Since we are optimizing over a graph G, we expect that the evaluation function b Q should take into account the current partial solution S as it maps to the graph.

x v = 1 for all nodes v 2 S, and the nodes are connected according to the graph structure.


the full time graph, and the summary time graph is acyclic.

then the full time graph can be recovered from the joint distribution of X t . the true causal summary time graph is identifiable.


An example of this construction is provided again by linear functions.

we choose X = IR d , where d ∈ IN and C q (x , t q ) = (Q x , Q q t q ), x ∈ X , t q ∈ X q , where Q are p × d matrices.


Such a modification, though simple, has a profound impact: the NAG algorithms attain faster convergence than VGD.

nAG achieves O(1/k 2 ) convergence for smooth convex functions, and linear convergence with a better constant term for strongly convex functions [11].


Then, a reasonable ranking rule based on the binary answers is r(x) = y = 1 + min {k : Although the definition can be flexibly applied even when f b is not consistent, a consistent f b is usually desired in order to introduce a good ranking rule r. Furthermore, the ordinal information can help to model the relative confidence in the binary outputs.

when k is farther from the rank of x, the answer f b (x, k) should be more confident.


The larger cluster is clearly a bent-double cluster: 89 of the 150 objects are more likely to belong to this cluster under the model and 88 out of the 89 objects in this cluster have the bent-double label.

the unsupervised algorithm has discovered a cluster that corresponds to "strong examples" of bent-doubles relative to the particular feature-space and model.


At each round, we either add a new predictor tree or a new state-space trajectory to the model, depending on which results in a greater decrease in the objective.

with the following definitions: (y t f k 1 (x t , s t )) j (x t ), k,j = 1 T T X t=1 (y t f k 1 (x t , s t ))s t,j .


LWMB splitting is similar to the splitting operation presented in [9] for factor-graph models, but operates by partitioning a group of MLN domain elements rather than a group of variational parameters (as in [9]).

we partition a single domain ∆ ∈ ∆ into two disjoint domains (∆ (1) ∪ ∆ (2) = ∆ and ∆ (1) ∩ ∆ (2) = ∅).


It first compresses all the visual region features to an integrate vector v G and then feeds it into the same latent space with a three-layer neural network.

the output of the mid-layer is forced to be close to the corresponding class semantic feature.


The basic assumption is that neuronal development 1 is a Markov process governed by local rules [4].

we assume that (i) each step in dendritic outgrowth only depends on the preceding step and on current local conditions; and (ii) dendrites do not undergo geometrical or topological modification after their formation (see, however, Discussion).


While well grounded in statistics, this approach does not explicitly promote distinctiveness.

the differences among the captions of different images are not explicitly taken into account.


Our goal in this paper is to address this stark contrast by providing a theoretical analysis of the performance of affine policies on synthetic instances of the problem generated from a probabilistic model.

we consider random instances of the two-stage adjustable problem (1) where the entries of the constraint matrix B are random from a given distribution and analyze the performance of affine policies for a large class of distributions.


Our second contribution drills down into the information-theoretic and information-geometric parts of (3).

from the former standpoint, we completely specify the parameters of the supervised game, unveiling a key parameter left arbitrary in [30] (explicitly incorporating the link function of proper composite losses [32]).


Proof: s = 0 implies [c, c + s 1] = ;, and therefore xp,c exactly consists of all updates until clock c 1.

⇤ Our key tool for convergence analysis is to define a reference sequence of states x t , informally referred to as the "true" sequence (this is different and unrelated to the SSPtable server's view): , where u t := u t mod P ,bt/P c . we sum updates by first looping over workers (t mod P ), then over clocks bt/P c. We can now bound the difference between the "true" sequence x t and the noisy views xp,c : for certain constants F, L. Then, under suitable conditions (f t are L-Lipschitz and the distance between two points )).


If we assume that p j ≤ c/m for a universal (numerical) constant c, then Theorem 5 guarantees that which is the convergence rate of ADAGRAD except for a small factor of min{ √ log T /T, p j } in addition to the usual p j /T rate.

optimizing by choosing η = R ∞ , and assuming p j 1 T log T , we have convergence guarantee which is minimax optimal by Proposition 1.


It follows that Q satisfies the differential privacy constraints and the resulting f -divergence is also preserved.

using the fact that and thus the corresponding f -divergence is invariant: We can naturally define equivalence classes for staircase mechanisms that are equivalent up to a permutation of columns and merging/splitting of columns with the same pattern: [Q] = {Q ∈ S ε | exists a sequence of permutations and merge/split of columns from Q to Q} .


The first experimental results, obtained by using real speech recognition tasks, showed the effectiveness of VBEC.

when the training data size was small, VBEC significantly outperformed conventional methods.


Instead, a family of more convenient sparse penalized regression cost functions are reviewed in Section 2.

we discuss conventional Type I sparsity penalties, such as the ℓ 1 norm and the ℓ 1,2 mixed norm, and a Type II empirical Bayesian alternative characterized by dictionary dependency.


Let b M be constants such that where m k is the multinomial coefficient and a(k, m, M m . Thus, φ M is an M -degree polynomial approximation to φ logit (yx • θ) with the d+M d monomials of degree at most M serving as sufficient statistics derived from yx.

we have a exponential family model with where k is taken over all k ∈ N d such that j k j ≤ M . We next discuss the calculation of the b and the choice of M . Choosing the polynomial approximation.


Our results show that accumulating knowledge over the course of learning multiple tasks can be beneficial for the future even if these tasks are not i.i.d.

for the situation when the observed tasks are sampled from the same task environment but with possible dependencies we prove a theorem that generalizes the existing bound for the i.i.d.


Next, we learn to synthesize realistic 2D images given projected 2.5D sketches that encode both the viewpoint and the object shape.

we learn a texture network G texture that takes a randomly sampled texture code z texture and the projected 2.5D sketches v 2.5D as input, and produces a 2D image x = G texture (v 2.5D , z texture ).


The counts n ¬ij jk• , n ¬ij •kxij , n ¬ij •k• act as fields through which z ij interacts with other variables.

averaging both sides of (8) by p(z ¬ij |x, α, β) gives us the Callen equations, a set of equations that the true posterior must satisfy: Since the latent variables are already weakly dependent on each other, it is possible to replace (9) by a set of mean field equations where latent variables are assumed independent and still expect these equations to be accurate.


(1993), we can prove the following basic result: then no RBM network of size m, whose weights are bounded in magnitude by C, can compute a function which represents ndimensional IP with margin δ.

for C and 1/δ bounded by polynomials in n, for n sufficiently large, this condition is satisfied whenever m < 2 (1/9− )n for some > 0.


SVDM optimizes an objective that is a combination of dimensionality reduction and classification.

a linear mapping from input to feature space and a linear classifier applied to feature space, are trained simultaneously.


We exploit diverse human commonsense knowledge for reasoning over large-scale object categories and reaching semantic coherency within one image.

we present Hybrid Knowledge Routed Modules (HKRM) that incorporates the reasoning routed by two kinds of knowledge forms: an explicit knowledge module for structured constraints that are summarized with linguistic knowledge (e.g.


Notice that the weights in the boosting algorithm correspond to the derivative of the potential, because −φ (z) = min{1, exp(−z)} ∈ [0, 1].

the weak learning step is essentially a gradient descent step.


This literature highlights the point that an effective exploration scheme is critical to the design of any efficient reinforcement learning algorithm.

popular exploration schemes such as ✏-greedy, Boltzmann, and knowledge gradient can require learning times that grow exponentially in the number of states and/or the planning horizon.


sample drawn from the mixture distribution in (2).

we replace the empirical averages in (3) and (4) with the expectations with respect to the mixture distribution.


Then the algorithm move to the node with the third smallest label and so on.

the algorithm is summarized in Algorithm 1 in Appendix D. Algorithm 1 returns a list r * (s) per node s ∈ V, which contains information about distance to the smallest reachable labels from s. each list contains pairs of distance and random labels, (d, r), and these pairs are ordered as where {•} (l) denotes the l-th element in the list.


L 3 is unique in being universally rooted.

for any integer r > 1 other than r = 3, we constructively demonstrate a hypergraph G(V, E) with |V | = r + 1 variables for which Theorem 21 examines L 0 r+1 (∇G) and L i r+1 (∇G), which by Theorem 17 are the uprooted equivalents of L r (G) and L r (G i ).


1, A c is the derivative of f c (I) with respect to the input I at the sample I 0 : Eq.

1 indicates that the output score of class c is affected by the element values of A c . the values of A c indicate the importance of the corresponding pixels of I 0 to generate the class score.


Here, we show that privileged information can naturally be treated as noise in the latent function of a Gaussian process classifier (GPC).

in contrast to the standard GPC setting, the latent function is not just a nuisance but a feature: it becomes a natural measure of confidence about the training data by modulating the slope of the GPC probit likelihood function.


To make further progress, we introduce a second layer of approximations based on belief propagation.

we approximate the required marginals in the gradient for λ MP with the ones obtained with BP.


We have shown how to use these bounds to infer the parameters of a DPP, and demonstrated their use for expensive-butexact MCMC and cheap-but-approximate variational inference.

these bounds have some degree of freedom -the pseudo-inputs -, which we optimize so as to tighten the bounds.


At the same time this is likely to come with a large number of useless features, which makes it suitable for sparsification.

we include the player hand value, indicator functions for each possible player hand value and dealer hand value, and a large description of the cards not dealt yet (corresponding to the history of the game), under the form of indicator functions for various ranges.


The construction method for the nonconvex penalties provides us with a new approach for solving the corresponding penalized regression model.

to solve the nonconvex penalized regression problem: we equivalently formulate it as Here D(w, a) is either KL(a, w) or KL(w, a).


This time, however, we use the fact that action-sets depend upon the global state (rather than events) to enforce the desired dynamics.

we add special state-dependent actions that, based on their availability (or lack thereof), affect each agent's local reward.


Our approach involves a "discriminative" approach to the problem of speech separation.

rather than building a complex model of speech, we instead focus directly on the task of separation and optimize parameters that determine separation performance.


However, the challenge of ALM is that it produces a series of bound-constrained quadratic problems that, in the traditional sense, are harder to solve than linear system produced by IPM or Simplex methods [17].

in a Projected-CG approach [18], one needs to solve several linear systems via CG to find solution to the bound-constrained quadratic program, while there is no guarantee on how many iterations it requires.


For weakly correlated features, the bounds are tighter.

for normalized features and R 1 < 1, Theorem 2 implies: This shows that the best conditioning of the optimization problem is obtained for uncorrelated and normalized features.


In fact, sufficient conditions for the convergence and correctness of Gaussian belief propagation (GaBP) are known for these models.

gaBP converges to the optimal solution if the positive definite matrix A is walk-summable, scaled diagonally dominant, or log-concave decomposable [22; 7; 8; 9].


Short-term coding: r(θ) ∼ I (f (θ), Σ(θ)), where r i ∈ {0, 1} and I(µ, Σ) is the maximum entropy distribution consistent with the constraints provided by µ and Σ, the Ising model [16].

for short-term population coding, we assume the population acitivity to be binary with each neuron either emitting one spike or none.


This means that the centering imposed in MVU via constraints is equivalent to the centering in HSIC by means of the dependence measure tr HKHL itself.

mVU equivalently maximizes tr HKHI, i.e.


For each data point x (i) , we actively acquire the features in a sequential order.

at t = 0 we start with an empty acquired set O 0 := ;.


The noise assumption is compatible with the manifold assumption within the manifold+noise model.

we can assume that the functions of interest vary along the manifold and are constant in the orthogonal direction.


Another set of techniques, based on quantization, have become popular recently due to their strong performance on real-world data.

product quantization (PQ) [12,20] and its variants have regularly claimed top spots on public benchmarks such as GIST1M, SIFT1B [20] and DEEP10M [3].


In order to evaluate the generality of our framework, we simulate data from three different instances of our model: those given by Gaussian, Bernoulli (Ising), and Poisson node-conditional distributions.

we assume the true conditional distribution, P (Y |X), follows (7) with the parameters: ✓ we permit both the mean, ✓ s (X) and the covariance or edge-weights, ✓ st (X), to depend on the covariates.


Finally, when we described the associative memory in Section 3, we did not specify the memory key generation procedure.

after answering the current question, we append the computed attention map to the memory.


For stochastic optimization problems, another important evaluation criterion is the confidence level of the objective value.

it is of great interest to find (N, δ) as a monotonically decreasing function in both N and δ ∈ (0, 1) such that the solution x N +1 satisfies Pr (φ(x N +1 ) − φ(x * ) ≥ (N, δ)) ≤ δ.


We conjecture the techniques developed here will help in characterizing these issues.

the most natural question is how to develop algorithms that satisfy the mixed stability criterion: the algorithm should give "nearly the right answer to nearly the right problem" (as in [Trefethen and Bau III, 1997]).


This is difficult to evaluate since it conditions an entire latent spike train on an entire observed spike train.

the posterior distribution of spiketimings of the latent neurons depends on both past and future of the observed neurons' spike train.


In practice, we may cluster the positive examples into p groups, where p is much smaller than the number of positive examples, and assign all positive examples in each group with a common prototype.

we have j.i / D j.k/ if x i and x k belong to the same cluster.


In this paper we use the winner-take-all interaction function, inspired by studies in computational neuroscience.

we use the hard winner-take-all function: In the case of multiple winners, ties are broken by index precedence.


We compare the proposed bilevel distance metric learning model with the following methods: the base-line KNN [14], LMNN [16], FANTOPE [7], CAP [5] and RML [10].

as baselines, we consider the most relevant technique from the literature, i.e., k-nearest neighbor method (KNN).


Since our motivation is an application in bioinformatics, our notation and terminology will be drawn from that area.

genes are features, samples are examples, and pathways are factors.


A curve γ is geodesic if it satisfies the equation ∇ γ γ = 0.

geodesics are curves with zero acceleration.


It turns out that there is a very subtle bug in the online learning algorithm.

we were using standard stochastic gradient ascent to estimate Σ v,11 (and Σ v,22 ), and on every step there is a small chance that the gradient update overshoots zero, causing Σ v,11 to become negative.


The above theorem still requires us to minimize over F. Our next results shows that the solution is finite dimensional.

we establish a representer theorem for our problem.


's work [2], this notion of structure is distilled further by the use of "atoms."

given a set A of very "simple" objects, called atoms, we can say that a vector x is simple (with some low-dimensional structure) if it can be written as a linear combination of few atoms: x = LZ~I C,8;, where k is small relative to the ambient dimensionality.


To show the interpretability of LDR, we visualize representative individuals, each of which suffers from an inferred sub-risk.

for each inferred sub-risk k under risk j, we find the representative by evaluating a weighted average of all uncensored observations as i w ijk x i / i w ijk , where , λ ijk ∼ Gamma(r jk , e x i βjk ), and rjk and βjk are the estimated values of r jk and β jk , respectively.


However, not all paths carry the same amount of gradient.

the length of the paths through the network affects the gradient magnitude during backpropagation [1,8].


. . , k t < k are the episodes where (x, a) was taken at step h. Proof of Lemma 4.3.

for each fixed (x, a, h) ∈ S × A × [H], let us denote k 0 = 0, and denote k i is the episode of which (x, a) was taken at step h for the ith time (or k i = K + 1 if it is taken for fewer than i times).


We first review a well-known method for active learning in the realizable case, refered to as CAL after its discoverers Cohn, Atlas, and Ladner [8].

some n of this size suffices to guarantee that, with probability 1 − δ, the returned classifier ĥ has er( ĥ) ≤ ǫ.


Any long-run learning model can model sequential data by being applied to all of the data observed up to a particular point.

after observing n datapoints, one simply applies the model, regardless of whether n is "the long-run."


Let (h, g) be a selective classifier selected after observing the training set S m . We say that (h, g) is a weakly optimal selective classifier if, for any 0 < δ < 1, with probability of at least 1 − δ over random choices of S m , R(h, g) ≤ R(h * , g).

with high probability our classifier is at least as good as the true risk minimizer over its region of activity.


In the field of combinatorial optimization, it is well-known that submodular maximization is NP-hard and the approximation factor of (1 − 1/e) (≈ 0.63) achieved by the greedy algorithm [19] is the theoretical limit of a polynomial-time algorithm for positive and nondecreasing submodular functions [3].

in the worst case, any polynomial-time algorithm cannot give a solution whose function value is more than (1 − 1/e) times larger than the optimal value unless P=NP.


An integrator is said to be a Kth-order local integrator if for any smooth and bounded function f , the corresponding Kolmogorov operator P h satisfies the following relation: The second approximation error is manifested when handling large data.

the SGLD and SGHMC use stochastic gradients in the 1st and 2nd-order LDs, respectively, by replacing in F and L the full negative log-posterior U with a scaled log-posterior, Ũl , from the l-th minibatch.


Then, V (α; w, λ, I, J) ≤ U (α; w, λ, I, J) and V (0; w, λ, I, J) = U (0; w, λ, I, J).

u is an upper bound on V and the bound is exact at α = 0.


We first examine the case when d = 1 and M = 1.

when θ i ∈ R and when each θ i only corresponds to one y i .


There are several advantages by using our new definition: (1) The new definition leads to a unified analysis for both VGD and NAG.

if we follow the same notion as [16], we need to redefine t = kη for VGD, which is different from t = k √ η for NAG; (2) The new definition is more flexible, and leads to a unified analysis for both gradient-type (VGD and NAG) and coordinate-gradient-type algorithms (RCGD and ARCG), regardless of their different step sizes, e.g η = 1/L for VGD and NAG, and η = 1/L max for RCGD and ARCG; (3) The new definition is equivalent to [16] only when h = √ η.


The RMSProp optimization algorithm has a significant advantage for backpropagation-based planning because of its ability to perform gradient normalization that avoids exploding gradients and additionally deals with piecewise gradients [Balduzzi et al., 2016] that arise naturally as conditional transitions in many nonlinear domains (e.g., the Navigation domain of Figure 1 has different piecewise transition dynamics depending on the starting region).

instead of naively updating action a itj through equation 1, RMSProp maintains a decaying root mean squared gradient value G for each variable, which averages over squared gradients of previous epochs and updates each action variable through Here, the gradient is relatively small and consistent over iterations.


Second, though standard RL only updates the chosen option's value and DBM updates the posterior probability of all options, assuming maximization decision rule (Blakely, Starin, & Poling, 1988), both models will lead to qualitatively similar choice preference.

the mostly rewarded choice will have an increasing value in RL model, and increasing reward probability in DBM, while the often-unrewarded choice will lead to a decreasing value in RL model, and decreasing reward probability in DBM.


To study the consequences of Theorem 3, observe that when the covariance matrix Σ is generated randomly, the amount of correlation is much larger than in the idealized case that Σ is diagonal.

for a fixed value of tr(Σ), the quantity tr(D −1 σ )/ |||R||| F , is much smaller in in the presence of correlation.


We also note that the model we evaluated was chosen based on overall performance on a withheld validation set, using a single seed.

we did not run multiple replicas with the best hyperparameter settings (as was done in other models, such as the Sparse DNC, which demonstrated performance fluctuations with a standard deviation of more than ±3 tasks passed for the best choice of hyperparameters).


Given a set of images {I j } K j=1 , we first extract their features {v j } K j=1 with a pretrained CNN, and then average {v j } K j=1 to obtain v. The image encoder converts v into a context vector h v ∈ R D which is used as the visual input for all subsequent modules.

the image encoder is parameterized as a fully-connected layer, and the visual features are extracted from the last convolution layer of a DenseNet [15] or VGG-19 [29].


With the opponent model τ , the similarity between opponent's different policies τ 1 and τ 2 can be measured using a KL divergence, denoted by D KL ( τ1 , τ2 ) ≈ E (s,a) log τ1(s,a) τ2(s,a) , whereby opponent's different policies with same belief can be further distinguished by utilizing the corresponding opponent model.

during online interaction, line 5 in Algorithm 1 estimates the opponent's online policy τo , thus the posterior probability (unnormalized) of the opponent using policy τ can be defined as follows: p(τ Here, τi is the previously learned approximate policy of the known opponent's policy τ i stored in T and the KL divergence is measured using observed state-action pairs.


Two proposal distributions R(w, x) and R(w , x) are the same provided that π(x) = π (x).

the specific values of Φ(x, y, h) • w and Φ(x, y, h) • w are irrelevant, and only their ordering matters.


We address the other two issues in this section.

we attempted to validate that the behavior of the method on simulated data is consistent with that on real data, and that the fitted-hyperparameter samples can be used as a proxy for samples from the posterior.


We just need to change the payoff function g to account for ψ(x).

we consider the following two-players zero-sum game, min x max y { x, y −f * (y)+ψ(x)}.


The analysis allows for an arbitrary number of coding units, thus including both under-and over-complete representations, and provides a number of important insights into optimal coding strategies.

we show how the form of the code adapts to the number of coding units and to different data and noise conditions to achieve robustness.


The experimental results demonstrated that CPLST is the best among the LSDR approaches when coupled with linear regression or kernel ridge regression.

cPLST is consistently better than its feature-unaware precursor, PLST.


This proves that σ (D) and σ (D) are independent when D is drawn from X(n, 1 / 2 ).

this reasoning easily extends to s XORs in Q s and we have that Evaluating this using the union bound and pairwise independence shows p one,s (σ ) > c(α) 2 −s .


One case of particular interest is the so-called box sum, in which the convolution kernel is a constant function within a region.

h(j) = k over some interval and h(j) = 0 outside that interval.


In addition to elucidating possible computational mechanisms in the brain, implementing "usable computation" with the dynamics of a neural network holds a number of advantages over conventional digital computation, including massive parallelism and fault tolerance.

by following such dynamical systems approach, we can exploit the rich behavior of physical devices such as transistors to directly emulate these dynamics, and obtain more dense and power efficient computation [12].


The construction in our result is ultimately rather simple, and relies on powerful results developed in complexity theory over the last decade.

we exploit striking results on the complexity class associated with computational protocols known as Arthur-Merlin games.


We propose a novel Adaptive Markov Chain Monte Carlo algorithm to compute the partition function.

we show how to accelerate a flat histogram sampling technique by significantly reducing the number of "null moves" in the chain, while maintaining asymptotic convergence properties.


To address the task of following natural language instructions, we rely on two models: an instructionfollower model of the kind considered in previous work, and a speaker model-a learned instruction generator that models how humans describe routes in navigation tasks.

we base our follower model on the sequence-to-sequence model [1], computing a distribution P F (r | d) over routes r (state and action sequences) given route descriptions d. The follower encodes the sequence of words in the route description with an LSTM [23], and outputs route actions sequentially, using an attention mechanism [5] over the description.


Therefore, we rely on a larger outside dataset for training, as did by all recent highperformance face verification algorithms [4,2,21,20,13].

we use the CelebFaces+ dataset [20] for training, which contains 202, 599 face images of 10, 177 identities (celebrities) collected from the Internet.


Then, posterior inference via the Bayes' theorem is equivalent to solving the following optimisation problem [15]: where P is the space of probability distribution 1 , q(Θ) is the required posterior (here and afterwards we will drop the dependency on X for notation simplicity).

the Bayesian posterior p(Θ|X) is identical to the optimal solution to (1).


What happens when we change the activation function?

two widely used activation functions for neural networks are the sigmoidal activation function, σ sig (z) = 1/(1 + exp(−z)), and the rectified linear unit (ReLU) activation function, σ relu (z) = max{z, 0}.


Clearly there is a one-to-one correspondence between the two plots as both are based on the same contingency tables [2].

precision associated with an ROC point is proportional to the angle between the line connecting the point with the origin and the x-axis.


Any local maximizer x * must satisfy the Karush-Kuhn-Tucker (KKT) condition [14], i.e., the firstorder necessary conditions for local optimality.

since x * i , µ i and β i are all nonnegative for all i's, Hence, the KKT conditions can be rewritten as: According to x, the vertices set V can be divided into three disjoint subsets, The Equation ( 9) characterizes the properties of the solution of ( 5), which are further summarized in the following theorem.


Roughly speaking, our analysis indicates that for streaming PCA, the delay ⌧ k 's are allowed to asymptotically scale as Moreover, our analysis also indicates that the asynchrony has very different behaviors from momentum.


Deep features in standard CNNs must eventually transition from general to specific along the network, and the transferability of features and classifiers will decrease when the cross-domain discrepancy increases [3].

the shifts in the data distributions linger even after multilayer feature abstractions.


Our objective can be interpreted similarly to that in [Das16].

similar points i and j should be located lower in the tree as to maximize |non-leaves(T [i ∨ j])|, the points that get separated at high levels of the hierarchical clustering.


the first parameter, while the update of the EM-algorithm is given by w k+1 = argmax w∈W Q(w, w k ).

steepest gradient ascent moves in the direction that most rapidly increases Q(w, w k ) w.r.t.


This gives a theoretical justification of Young's idea and other followups under our framework.

f 1 B is similar to rule studied by Procaccia et al.


Recent studies in developmental and cognitive psychology suggest that successful learning in toddlers lies in the quality of visual data they collect from their everyday activities [7].

while most images in ImageNet and other computer vision datasets are photographs that were taken with consumer cameras by adults, the visual information cast on a toddler's retina is from a first-person perspective.


In particular, choosing π(d|x) to be the same distribution as the log-likelihood for our statistical model, the s samples from the extended state space D may be thought of as pseudo-data, from which we can deterministically calculate an estimate of the Expected Fisher Information to use as the covariance of a proposal distribution.

each sampled pseudo-data can be used deterministically to give a sample of ∂L dθ given the current θ, all of which may then be used deterministically to obtain an approximation of the covariance of tangent vectors at the current point.


Formally, sub-structure mining is to find the complete set, but no-duplication, of all significant (generally frequent) sub-structures from dataset.

we apply combination of a sequential pattern mining technique, PrefixSpan [9], and a statistical metric pruning (SMP) method, Apriori SMP [8].


In this paper, belief propagation is extended to represent factors with tree approximations, by way of the expectation propagation framework.

each factor sends a "message" to all pairs of nodes in a tree structure.


Then for i ∈ {0, 1}, Recall the discussion of expectation, variance, pairwise independence, and 3-wise independence in Section 2.

when a number of random variables are 3-wise independent, the conditional variance of their sum (conditioned on one of these variables) equals the sum of their individual conditional variances.


The linear time and space complexity then follows from the linear time and space inference complexity of SPNs.

we reduce the original moment computation problem for edge (k, j) to a joint inference problem in S with a set of weights determined by α α α.


For any r ≥ 1 and series of vertices v 0 , ..., v m , let W r ((v 0 , ..., v m )) be 1 if v 0 , ..., v m is an r-nonbacktracking walk and 0 otherwise.

also, for any r ≥ 1, series of vertices v 0 , ..., v m and c 0 , ..., c m ∈ R m+1 , let W (c0,...,cm)[r] ((v 0 , ..., v m )) = (i0,...,i m )∈(0,...,m) W (c0,...,cm)[r] ((v 0 , ..., v m )) is the sum over all subsequences of (v 0 , ..., v m ) that form r-nonbacktracking walks of the products of the negatives of the c i /n corresponding to the elements of (v 0 , ..., v m ) that are not in the walks.


Previous work has shown that combining multiple heuristics for NP-hard problems into a portfolio can dramatically reduce average-case running time [8,9,19].

algorithms based on chronological backtracking often exhibit heavy-tailed run length distributions, and periodically restarting them with a fresh random seed can reduce the mean running time by orders of magnitude [8].


However, the teacher can ensure that the learner achieves better performance in a worst case sense by providing demonstrations tailored to the problem at hand.

assume that the teacher uses π T = π 0 , the policy shown in Figure 1a, which moves to the central cell as quickly as possible and then alternates between cells 3 and 4.


Moreover, vS has been found to mediate rewards signal engaged in mutual contribution, altruism, and social approval [18].

the left vS has been found to be associated with both social and monetary reward prediction error [13].


The graphs show the performance advantage of both multi-level HSM agents over the other agents.

we find that the flat memory-based agent does considerably worse than the other three, as expected.


The contribution of this paper is a new way to include constraints into multivariate GPs.

we develop a method that transforms a given GP into a new, derived, one that satisfies the constraints.


We then have to solve the prediction problem, i.e.,  We now notice that if the scalar c j = c(j), where c is a non-increasing function of rank position j, then ( 17) can be solved simply by sorting the values of h i , ✓i in decreasing order.

1 the matching problem becomes one of ranking the values h i , ✓i.


when the prediction task is regression or multi-class classification) this limits the benefit landscape that can be expressed, but in the important special case of binary classification, the following Proposition establishes that this restriction is without loss of generality 4 .

we can attach an arbitrary combination of benefit values to the four possible (y, ŷ)-pairs (i.e.


In practice we found that it may not necessary to wait CG to converge.

even if we set the maximum iteration K in CG to a small fixed number (e.g., 10 in our experiments, though with thousands of parameters), the performance does not deteriorate.


More recent work employs the method-of-moments, which permit the means of the mixture components to be arbitrarily close, provided that the sample size is sufficiently large (Kalai et al., 2010;Belkin and Sinha, 2010;Hsu and Kakade, 2013;Hardt and Price, 2015).

hardt and Price (2015) characterize the information-theoretic limits of parameter estimation for mixtures of two Gaussians, and that they are achieved by a variant of the original method-of-moments of Pearson (1894).


. . , x n }, let S denote the n × n similarity matrix, which is both symmetric and non-negative.

s i,j = S j,i and S i,j ≥ 0, where S i,j is the element of S in the i th row and the j th column (i, j = 1, . . . , n).


refining a partition can only reduce the margin.


2) where agent i with type θ i arrives in period t , and with the expectation taken over future types given current reports θ ≤t .

when truth-telling is an -BNE we say that the mechanism is -BNE incentive compatible and no agent can improve its expected utility by more than > 0, for any type, as long as other agents are bidding truthfully.


In our first experiment, we check whether our intuitions about higher-order statistics described in the introduction are captured by the proposed method.

we expect that arbitrarily oriented lines can only be predicted using third-order statistics.


We measure the algorithm's performance on 3 by its cumulative loss 3 $ S 3d fe F6 X ` gD 6 $ b § " $ b9 In our analysis, we will write h and i j ` § @9 A9 @9 § #i 3 when we want to stress the fact that the cumulative loss and the hypotheses of the on-line algorithm are functions of the random sample ( 3 .

throughout the paper i will denote the (deterministic) initial hypothesis of an arbitrary on-line algorithm and, for each k ml U nl £ , i o will be a random variable denoting the -th hypothesis of the on-line algorithm and such that the value of i j # f( S6 8 § A9 A9 @9 B § C( 3 $ does not change upon changes in the values of ( p fq F6 ¤ § @9 A9 @9 § C( 3 .


In this work, we move a tiny step towards a theory and a systematic approach that characterize the representations learned by deep nets.

we consider a simpler problem: How similar are the representations learned by two networks with identical architecture but trained from different initializations.


Even for small t the curves are close to linear, showing that the local linear convergence rate kicks in essentially from the start of trade in our simulations.

there exist some ĉ and γ such that, empirically, we have Ê[F (r t )] − F ≈ ĉγ t , or equivalently, log( Ê[F (r t )] − F ) ≈ log ĉ + t log γ.


In this paper, we show that in many cases a powerful multiplicative low-rank approximation in Frobenius norm can be obtained by applying a simple truncated SVD procedure on a crude, easyto-find spectral norm approximate.

given the spectral norm approximation condition , where A k is the best rank-k approximation of A in Frobenius and spectral norm.


Although Binary Dropout does not prevent these networks from overfitting, SBP decides to remove all neurons of the neural network and provides a constant prediction.

in this case SBP chooses the simplest model that achieves the same testing error rate.


Second, unlike the average cost objective, the maximum likelihood objective is sensitive to the magnitude of the cost.

the logarithm of Equation 1 corresponds to a soft minimum over all likely trajectories under the current policy, with the softness of the minimum inversely proportional to the cost magnitude.


In particular we compare our approach to the baselines in a missing data scenario, where part of the sequence is missing.


In what follows, we let Q ⊂ m 4 be the set of all quadruples {p, q, r, s} ∈ m 4 such that only one of the three path pairs in (2.4) gives an empty set when the union of its two component paths is taken.

q contains all S ∈ m 4 such that T |S is one of the configurations in Figure 2.2(a)-(c).


Again, the MRes cascade was faster and more accurate than the others.

it was faster than other methods, while detecting/recognizing 6% more traffic signs.


When using the smoothed weighted max norm, we need to choose the amount of smoothing to apply to the marginals, that is, we need to choose ζ in our definition of the smoothed row and column weights, as given in (3).

alternately, we could regularize simultaneously over all possible amounts of smoothing by considering the local max norm with and same for C. R and C are line segments in the simplex-they are larger than any single point as for the uniform or weighted trace norm (or smoothed weighted trace norm for a fixed amount of smoothing), but smaller than the entire simplex as for the max norm.


Experimental results show that-despite the lack of any biological prior knowledge-our model performs comparably to existing approaches, and in fact learns image features that resemble findings from several previous studies.

its maximally excitatory stimuli have center-surround structure, similar to receptive fields in the early human visual system.


The VAE framework is a common workhorse in the stable of probabilistic inference methods, and it is extended to the temporal setting by [2,6,8,16,19].

deep neural networks can parameterize the transition and emission distributions of different variants of deep state-space models [8,16,19].


we attribute a lie to buyer k if (1) the reserve was larger than the market value, but buyer k won the auction anyway, or (2) buyer k had the largest valuation, but nobody cleared the reserve.


However, the data complexity (fixed formula-size, variable domain-size) is polynomial.

for k variables in a formula where the domain-size of each variable is d, the complexity is clearly O(d k ) to go over every grounding.


In the appendix, we extend the analysis so that convergence still holds under our setting.

we prove the following theorem in Appendix 7.2: Theorem 1.


Linear averaging weighs each iteration σ t in the average strategy by t. It does not affect regret or influence the selection of strategies on an iteration.

with linear averaging the new definition for average strategy becomes σT i (I) = t∈T (tπ σ t i i σ t i ) t∈T (tπ .


In our own experiments, shown in Section 6, we use an implementation of TextonBoost [20] and augment it with quadratic higher order potentials.

we use TextonBoost to generate per-pixel scores c, and then minimize f @AA a c¡e A C j jAR j jjR j nAj, where the regions R j are regions of pixels that we expect to be of the same class (e.g., by running a cheap region-growing heuristic).


To accommodate for all kinds of observed random variables described above, we introduce an auxiliary Gaussian variable y d n , such that when conditioned on the auxiliary variables, the latent variable model behaves as a standard IBP with Gaussian observations.

we assume y d n is Gaussian distributed with mean z n B d and variance σ 2 y , i.e., and assume that there exists a transformation function over the variables y d n to obtain the observations x d n , mapping the real line into the observation space.


A second reason for using this model is that the independence assumptions of naive Bayes are too strong.

with a document of length n, the classifier "assumes" that it has n completely independent pieces of evidence supporting its conclusion about the document's label.


For example, consider two disjoint groups k and l such that their lifetimes and members do not overlap.


We study Nyström type subsampling approaches to large scale kernel methods, and prove learning bounds in the statistical learning setting, where random sampling and high probability estimates are considered.

we prove that these approaches can achieve optimal learning bounds, provided the subsampling level is suitably chosen.


The obvious downside to augmentation lies in the greater computational cost it entails.

the training set must contain sufficient information to represent the output variable for many combinations of operation and target times that can be provided as input.


An important property of differential privacy is that the privacy guarantees degrade gracefully if the same sensitive data is used in multiple private computations.

if we apply an αdifferentially private procedure k times on the same data, the result is kα-differential private as well as (α , δ)-differentially private for α = kα(e α − 1) + 2k log(1/δ)α [7,8].


We measure the distance between two distributions D and D ′ using a hypothesis class-specific distance measure.

let H be a hypothesis class for instance space X , and A H be the set of subsets of X that are the support of some hypothesis in H. for every hypothesis h ∈ H, {x : x ∈ X , h(x) = 1} ∈ A H . We define the distance between two distributions as: For our purposes, the distance d H has an important advantage over more common means for comparing distributions such as L 1 distance or the KL divergence: we can compute d H from finite unlabeled samples of the distributions D and D ′ when H has finite VC dimension [4].


In the adapted state, we assume a simple functional description for the variance of the measurement noise around the adapter.

we use a constant plus a difference of two Gaussians,   [14] (see also [11]).


We consider a standard setting of stochastic multi-armed bandit selection.

there is a bandit with a set B of n arms, where the i-th arm is associated with a Bernoulli distribution with an unknown mean θ i ∈ (0, 1].


As a result, many techniques that [6] builds upon are not applicable.

we will focus on the case when a good initialization V 0 is given: we analyze the later stage of the convergence, which is typical in the literature (e.g., [21]).


We studied Neural Networks through the lens of the Hessian operator.

we studied large batch size training and its connection with stability of the model in the presence of white-box adversarial attacks.


After this, in the fine-tuning stage, we build a big batch using all the data to minimize the loss L(Θ) defined in (4) with a gradient descent method.

we use Adam [18], an adaptive momentum based gradient descent method, to minimize the loss, where we set the learning rate to 1.0 × 10 −3 in all our experiments.


Thus, how successful a lexicographic strategy is in a comparison task consisting of a partial ordering of cue profiles depends on how well the cue ranking minimizes the number of incorrect comparisons.

the accuracy of TTB relies on the degree of optimality achieved by the ranking according to decreasing cue validities.


A function g is L-Lipschitz when kg(x) g(y)k  L kx yk for all x, y, and it is H-smooth when it is differentiable and its gradient is H-Lipschitz.

we consider optimization algorithms that use either a stochastic gradient or stochastic prox oracle (O grad and O prox respectively): where prox The prox oracle is quite powerful and provides global rather than local information about f . querying the prox oracle with = 0 fully optimizes f (•; z).


In our case, since the groups are non-overlapping, we can apply the proximal operator to each group independently.

for a single group, this translates to updating the parameters as where θn l is the solution obtained from the loss-based gradient step.


We assume that the models may be mis-specified.

let k * be the best single model defined as: then f k * = g. We are interested in an estimator f of g that achieves a small regret This paper considers a special class of model combination methods which we refer to as model averaging, with combined estimators of the form where ŵk ≥ 0 and k ŵk = 1.


In addition, this paper offers fundamental contributions to the policy gradient literature.

the kind of protocol we propose can be viewed as extending policy gradient methods to a context involving a team of agents optimizing system behavior through asynchronous distributed computation and parsimonious local communication.


We use this approach to refine the candidate solutions produced by tree search.

we use a 2-improvement local search algorithm [2,13].


However, the LP-relaxation may generate fractional solutions.

whenever (λ 12 + λ 23 + λ 31 )/2 > λ i * j * the configuration that assigns to every node both of its allowed labels in equal proportion-µ = 1/2-is optimal.


Recent work on Boltzmann machine models and their generalizations to exponential family distributions have allowed these models to be successfully used in many application domains.

the Replicated Softmax model [6] has been shown to be effective in modeling sparse word count vectors, whereas Gaussian RBMs have been used for modeling real-valued inputs for speech and vision tasks.


The approach can flexibly incorporate different likelihood functions, express prior assumptions about the relative size of different classes and incorporate partial labels for semi-supervised learning.

we instantiate the framework to unsupervised, multi-class kernelized logistic regression.


The first set P uB contains all sub-Gaussian distributions with unbounded support, and bounded mean and variance.

we assume that there exist τ > 0 and Γ uB > 0 such that for any probability measure µ ∈ P uB and any vector a ∈ R h , The second class P B of distributions contains all sub-Gaussian distributions with bounded support X = {x : x ≤ Γ B } ⊂ R h (note that this set in fact includes all distributions with bounded support).


There exist ζ > 0 and R > 0, such that The above assumption is fairly standard [7,19] in non-parametric regression.

the bigger ζ is, the more stringent the assumption is, since for ζ = 0, we are assuming f H ρ < ∞, while for ζ = 1/2, we are requiring f H ∈ H ρ , since [25,19] Finally, the last assumption relates to the capacity of the hypothesis space.


If this limit exists, then it equals the intrinsic dimension of M . Alternatively, Kégl [4] suggests using the easily proved inequality N M (ǫ) ≤ P M (ǫ) ≤ N M (ǫ/2) to express the capacity dimension in terms of packing numbers as .

now, a simple geometric argument shows that, for any µ supported on there exists a decreasing sequence {ǫ k }, such that for sufficiently large values of k (i.e., in the high-rate regime) the ratio − log k/ log e * ∞ (k|µ) can be approximated increasingly finely both from below and from above by quantities involving the packing numbers P M (ǫ k ) and P M (2ǫ k ) and converging to the common value d cap (M ).


Importantly, the advantage of the CBN model is significant as the units of improvement are in bits/instance.

an improvement of 2 bits/instance translates into each test instance being, on average, four times as likely.


At high level, the algorithm proceeds through several rounds, in each performing the following operations (see also Figure 1 Concept space localization: Solve for hypothesis vector w k+1 by minimizing hinge loss subject to the constraint that w k+1 lie within a radius r k from w k ; that is, [1, 10,16] show that by setting the parameters appropriately (in particular, b k = Θ(1/2 k ) and r k = Θ(1/2 k )), the algorithm will achieve error using only k = O(log 1/ ) rounds, with O(d) label requests per round.

a key idea of their analysis is to decompose, in round k, the error of a candidate classifier w as its error outside margin b k of the current separator plus its error inside margin b k , and to prove that for these parameters, a small constant error inside the margin suffices to reduce overall error by a constant factor.


For each trial, the network was simulated for 100 ms and its estimates of s 1 and s 2 were read out using a suboptimal decoding strategy.

to get an estimate of s 1 , we considered only the row of neurons in the output layer whose preferred vertical locations were closest to the vertical location of the first stimulus and then we fit a Gaussian function (with amplitude, peak location and width parameters) to the activity profile of this row of neurons and considered the estimated peak location as the model's estimate of s 1 .


Do non-maximal suppression within each of the computed sensitivities.

suppress the sensitivity A if there is a strictly more negative value for some vertex l k in the neighbourhood of l A , or some l 9 in the neighbourhood of l 2 .


Interestingly the choice of the STRESS or SSTRESS measure in MDS has a more natural interpretation when viewed within the framework of Sammon's mapping.

sTRESS corresponds to the use of the standard Euclidean distance metric whereas SSTRESS corresponds to the use of the squared Euclidean distance metric.


Methods We follow the training paradigm described in Section 3.4, but without the final fine-tuning stage, as ground truth 3D shapes are available on this synthetic dataset.

the 2.5D sketch estimator is trained using ground truth depth, normal and silhouette images and a L2 reconstruction loss.


However, the sampling procedure used in MCTS algorithms is not well-suited for other kinds of problems [Nakhost, 2013], such as deterministic single-agent problems where the objective is to find any solution at all.

if the reward is very sparse-for example the agent is rewarded only at the end of the task-MCTS algorithms revert to uniform search.


While classical approaches consider finite, albeit potentially huge, output spaces, in this paper we discuss how structured prediction can be extended to a continuous scenario.

we study a structured prediction approach to manifold valued regression.


Compared to the state of the art [5,10], the polynomial code provides order-wise improvement in terms of the recovery threshold.

the recovery thresholds achieved by 1D MDS code [5,10] and product code [10] scale linearly with N and √ N respectively, while the proposed polynomial code actually achieves a recovery threshold that does not scale with N . Furthermore, polynomial code achieves the optimal recovery threshold.


In this section, we show that the simple 1 regularized logistic regression classifier attains fast classification error rates.

we are interested in the M -estimator [21] below: which maximizes the penalized log-likelihood of the logistic regression model, which also corresponds to the conditional probability of the response given the covariates P(Y |X) for the conditional Gaussian model.


Therefore in this paper, we take a limited view and ask the question: Which global minima (of the empirical risk, of course) are accessible to a particular optimization algorithm with a particular set of hyper-parameters?

how the different optimization algorithms with different hyper-parameters pick out the different set of global minima?


Motivated in part by the spike-sorting problem, we consider the case where we are given a signal that is the sum of known waveforms whose timing and amplitude we seek to recover.

we suppose our signal can be modeled as: where the waveforms f n are known, and we seek to estimate positive amplitudes a n,j and event times τ n,j .


Let Q(•) denote the marginal probability defined by The mutual information between X and Z is the expected Kullback-Leibler (KL) divergence between Q(• | X) and Q(•): We view the problem of privacy as a game between the adversary controlling P and the data owners, who use Q to obscure the samples X.

we say a distribution Q guarantees a level of privacy I * if and only if sup P I(P, Q) ≤ I * .


So a partition {S 1 , S 2 } of S that minimizes the description length (DL) is in We will denote by 2 S the set of all subsets of S. A set function f : 2 S → R assigns a (real) number to every subset of S. We say that f is In [1], Queyranne gives a polynomial time algorithm that finds a set A ∈ 2 S \ {S, φ} that minimizes any symmetric submodular set function (specified in the form of an oracle).

queyranne's algorithm finds a non-trivial partition {S 1 , S \ S 1 } of S so that f (S 1 ) (= f (S \ S 1 )) minimizes f over all non-trivial subsets of S. The problem of finding non-trivial minimizers of a symmetric submodular function can be thought of a a generalization of the graph-cut problem.


Powering spreads the values apart and accordingly, A q 's lower singular values are relatively much smaller than its top singular values (see example in Figure 2a).

q = O( log d ) is sufficient to increase any singular value ≥ (1 + )σ k+1 to be significantly (i.e.


In the limit, Λn → Λ Y = Λ X + Λ Z , a sum of a rank-k matrix and a diagonal matrix.

the non-diagonal entries of Λn approach those of Λ X .


In this paper we point out a source of information in image sequences that is usually neglected by factorization algorithms: temporal coherence.

in a video sequence, the camera location at time t + 1 will probably be similar to its location at time t. if we randomly permute the temporal order of the frames, we will get a very unlikely image sequence.


The results, which match and slightly improve the recent winners in the 2010 VOC challenge, are reported in table 2.

our method produces the highest VOC score average over all classes, and also scores first on 9 individual classes.


In this paper, we employ such semantic relatedness among image categories for large-scale image categorization.

a category hierarchy is utilized to properly define loss function and select common set of features for related categories.


As before, we characterise the accuracy of this model by the average Hamming distance between predicted vectors and true vectors, obtaining about 89% accuracy.

the inclusion of the relations between AUs by means of the MRF was able to slightly improve the accuracy obtained independently for each AU from the SVM.


As this data set is difficult to visualize without taking up large amounts of space, we have included the results in the supplementary material which accompanies our NIPS submission.

we provide a first figure which shows the data mapped to b = 2 dimensions, with certain authors (or groups of authors) color coded-the choice of authors and their corresponding color codes follows precisely those of (Song et al., 2008).


Assume in (1) each f i is L-g-smooth, and f is µ-strongly g-convex, then if we run Algorithm 1 with Option I and parameters that satisfy then with S outer loops, the Riemannian SVRG algorithm produces an iterate x a that satisfies Algorithm 1: RSVRG (x 0 , m, ⌘, S) Parameters: update frequency m, learning rate ⌘, number of epochs S initialize x0 = x 0 ; for s = 0, 1, . . . , S 1 do The proof of Theorem 1 is in the appendix, and takes a different route compared with the original SVRG proof [18].

due to the nonlinear Riemannian metric, we are not able to bound the squared norm of the variance reduced gradient by f (x) f (x ⇤ ).


Loosely speaking, we first show (Lemma 10) that the Bayes error R * can be well approximated using 1-NN classifiers defined by the true (as opposed to empirical) majority-vote labels over fine partitions of X .

this holds for any partition induced by a γ-net of X with a sufficiently small γ > 0.


It is important to note that conditional means can be efficiently calculated using techniques from numerical linear algebra [9].

it can be seen from equation ( 1) that computing the conditional mean iC is equivalent to computing the product of a matrix inverse and a vector.


The MRF also contains pairwise potentials {φ ij } that can encode dependencies between the landmark assignments.

we may want to encode geometric potentials, which enforce a preference for preservation of distance or orientation for pairs of markers x i , x j and their assigned targets x k , x l . Finally, as the goal is to find a 1-to-1 mapping between landmarks in the source and target images, we also encode a set of mutual exclusion potentials over pairs of variables, enforcing the constraint that no two markers are assigned to the same candidate x k . Our task is to find the MAP solution in this MRF.


3 Mode collapse refers to the problem that the Generator characterizes only a few images to fool the discriminator in GAN.

multiple latent codes are mapped to the same output in the image space.


Our approach can be generalized to feedforward networks of arbitrary depth.

we present in Section 4.1 results for two hidden layers.


This has become known as "asymmetric hashing" [2,4], but even with such a-symmetry, both mappings are based on the same fractional mapping f (•).

the asymmetry is in that one side of the comparison gets thresholded while the other is fractional, but not in the actual mapping.


In general, there are numerous reasons for the target matrix L 0 to be isomeric.

the widely used assumptions of low-rankness, incoherence and uniform sampling are indeed sufficient (but not necessary) to ensure isomerism, as shown in the following theorem.


The finite-sample bound (9) suggests that improvements to the upper bound on Z may translatable into improvements in the probabilistic, sampling bound.

if we define a proposal that uses the search tree S and its bound U S , we can improve our sample-based bound as well.


For convenience and generality we state our bounds on the estimation error for arbitrary data dependent functions (and not just losses of models).

let M : Z n ×Z → R be an algorithm that is given a dataset S and a point z as an input.


to compute P (s (t) , E), given a sample, we need to go over each ground formula in M and check if it is satisfied or not.


To further reduce the regret, we can break the ties according to estimated regret.

with samples θ (1) ij (t), the normalized Copeland score for each arm i can be estimated as ζi (t) = 1 Then the maximum normalized Copeland score is ζ * (t) = max i ζi (t), and the loss of comparing arm i and arm j is rij (t) = ζ * (t) − 1 2 ζi (t) + ζj (t) .


As defined, the temporal model imposes the restriction that the prior π of population behavior at t = 0 and the density f of behavioral evolution are both independent of treatment assignment Z.

regardless of how agents are assigned to games, the population behavior in the game will evolve according to a fixed model described by f and π.


We assume that the cumulative intensity of the counting process is modeled by a time-varying coverage function, i.e., the observation pair in the time window [0, T ] for some T > 0, and df (S, t) = a(S, t)dt.

the timevarying coverage function controls the propensity of occurring events over time.


In the following, we will drop the subscript referring to the gold standard structure in the definition of ∆ to indicate that the feedback is in general independent of gold standard outputs.

we allow ∆ to be equal to 0 for several outputs.


Model scores can then be used as approximate surrogates for probabilities, obviating the computation normalizer computation.

given a model of the form p η (y | x) = e η T T (y, x)−A(η, x) (1) with A (η, x) = log y∈Y e η T T (y, x) , we seek a setting of η such that A(x, η) is close enough to zero (with high probability under p(x)) to be ignored.


We use ShapeNet [Chang et al., 2015] renderings for network training and testing.

we render each object in 20 random views.


In order to estimate the value of policy µ = µ i , we follow the temporal difference learning (TD) approach.

given a new estimate of the expected future returns R t at time t after taking action a t in state s t according to some policy µ, Q(s t , a t ) (dropping the dependence on µ) is updated as follows where α > 0 is a problem-dependent learning rate parameter.


In addition, the OLS estimation for tensor regression (2) can be rewritten as the following nonconvex problem in terms of low-rank tensor parameters o .

the number of parameters for a general tensor Nevertheless, solving the tensor regression problem (5) is still expensive in terms of both computation and memory requirements, for typical settings, when n R • P D d=1 p d . the per iteration complexity is at least linear in n for popular algorithms such as block alternating minimization and block gradient descent [27,28].


These findings suggest that the speed at which a contagion explodes is bounded by a certain quantity, and thus that the process needs a certain amount of time to become super-critical.

this intuition is made formal in the following corollary: Then, the regime of the contagion is sub-critical before ln n and .


Its rows are not only V-shaped, but also convex.

c y,k+1 − C y,k ≥ C y,k − C y,k−1 for 1 < k < K. The convex rows encode a stronger preference in making the prediction "close."


A DP mixture model in the stick-breaking representation can be viewed as possessing an infinite number of components with random mixing weights [4].

the generative model of a DP mixture assumes: i=1 that are independently drawn from a prior p η (η i |λ) with hyperparameters λ.


Since we do not have access to the oracle editor p * edit , we train the editor p edit to directly maximize L(p edit , pret ).

we solve the optimization problem: In our experiments, we let p edit be a standard sequence-to-sequence model with attention and copying [12,36] (see Appendix B for details), but any model architecture can be used for the editor.


The optimal tradeoff is indeed captured by the value of λ provided in Theorem 2.

as the number of training samples N increases, variance diminishes and λ approaches 1, placing increasing weight on EO.


For instance, consider a single-hidden layer S-H-F MLP (with S-input H-hidden F -output nodes); there are n A =F (H + 1) terminal parameters " A (including threshold parameters) on direct connections to F terminal nodes, each of which has CA(=H + 1) direct connections, and the rest of n B =H(S + 1) parameters are not directly connected to any terminal node; hence, n B hidden parameters " B .

model's parameters " (n=F CA + nB in total) can separate as: where " A k is a vector of the kth subset of CA terminal parameters directly linked to terminal node k (k = 1, • • • ,F ).


This variant, described in Figure 1, queries all labels (and stores all examples) during an initial stage of length at least (16d)/λ 2 , where λ denotes the smallest nonzero eigenvalue of the process covariance matrix E[X 1 X ⊤ 1 ].

when this transient regime is over, the sampler issues a query at time t based on both the query counter N t−1 and the margin ∆ t . if evidence is collected that the number N t−1 of stored examples is smaller than our current estimate of 1/∆ 2 t , that is if ∆ 2 t ≤ (128 ln t)/(λN t−1 ), then we query (and store) the label of the next instance x t+1 .


Informally, algorithmic stability measures the change in loss when the inputs to a learning algorithm are perturbed; a learning algorithm is stable if small perturbations lead to proportional changes in the loss.

a learning algorithm should not be overly sensitive to any single input.


This allows us to analyze the functional connection structure in a neural population at the same time as its relationship with further continuous signals of interest.

this approach makes it possible to include a stimulus as a continuous variable into the framework of maximum-entropy modeling.


In this case we can precisely characterize behavior at every step of serial reproduction.

we show that this defines a simple first-order autoregressive, or AR(1), process, allowing us to draw on a variety of results characterizing such processes.


We want to show that as m → ∞, the semi-supervised algorithm in Figure 1 is wellbehaved.


A common identifiability issue, also shared by the BIP in (1), is the scaling ambiguity.

if (w \ , x \ ) solves a BIP, then so does (cw \ , c 1 x \ ) for any nonzero c 2 R. In this paper, we resolve this scaling ambiguity by finding the point in the solution set closest to the origin with respect to the `1 norm.


On the other hand, for some of α A , the second assumption does not necessarily hold because we do not know the optimal α A beforehand.

if the point i ∈ A which was located inside the margin before the update moved to M during the update (i.e.


Another characteristic shared by these three conversions is that the training data does not play a part in determining how the online hypotheses are combined.

the training data is not used in any way other than to generate the sequence h 0 , .


When coupled with kernel ridge regression, the comparison between PLST and kernel-CPLST in terms of the Hamming loss is shown in Table 5. kernel-CPLST performs well for LSDR and outperforms the feature-unaware PLST in most cases.

in five out of the eight datasets, kernel-CPLST is significantly better than PLST regardless of the number of dimensions used.


To further boost the computational efficiency at each stage of the convex relaxation, we apply the pathwise optimization [10] for all algorithms in practice.

at each stage, we use a geometrically decreasing sequence of regularization parameters { , where [0] is the smallest value such that the corresponding solution is zero, ↵ 2 (0, 1) is a shrin- .


Thus, an effective teacher will modify its demonstrations when showing and not simply doing a task.

as in Equation 4, we can define a teacher that selects trajectories that best convey the reward function: showing depends on a demonstrator's inferences about an observer's inferences about doing.


For Lemma 5.1 to be practically useful, it is essential that f1 and f2 be efficiently optimizable over C. We discuss two general curvature-dependent approximations that work for a large class of combinatorial constraints.

we use Theorem 3.1: we decompose f into f κ and a modular part f m , and then approximate f κ while retaining f m , i.e., f = f κ + f m . The first approach uses a simple modular upper bound (MUB) and the second relies on the Ellipsoidal approximation (EA) we used in Section 3.


In other words, the length of target sequences of RNN b is 90 and fixed regardless of the content of training documents.

rNN b has trouble with the r2f label ordering, where training is unstable.


This paper studies the problem of learning a low-dimensional Euclidean metric from comparative judgments.

consider a set of n items with high-dimensional features x i 2 R p and suppose we are given a set of (possibly noisy) distance comparisons of the form sign(dist(x i , x j ) dist(x i , x k )), for a subset of all possible triplets of the items.


Now, what happens when P [X 1 = 1|X 2 = 1] = 0.99?

what can we say about P [X 1 = 1|X 2 = 1, . . . , X n = x n ] given information about probabilities P [X i = x i , X j = x j ].


The CONSISTENCY problem has been studied before as the membership problem for the marginal polytope.

[8] shows that finding the MAP assignment for Markov random fields with pairwise potentials can be cast as an integer linear program over the marginal polytope -that is, algorithms for the CONSISTENCY problem are useful subroutines for inference.


Typical results study R T (w) against all w with a bounded norm in some geometry.

for an invariant update, we relax this requirement and only put bounds on the predictions w x t . for some pre-chosen constant C we define K t def = w : |w x t | ≤ C . We seek to minimize regret to all comparators that generate bounded predictions on every data point, that is: Under this setup, if the data are transformed to M x t for all t and some invertible matrix M ∈ R d×d , the optimal w * simply moves to (M −1 ) w * , which still has bounded predictions but might have significantly larger norm.


Hannan consistency (HC) is a commonly studied property in the context of online learning in repeated (single stage) decisions.

rM and variants of Exp3 has been shown to be Hannan consistent in matrix games [15,16].


Conversely Nelson and Nguy ễn [NN13] showed that this is almost asymptotically optimal.

every distribution over n × m matrices satisfying (1) with m = Θ(ε −2 lg(1/δ)), and such that every column has at most s non-zero entries must satisfy s = Ω((ε lg(1/ε)) −1 lg(1/δ)).


Our characterization is radically different from existing results, such as in [18,8,19,9,10,11,12,13].

we unify primal-dual methods [20,21], smoothing (both for Bregman distances and for augmented Lagrangian functions) [22,21], and the excessive gap function technique [1] in one.


This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575.

it used the Bridges system, which is supported by NSF award number ACI-1445606, at the Pittsburgh Supercomputing Center (PSC).


We use the independent-sites parent-independent mutation model over multinomial vectors in [1] as our likelihood model.

this model assumes that each point on the tree is associated with a D dimensional multinomial vector, and each entry of this vector on each branch of the tree evolves independently (thus independent-sites), forward in time, and with mutations occurring at rate λ d on entry d. When a mutation occurs, a new value for the entry is drawn from a distribution φ d , independently of the previous value at that entry (thus parentindependent).


This image is distorted by variations between the properties of individual target cell and neurotropin circuits that are introduced during fabrication.

poor power distribution on the Neurotrope1 chip creates a systematic gradient in target cell excitability that warps a growth cone's impression of the relative coactivity of its neighbors, attracting it preferentially toward the more excitable target cells on the right side of the array.


However, their method requires multiple passes over the data, thus doesn't fit directly in the streaming k-means setting.

the above mentioned paper analyzes the stochastic k-means method only for highly accurate initial set of iterates which requires a large burn-in period of t = O(N 2 ) and hence needs O(N ) passes over the data, where N is the number of data points.


The number of candidate actions available in a given state is the power set of the tasks that are eligible to start.

with N eligible tasks there are 2 N possible actions.


Note that the IB update rule alone sets some constraints on the performance.

if we require the error probability to be low, then the expected number of samples is necessarily larger than a certain quantity depending on the model parameters.


Next, we iteratively removed the users with the smallest number of ratings until we met some desired density level.

we executed the above process until we obtained two sets B and G such that the number of likes between the two parties is at least 2(min{|B|, |G|}) Random baselines.


Also, let µ max denote the highest estimate of the predictive distribution thus far.

µ max is the highest valuation for the data provided by the individual.


The main difference in our work is that attentions are treated as confidences of the logical rules and have semantic meanings.

neural LP learns a distribution over logical rules, instead of an approximation to a particular rule.


His revenue resulting from the cleared bid will be the good's spot price (utility) π t,k .

the payoff obtained from good k at period t is (π t,k − λ t,k )1{x t,k ≥ λ t,k } where 1{x t,k ≥ λ t,k } indicates whether the bid is cleared.


Counterfactual fairness also provides an answer to some problems on the incompatibility of fairness criteria.

consider the following problem raised independently by different authors (e.g., [7,22]), illustrated below for the binary case: ideally, we would like our predictors to obey both Equality of Opportunity and the predictive parity criterion defined by satisfying as well as the corresponding equation for Ŷ = 0.


We address this issue for multiple popular ERM problems including kernel SVMs, kernel ridge regression, and training the final layer of a neural network.

we give conditional hardness results for these problems based on complexity-theoretic assumptions such as the Strong Exponential Time Hypothesis.


Using F to denote the first node ordered in the contamination set of (s → t) (and L for the last), we can compute C[1, N ] via the following recursive update principle where the maximization is with respect to the structure score (e.g., BIC).

the best chain in a subsequence [i, j] in the ordering is the maximum of three alternatives: edges whose contamination boundaries are exactly i and j (no split); two chains that are joined at some node i < k < j (split); a gap between i and j when there is no positive edge whose contamination is in Finally, for lack of space we only provide a brief description of our topological node ordering.


That said, combining the two related approaches may be fruitful.

there may be practical benefit to learning many forms of models (neural networks, tree-based models, etc.)


For any p > 0, there exist constants (C 0 > 0, ρ s > 0, m 0 ) with the following property: if m > m 0 , (A 0 , E 0 ) ∈ R m×m × R m×m with the singular spaces of A 0 ∈ R m×m distributed according to the random orthogonal model of rank and the signs and support of E 0 ∈ R m×m distributed according to the Bernoulli sign-and-support model with error probability ≤ ρ s , then with probability at least 1 − Cm −p and the minimizer is uniquely defined.

matrices A 0 whose singular spaces are distributed according to the random orthogonal model can, with probability approaching one, be efficiently recovered from almost all corruption sign and support patterns without prior knowledge of the pattern of corruption.


Inspired by the efficient dual averaging method [24] for solving lasso and group lasso [25] logistic regression, we extend the results to the multinomial case.

the loss minimizing sequence of parameter matrices W t can be achieved by the following online update scheme.


To test this, we synthetically vary the class overlap in our common test setting of CIFAR-10.

we perform 6-class classification on CIFAR-10's animal classes (bird, cat, deer, dog, frog, horse).


[16] in the context of kernel Fisher's discriminant analysis contains the key idea that we will follow: noise and distracting signal aspects with respect to which we want to make our feature extractor invariant is added to the denominator of a Rayleigh coefficient.

our prior knowledge about the noise type helps to re-design the optimization of CSP feature extraction.


We demonstrated the elegance and unity of our approach by showing how to easily apply a Variational Bayes analysis of temporal ICA.

our Bayes ICA approach successfully extracts independent processes underlying EEG signals, biased towards preferred frequency ranges.


This question is difficult to answer in general, since it depends on the property of the statistical model.

for general statistical models it is often not clear whether the confidence intervals contract sufficiently quickly.


The stronger distributional assumptions made in the generative approach result in a less robust learner, but make it possible to learn linguistic exceptions without negative evidence.

generative models can exploit the "implicit negative evidence" provided by the absence of a sentence: the assumption that sentences are generated from the target probability distribution means that not observing a sentence provides weak evidence that it does not belong to the language.


We are required to produce a prediction, ŷt ∈ [k], as the label of x t . In response, we obtain only 1 bit of information: whether ŷt = y t or not.

when ŷt = y t , the identity of y t remains unknown (although one label, ŷt , is ruled out).


However, for the purposes of a structured cascade, we are only interested in computing the max marginals θ (x, y j ).

we are only interested in knowing whether or not a configuration y consistent with y j that scores highly in each sub-model θ p (x, y) exists.


To evaluate the performance of our optimal discriminant clustering (ODC) algorithm, we conducted experimental comparisons with other related clustering algorithms on several real-world datasets.

the comparison was implemented on three face datasets, the "SRBCT" gene dataset, and four UCI datasets.


To produce Figures 1a and 1b, we reimplemented the toy experiment of Bishop [5,Section 3.5.1].

we generated a learning sample of 15 data points according to y = sin(x) + ✏, where x is uniformly sampled in the interval [0, 2⇡] and ✏ ⇠ N (0, 1 4 ) is a Gaussian noise.


To aid the presentation below we simplify this model a bit further.

set b = 0 and observe that (7) reduces to where γ k = −r − 2c ψ k . we modify the parameterization to θ = {γ k } k=1 ∪ {θ P AQ } such that θ P AQ denotes the matrix of parameters in (8).


We then apply this result to obtain several new results related to subset ranking.

we first obtain calibrated, score-based surrogates for the Precision@q loss, which includes the winner-take-all (WTA) loss as a special case, and the expected rank utility (ERU) loss; to the best of our knowledge, consistency with respect to these losses has not been studied previously in the literature.


It is seen that the margin l 1 and l 2 models perform better than cross-entropy across the entire range of noise levels, while the margin l ∞ model is slightly worse than cross-entropy.

the margin l 2 model achieves a evaluation accuracy of 96.4% at 80% label noise, compared to 93.9% for cross-entropy.


Equation ( 3) provides an alternative characterization of the autoregressive model as a transformation f from the space of random numbers u to the space of data x.

we can express the model as x = f (u) where u ∼ N (0, I).


In DL, features are learned in a compositional hierarchy.

low-level features are learned to encode low-level statistical dependencies (e.g., "edges" in images), and higher-level features encode higher-order dependencies of the lower-level features (e.g., "object parts") [14].


These graphs are varying in sizes from small to large.

the number of nodes in Cora and Citeseer are of scale O( 103 ), while Pubmed and Reddit contain more than 10 4 and 10 5 vertices, respectively.


By using both gist and sparse codes features, we can achieve dramatic improvements in both max-margin and MLE methods.

the max-margin scene model achieves an accuracy of about 0.83 in scene classification, and the likelihood-based model obtains an accuracy of about 0.80.


Our experimental results suggest that MCP is advantageous for problems with sparse stochasticity.

mCP has allowed us to scale to larger environments than were otherwise possible for the robot-helicopter coordination problem.


If the white noise has unit intensity, the autocorrelation of f is W p where This is the autocovariance associated with the Bayesian interpretation of p-th order smoothing splines [23].

when p = 2, one obtains the cubic spline kernel.


Algorithm 1: PDDP algorithm The major computational effort is devoted to GP inferences.

the complexity of one-step moment matching (2.2) is O (N ) 2 n 2 (n+m) [14], which is fixed during the iterative process of PDDP.


Rather than trying to learn a robust function that also achieves high classification accuracy, we consider the problem of learning any function that is robust to adversarial perturbations with respect to any specified metric.

given a dataset of d-dimensional points and a metric of interest, can we learn features-namely functions from R d → R-which 1) are robust to adversarial perturbations of a bounded magnitude with respect to the specified metric, and 2) have significant variation across the datapoints (which precludes the trivially robust constant function).


A central issue in the Þeld of human speech perception is how lexical knowledge inßuences the perception of speech sounds.

how does knowing the word you are hearing inßuence how you hear the smaller units that make up the word (i.e., its phonemes)?


However, when p(y t |x t ) is twice differentiable, it is straightforward to find an approximating Gaussian model based on a Laplace approximation by simple numerical optimization [12,33,30], and use the obtained model as a basis of twisted SMC.

we use where p(y t |x t ) ≈ p(y t |x t ), t = 1, . . . , T are the Gaussian approximations obtained using Laplace's method.


it measures how well approximate L-risk minimizers t approximate the true minimizer t * L,Q , and second it equals the calibration function of the P-instance LP , i.e.

δ max, LP,L (ε, P( the self-calibration function can be utilized in Theorem 3.1.


In addition, the non-zeros of ∆ j are restricted to Active( Ã) defined in (7).

we restrict the minimization (8) to while all other elements are set to zero for the entire treatment of the j-th block.


We propose to use a primal-dual algorithm for the inner problem and show experimentally that this approach yields faster convergence than the FISTA method of [5] which was applied in [14].

let us restrict our attention to the case where R(f we apply the RatioDCA algorithm to (2) with S = S 1 which is what we need, e.g., for the tight relaxations of the Cheeger cut, normalized cut and truncated Cheeger cut families.


(iii) The columns of E 3 fall in the column span of C and tr(E (iv) The rows of E 4 fall in the row span of C and tr(E and 1 + 2 + ˜ 2 + 3 + 4 = , then X is a -approximation embedded matrix for X.

we have The proof can be referred to [17].


G is defined as For any hypothesis class H, target hypothesis h ∈ H, distribution P , sample S m , and real r > 0, define Finally, for any h ∈ H we define a ball in H of radius r around h [5].

with respect to class H, marginal distribution P over X , h ∈ H, and real r > 0, define The concept of perfect classification was introduced in [1] within a realizable selective classification setting.


Theorem 3.1 is stated as a bound on the probability that the norm of the error matrix is greater than some target error " given the number of samples s. In practice, the target error " is typically not known in advance, but rather is the quantity to minimize, given the matrix A, the number of samples s, and the target confidence .

for any given distribution p on the elements of A, define Our goal in the rest of the paper is to seek the distribution p minimizing " 1 .


In general, a graph-based method wants to find a function to simultaneously satisfy two conditions [2]: a) it should be close to given labels on the labeled nodes, and b) it should be smooth on the whole graph.

the work in [8] develops a regularization framework to carry out the above idea by defining the global and local consistency terms in their cost function.


Φr ALP will also satisfy the constraints of the exact LP, illustrated by the dark gray region; this implies that Φr ALP ≤ J * .

the approximate cost-to-go function is necessarily a point-wise lower bound to the true cost-to-go function in the span of Φ.


When the Gibbs sampler chooses to resample a cluster C that covers D (the input of f ), we can update the Rao-Blackwellized estimator for free.

the Gibbs sampler computes the cluster belief ) in order to compute the transition density p C∩E|E\C (• | w n−1 E\C ).


In general, training a BNN is more computationally efficient than a GP [22], while still providing coherent uncertainty measurements.

predictive distributions can be calculated by taking averages over samples of W from an approximated posterior distribution over the parameters.


For any pair of jobs (i, j), define d(i, j) as the amount of job i that has been executed before the completion time of j.

d(i, j) is the amount of time by which i delays j.


We showed that these two components are, in fact, capable of learning complicated high-level representations in large scale experiments on unlabeled images pulled from YouTube.

we found that higher level simple cells could learn to detect human faces without any supervision at all, and that our complex-cell learning procedure combined these into even higher-level invariances.


To provide some intuition for our model, we provide the following concrete example where the distribution of y ∈ R is a Gaussian function of the coordinate η ∈ M along the swissroll, which is embedded in a high-dimensional ambient space.

we sample the manifold coordinate, η ∼ U (0, 1).


In order to incorporate the label information, we add a semi-supervised training phase.

we set the number of clusters to be the same as the number of class labels and after executing the reconstruction and the adversarial phases on an unlabeled mini-batch, the semi-supervised phase is executed on a labeled mini-batch, by updating the weights of the encoder q(z|x) to minimize the cross-entropy cost.


When the likelihood belongs to an exponential family, Bayesian posterior computation is fast and easy.

it suffices to find the sufficient statistics of the data, which require computing a simple summary at each data point and adding these summaries across data points.


Finally, we comment that Theorem 3 can be used to obtain so-called oracle inequalities.

let q * be the optimal distribution minimizing L(q h), which can only be computed if the underlying distribution µ(x, y) is known.


While FastRNN controls the condition number of gradient reasonably well, its expressive power might be limited for some datasets.

this concern is addressed by a novel architecture, FastGRNN, that uses a scalar weighted residual connection for each and every coordinate of the hidden state h t . where 0 ≤ ζ, ν ≤ 1 are trainable parameters that are parameterized by the sigmoid function, and σ : R → R is a non-linear function such as tanh, sigmoid and can vary across datasets.


The two plots, however, exhibit some subtle but important differences that illustrate the qualitatively greater power we obtain from a larger state set.

the 81-state model has four short downward spikes rather than three in the time interval from 1.045 to 1.05.


The results show that our method is much faster than the competing method.

learning features that are 6x overcomplete takes 1 hour using our method, whereas [26] requires 2 days.


In particular, we test CYCLADES on a classification problem for text based data.

we run classification for the URL dataset [15]  Table 2: Ratio of the time that CYCLADES consumes for partition and allocation over the time that HOGWILD!


Our approach uses auxiliary parameters to speed up the deterministic approximation of the target distribution.

we propose Parameter-eXpanded Variational Bayesian (PX-VB) method.


We consider standard convolutional neural networks and train models on datasets of varying complexity.

we study the MNIST [34], CIFAR-10 [33], and SVHN [40] datasets.


The primary objective here is to bound the number of pairwise comparisons needed to correctly determine the ranking when the objects (and hence rankings) satisfy certain known structural constraints.

we suppose that the objects may be embedded into a low-dimensional Euclidean space such that the ranking is consistent with distances in the space.


While quantitative evaluation such as measuring edit distance is helpful for comparing results, it is also illuminating to consider the plausibility of the learned parameters in a historical light, which we do here briefly.

we consider rules on the branch between la and vl, for which we have historical evidence.


This restriction sets mutagenetic trees apart from standard Bayesian networks in that it allows for an evolutionary interpretation of the tree topology.

the model implies the existence of certain mutational pathways with distinct probabilities.


This differential-difference equation describes the time evolution of the conditional mass.

the differential term ˜ t , i.e., the instantaneous rate of change in the probability mass, is equal to a first order difference equation on the right-hand-side.


However, contrary to our framework, in MIML noisy labeling is not allowed.

all the labels being assigned to the bags are assumed to be true.


The detailed balance property underlies most MCMC sampling algorithms.

if one desires to sample from a target distribution, π(z), satisfying detailed balance for an ergodic Markov chain guarantees that simulations of the chain will uniquely converge to the target distribution of interest.


In this version of the paper we prove this result for k > n/d 1/4 .

the full version of this paper [ACD + ] contains the statement and the result that holds for all d > Ω( there exists a perfect matching in G that does not contain an induced matching of size larger than O(n/ √ d).


We give a smoothed analysis which shows that the worst case is brittle, however.

we consider a model in which the contexts x t i are chosen at each round by an adaptive adversary, but are then perturbed by independent Gaussian perturbations in each coordinate, with standard deviation σ.


We introduced a new method for modelling relational data which is able to both discover meaningful structure and generalize well.

our results illustrate the predictive power of distributed representations when applied to modelling relational data, since even simple tensor factorization models can sometimes outperform the more complex models.


Basically, each layer is processed independently based on the initial model input and output, without taking into account the retraining result from the previous layer.

denoting Y ( −1) and Y ( ) as the input and output of the -th layer of the initial trained neural network (see equation ( 1)), we propose to relearn the coefficient matrix W via the convex program The optimization in (5) can be independently applied to every layer in the network and hence computationally distributable.


In this paper, we propose an approach to learning such factorized representations inspired by sparse coding techniques.

we show that structured sparsity allows us to address the multiview learning problem by alternately solving two convex optimization problems.


We define a weighted and directed graph G UC on all clusterings C. Thus, N = B n ≤ n n is the n th Bell number.

when C is obtained by a MERGE of two clusters in C, G UC contains a directed edge G UC contains an edge from C to the clustering obtained from breaking C i into singleton clusters of all its elements.


We hope restrictive assumptions considered in this paper, such as binary ratings and rating being shared across the same group, be relaxed in the future endeavors.

it would be interesting to characterize the optimal sample for feature-based side information models [14,45].


In other words, ⇡(i, s, k) corresponds to a choice probability when restricting U to the first k products, a 1 , .

. . , a k . With this notation, we have for all i 2 [n], P(a i |S) We compute ⇡(i, s, k) iteratively for k = 1, . . . , n. in order to compute ⇡(i, s, k + 1), we use the correctness of the sampling procedure.


To the best of our knowledge, this is the first generalization to exponentially large structured output spaces.

our contribution lies in reducing the greedy augmentation step to inference with structured, efficiently solvable HOPs.


The target speech recognition system in this work consists of CTC-trained AM, RNN LM, and beam search decoder.

we have developed two models; one is based on character as the output units of AM and LM, and the other is based on word piece units [22].


Since U s (t) is varying over time, we can associate each u ∈ U with a |V|-dimensional vector τ u of change points.

the s-th coordinate of τ u records the time that source node s covers u.


We generate data with precisely the property that coarser categories are distinguishable using feature dimensions distinct from those needed to discriminate their subclasses.

we sample 2000 points from each of four 4D Gaussians, giving four leaf classes {a, b, c, d}.


Since our algorithm does not depend on ℓ, our analysis is actually true for any semimetric ℓ that satisfies Assumptions 1-4, thus Theorem 2 and Corollary 2 hold for the best possible choice of such a ℓ.

we can think of problems for which there exists a semimetric ℓ such that the corresponding near-optimality dimension d is 0.


This can be problematic when using symbolic mismatch with kernel-based algorithms since the optimization problems become non-convex and the underlying theory is invalidated.

kernel-based classification methods require Mercer's condition to be satisfied by a positive semi-definite kernel matrix [12].


Our goal is to compute the top k principal components of the data: the k-dimensional subspace that offers the best squared-error estimate for the points.

we assume a probabilistic generative model, from which the data is sampled at each step t. where A ∈ R p×k is a fixed matrix, z t ∈ R k×1 is a multivariate normal random variable, i.e., and w t ∈ R p×1 is the "noise" vector, also sampled from a multivariate normal distribution, i.e., Furthermore, we assume that all 2n random vectors (z t , w t , ∀1 ≤ t ≤ n) are mutually independent.


We assume that our bandit algorithm receives a context x t ∈ X at each round t, and that there exists a function f ∈ F, in some known concept class F, such that f (x t ) = +1 if an event occurs at round t, and f (x t ) = −1 otherwise.


Recent papers further show that this problem can be formulated and solved as a compressed sensing problem using random queries [14,18].

[14] gives an algorithm using O(s 2 n) queries based on mutual coherence, whereas the Restricted Isometry Property (RIP) is used in [18] to give a query complexity of O(sn 4 ).


We say that a policy ⇡ covers the function f when the agent executing ⇡ always achieves its goal upon termination.

f (dom( ), ) = f (X, ) for all scenarios ⇠ , where is the history when the agent executes ⇡.


Pairwise distance computations are fundamental to many important computations in machine learning and are some of the most expensive for large datasets.

we consider the class of all-query problems, in which the combined interactions of a reference set ℛ of points in ℝ is computed for each point in a query set of size O( ).


A column vector of ones of arbitrary dimension will be denoted by e. For A ∈ ℜ m×n and B ∈ ℜ n×k , the kernel K(A, B) maps ℜ m×n × ℜ n×k into ℜ m×k .

if x and y are column vectors in The identity matrix of arbitrary dimension will be denoted by I.


Strongly Convex Case: Our second algorithm for strongly convex function f t (x) gives us the improved upper bounds compared with the previous work in [10].

we have The improved bounds match the regret order of standard OCO from [9], while maintaining a constraint violation of reasonable order.


We identify the resistance diameter of a graph G as R G := max p,q∈V r G (p, q); thus, from the previous theorem, we may also conclude that We complete this section by showing that there exists a family of graphs for which the above inequality is nearly tight.

we consider the "flower graph" (see Figure 4) obtained by connecting the first vertex of a chain with p − 1 vertices to the root vertex of an m-ary tree of depth one.


Thus, K is the P -invariant Krylov space generated by P and R. An incremental variant of the Krylov-based approach is called Bellman error basis functions (BEBFs) [9].

given a set of basis functions Φ k (where the first one is assumed to equal R), the next basis is defined to be In the model-free reinforcement learning setting, φ k+1 can be approximated by the temporal-difference (TD) error φ k+1 = r + γ Qk (s , π k (s )) − Qk (s, a), given a set of stored transitions in the form (s, a, r, s ).


The intuition behind the algorithm is that since the difference in payoff obtained by using S 2 instead of S 1 is given by (s 2,t − s 1,t ), it is sufficient to emulate Algorithm 1 on this sequence.

we set x t = t−1 j=1 ρ t−1−j (s 2,j − s 1,j ) and predict ḡ(x t ) (note that since |s 1,t − s 2,t | ≤ 2, we need to use g( 1 2 x) in the definition of ḡ to scale the payoffs).


As in HMC, the idea is to translate the task of sampling from the posterior distribution to simulating from a continuous dynamical system which is used to define a Markov transition kernel.

over any interval h, the differential equation defines a mapping from the state at time t to the state at time t + h. One can then discuss the evolution of the distribution p(z, t) under the dynamics, as characterized by the Fokker-Planck equation for stochastic dynamics [14] or the Liouville equation for deterministic dynamics [20].