike LDA, the φ j cannot be updated in parallel.


For any T 1 ⊂ T 2 and any given example x / ∈ T 2 , we have Similarly, we have g(T 2 ∪ x) − g(T 2 ) = wq x r x − S x,x r 2 x − 2r x j∈T2 S x,j r j .

we have which completes the proof of (P1).


This is implemented by training on examples available for nodes in the current subtree (i.e., the tasks below the current node), while similarity to the parent classifier is induced through regularization.

for each node n, one solves the following optimization problem, where p is the parent node of n (with the special case of w * p = 0 for the root node), is an appropriate loss function (e.g., the hinge-loss).


Fortunately, the active learning method can immediately tell whether it has been misled, by comparing the true labels with its optimized guesses.

one can then adjust the active selection strategy to avoid such over-fitting in the next iteration, whenever a mismatch between the labeled and unlabeled data has been detected.


We note here that what lacks from GBVS described as above is any notion of a multiresolution representation of map data.

because multiresolution representations are so basic, one may extend both the graph-based activation and normalization steps to a multiresolution version as follows: We begin with, instead of a single map A : [n] 2 !


Optimization problem (3) ensures that selectrons spike for inputs that, on the basis of their empirical sample, reliably lead to neuromodulatory rewards.

spikes encode expectations about rewards.


Although DNN models normally require a vast number of parameters to guarantee their superior performance, significant redundancies have been reported in their parameterizations [4].

with a proper strategy, it is possible to compress these models without significantly losing their prediction accuracies.


If f is non-smooth yet convex and well structured, which is not suitable for many existing optimization approaches requiring smoothness, its saddle-point representation ϕ is often smooth and convex.

convex-concave saddle-point problems are, therefore, usually better suited for first-order methods [6].


However, a stratification-based approach suffers in estimation when the count distributions of the point processes under consideration are flat, since in this situation the spike train realizations tend to exist in separate Euclidean partitions, and given a finite set of realizations, it becomes difficult to populate each partition sufficiently.

other methods should be investigated that allow two spike trains to interact irrespective of their spike counts.


This neural network function can also be viewed as a PNN whose lines are determined by input weight vectors to neurons.

in this case r * ≤ k * where r * is the number of lines of the original network.


The second part of our proof to Theorem 1 shows that, with high probability, the good neighborhoods have, through random exploration, accurately estimated the probability of liking each item.

we correctly classify each item as likable or not with high probability, which leads to a lower bound on P(X ut = 1).


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

the sample complexity of our algorithm is optimal when k = d O(1) .


Unfortunately, solving a classification problem with a bounded loss function is computationally intractable, because the problem is no longer convex and even state-of-the-art solvers like, for example, CPLEX [20] fail to found an exact solution, when the training set size exceeds few tens of samples.

we propose here to find an approximate solution through well-known algorithms like, for example, the Peeling [6] or the Convex-Concave Constrained Programming (CCCP) technique [14,21,22].


The quality measures used in information retrieval are particularly difficult to optimize directly, since they depend on the model scores only through the sorted order of the documents returned for a given query.

the derivatives of the cost with respect to the model parameters are either zero, or are undefined.


We assume in addition that f is σ-strongly convex.

based on Lemma 2, the function f is differentiable.


Then, the f -spectral distance can be expressed as S f (x, y) = ||Ψ(x) − Ψ(y)|| 2 2 , where Ψ(x) is the x th row of Ψ.

ψ represents an Euclidean embedding of G where each node x is represented by the vector Ψ(x).


Access to this activity by those layers can be modeled as instantaneous sampling of a selected brief window of temporal behavior of PNs [7].

the recognition system in our model utilizes such snapshots from the spiking activity in the excitatory population x i (t).


Note that a linear combination of several neural net functions in F nn may not remain in F nn .

we apply the duality framework to F nn 's convex hull, which we denote by conv(F nn ), containing any convex combination of neural net functions in F nn .


On the other hand, for CC both BA and WS are known to be more realistic models than ER random graphs.

we compare the graphs generated by FRFs to those generated by the BA, ER, and WS models for the same networks.


However, since the total iteration only depends on log m, when m = poly(N, 1/δ ) but the input data is actually structured (with small k, l and large δ), then SGD can actually achieve a small generalization error, even when the network has enough capacity to fit arbitrary labels of the training examples (and can also be done by SGD).

we prove that SGD has a strong inductive bias on structured data: Instead of finding a bad global optima that can fit arbitrary labels, it actually finds those with good generalization guarantees.


Notice that for all k ∈ K, k ≥ k, the QN-step is adopted.

we obtain that as {p k i } is bounded (Proposition 8 in Supplement F).


It is straightforward to see that, on the same event, we havek ŵmle wk 2  C p d/n for some absolute constant C > 0.

the necessary and sufficient conditions on SNR for approximate recovery of w lie between C 0 d/ log log(n) and n C 00 (for absolute constants C 0 , C 00 > 0).


The same monotonicity argument as in [3] shows that we reach an objective value -close to the optimal one within O( 1 ) steps.

mLSP training is roughly comparable in computational complexity to SSVM training.


However, when long-term conflict increases this is taken to indicate that the current decision strategy is not working.

increased long-term conflict biases the LC to the tonic mode so as to increase response volatility.


Moreover, by a reduction from the estimation problem to the testing problem, these boundaries for testing imply the ones for estimating µ 2 − µ 1 as well.

there exists a significant gap between the computational and information-theoretic boundaries for small α.


Mixture models constitute some of the most popular models for MFVB application [1,2] and are often used as an example of where MFVB covariance estimates may go awry [5,6].

we will consider in detail a Gaussian mixture model (GMM) consisting of a K-component mixture of P -dimensional multivariate normals with unknown component means, covariances, and weights.


Now, for an uninfected j ∈ [n], to compute the corresponding row of Y , y T j ∈ R k , we only need k independent linear equations.

if M is non-degenerate, to compute y T j we only need k entries of the j'th column of M with row indices in L. Casting the condition in terms of the sampling graph G Ω , y T j can be computed and vertex v j ∈ V be marked as infected if there are at least k edges from v j to infected vertices in L. Analogously, x T i can be computed and the vertex u i ∈ U be marked as infected if there are at least k edges from u i to previously infected vertices R. Observe that M = XY T = XW W −1 Y T , for any invertible matrix W ∈ R k×k .


For an RKHS based on a Gaussian kernel, the entropy condition (10) holds for any γ > 0 [23].

furthermore, (9) trivially holds via the Cauchy-Schwarz inequality: by Theorem 2(a), ŵ H = ĝn H = O P ( g 0 H ), so the penalty term λ n ŵ 2 vanishes at the same rate as λ n . We have arrived at the following estimator for the KL divergence: log G is an RKHS.


The only clear discrepancy between the model predictions and the data is the cluster of points at the lower left, labeled as Decisions 6-13 in Figure 2a.

these are all cases in which effect x appears in all options and therefore these decisions provide no information about a decision maker's preference for x. the models assign the same ranking to this group as to the group of decisions in which there is only a single option (Decisions 1-5).


Typically, standard approaches train small networks (order of N = 20), whereas ESN typically relies on large networks for precision (order of N = 100).

the RLS-based ESN online learning algorithm is typically more expensive than standard techniques.


Often each of these individual classifiers could be quite complex, e.g., producing labelings over pixels in an entire image.

our method is applicable to many tasks that have different but correlated outputs.


The communities being served by the different hospitals may be different in socioeconomic status, demographics, genetic backgrounds, and exposure to climate and environmental hazards.

different disease sub-types are likely to appear in different ratios in the different hospital.


after regressing out all the other groups, indexed by S\g.

the following question is central to selective inference: Question g,S : What is the magnitude of the projection of µ onto the span of P ⊥ X S\g X g ?


For simplicity, we will also restrict our attention to linear encoding functions, though generalizations to the nonlinear case are straightforward.

as before, let K i denote the matrix implementing the transformation (K i θ) t = i,t (θ), the projection of the stimulus onto the i-th neuron's stimulus filter.


Assuming uniformly distributed variables z n and independence between z n and x n implies that w T k x n + b k = 0.

using the second-order expansion of the log-sum-exp function ϕ(u) = log( K k=1 exp(u k )) around 0 leads to the following approximation of the terms depending on (w, b): where K is the usual centering projection matrix, and X = (x 1 , .


For a given f and a radius f ∅ = f (y) this surface area is given by the equation where I denotes the set of all multi-indices of inner nodes, n I the number of leaves of the subtree under I and B [a, b] the beta function.

if the partition function of the radial distribution can be computed easily, so can the partition function of the multivariate L p -nested distribution.


Mondrian forests excel in this setting: at iteration N + 1, each Mondrian tree T ⇠ MT ( , D 1:N ) is updated to incorporate the next labeled example (x N +1 , y N +1 ) by sampling an extended tree T 0 from a distribution MTx( , T, D N +1 ).

using properties of the Mondrian process, we can choose a probability distribution MTx such that T 0 = T on D 1:N and T 0 is distributed according to MT ( , D 1:N +1 ), i.e., the distribution of Mondrian trees trained on a dataset in an incremental fashion is the same as that of Mondrian trees trained on the same dataset in a batch fashion, irrespective of the order in which the data points are observed.


We denote by Û (π) the utility of the policy π in an MDP whose first-order transition probabilities are given by P θ (and similarly V π the value function in the same MDP).

we have 2 for all π, then finding the optimal policy in the estimated MDP that uses parameters P θ (using value iteration or any other algorithm) will give a policy whose utility is within 2ε of the optimal utility.


One nuance is that we introduce b The term b V +1 extends the vocabulary to V + 1 terms, where the V + 1th term never appears in the documents.

dirichlet(γb k,1:V +1 ) is always well defined.


Note also that scaling does not change the centers selected by Relative Margin.


For any ε ∈ (0, 1] and µ ∈ (0, 1 The complete proof of Theorem 2 is given in the supplementary materials.

the main idea is to show that ρ z is bounded from below by a positive constant with high probability, and then bound the excess misclassification error R(sign choosing C such that lim n→∞ C/n = 0, which is equivalent to lim n→∞ (1 + C κ,H ) 2 /n = 0, then R(sign(f z )) can be arbitrarily close to the Bayes error R * , with high probability, as long as n is sufficiently large.


Parameters were simultaneously adjusted for both attentional conditions; that is, the total fit error was the sum of the error obtained with the baseline set of parameters on the poorly attended data, and of the error obtained with the same parameters plus some attentional perturbation on the fully attended data.

no bias was given to any of the two attentional conditions.


In a PoE, linear combinations of an input vector are penalized according to their negative logprobability and act as constraints.

configurations of high probability have most of their constraints approximately satisfied.


To solve this optimization problem, we introduce the Accelerated Parallel ProximaL gradiEnT (APPLET) method, which tackles the non-smoothness of overlapping-group-lasso penalty via proximal gradient [20,9], and the huge number of training samples by Map-Reduce parallel computing [10].

the contributions made in this paper are: (1) We incorporate the semantic relationships between object classes, into an augmented multi-class logistic regression formulation, regularized by the overlapping-group-lasso penalty.


where the homogeneity measure among nodes in e, Hom w (x, e), is also the inner product of the parameter vector w and the feature vector ϕ e (x) and takes values of both signs such that a large positive value means strong homogeneity while a large negative value means high degree of nonhomogeneity.

note that the proposed discriminant function for higher-order correlation clustering is decomposed into two terms by assigning different parameter vectors to the pairwise edge set E p and the higher-order edge set E h such that w = [w T p , w T h ] T . in addition to the pairwise similarity between neighboring superpixels, the proposed higher-order correlation clustering considers a broad homogeneous region reflecting higher-order relations among superpixels.


We generate four hard negatives: very small in size relative to the foreground, the resulting hard negatives can have high overlap with the foreground, which will introduce undesirable noise (false negatives) when training the detector.

we shrink any hard negative that overlaps with the foreground estimate by more than 50%, until its overlap is 50% (we adjust the boundary that does not coincide with any of the foreground estimation boundaries).


Moreover, any constraint that is turned off will mostly stay off in the later steps.

for the bundle method we developed, whenever the size of B is larger than a given constant b, we will keep the active points of B that correspond to the first b largest α values, and drop the remaining ones.


Given a test data point x ∈ R m its non-linear projection onto M is x = f (g(x)).

w can be written as x , where we have made the approximation that U (x − x ) ≈ 0.


(b) The dependence of the upper bound on neural networks in [3] is characterized by the total number of parameters of neural networks, which can be quite loose by considering recent work, e.g., [20,27].

the goal of this paper is to address the above issue (a) by developing a lower bound on the minimax estimation error of d Fnn (µ, ν) (see Section 2.2 for the precise formulation) and to address issue (b) by developing a tighter upper bound than [3].


Now each steganographic algorithm has an associated training  Due to the non-convexity of the models in the training scheme, we cannot guarantee that two separate parties training on the same images will converge to the same model weights, and so learn the same embedding and decoding algorithms.

prior to steganographic communication, we require one of the communicating parties to train the scheme locally, encrypt model information and pass it to the other party along with information about the set of training images.


Second, a previous related study [8] has shown that, when considering large neuronal populations with a uniform distribution of stimulus preferences (such as an hypercolumn in V1 containing all stimulus orientations) the tuning width of individual neurons which is optimal for population coding depends crucially on the number of stimulus features being encoded.

it is interesting to investigate how the optimal arrangement of corticocortical synaptic systems depends on the number of stimulus features being encoded.


This is because for each natural image there are many non-natural images mapped to the same feature vector 2 .

a natural image prior is necessary to constrain the generated images to the manifold of natural images.


Our first task is to compute the coefficients P in the objective function.

note that, P ( 0 H P P FQ #B &P FQ s( 0 © P P Q W P Q , as basis B has scope restricted to S . Here, P P Q represents the marginal of the state relevance weights P over S s . the coefficients P can be pre-computed efficiently if P is represented compactly by its marginals P P S c pQ .


We found RADAGRAD to have a higher impact with convolutional layers than with dense layers, due to the higher correlations between weights.

for computational reasons, RADAGRAD was only applied on the convolutional layers.


This is rarely reasonable, while using instead directly the distribution of interest is more natural.

the experiments we propose on the next section focus on the influence of this concentrability coefficient.


we are only allowed to go in a direction which is a linear function of g).

n , we have The matrix M * which minimizes the probability of v T g to be positive satisfies The numerator of the derivative of this quantity is g g T M T CM g g T − 2 CM g g T M g g T . The first term is in the span of g and the second one is in the span of CM g. for this derivative to be 0 for all g, one must have M ∝ C −1 and we obtain the same result as in the Bayesian case: the natural gradient represents the direction minimizing the probability of increasing the generalization error.


Exact Pareto Distributions In the first experiment, we consider K = 3 arms with the distributions P k (x) = 1−x −α k , where α = [5, 1.1, 2].

the most heavy-tailed distribution is associated with the arm k = 2.


Indeed, this would implicitly shape the gate variable to include a linear operator on the features.

we chose a bilinear potential: where hi s = (h i s , 1) and K i,j s,s ∈ R (Cs+1)×(C s +1) being C s the size, i.e.


For small values of η the two networks relax to a state of a mutual agreement, R → 1 for η → 0.

with increasing learning rate η the angle between the two weight vectors increases up to the analytic solution shows, well supported by numerical simulations for N = 100, that two neural networks can synchronize to each other by mutual learning.


If this is the unique solution, it contradicts the index sum formula.


12, we alternating between optimization of messages {λ k } and energy functions {f α }.

optimization with respect to λ k for fixed F decomposes into minimizing A(λ k , θ k F ) independently for each y k , which can be done by running message-passing updates as in Section 5 using the parameter vector θ k F . the rest of this section is concerned with how to optimize with respect to F for fixed messages.


2 allows us to limit attention to the marginal distribution over other agents' actions given the state.

let z 0:T i = {s t , o t i , n t i,l , a t i , a t j , .


In reality, we expect the mapping to be more complex, because acoustically similar sounds can be produced by very different sources.

to accurately identify the underlying semantics, we would need to effectively use more (and perhaps, deeper) structure, such as the sound context, while making inferences.


Thus, if the initial estimate µ 0 = µ(0) is below the target value u r , γ will be small and hence the time constant τ will be small as well.

each spike will greatly increase the estimate and therefore speed up the approach of this estimate to the true value.


In a convolutional layer, activations of the same feature map at different locations (pixels) should be normalized in the same way.

we jointly normalize all the activations in a mini-batch over all locations (pixels) by following BN.


We use ideas from the Markov Chain Monte Carlo (MCMC) literature to satisfy the first condition and guarantee that {⇡ t ✓ } 1 t=0 will asymptotically converge to p d . Specifically, we require the transition operator T ✓ (•|x) to satisfy the detailed balance condition: for all x and x 0 .

this condition can be satisfied using Metropolis-Hastings (MH), where a sample x 0 is first obtained from a proposal distribution g ✓ (x 0 |x) and accepted with the following probability: the resulting MH transition kernel can be expressed as , and it can be shown that p d is stationary for T ✓ (•|x) [25].


The set of models, M j,l−1 , can be divided into two classes, the intentional models, IM j,l−1 , and subintentional models, SM j .


However, unlike previous color constancy methods that explicitly model statistical co-dependencies between pixels-for example, by modeling spatial derivatives [4,5,6], or learning functions on whole-image histograms [21]-we retain the overall parametric "form" by which we compute the illuminant in (4).

even though L[x, y] itself is learned through knowledge of co-occurence of chromaticities in natural images, estimation of the illuminant during inference is still achieved through a simple aggregation of per-pixel distributions.


On the other hand, if Assumption 2 holds, then the features depend both on x i and class label y; therefore, generally speaking, distribution over those features does not factorize by solely conditioning on x i -there will be remnant dependencies due to y.

if Assumption 2 is true, then Assumption 1 cannot be true in general, unless the data is generated according to a very specific model shown in the rightmost model in Fig.


However, these indirect supports are not conclusive.

we need to accurately estimate κ to make the assumption more reliable.


However, L2H significantly outperforms both L-LDA and L2F.

if incorporating labels into a model, learning an additional topic hierarchy improves predictive power and generalizability of L-LDA.


Each extension length considered must add an e-test else the algorithm stops, and so the maximal length of each e-test in Q V is upper bounded by the number of states.

our algorithm returns Q V in polynomial time.


When S w is singular, any positive S b makes Fisher's criterion infinitely large.

these naive attempts to calculate the (pseudo or approximate) inverse of S w may lead to arbitrary (meaningless) results.


This spatial correlation is measured with respect to the physical distances between the electrodes placed in the brain.

the same definition of the spatial factor function described in Section 3 is applicable.


Intuitively, m G (u, v) "aligns" the orbits of u and v before taking the inner product.

note also that  letting L : V → R is the Fenchel dual of the loss L, the dual problem of Eq.


Since for any (q, q 0 ) 2 [q min , q max ] 2 , s q and s q 0 differ only by a set of measure zero, the particular choice of q in that interval has no significant consequence.

in what follows, when it is not unique, we will choose the quantile value to be the maximum, q max .


As observed in [9] and repeated here, hyperalignment leads to increased between-subject accuracy and outperforms within-subject accuracy.

we are extracting more common structure across subjects.


It is easy to verify that ψ(x, Y ) is convex in x and concave in Y , and the constraint sets are closed convex for both x and Y .

(13) has a saddle point, and the min-max can be exchanged.


The eigenvalues of the Kronecker product S ⊗ X are of the form µ i (S)µ j (X) (see [8,Theorem 4.2.12]).

the spectrum of the Hessian is determined by the eigenvalues of X: A difficulty in the analysis of the unregularized case is that the objective function is only convex, but not strictly convex.


When these values are known, we can get the aggregated score η g(x i , k) of a weighted majority voting (WMV), and estimate the true labels by the rule: ŷi = argmax d∈[D] η g(x i , d).

reliable workers contribute more to the decisions.


1: for all t ≤ H − 1 do 2: ξ t ← GENSAMPLES(T, R) 3: for all s ∈ S do 4: creg H (s) ← 0 5: while t >= 0 do 6: for all s ∈ S do 7: for all ξ t q ∈ ξ t , a ∈ A do 8: It can be noted that MINIMIZECER() makes only H • |S| calls to the LP in GETCER() function, each of which has only |A| continuous variables and at most [1 + max t |ξ t |] number of constraints.

the overall complexity of MinimizeCER() is polynomial in the number of samples given fixed values of other attributes.


Intuitively, such a restriction on f implies that the second-order cross-moments contains the information of ⇤ .

we name this type of link functions as the second-order link.


For any teaching sequence of length 2, the individual intervals of the version space are of size shows that for all h 1θ11 ∈ V 1 (2), the decision boundary is parallel to the true decision boundary and the test error is E (x,y)∼p(x,y) , the decision boundary is orthogonal to the true decision boundary and the test error is 1/2.

we have R . The intuition is that a pair of teaching instances lead to a version space V (2) consisting of one interval per dimension.


If the decision maker has the perfect observation model P (Z|S), she could compute P (S|Z) by estimating P (S) and P (Z) beforehand by counting the total number of occurrences of each state without considering any observations, and the total number of occurrences of observation z, respectively.

accuracy and confidence are equal if the decision maker has the true model for observations.


We then denote the symmetric part of W by so that the eigenvalues of W k are all real.

note that the effect of W on the decay properties of the network is determined by the associated quadratic form, which is equivalent to that of the symmetric part of W , i.e., W . the eigenvalues of W k , especially those with the greatest magnitudes (absolute values), would describe the characteristics of the weights in nonlocal blocks.


If nature is adversarial, the learner will always make a "mistake", but if nature is regular or simple, there is hope that a learner may incur only a few mistakes.

a central goal of online learning is to design algorithms whose total mistakes can be bounded relative to the complexity of nature's labeling.


In fact, in the results section we will show that a learning rule with G xyy 3 (s, s ) ≥ 0 for all s, s > 0 and F xy 2 (s) ≤ 0 for s > 0 and all other terms set to zero is sufficient to explain the results from protocols P1 and P2.

in our learning rule an isolated pair of spikes in configuration post-before-pre will lead to depression.


It is capable of accumulating gradients from higher layers via back-propagation.

if the trajectory can be estimated by a parametric model, we can learn the model parameters using back-propagation as well.


This makes it very unlikely that those moves will be accepted, which typically traps the sampler in one mode.

the local structure of the Gibbs sampler is likely to be inadequate for problems of the type considered here, in which several probable assignments will produce well-separated modes.


If the conditions of Lemma 1 hold for X, β * , then for m large enough they will hold for X m , β * .

furthermore, as m → ∞ where the stochasticity on the left is due to W m , w m and on the right is due to w. with respect to the bounds ratio λ u /λ l , the construction of Eq.


For every i = 1, . . . , k, we have a matrix X i ∈ R T ×m where each column of X i contains a week of energy usage (measured every hour) for a particular house and for this particular type of device.

for example, the jth column of X 1 , which we denote x (j) 1 , may contain weekly energy consumption for a refrigerator (for a single week in a single house) and x (j) 2 could contain weekly energy consumption of a heater (for this same week in the same house).


ImageNet is a notoriously large data set which can be quite a feat of engineering and infrastructure to run experiments upon it, requiring many resources.

as well as using the full ImageNet data set, we devised a new data set -miniImageNet -consisting of 60, 000 colour images of size 84 × 84 with 100 classes, each having 600 examples.


For example, "Boston Globe" is a newspaper, and so it is not a natural combination of the meanings of "Boston" and "Globe".

using vectors to represent the whole phrases makes the Skip-gram model considerably more expressive.


The overestimation that results from the single estimator approach can have a large negative impact on algorithms that use this method, such as Q-learning.

we look at an alternative method to approximate max i E{X i }.


The innovations noise σ 2 i in ( 4) is involved in setting the smoothness of the time series, as shown in (5).

(4) would need to be augmented to explicitly capture departures from the AR model.


First, we show that the mean of the stationary distribution has low error.

if we ran for a really long time and averaged over many samples, the error would be low.


As in the Dirichlet-multinomial case, it is easily seen that if for each clique C ∈ C, the marginal distribution is defined by (11), then these distributions are consistent as they yield the same distribution (11) over the separators.

the unique Markov distribution over G with Beta-Bernoulli distribution over the cliques is defined by [8] Pr(Z where Pr(Z B ) given by (11) for each complete set B ⊆ G. The prior over ξ 1:n is thus given, for ξ 1:n ∈ E K , by where for each complete set B ⊆ G, n j,B is the number of items having feature j, j = 1, . . . , K in the set B and n B is the whole set of objects in set B.


Although this result is encouraging, in real problems we won't have convenient access to binary indicators for what matters.

our challenge evaluation domain gave the robot access to only a high-dimensional observation space.


The prediction and reconstruction tasks in our framework are designed to compete with each other.

η = α β influences which task has higher priority in the overall objective.


Thus, we implement a Rao-Blackwelized particle filter [6]: each particle corresponds to one sampled γ vector and contains the analytically computed state estimates for each of the objects, conditioned on that sampled γ vector.

taken together, the particles used for tracking (in our case we use 50, but see Section 3.4 for discussion) approximate the joint probability distribution over γ and S. In practice, we sample γ with the following iterative procedure.


Then, outliers tend to have smaller densityratio values (i.e., close to zero).

density-ratio approximators can be used for outlier detection.


In this setting the goal is to identify a vector w with low generalization error (E[ (w ) − (u)]).

we can solve this by running an OCO algorithm for T rounds and setting w to be the average value of w t . By online-to-batch conversion results [3,4], the generalization error is bounded by the expectation of the regret over the t divided by T . OCO algorithms can be used to solve stochastic optimization problems while also performing well in non-i.i.d.


Here we use a standard quadratic form with a trade-off factor.

given a matrix of word dependencies C, we can use the prior: for some power ν.


While the resulting model is more flexible than traditional parametric models, we find that it is still not as accurate as robust single-layer models.

we formulate a hybrid model that combines a base affine motion with a robust Markov random field (MRF) model of deformations from affine [6].


For practical applications, the elliptic symmetric assumption on P (X) in SIR cannot be fully satisfied, because many real datasets are embedded on manifolds [1].

sIR cannot select an efficient subspace for predicting the response Y because the local geometry of the predictors X is ignored.


6, and (3) The final inverse FFT is also shared.


For those classes y with very small P (y|m O ), their contributions to the value of classifier (10) are negligible.

when computing (10), we ignore the components whose P (y|m O ) is below some small threshold (equivalent to setting the contribution from this component to 0).


There is also physiological evidence on how these features could be extracted with separate sets of dedicated detectors working in parallel across the entire space [6].

in schematic diagrams as well as computational models on visual saliency [13,3,12], image segmentation [5], object recognition [10,4,20], and scene classification [12,11], it is routinely assumed that features at all scales, colors, and orientations are processed and available simultaneously.


Proof: Choose an arbitrary index set χ ⊂ {1, 2, . . . , P } such that |χ| = K. Let F χ be the sub-matrix formed by chosen K rows of F indexed by χ.

then, each row of Ã is generated by the chosen K rows of F . In the next lemma, we show how the row sparsity of F can be constrained to be at most N P (P −K+M ) by appropriately choosing the appended vectors z 1 , .


Equivalently, this means that multiple visits to a state can be treated as visiting different states, which leads to the Assumption 1 without loss of generality (by adding dummy states).

we can partition S according to the stage each state belongs to, and let S t be the set of states belong to t th stage.


However, the function ψ k is unfortunately nonconvex since the function returning the k-th largest coordinate is nonconvex for k ≥ 2.

finding a globally optimal solution is computationally intractable.


Later we will show that the sequence of iterates obtained from discretizing a suitable ODE never escapes this sublevel set.

the assumptions that we introduce need to hold only within a subset of R d . Let this subset be defined as that is, the set of points at unit distance to the initial sublevel set (2).


Roughly speaking, these algorithms construct a tree that can approximate a nearest neighbor search and have a controllable tradeoff between approximation error and search speed.

they can be used to approximate a nearest-neighbor classifier fast.


Suppose that in the upper row, the k-th neuron has the best-matching tuning function for a given s 1 .

on average, the k-th neuron has the highest firing rate in response to s 1 .


The segments found by HSLDS correspond to periods of time in which the latent dynamics seem to evolve according to different linear dynamical laws, suggesting that the observed firing pattern of the network has changed as a whole.

by construction, the HSLDS model can subdivide the network activity into different firing regimes for each trial specifically.


The sum-to-one constraint in (5) that is associated with leaf l can therefore be written as Furthermore, for all k ∈ K l the q B k(l) , as defined in (9), is a function of the same λ l . Accordingly, Now, consider l's leaf-node sibling, l . For example, in Figure 1, node l = a and l = b.

the two leaves share the same path from their parent to the root in T . Hence, using Property 1, it must be the case that B k(l) = B k(l ) for k ∈ K l . The two sum-to-one constraints-one for each leaf-therefore imply that q l = q l . Using (11), it now follows that we can replace λ l with f (λ l ) in all q B k expressions.


(The form of the linear EPSR update equation is identical to the form of the update in linear PSRs with s-tests given in Equation 3).

a linear EPSR model is defined by Q V and the set of weight vectors, m aoe for all a ∈ A, o ∈ O, e ∈ {Q V ∪ φ}, used in Equation 5.


Also, the channels Xi(t) appearing in this cost function are sampled at a clock rate on the order of fo, as opposed to the original sampling rate of the speech.

the few cross-correlations that are required can be computed with many fewer operations.


In this section we describe experiments we performed that compared PRank with two other online learning algorithms applied to ranking: a multiclass generalization of the perceptron algorithm [2], denoted MCP, and the Widrow-Hoff [9] algorithm for online regression learning which we denote by WHo For WH we fixed its learning rate to a constant value.

the hypotheses the three algorithms maintain share similarities but are different in their complexity: PRank maintains a vector w of dimension n and a vector of k -1 modifiable thresholds b, totaling n + k -1 parameters; MCP maintains k prototypes which are vectors of size n, yielding kn parameters; WH maintains a single vector w of size n. MCP builds the most complex hypothesis of the three while WH builds the simplest.


Also, observe that the minimizer of f is x * = 1 H and f (x * ) = 0, while the minimizer of g is any vector x : i x i = n 1/2+δ and g(x ) = n 1+δ − n 1/2+2δ = Θ(n 1+δ ).

the ratio between these two functions is unbounded.


For simplicity of notation, we also let i denote the product ranked at position i.

1 2 • • • n is the list of the products written by increasing order of their ranks.


We now compare the proposed sampling distribution pk with the optimal sampling solution in hindsight.

we know that if the true (gradient) vector c ∈ C k would be given to us, then the corresponding optimal probability distribution would be p (c) = for this c we can now analyze the ratio V ( pk ,c) V (p (c),c) .


Our results are consistent with recent work [30] noting benefits of adding noise to gradients when training deep networks.

quantization can be seen as a source of zero-mean noise, which happens to render communication more efficient.


A sufficient number M of random projections is determined by a nonlinear procedure (i.e., sequential computation of Isomap residual variance) so that conventional In this manner, ML-RP helps determine the number of rows that Φ requires in order to act as an operator that preserves metric structure.

mL-RP can be viewed as an adaptive method for linear reduction of data dimensionality.


(b), the decision boundaries for class 1 and 2 have a large inter-class angular margin with each other due to the additionally injected class 5 (which doesn't take part in the final recognition evaluation) and the constrained region of class 1 is much tighter than before (other classes have the same phenomenon), yielding more compact and separable features which are highly beneficial to the original 4-class recognition.

artificially injected virtual classes (e.g.


In the limit, as the scale of W approach +∞, the loss term (Θ T f (g), y) becomes negligible compared to the score term g T W x.

the solutions to loss-augmented inference and inference problems become almost identical, except when an element of W x is very close to 0.


Also, once the consistency limits in (b) and (c) hold, the SE equations in Algorithm 2 reduce to the SE equations for the non-adaptive GAMP method running with the true parameters.

we conclude there is asymptotically no performance loss between the adaptive GAMP algorithm and a corresponding oracle GAMP algorithm that knows the correct parameters in the sense that the empirical distributions of the algorithm outputs are described by the same SE equations.


The clique problem can be reduced to the maximum likelihood problem of Definition (1).


The time stamps assigned to nodes by a cascade induce a directed acyclic graph (DAG) by defining node j as the parent of i if t j < t i .

it is meaningful to refer to parents and children within a cascade [3], which is different from the parent-child structural relation on the true underlying diffusion network.


Apparently, the discriminant function (12) represents a strong coupling between the generative model and the discriminative margin constraints.

crowdSVM jointly considers these two factors when estimating true labels.


Further more, set B = D(A) and Ŷ = D( X), we have Lemma 2.

given a column centered data matrix A, set B = D(A) and Then, X is the optimal solution to Problem (1) iff we can determine whether X is the membership matrix of A by solving Problem (5).


Note that X = GS where G(: C(v,:) 1 D(v,:) 1 .

we have 1 T S = 1 T if W ≥ 0, meaning the columns of X all lie on the simplex spanned by the columns of G, and the vertices of the simplex correspond to the anchor words.


The graph laplacian depends only on nearest neighbor relations and in MCU these are assumed to be unchanged as the points are embedded from the original space to the low dimensional manifold.

l 1 and L 2 can be obtained using the graph of data points, {x 1i } and {x 2i }, in the original space.


More precisely, the features were chosen as the power around 12Hz and 18Hz extracted at 3 electrodes over the sensori-motor cortex (C3, C4 and Cz).

6 features are extracted during the movement and 6 during the rebound.


. . , x T is the solutions generated by A.

the upper bound in Theorem 2 cannot be improved in general.


( Since the theory is short scale insensitive, we can generate random probability densities chosen from the prior by replacing ¢ with its Fourier series and truncating the latter at some  3) enforces the amplitude of the k'th mode to be distributed a priori normally with the standard deviation Coded in such a way, the simulations are extremely computationally intensive.

monte Carlo averagings given here are only over 500 runs, fluctuation determinants are calculated according to Eq.


Quantitatively, each NBP marginal is 0.5 to 1.5 bits/instance less accurate than the exact ones.

the accuracy of NPNBP in this case is approximately twice that of NBP per variable, amounting to a substantial per sample advantage.


This estimates to which extent the last k experiences help to learn a better model as evaluated over the complete data.

if ζ(s, a) is small, then the last k visitations in the data-set D s,a did not have a significant effect on improving T and in turn s, a does not require further exploration.


The constrained gradient algorithm chooses tests that give an early indication of being linearly independent.

the constrained gradient finds most, or all, core tests extremely quickly, but can also choose tests that are not linearly independent.


For example, for a CoffeeScript program with 20 tokens, its corresponding JavaScript program may contain more than 300 tokens.

the model needs to predict much fewer tokens for a CoffeeScript program than a JavaScript program, so that even seq2seq models can achieve a reasonably good accuracy.


In particular, note that if ✓ t = 0 for ↵ t < 1, then the corresponding term in P (✓) will take the value 1, overwhelming the likelihood term.

any feasible solution with the above property could be considered 'optimal'.


UoI-based methods leverage stochastic data resampling and a range of sparsity-inducing regularization parameters/dimensions to build families of potential features, and they then average nearly unbiased parameter estimates of selected features to maximize predictive accuracy.

uoI separates model selection with intersection operations from model estimation with union operations: the limitations of selection by intersection are counteracted by the union of estimates, and vice versa.


The relation between contrast and Bayes risk becomes more convenient when we slightly modify the above definition (3) by a unit upper bound and by adding a lower bound on the -scaled density differences: with scale factor ¡ 9 A@ 5B DC .

for an infinite scale factor the (expected) contrast F ¡ ¥ ¡ £ ¦¥ F ¡ ¤£ § G ¥ © approaches the negative Bayes risk up to constant shift and scaling: Thus the scale factor defines a subset of the input-space, which includes the decision boundary and which becomes increasingly focused in their vicinity as & .


A best response is a strategy such that u i (BR(x −i ), [17] is a strategy profile in which no player can improve her utility by unilaterally deviating from her strategy.

for each player i, a NE The sequence form [12,23] of a game is a compact representation applicable only to games with perfect recall.


by a concatenation of v + 1 activation vectors of length 56, with the additional vector being the representation of the output.

we obtain a state vector of length s = 56(v + 1).


After T iterations, since φ 0 is a convex combination of functions in Φ, and since all the f (t) are linear functions, we have Since A is a no-regret algorithm, assume that A ensures that Regret Φ (T ) = O(T 1−c ) for some constant c > 0.

when T = Ω( 1 ε 1/c ) the lower bound (2) on the regret cannot hold unless we have already found an ε-approximate fixed point of φ 0 .


Otherwise, the resulting policy has a sparse and smaller supporting set.

a sparsemax policy has advantages over a softmax policy for modeling sparse and multi-modal behaviors of an expert whose supporting set varies according to the state.


Thus, the mental look back into the past seems to occur exactly when credit assignment is to be performed.

it is plausible that hippocampal replay could be a way of doing temporal credit assignment (and possibly BPTT) on a short time scale, but here we argue for a solution which could handle credit assignment over much longer durations.


However, they act on individual parameters.

similarly to [Jaderberg et al., 2014b, Denton et al., 2014 and to other parameter regularization techniques [Krogh andHertz, 1992, Bartlett, 1996], these methods do not perform model selection; the number of layers and neurons per layer is determined manually and won't be affected by learning.


Thus, we only use |S i | /2 independent pairs.

t i is updated as T i + |S i | /2 at each round.


The key advantage is that the acceptance probability in (2) only depends on d(y j , C) 2 and d(x j 1 , C) 2 since min ✓ p(y j ) p(x j 1 ) q(x j 1 ) q(y j ) , 1 Critically, in each of the k 1 iterations, the algorithm does not require a full pass through the data, but only needs to compute the distances between m points and up to k 1 centers.

the complexity of K-MC 2 is O mk 2 d compared to O(nkd) for k-means++ seeding.


In our web collection, C, there are strong indicators of content in the form of caption words -if an object is described in the text associated with an image then it is likely to be depicted.

for the images, I m ∈ M , in our matched set we run only those detectors for objects (or stuff) that are mentioned in the associated caption.


Allowing multiple near-optimal policies, Tree-RL offers more diversity in search paths and is able to find multiple objects with a single feedforward pass.

tree-RL can better cover different objects with various scales which is quite appealing in the context of object proposal.


The DI values generally increased as distant correlations appeared more frequently, i.e., more "auditoriness" of the inputs grew.

the topographic disorder of auditory maps results from distant correlations presented by natural auditory signals.


The assumption of −α regularly varying in the tail is thus the weakest possible assumption that ensures that the (properly rescaled) maximum of samples emitted by a heavy tailed distributions has a limit.

the very related assumption of approximate Pareto is almost minimal, but it is (provably) still not restrictive enough to ensure a convergence rate.


We observed that each reversible block has a computation depth of two original residual blocks.

we reduced the total number of residual blocks by approximately half, while approximately doubling the number of channels per block, since they are partitioned into two.


With the simple encoding scheme E = I K−1 , we can see that f (x, k) = (u, −θ), x (k) .

when the absolute cost matrix is taken and a modified perceptron learning rule 2 is used as the underlying binary classification algorithm, the PRank algorithm is a specific instance of our framework.


Although [26] has recently proposed an approximate iterative method, we provide an algorithm in Appendix K that is able to efficiently compute the exact solution.

by combining this result with Propositions 2 and 3 we are able to efficiently compute the polar Ω • p and hence apply Algorithm 1 to solving (23) with respect to W . To investigate the effectiveness of these computational schemes we considered three applications: group lasso, path coding, and latent fused lasso.


The sum of 128 terms in (4) can be expressed as a single look-up table, whose keys are 7-bit long binary vectors, the " th bit corresponding to whether § % .

the evaluation of £ from equation (1) on thousands of pieces of music can be done in less than a second on a modern PC.


These results are supported by preliminary theoretical analysis of a variant of U oI Lasso (see [4]).

the relative values of B 1 and B 2 express the fundamental balance between the two basic operations of intersection (which compresses the feature space) and union (which expands the feature space).


Note that A diag(π) is an arbitrary d × d matrix whose entries sum to 1, which has d 2 − 1 degrees of freedom, whereas µ(θ) is a symmetric matrix whose entries sum to 1, which has d+1 2 − 1 degrees of freedom.

s Θ (θ) has dimension d 2 and therefore the model is non-identifiable.


The standard procedure is to (re)generate entries of R on-demand [7] using pseudo-random numbers [16].

we only need to store X ∈ R k . When a stream element a t = (i t , I t ) arrives, one updates the entries of X: ( By property of stable distributions, the samples x j , j = 1 to k, are also i.i.d.


In contrast, the events in the former category are less likely to inspire new links but found in abundance.

it is very suitable for real-world sparse data.


Recall that at each iteration t, the inner iteration is at most 2.

the time complexity for any given outer iteration is O(dmn).


The circuits were fabricated on the AMS 0.6µm 2P3M CMOS process, which allows a power supply voltage of five volts.

the states of neurons {s i } and the corresponding weights {w ij } are designed to be represented by voltage in [1.5, 3.5]V and [0,5]V respectively, with both arithmetical zeros at 2.5V.


2 leads to a nice property that maximization over θ, given β can be done in closed form.

it just amounts to taking a (scaled) average of all β k 's which are children of the loss function in Eq.


For example, consider the alternate objective function f (w) min( w 0 , N ), leading to the optimization problem min While the global minimum remains unchanged, we observe that all local minima occurring at non-degenerate basic feasible solutions have been effectively removed.

6 In other words, at any solution w * with N nonzero entries, we can always add a small component αw ′ ∈ Null(Φ) without increasing f (w), since f (w) can never be greater than N . we are free to move from basic feasible solution to basic feasible solution without increasing f (w).


As we can see, the decision boundaries obtained by the Gaussian process classifier is heavily affected by outliers, while those obtained by the Student-t process classifier are more stable.

as expected, Student-t process classification is more robust against outliers compared to Gaussian process classification, thanks to the heavy-tailed structure of the Student-t distribution.


Let X = {x i } n i=1 be the data set, F be the feature space, and φ : X → F be the associated feature map of K. Then, the degree of smoothness of φ on the data graph can be captured by (Zhou et al., 2004): where w ij is the similarity of x i and x j , d ii = n j=1 w ij , and • F is the distance metric in F. The smaller the value S(φ), the smoother is the feature map φ.

the SDP in (5-9) seeks the smoothest feature map that embeds the data on the unit hypersphere and at the same time respects the pairwise constraints.


This is consistent with improved classification performance -as the representation of basis distributions gets more holistic, the more unlike they become when compared to bases of other classes.

there is a lesser chance that the bases of one class can compose an image in another class, thereby improving performance.


Indeed, Vavasis has shown that NMF is NP-Hard [27]; see [4] for further worst-case hardness results.

we must instate additional assumptions on the data if we hope to compute nonnegative matrix factorizations in practice.


In this experiment, there are only 25 items and many of them are quite distinctive and clearly at the ends of any continuum of actions.

participants are likely to remember not only having rated items previously, but also the ratings that they assigned.


As mentioned before, μP is the "best" possible estimator to estimate µ P if nothing is known about P. However, depending on the information that is available about P, one can construct various estimators of µ P that perform "better" than µ P . Usually, the performance measure that is used for comparison is the mean-squared error though alternate measures can be used.

our main objective is to improve upon KME in terms of the mean-squared error, i.e., construct μP such that E P μP − µ P 2 ≤ E P μP − µ P 2 for all P ∈ P with strict inequality holding for at least one element in P where P is a suitably large class of Borel probability measures on X .


Specifically, the log likelihood of the training data for the CPS model can be written as below, It is not difficult to prove that L(θ) is concave with respect to θ.

we can use simple optimization techniques like gradient ascent to find the globally optimal θ.


It is obvious that the resulting c λ t (s) is also continuous, piecewise linear.

since c N = 0, the assumption of continuous and piecewise linear value functions holds by backward induction.


This is because the proposition is true for all values of µ satisfying µ L 8 n , so we can choose the smallest possible value of µ = 8L n . We have observed in practice that the IAG method with a step size of α k = 1 2nµ may diverge, even under these assumptions.

for certain problems the SAG iterations can tolerate a much larger step size, which leads to increased robustness to the selection of the step size.


. . , ξ t,mt } be the samples in the t-th iteration with size m t ≥ 1.

the stochastic partial gradient is computed as gi ).


Human retina, for example, conveys signals from ≈125 million photoreceptors to the rest of the brain via ≈1 million ganglion cells [1] suggesting a hundred-fold dimensionality reduction.

biologically plausible dimensionality reduction algorithms may offer a model of early sensory processing.


Despite being worst-case optimal, G-allocation is minimizing a rather loose upper bound on the quantity used to test the stopping criterion.

we define an alternative static allocation that targets the stopping condition in Eq.


We also noticed that the explanations in SICK-E, even when wrong, were generally on-topic and valid statements, while the ones in MultiNLI were generally nonsense or off-topic.

transfer learning for generating explanations in out-of-domain NLI would constitute challenging future work.


Therefore, if eventually θ 1 is the true parameter vector satisfying C 1 and C 3 , then θ 2 satisfies these conditions as well.

the true vector cannot be accurately estimated.


As before, we make the approximation that the posterior is Gaussian, q(f |m, C) = N (f |m, C) where m ∈ R N is the mean posterior latent function, and C ∈ R N ×N is the posterior covariance.

since the likelihood is isotropic and factorizes over the N observations we have the following expectation under our variational inference framework: the linearization is one-dimensional, that is g(f n ) ≈ a n f n + b n . Using this we can derive the approximate gradients, where A = diag([a 1 , .


The disaggregation performance was evaluated by the normalized disaggregation error (NDE) For the energy data we are also particularly interested in recovering the total energy used by each appliance [16,10].

another objective of the disaggregation is to estimate the total energy consumed by each appliance over a period of time.


1 Equivalently, min  In contrast with optimal control and planning tasks, softened distance functions, d soft (s), and costto-go functions, h soft (s), in predictive IOC are based on many paths rather than a single (best) one.

unlike in A* search, each sub-optimal path cannot simply be ignored; its influence must instead be incorporated into the softened distance calculation (4).


Consider the typical situation in forward selection where we have a current working set I and we are interested in choosing the next basis vector, x i .

the method of Smola and Bartlett (2001) evaluates each given x i / ∈ I by trying its complete inclusion, i.e., set I = I ∪ {x i } and optimize π(α) their selection criterion for the instance x i / ∈ I is the decrease in π(α) that can be obtained by allowing both α I and α i as variables to be non-zero.


In particular, if the data are characterized by functions that are continuous, values of the trajectory that are close to each other tend to be highly redundant and convey similar information.

if the value of the process at a particular instant has high discriminant capacity, one could think of discarding nearby values.


Besides, kernel methods have the important computational advantage that no nonlinear optimization is involved.

the use of kernels provides elegant nonlinear generalizations of many existing linear algorithms.


The data-dependent mean w is not necessarily differentially private, even approximately.

under suitable assumptions, SGLD can be used to optimize a data-dependent mean and still yield a valid PAC-Bayes bound.


Because the approximated posterior of µ 2 does not depend on the sequence X, the sequence variational lower bound L(θ, φ; X) can be decomposed into the sum of L(θ, φ; x (n) |μ 2 ), the conditional segment variational lower bounds, over segments, plus the log prior probability of μ2 and a constant.

instead of sampling a batch at the sequence level to maximize the sequence variational lower bound, we can sample a batch at the segment level to maximize the segment variational lower bound: This approach provides better scalability when the sequences are extremely long, such that computing an entire sequence for a batched update is too computationally expensive.


Specifically, we have the likelihood of "observing" the topic assignments z d for document d 1 as .

following Homes & Held [8], the likelihood for η d k conditioned on dn is the number of words assigned to topic k in document d. we have the conditional distribution where This is a posterior distribution of a Bayesian logistic model with a Gaussian prior, where z k dn are binary response variables.


To train the deep learning network h(y; θ) with respect to θ, a gradient-based optimization algorithm is used such as the stochastic gradient descent (SGD) [31], momentum, Nesterov momentum [32], or the Adam optimization algorithm [33].

for any gradient-based optimization method, it is essential to calculate the gradient of ( 7) with respect to θ as follows, it is sufficient to calculate the gradient of the empirical risk (8) to approximate (9) for any gradient-based optimization.


This property suggests to use the logarithm of the eigenvalues of S as the natural quantities since multiplying all eigenvalues with a constant results in a vertical shift of the log-spectrum.

the properties of the optimal solution only depend on the shape of the log-spectrum.


Consider a variational distribution q that lies between the input p m and p n . The log Bhattacharyya affinity with β = 1/2 can be written as follows: log B(p m , p n ) = log q(x) p m (x)p n (x) q(x) dx ≥ −D(q p m )/2 − D(q p n )/2.

b(p m , p n ) ≥ exp(−D(q p m )/2 − D(q p n )/2).


Our general findings is that in Euclidean spaces, one can achieve the same regret bounds when movement cost is applied.

the SMB algorithm can achieve the optimal regret rate.


Consider the case where the population contains many liberal users with very low anonymity levels yet one single paranoid user (user i) wants to have a maximal anonymity with δ i = n. In the k-anonymity framework, that user will require n − 1 identical copies of his data in the released database.

a single paranoid user will destroy all the information of the database which will merely contain completely redundant vectors.


(2009) is because their fully-trained (non-random) networks were overfitting more on the smaller Caltech-101 dataset than ours on the larger ImageNet informative, however, because the performance at each layer is based on different random draws of the upper layer initialization weights.

the fact that layers 5, 6, and 7 result in almost identical performance across random draws suggests that multiple runs at a given layer would result in similar performance.


The score is taken to be the total number of edge directions guaranteed to be learnt in the worst case.

the score P (v) is: The node with the highest score among the color class c is used for the intervention I i .


Due to axis symmetry, and E[x i x 3 j ] are all zero for distinct indices.

the MEL estimates for a and b are identical to the Gaussian case given above.


The non-convexity of F makes it hard to directly convert the problem into its dual form.

our approach for training GANs with non-convex discriminators is based on repeatedly linearizing and dualizing the discriminator locally.


The goal is to find a decision-policy π which minimizes the total expected loss A decision policy is characterized by how τ and δ are generated as a function of the data observed so far.

finding the optimal decision-policy is equivalent to finding the random variables τ and δ that minimize l(τ, δ; θ, D) .


. . , p n ) T . Each parameter p i of Q, with 1 ≤ i ≤ n, approximates P(z i = 1|y, X), i.e., the posterior probability that the i-th training instance is an outlier.

these parameters can be used to identify the data instances that are more likely to be outliers.


. . , λ d ), possibly with complex entries.

we can recover Λ 2 + Λ = Q −1 (AA + A)Q.


However, these methods deal with deterministic estimates of the gradient and are not based on linear transformations of the gradient.

this second line of research is also significantly different from what we do here.


However, most open datasets for text classification are quite small, and large-scale datasets are splitted with a significantly smaller training set than testing [21].

instead of confusing our community more by using them, we built several large-scale datasets for our experiments, ranging from hundreds of thousands to several millions of samples.


The optimality result for the vector-valued graphical model [4] implies that the optimal rate for estimating , which matches our result in (3.6).

the rate in (3.6) obtained by our Tlasso estimator is minimax-optimal since it is the best rate one can obtain even when ⌦ ⇤ j (j 6 = k) are known.


Clearly, for 1 − p > p, R(steak) > R(salmon), and the diner would be rational in switching his preference.

through our inferential machinery, we retrieve the anecdotal explanation for the diner's behavior: if he believes that he is more likely to be in a good restaurant (with probability (1 − p)) than not, he will prefer steak.


On an average, the cost of solution by SMO forms 85 to 95% of the total computational time.


To create a more economical representation, we propose replacing the link attributes by a smaller set of attributes that "summarize" the information from link matrix L , possibly in combination with the term matrix T . The most obvious attributes of a document are what terms it contains.

one simple way to represent the "attributes" of a document's inlinks is to aggregate the terms in the documents that link to it.


This potential encourages the assignment y = 1 to agree with the assignment x = 1 and y = 0 to agree with x = {2, . . . , L}.

phrased differently, this potential favours beliefs such that: one may replace the above potential using a single consistency constraint.


Although we can directly solve the IPP using any off-the-shelf mathematical optimization software, IPP solvers are not as mature as Integer Linear programming(ILP) solvers.

for efficiency reasons, we propose to convert the IPP to an ILP using the classic method outlined in [25] (we skip the details for lack of space).


ImageNet contains 1,000 highly specific classes that typical people may not be able to identify, such as "Chesapeake Bay retriever".

we combined some of these fine classes to form six coarse classes we were confident would be familiar to our experiment subjects ({dog, cat, broccoli, cabbage, spider, snake}).


Each observation of x i in context x 1:i−1 corresponds to a customer in restaurant x 1:i−1 who is served dish x i , and each table in each restaurant u, being a draw from the base distribution G σ(u) , corresponds to a customer in the parent restaurant σ(u).

the numbers of customers and tables have to satisfy the constraints where c x us = 1 if s = x i and u = x 1:i−1 for some i, and 0 otherwise.


For instance, the height of first peak is determined by the total energy density of the universe, while the third peak is related to the amount of dark matter.

by fitting models of the CMB power spectrum for given values of the seven parameters, we can determine how the parameters influence the shape of the model spectrum.


In this figure, the layer labeled "Composite" corresponds to the time-tempo variables, (tn, sn)t, for the composite rhythm, while the layer labeled "Update" corresponds to the interpretation variables ( Tn, 0" n) t. The directed arrows of this graph indicate the conditional dependency structure of our model.

given all variables "upstream" of a variable, x, in the graph, the conditional distribution of x depends only on the parent variables.


While the quantitative evaluation provides some indication of transfer quality, it does not capture all the aspects of this generation task.

we also perform two human evaluations on 500 sentences randomly selected from the test set 2 .


The difficulty is that the set {a | | a, G | ≤ λ r } is possibly hard to characterize, and even if we could characterize this set, it may not be amenable enough for the optimization solvers to leverage in order to provide a speedup.

we propose an alternative characterization of the fixed subspace: Definition 1.


This phase transition (Lemma 7) tells us that a tight community C contains w.h.p.

a giant connected component with a constant fraction of the nodes from C. a single node from a tight community influences a constant fraction of its community in expectation.


( 4)) of the ith component, and zi 1:K,P is parameterized by the initial pose z i 0,P and the transition variables β i 1:(T +K) .

in our model, we treat z 1:N 0,P , β 1:N 1:(T +K) ,   [31] 341.2 -Brabandere et al.


In order to do this well, it is essential to use previously recorded events to form a good policy on the first round of interaction.


We show for mixtures of multiple sound sources that there exists a strong correlation between the relative strength of target and interference and estimated ITD/IID, resulting in a characteristic clustering across frequency bands.

we employ a nonparametric classification method to determine decision regions in the joint ITD-IID feature space that correspond to an optimal estimate for an ideal mask.


We emphasize that just as in the non-adaptive setting, at no time can an algorithm stop and declare that it is TPR-, ⌧ or FWPD-, ⌧ for any finite ⌧ 2 N. This is because there may be an arm in H 1 with a mean infinitesimally close to µ 0 but distinct such that no algorithm can determine whether it is in H 0 or H 1 .

the algorithm must run indefinitely or until it is stopped externally.


The hypotheses are linear classifiers X I-t (w,ifJ(x))/C =: (w,x)/C in some fixed feature space K ~ £~ where we assume that a mapping ¢ : X -+ K is chosen a priori 1 .

since all we need for learning is the real-valued output (w, Xi) /C of the classifier w at the m training points in Xl, ... , Xm we can assume that w can be expressed as (see [9]) it suffices to learn the m expansion coefficients a E IR m rather than the n components of w E K. This is particularly useful if the dimensionality dim (K) = n of the feature space K is much greater (or possibly infinite) than the number m of training points.


The obvious choice is the so-called defining representation, whose elements are the familiar permutation matrices ρ def (σ) = P (σ) [P (σ)] p,q = 1 σ(q) = p 0 otherwise, since the corresponding loss function is The squared Frobenius norm in this expression simply counts the number of mismatches between the observed but noisy permutations τ ji and the inferred permutations σ j σ −1 i .

furthermore, by the results of the previous section, letting P i ≡ P (σ(i)) and P ji ≡ P (τ ji ) for notational simplicity, (4) can be written in the form V ⊤ LV with similarly to the rotation case, synchronization over S n can be solved by forming V from the first d ρdef = n lowest eigenvectors of L, and extracting each P σi from its i'th n × n block.


Moreover, each corrupted row/column item has perturbed feature x i = u i + ∆u i and y i = v i + ∆v i , where ∥∆u∥ ∞ ≤ ξ 1 and Theorem 3 suggests that, if features have good quality in the sense that items with corrupted features are not too many, for example g(n), h(n) = O(log n), then sample complexity of DirtyIMC can be O(n log n √ log n) = o(n 3/2 ) as well.

both Theorem 2 and 3 provide concrete examples showing that given imperfect yet informative features, the sample complexity of our model can be asymptotically lower than the lower bound of pure matrix completion (which is Ω(n 3/2 )).


We parameterise the costs with γ and µ as follows: The matrix of pairwise distances as defined above will, in general, not be a proper kernel (i.e., it will not be positive definite).

we use it to build a new representation of the data (see below).


We assume that the posterior density would be unimodal if not for the effect of outliers that we do not wish to explicitly model.

we seek for a point estimate at the outset, rather than a generic approximation of the entire posterior.


By the triangle inequality, Now, using (4) and (5), Thus, ( 6) is bounded as Next, a bound on (7) follows simply by the norm comparison inequality (5): The statement follows by rearranging the terms.

, where η = (e 2 − 1)η, then for all i, for all i Proof.


4 Our algorithm proceeds by estimating subtree widths of near-optimal subtrees.

we encounter a two-pronged exploration-exploitation trade-off: samples from a given subtree reveal information not only about payoffs but also about the width, whereas in Lipschitz MAB we only need to worry about the payoffs.


The interval defines constraints for the parameters because the latter quantifies the deviation of © ¡ from the benchmark 1 ¡ .

the return maximization task can be stated as a constraint optimization problem with ¤ ¡ f as the actual return of asset This problem can be solved as a penalized maximization task with © is defined as a type of -insensitive error function: Summarizing, the construction of the allocation scheme consists of the following two steps: (i.)


A person who signs the petition forwards the petition, on average, to 20.37 other addressees.

by linearity of expectation, we can estimate that the ≈ 172,832.38 signers in the unobserved tree sent a total of ≈ 3,520,595.58 chain-letter e-mails in the Iraq-War petition.


We explain why Neon2 online weak works as follows.

starting from a randomly perturbed point x 1 = x 0 + ξ, it keeps updating x t+1 ← x t − η (∇f i (x t ) − ∇f i (x 0 )) for some random index i ∈ [n], and stops either when T iterations are reached, or when x t+1 −x 0 2 > r. we have x t − x 0 2 ≤ r throughout the iterations, and thus can approximate .


The first set of experiments was conducted using the single-site features for all the three methods.

no neighborhood data interaction was used for both the logistic classifier and the DRF, i.e.


To represent various log probability distributions log(P * (y|x)), the range of log (f s (z)) ∈ R M should be sufficiently large.

we investigate the range of log (f s (z)).


Moreover, the constant are inside bigO is also large, because of complex transformations.

for datasets with high values of d (non-zeros) this scheme is very slow.


However, previous functional approaches to dendritic processing were limited because they studied dendritic computations in a firing rate-based framework [3,4], essentially requiring both the inputs and the output of a cell to have stationary firing rates for hundreds of milliseconds.

they ignored the effects and consequences of temporal variations in neural activities at the time scale of inter-spike intervals characteristic of in vivo states [5].


Because I/O devices usually access memory in units of blocks, many densely packed values might be read faster than a few numbers of largely distributed values.

the implementability of an efficient algorithm in terms of both FLOPs and memory access time would be more important.


For the PhysioNet dataset, we can not access the complete samples.

we measure the filling accuracies of our proposed method and some other imputation methods indirectly.


Since it is enough to show that the following holds an arbitrary 1 ≤ j ≤ m − k to prove Theorem 3: Then, (8) follows from the following inequalities: Pr The inequality (9) holds since ∥d j ∥ = 1 by the definition and Assumption 4.


( 9), all elements of g(z) should be non-negative to limit output in [0, 1].

g(z) should be non-negative: [g(z)] i ≥ 0.


When facing the large-scale data but with sparse labels, these models might have computational complexity issues or parameter estimating issues.

in machine learning community, numerous studies have been focusing on optimizing the parameters of these models [15][16], designing efficient algorithms [17] [18], providing sharp minimax bounds [19] and proposing novel aggregation models [9][20] [21].


. . , π θ ) L times, and then averaging bidder i's resulting utilities.

we approximate (π) by ˆ (π) ≈ ÛL (π * N , π −i ) − ÛL (π, π −i ).


As outlined in Section 3.3, our selection of the fixed sets ensures that a block update restricted to the fixed set would not change any values since these variables in fixed sets already satisfy the coordinatewise optimality condition.

while our algorithm only explicitly updates the free set block, this is equivalent to updating variables in fixed and free blocks consecutively.


Integrating a log-concave function over some of its arguments produces a log-concave function in the remaining arguments (Prekopa, 1973).

the Bhattacharyya affinity is log-concave in the parameters of jointly log-concave distributions.


Using the form of ( 9), observe that min s∈R m + y(x, s) is equivalent to min s∈R m + m j=1 2λ j ρs j + s 2 j + 2s j c j (x).

for fixed x, this is strictly convex in s. its unconstrained minimum can only be its stationary point, which satisfies 0 = 2λ j ρ + 2s * j (x) + 2c j (x), for j = 1, . . . , m. Accounting for the nonnegativity constraint, we obtain the following optimal slack as a function of x: Above we write s * as a function of x to convey that x remains a "free" quantity in y(x, s * (x)).


As a result, Patients 1, 2 are scheduled for next screening at date 4 but Patient 3, who has a lower ex ante survival probability, is scheduled for next screening at date 3.

the policy is personalized to the ex ante risk.


Each term inside the summation depends on ✓S t and ✓at t , which in turn depend on all the model parameters in ⇥ via the computational graph defined by Algorithm 2 (see Appendix E for a diagram of the graph).

we can learn the model parameters in ⇥ by sampling an episode in the data, computing the corresponding stochastic gradient in (7) by back-propagation on the computational graph given in Algorithm 2, and updating ⇥ by stochastic gradient/mirror descent.


Four layer transducers produced results with higher spread in accuracy -possibly because of the more difficult optimization involved.

the best average PER we achieved (over 3 runs) was 19.8% on the TIMIT test set.


The following observation will also be useful to prove error bounds for DPDA-D iterate sequence.

for each i ∈ N , the definition of R k i in (17) implies that R k i (x) ∈ B 0 for all x; hence, from (19), we trivially get the following bound on µ k : Moreover, for any µ and λ we have that Theorem 3.2.


It is easy to see that for K * = K + this condition is satisfied provided that K((x 1 , x 1 ), (x 2 , x 2 )) ≥ 0 for all x 1 , x 1 , x 2 , x 2 ∈ X .

in this case to solve (1) it is sufficient to optimize the variables (α ij ) m i,j=1 , introducing the additional constraints α ij ≥ 0 to enforce k ∈ K + .


However, the computational complexity of jointly sampling all variables in a block typically increases with the treewidth of the Markov network projected on the block.

in practice, given time and memory resource constraints, the main issue in blocked Gibbs sampling is finding the right balance between computational complexity and accuracy.


Each utterance is transcribed at phone level.

phone is considered as a source of variability or source of information.


We prove that the difference between the two bounds decreases as O(k − 1) + log m/m, where k is the number of topics in the model, and m is the number of words in the document.

the advantage of CVB over VB is lost for longer documents.


The value w t (x, y) can be interpreted as a probability, and so we have that w t (x, y) ≤ 1.


A standard approach is to represent the collection of objects as a set of abstract points, and define distances among the points to represent similarities -the closer the points, the more similar they are.

clustering is centered around an intuitively compelling but vaguely defined goal: given an underlying set of points, partition them into a collection of clusters so that points in the same cluster are close together, while points in different clusters are far apart.


To achieve this goal, we show that the second order smoothness measures the linearity of the function, and the gradient field of a linear function has to be a parallel vector field.

we propose to find a function which minimizes the empirical error, and simultaneously requires its gradient field to be as parallel as possible.


It is worth to mention that although it seems that the layer-wise error is scaled by a quite large product factor, S l = L k=l+1 Θk F when it propagates to the final layer, this scaling is still tractable in practice because ultimate network output is also scaled by the same product factor compared with the output of layer l. For example, we can easily estimate the norm of ultimate network output via, Y L F ≈ S 1 Y 1 F . If one pruning operation in the 1st layer causes the layer-wise error √ δE 1 , then the relative ultimate output error is .

we can see that even S 1 may be quite large, the relative ultimate output error would still be about F which is controllable in practice especially when most of modern deep networks adopt maxout layer [23] as ultimate output.


3.5 [19]) or moments accounting [2] can be used to relate the privacy guarantee of SGD to that of gradient mean estimate at each instance t. We also note that in SGD, we often sample the clients, standard privacy amplification results via sampling [2], can be used to get tighter bounds in this case.

akin to [33], in the rest of the paper we just focus on the MSE and privacy guarantees of DME.


For example, Srebro [1] showed that finding a maximum-likelihood chordal (or triangulated or decomposable) Markov network is NP-hard even for networks of treewidth at most 2, in sharp contrast to the treewidth-1 case [2].

various approximative approaches and local search heuristics have been proposed [3,1,4,5,6,7,8,9,10,11].


We first show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems.

we establish that a class of rank minimization problems have closed form solutions.


For example, for the quadratic function and the Hessian ∇ 2 F (x) = A T A.

hence the parameter σ and L i satisfy (choose y = x + te i , where t ∈ R, e i ∈ R n is the unit vector), xA T Ax ≥ σ x 2 and tA T i Ae i ≤ L i |t|, ∀x, t. the σ is the smallest eigenvalue of A T A and L i = A i 2 , where A i is the ith column of the matrix A.


Eqns 9 and 10 are generalized Sylvester matrix equations of the form AXB ⊤ + CXD ⊤ = E whose unique solution X under certain regularity conditions can be exactly obtained by an extended version of the classical Bartels-Stewart method [9] whose complexity is O((p+q) 3 ) for p×q-sized matrix variable X. Alternatively, one can solve the linear system [10]: where ⊗ denotes Kronecker product and vec(X) vectorizes X in a column oriented way (it behaves as the matlab operator X(:)).

the solution to Eqns (9,10) are as follows, These linear systems are of size nm r × nm r and dm c × dm c respectively.


In practice, the incomparability information might be difficult to obtain.

we assume the underlying incomparability structure, is unknown and inaccessible.


Indeed, the performance of a kernel method depends crucially on the kernel used, where different choices of kernels often lead to quite different results.

substantial efforts have been made to design appropriate kernels for the problems at hand.


. . , z m ), our starting stable polynomial (this stability follows from Prop.

8) is which can be expanded as h(z, w) is real stable in 2m variables, indexed below by S and R where R := S\T .


This is a finitesample guarantee, as compared to the asymptotic version of [OD12].

absolutediscounting becomes competitive with respect to the family of classes defined by permutations of power-laws, inheriting Good-Turing's behavior [OS15].


(If we instead fixed its value, the proposition above would not hold.)

the time complexity of computing conditional messages can be reduced by sampling more variables, but only up to a point: the time complexity of computing the conditional message must be o(|X B∩C |).


the average degree of the the vertices in L is at most Setting α > 2 e 3 d yields e 3 α 2 d < 1 2 , and it can be verified that (2πd) < 2 for all such α.