clusters where each training example and its k-nearest points have same label and where points differently labeled are distanced by an additional safety margin.

given n input examples x 1 , ..., x n in ℜ D and corresponding class labels y 1 , ..., y n , the technique in [9] learns the D × D transformation matrix L that optimizes the following objective function: where η ij ∈ {0, 1} is a binary variable indicating whether example x j is one the k-closest points of x i that share the same label y i , c is a positive constant, y il ∈ {0, 1} is 1 iff (y i = y l ), and h(s) = max(s, 0) is the hinge function.


[2005] observed that for suitable cost-weighted losses, one may use (4) to solve density sublevel set estimation.

let : {±1} × R → R be any classification-calibrated loss (e.g., hinge or logistic) in the sense of [Zhang, 2004, Bartlett et al., 2006.


In these experiments we used a model with both convolutional and feed-forward layers.

the model used for these experiments includes three convolutional layers with max pooling followed by a fully-connected layer with 32 hidden units; all non-linearities were ReLU activations with batch normalization.


The Thurstonian view described in the finite model can be easily extended to the nonparametric one, where a partial ranking (X ρ 1 . . . X ρ m ) can be generated as the first m items to arrive in a race.

for each atom X k let z k ∼ Exp(w k ) be the time of arrival of X k and X ρ i the ith item to arrive.


Our first set of results evaluate Algorithm 1 against other stochastic optimization baselines assuming a complete knowledge of problem parameters.

we epoch i is terminated once This ensures that θ * remains feasible throughout, and tests the performance of Algorithm 1 in the most favorable scenario.


We also provide an algorithm for sampling CPDAGs efficiently from their posterior given the learned tree.

not from the full posterior, but from a reduced space of CPDAGs encoded in the learned tree.


We repeated the experiment by sampling data from a model with fifteen labels, consisting of five independent copies of the model above.

of other labels, and dependent on each other as specified by Figure 1.


There is a long history for proving hardness results for the class of algorithms that predict with linear combinations of instances [KW97a,KWA97].

in [WV05] it was shown for the Hadamard matrix and the square loss, that the average loss is at least 1 − k n even if an arbitrary expansion is used.


Either of the following assumptions would suffice: (A2) p is continuous not only on X • but also on ∂X (i.e., p(x) → 0 as dist(x, ∂X ) → 0).

(A3) p is supported on all of R D . the support of p has no boundary.


Therefore, we scale-up each of these steps.

to scale-up step one, based on the recently proposed MLN approximation approach [19], we design an informed proposal distribution using a "compressed" representation of the ground MLN.


Similar assumptions have been considered in other work [3,30,14,13,29,23].

[3] which studied finding causal relation between two variables related to each other by an invertible function, assumes that " the distribution of the cause and the function mapping cause to effect are independent since they correspond to independent mechanisms of nature".


Previous proposals concerning the dynamics of memory networks have fallen short of incorporating some key physiological constraints in a unified way.


2  Given the above system model, we formulate the distributed matrix multiplication problem based on the following terminology: We define the computation strategy as the 2N functions, denoted by that are used to compute each Ãi and Bi .

for any integer k, we say a computation strategy is k-recoverable if the master can recover C given the computing results from any k workers.


. . , x T , the goal is to find a low-dimensional approximation of this data that minimizes the 2-norm approximation error.

we want to find a rank k projection matrix P and a bias vector b ∈ R n such that the following cost function is minimized: Differentiating and solving for b gives us b = (I − P ) x, where x is the data mean.


[9] for evaluating segmentation accuracy around boundaries.

we count the relative number of misclassified pixels within a narrow band ("trimap") surrounding actual object boundaries, obtained from the accurate ground truth images.


Although a developmental approach to the velocity estimation task is shown to be beneficial, it is not the case that all developmental progressions that lead to performance advantages on the disparity estimation task also lead to advantages on the velocity estimation task.

a coarse-to-multiscale developmental system outperformed non-developmental and random developmental systems on the velocity estimation task, but a fine-to-multiscale system did not.


For large data, when the computational budget is limited, that acceleration translates into significantly improved accuracy.

we are able to improve or match the state-of-the-art results reported for large datasets in the kernel literature with only a small fraction of their computational budget.


We now perform testing in the leave-one-out experiments.

we can test the zero-shot transfer ability of policy network on new type of robots.


Those with knowledge of the theory of stochastic linear bandits will recognise some similarity.

once the nonlinearity of the objective is removed, the problem is equivalent to playing K linear bandits in parallel, but where the limited resources constrain the actions of the learner and correspondingly the returns for each task.


We propose to learn an ensemble of models that correspond to different distances ρ.

our iterative method generates a collection of datasets, each corresponding to a different inter-dataset distance level ρ, and we learn a model for each of them.


Note that an interval may be defined over several epochs.

an epoch is defined between adjacent transition times while an interval is defined over the epochs during which a particular parent set is operative (which may include all epochs).


In large-scale problems, the linear-equation solution sequence {∆" k } is generated iteratively while seeking a trial step ∆" in the inner iteration process, and the parameter sequence {"i}, whose two consecutive elements are denoted by "now and " next , is produced by the outer iteration (i.e., epoch in batch mode).

the outer iterative process updates parameters by " next = "now + ∆" without taking any uphill movement: if the step is not satisfactory, then R is decreased so as to realize an important Levenberg-Marquardt concept: the failed step is shortened and deflected towards the Cauchy-step direction simultaneously.


The scheduling problem is to decide which nodes should transmit at a given time over a given frequency, so that (a) there is no interference, and (b) nodes which have a large amount of data to send are given priority.

it is well known that if each node is given a weight equal to the data it has to transmit, optimal network operation demands scheduling the set of nodes with highest total weight.


We empirically demonstrate that our learning model and landmark selection heuristic consistently offers significant improvements over the existing approaches.

for small number of landmark points, which is a practically important scenario as it is expensive to compute similarity function values at test time, our method provides, on an average, accuracy boosts of upto 5% over existing methods.


More recently, causal inference or cue conflict models have been developed that allow for the possibility of probabilistic equality [9,11,12].

there is some probability that the two sources are equal and some probability that they are unequal.


As a final experiment, we test our method on a suite of related vision datasets.

we train and test on MNIST [19], notMNIST 6 , Fashion MNIST [40], SVHN [29] and CIFAR10 [18] in this order.


Here, we study large batch size training through the lens of the Hessian operator and robust optimization.

we perform a Hessian based study to analyze exactly how the landscape of the loss function changes when training with large batch size.


For future work, it would be of interest to enumerate structures that allow for parallel gradient computations with no collisions at all.

it may be possible to bias the SGD iterations to completely avoid memory contention between processors.


Under the curvature condition in (8) on the boundaries between k and classes in A c , our result shows that the robustness to random and semi-random noise exhibits the same behavior that has been observed earlier for linear classifiers in Theorem 1.

r * S 2 is precisely related to the adversarial robustness r * 2 by a factor of d /m.


Specifically, the un-normalized Fredholm kernel from Eqn 6 can be rewritten as: Thus k F (x, z) is simply an inner product which depends on both the unlabeled data distribution p(x) and the "inner" kernel k H . This inner product re-weights the standard norm in feature space based on variances along the principal directions of the matrix Σ F . We show that for the model when unlabeled data is sampled from a normal distribution this kernel can be viewed as a "soft thresholding" PCA, suppressing the directions with low variance.

we have the following 3 and assume the distribution p X for unlabeled data is a single multi-variate normal distribution, N (µ, diag(σ 2 1 , .


We prove that ProxSVRG+ improves/generalizes several well-known convergence results (e.g., ProxGD, ProxSGD, ProxSVRG/SAGA and SCSG) by choosing proper minibatch sizes.

proxSVRG+ is ) times faster than ProxGD, which partially answers the open problem (i.e., developing stochastic methods with provably better performance than ProxGD with constant minibatch size b) proposed in [24].


Note the choice of decoupled parametrization is not unique.

the bases can be partially shared, or (a, B) can be further parametrized (e.g.


Another, slightly more ambitious experiment would involve perturbing the reliability of one of the landmark cues.

one could imagine a training phase, where the position of one landmark is jittered over time, such that each time the rat encounters it it is at a slightly different heading.


We consider relational data consisting of triplets that encode the existence of relation between two entities that we will call the subject and the object.

we consider a set of n s subjects {S i } i∈ 1;ns along with n o objects {O k } k∈ 1;no which are related by some of n r relations {R j } j∈ 1;nr .


However, to avoid module collapse, we must take sufficient gradient steps within the partial M-step to optimize both the module parameters θ, as well as the controller parameters φ.

between each E-step, there are many gradient updates for both θ and φ.


As shown in Table 3, GroupReduce can compress both the input embedding and softmax layer 5-10 times without losing much accuracy.

groupReduce compress 6.6 times on the language model trained on OBW benchmark, which saves more than 5 GB memory.


are all relatively small, while the values for β are extremely large, with a median around exp 30.

given the values of β , the prior over Φ is effectively a symmetric prior over Φ with concentration parameter β.


Following the argument used in [17] for a smoothed original estimate of the ROC curve, one may show that a smoothed bootstrap of the studentized statistic r n (α)/σ n (α) yields a better pointwise rate of convergence than 1/ √ n, the one of the gaussian approximation in the Central Limit Theorem.

for a given α ∈]0, 1[, if the bandwidth used in the computation


The subgradient algorithm can be applied to the non-smooth objective [11], however its convergence rate is rather slow, both in theory and in practice.

the algorithm requires O(1/✏ 2 ) iterations to obtain an ✏-accurate solution to the dual problem.


These conditional probabilities Y ad ¢b vS g T # define our new representation for the examples.

each point T is associated with a vector of conditional probabilities Y ¤d ¦b vS g T # , S ( F e¥ ¨ § © § ¨ § ¨¥ 8y .


Curvature and other properties of target manifolds have been studied extensively for different kernel functions [1,4].

one can show that the kernel function implies a metric on the original space.


[31] to provide a general Optimistic Values Theorem in Appendix B.

similar to online learning, we can guarantee that greedy-action selection according to upper-confidence values will converge to the optimal policy, if the confidence interval radius shrinks to zero, if the algorithm to estimate action-values for a policy converges to the corresponding actions and if upper-confidence estimates are stochastically optimal-remain above the optimal action-values in expectation.


We consider in this paper that the MDP is solved with 1) an approximate dynamic programming algorithm and 2) a different discount factor β < γ.

our analysis applies to approximate value and policy iteration with existing error bounds.


Since Theorem 1 proves that the localization error per data point is bounded by a constant given an OCC, in practice we only need to minimize the data reconstruction error in order to minimize the upper bound of the localization measure.

we need to solve the following problem: This optimization problem is quite similar to sparse coding [12], except that there exists the orthogonal constraint on the basis vectors.


However, as the postsynaptic neuron does not have direct access to the presynaptic membrane potentials, only to the spikes emitted by its presynaptic partners based on those potentials, computing the required function becomes a non-trivial inference problem.

neurons need to perform computations on their inputs in the face of significant uncertainty as to what those inputs exactly are, and so as to what their required output might be.


This yields a new Poisson process (due to theorem 1) and a related DP which depends on the source.

we consider three such operations: superposition, subsampling, and point transition.


In this paper, we discuss an alternative: a discriminative framework in which we assume that supervised side information is present, and in which we wish to take that side information into account in finding a reduced dimensionality representation.

we present DiscLDA, a discriminative variation on Latent Dirichlet Allocation (LDA) in which a class-dependent linear transformation is introduced on the topic mixture proportions.


We train binary n-choose-k models, experimenting with different training protocols that directly maximize expected gain under the model, as described in Section 4.1.

we train on the expected top-K gain for different values of K. Note that top-1 is equivalent to softmax regression.


While the analysis technique used here is inspired from [24,23], it is the first exploration for additive classification model with group sparsity.

the hypothesis error analysis develops the stepping stone technique from the 1 -norm regularizer to the group sparse 2,1 -norm regularizer.


Additionally, care has been taken throughout to analyze the behavior of the constants.

there are no constants in the analysis that grow exponentially with the dimension, and in fact, many have polynomial, or slower growth.


In order to verify that our proposed architecture is overall competitive with the RealNVP architecture, we compare   our models on various natural images datasets.

we compare on CIFAR-10, ImageNet (Russakovsky et al., 2015) and LSUN (Yu et al., 2015) datasets.


This allows application of general gradient-based optimization algorithms for inference.

we can optimize θ as a modular component within an offthe-shelf expectation maximization (EM) algorithm.


Next, we define this distribution which is essentially a truncated Geometric distribution.

given parameters ε ∈ (0, 1) and the polynomial growth rate ρ (with constant C) of the graph, define ϕ = ε 5C2 ρ , and Then, the distribution (or random variable) Q is defined over integers from 1 to K(ε, ρ) as (1) Initially, select x ∈ Σ n arbitrarily.


We propose a novel deep kernel learning model and stochastic variational inference procedure which generalizes deep kernel learning approaches to enable classification, multi-task learning, additive covariance structures, and stochastic gradient training.

we apply additive base kernels to subsets of output features from deep neural architectures, and jointly learn the parameters of the base kernels and deep network through a Gaussian process marginal likelihood objective.


The basic statistical properties of natural sounds, as observed through a linear filter, have been previously documented by Attias [13].

he showed that, as with visual images, the spectral energy falls roughly according to a power law, and that the histograms of filter responses are more kurtotic than a Gaussian (i.e., they have a sharp peak at zero, and very long tails).


Given two equivalence relations ∼ 1 and ∼ 2 , the refinement ∆(∼ 1 ) ∆(∼ 2 ) holds if and only if s ∼ 2 s ⇒ s ∼ 1 s and s ∼ 2 s ⇒ ∀a ∈ A, ∀C ∈ S/∼ 1 P a (s)(C) = P a (s )(C) .

∀s, s with s ∼ Γn+1 s , and ∀C ∈ S/∼ Γn , P a (s)(C) = P a (s )(C).


Slow Feature Analysis [1] combines many of the best properties of these methods to provide a good general nonlinear model.

it uses an objective function that minimises the first-order temporal derivative of the outputs; it provides a closedform solution which maximises this function by projecting inputs into a nonlinear http://www.loria.fr/equipes/cortex/ space; it exploits sphering (or PCA-whitening) of the data to ensure that all outputs have unit variance and are uncorrelated.


Our method works with normalization schemes which don't introduce correlations between examples.

we recommend layer normalization [3] as a drop-in replacement for batch normalization.


2 Sequential learning in GANs We first introduce our conditional GAN framework in the non-sequential setting, where all categories are learned jointly.

this first baseline is based on the AC-GAN framework [20] combined with the WGAN-GP loss for robust training [7].


Assuming specific conditions that relate m and p, we develop new semi-supervised methods that lead to improved estimation.

we propose estimators that reduce bias by estimating the Hessian of the regression function, improve the choice of bandwidths using unlabeled data, and estimate the regression function on level sets.


In many clustering tasks, the raw features of data are usually weak.

the given distance measure between data points, such as the Euclidean distance, is only valid in a small neighborhood.


If G = (V, E) is the network that we want to fit our model to, and uv denote the graph that we obtain by resampling the value of X uv in G uv according to the conditional distribution P (X uv | x Guv \ {x uv }; θ) predicted by our model.

g * uv is the result of performing just one iteration of Gibbs sampling on X uv using θ, where the configuration x Guv of G uv is used to initialize the (single-step) Markov chain.


For comparison, at least 3.7 × 10 5 iterations are needed to train a VGG-16 network from scratch [25].

mTZ is able to inherit information from the pre-trained models and construct a combined model with an increase in test errors of less than 0.5%.


τ i and ν i can take the following values: In the x jt -update (5), a Bregman divergence is addded so that exact PDMM and its inexact variants can be analyzed in an unified framework [23,11].


This ensures that the set of possible functional dependencies between input and response is the same in both directions.

due to the equivalence of conditional mean embeddings and vector-valued regressors [12], this is analogous to the requirement for constraining the assumed functional class in structural equation models in order to ensure identifiability [25].


In a setting where losses are allowed to drift, we characterize -in a nearly complete manner-the power of adaptive adversaries with bounded memories and switching costs.

we show that with switching costs, the attainable rate with bandit feedback is � Θ(T 2/3 ).


Choosing M. For the mass matrix, we take inspiration from the connection between SGHMC and SGD.

the literature [23,24] shows how normalizing the gradient by its magnitude (estimated over the whole dataset) improves the robustness of SGD.


To allow our prediction to move between two distant locations efficiently, we first relax the regression problem to a classification task.

we evenly divide the input video into multiple anchor segments under multiple scales, and train a FC layer on thef cv to predict the best anchor that produces the highest Meteor score [32] of the generated caption sentence.


In the M-step, we apply the same shape and motion updates as in the previous section; additionally, we update Φ and Q in the same way as in Shumway and Stoffer's algorithm.

this reconstruction algorithm learns 3D shape with temporal smoothing, while learning the temporal smoothness term.


Instead, we focus on work related to the main novelty in this paper, the explicit modeling of structure on label counts.

given that we have prior knowledge of label count structure, or are modeling a domain that exhibits such structure, the question is how can the structure be leveraged to improve a model.


Despite not being involved in the dynamics per se, these fast connections are crucial in order to learn any temporal dynamics.

learning a detailed balance is a pre-requirement to learn dynamics with local plasticity rules in a spiking network.


All the above parameters and workers' actions stay unknown to our mechanism.

we regard workers as black-boxes, which distinguishes our mechanism from the existing peer prediction mechanisms.


This reference set of boxes may be a uniform sub-sampling of the space of windows as done in Sliding Window methods, or may itself be the output of another object proposal method such as Selective Search [39].

each greedy augmentation step under our formulation given by: argmax (7b) Intuitively speaking, the marginal gain of a new window y under our diversity function is the largest gain in coverage of exactly one of the references boxes.


In the proposed segmentation approach, the observations originate from a set of continuous-valued hidden trajectories, each representing a different movement template.

we assume that the observed time-series consists of a concatenation of segments (basic actions), each generated through a noisy transformation of one of the hidden trajectories, with possible time re-scaling.


Discrete-time finite-horizon LQR is a well-studied control method that optimizes a convex quadratic objective function with respect to affine state-transition dynamics from an initial system state x init .

lQR finds the optimal nominal trajectory τ ⋆ 1:T = {x t , u t } 1:T by solving the optimization problem From a policy learning perspective, this can be interpreted as a module with unknown parameters θ = {C, c, F, f }, which can be integrated into a larger end-to-end learning system.


Let the preference constraint (v i , g i ) ∈ V be selected at a given iteration, to enforce the constraint a∈A(gi) α a i + λ i = C, λ i ≥ 0, two elements from the set of variables {α a i |a ∈ A(g i )} ∪ {λ i } will be optimized in pairs while keeping the solution inside the feasible region α a i ≥ 0.

let χ 1 and χ 2 be the two selected variables, we restrict the updates to the form χ 1 ← χ 1 −ν and χ 2 ← χ 2 +ν with optimal choices for ν.


This number of samples is less than 10% of the full training set.

the cost is less than 10% of one pass over the dataset.


The algorithmic strategy will look for a sequence {J n , ι n } ∞ n=1 , where J n ⊂ Q is a set of covering prototypes, and ι n : J n → F S is a function that associates a feature to every prototype in J n . Starting with J 0 = ∅ and Γ 0 = ∅, the strategy needs to find, at step n > 0, a cover Ĵn for S, based on the distance metric D(Γ n−1 ).

it has to guarantee that ∀s ∈ S, ∃κ ∈ Ĵn with D(Γ n−1 )(s, κ) = 0.


There seem to be multiple opportunities for digging deeper into the connection between dropout training and adaptive regularization.

it would be interesting to see whether the dropout regularizer takes on a tractable and/or interpretable form in neural networks, and whether similar semi-supervised schemes could be used to improve on the results presented in [1].


If f (n) grows asymptotically faster than √ n, then one can "sparsify" f (n) so that it will be crossed with an arbitrarily small probability.

a boundary with the form in Equation ( 8) can be constructed to bound the threshold-crossing probability below any δ (part 2 of the Theorem).


Intuitively, this is due to the fact that the second term is a regularization term that enforces the rows of X to be the same, and can be seen as a generalization of the fused Lasso penalty [18].

this penalty seeks to fuse the rows of X.


We have traced an analytical connection from psychophysical results in monkey and man to a broad class of machine-learning algorithms, namely, density estimation in latent-variable models.

behavioral studies of multisensory integration have shown that primates estimate stimulus properties with the peak of the posterior distribution over the stimulus, conditioned on the two unisensory cues [3,4].


From the proof of Theorem 1, we can see that the key inequalities are ( 12), (13), and (14).

the regret-type bound in (12) depends on the algorithm.


In particular, 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 want to show that with probability at least 1 − δ, φ(x N +1 ) − φ(x * ) < (N, δ).


For a typical causal graph we can do even better.

we prove, in Section 4, that for a random causal graph a constant number of experiments is sufficient in expectation.


We expect that the consistency holds far beyond this case.

for the case of general Gaussian design matrices, consistency holds conditionally on a conjectured formula stated in [JM13] on the basis of the "replica method" from statistical physics.


∆ (Z) P Analogous expressions hold for ∆ (X) P and ∆ (Y ) P . Unlike in the two-variable case where either matrix or both can be centered, centering of each matrix in the three-variable case has a different meaning.

one requires centering of all three kernel matrices to perform a "complete" Lancaster interaction test, as given by the following Proposition.


We thus propose here to minimize the per-step loss only for the state, defined as a pair of the input x and the prefix ŷ<t , visited by the parametrized policy π θ .

we generate an entire trajectory (ŷ 1 , .


What is the optimal sampling schedule, and does it depend on the singular value spectrum of the data matric?

can improved theoretical bounds or empirical performance be obtained by carefully choosing how many columns to select in each round?


Specifically, we assume that there are parameters ρ, µ (possibly dependent on p such that the class of graph-structured activation patterns x is given as follows.

here out(C) = (u,v)∈E W u,v I{u ∈ C, v ∈ C} is the total weight of edges leaving the cluster C. the set of activated vertices C have a small cut size in the graph G. While we assume that the noise variance is 1 in (1), this is equivalent to the more general model in which Eξ 2 i = σ 2 with σ known.


Rather, they attempt to find 'good' maps by searching for filters which are maximally discriminative between different stimulus conditions.

they differ from the classical approach in that they do not assume the noise to be isotropic and uncorrelated, but make it hard to incorporate prior knowledge about the structure of maps, and can therefore be data-intensive.


Second, we provide a tighter upper bound of the Gibbs decoder distortion by randomizing the search space of the optimization problem.

instead of having a formulation over all possible structures and latent variables (usually exponential in size), we propose a formulation that uses i.i.d.


At the other extreme, we show that each processor can transmit ≤ 2.8n + 32 bits per iteration in expectation, while increasing variance by a only a factor of 2.

in the latter regime, compared to full precision SGD, we use ≈ 2.8n bits of communication per iteration as opposed to 32n bits, and guarantee at most 2× more iterations, leading to bandwidth savings of ≈ 5.7×.


TPSRs are a generalization of regular PSRs: a TPSR maintains a small number of sufficient statistics which are linear combinations of a (potentially very large) set of test probabilities.

a TPSR maintains a small number of feature predictions instead of test predictions.


Let ǫ ≥ 0 and α ∈ [0, 1] be given numbers.

then θ is an approximate global optimizer of U with value imprecision ǫ and residual domain the value U (θ) is within ǫ of a value which is greater than the values that U takes on at least a 1 − α portion of the domain.


In this work, we model the independence properties between the parameters for each location and for each task, by defining a Markov Random Field (MRF) over the parameters.

two sets of parameters are encouraged to have similar values if they are spatially close or semantically close.


Our work aims to generalize the typical gradient descent optimization framework to manifold optimization.

we will use the ILEA framework as our working example to generalize parallel optimization algorithms.


1(b), the waiting effect causes the variance to increase beyond the kinetic sampling prediction, agreeing with the trend of the simulation results.

the variance approaches 0.34/N in the extremely diverse limit, signi£cantly greater than the limit of 0.25/N in the absence of waiting effects.


For a typical causal graph needs only a constant number of experiments are required in expectation!

if the directed acyclic graph is random, based upon an underlying Erdös-Rényi model, then O(1) experiments in expectation are required to discover G. We prove this result in Section 4.


In addition to the unary model of [12], a model based on quadratic assignment is also presented, in which pairwise features are determined using the adjacency structure of the graphs.

if a pair of points (p 1 , p 2 ) in the template scene is to be matched to (q 1 , q 2 ) in the target, there is a feature which is 1 if there is an edge between p 1 and p 2 in the template, and an edge between q 1 and q 2 in the target (and 0 otherwise).


We show that the new feature provides high classification accuracy, even with a linear classifier.

we model an image as a Gaussian Markov Random Field (GMRF) whose nodes correspond to local features and consider the GMRF parameters as the image feature.


Both w −1 min and 1 − 2P π2-Loop [w(F ) < 0] depend on the choice of BP fixed point, however it is unlikely (unless a degeneracy) that these characteristics become large.

p π2-Loop [w(F ) < 0] = 0 in the case of attractive models [20].


From our results, we see that most LVMs outperform the popular FIM approach.

the BBMs outperform all other approaches on two of the three datasets, with a high of CPR = 79% in Eclipse Subversive.


Rejecting these examples has been investigated since the early days of pattern recognition.

chow (1970) analyses how the error rate may be decreased thanks to the reject option.


The focus of the current paper is therefore not to stress the usefulness of CG methods in practical applications (and we refer to the above mentioned references) but to examine its theoretical convergence properties.

we establish the existence of early stopping rules that lead to optimal convergence rates.


Various computational models have been proposed to describe the underlying feature analysis [see 2, for review].

a feedforward hierarchical model constrained by the anatomy and the physiology of the visual cortex was shown to agree well with human behavioral responses [16].


Changing the order of the square root and diagonal would give us the diagonal of |H|.

the equilibration preconditioner can be thought of as the Jacobi preconditioner of the absolute Hessian.


(Foygel et al., 2012) For any half-trek, π, let Right(π) be the set of vertices in π that have an outgoing directed edge in π (as opposed to bidirected edge) union the last node in the trek.

if the trek is a directed path then every node in the path is a member of Right(π).


Then for any δ > 0, we have We would like to characterize the achievable regret bounds for both switching and drifting distributions, in both full-information and bandit settings.

we would like to understand the interplay among the parameters Γ, V, Λ, and T , defined in (1).


[5] propose to perform transduction by requiring that the conditional class probabilities on training and test set match.

for classifiers generating a distribution of the form y i ∼ p(y i |x i , w) they require that the marginal class probability on the test set matches the empirical class probability on the training set.


In case of 3-way (and higher) c-NN problem, we have to take care of a more complicated event to create an indexing scheme.

during preprocessing we need to create buckets for each individual S 3 , and while querying we need to associate the query sets S 1 and S 2 to the appropriate bucket.


Previous works have also shown that using side information, e.g., the scene perspective, helps to improve crowd counting accuracy [5,6].

when extracting hand-crafted features (e.g., edge and texture statistics) [5][6][7][8][9] use scene perspective normalization, where a "perspective weight" is applied at each 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.


Here, our study reveals that through combining STF and STD properly, a neural system can hold stimulus information for an arbitrary time, serving for different computational purposes.

sTF tends to increase the lifetime of persistent activity; whereas, STD tends to decrease the lifetime of persistent activity.


Following [32] and [15], as we move toward our formal results we are considering an asymptotic model inspired by Johnstone's Spiked Model [22].

we are considering a sequence of increasingly larger data target matrices X n , and corresponding data matrices Y n iid ∼ f An,Bn (X n ).


Suppose that f (x) is ⌦ ln 1+✏ (x) for some ✏ > 0.

for UCB( ) algorithm, there exists an ✏ 0 > 0 such that the regret of UCB( ) is O(T ).


sVEB solves the following weighted least-square error (WLSE) problem to learn f t s: For labeled data (first term in eq.7), boosting weights, w i 's, and working responses, z i 's, are computed as described in equation 4.


We have presented a new boosting algorithm, PSDBoost, for learning a positive semidefinite matrix.

as an example, we use PSDBoost to learn a distance metric for classification.


In particular, we do not require any "low noise" conditions, nor that the loss function is strongly convex (it need only be weakly convex).

(6) applies, under no additional conditions, to the SVM objective.


Our main theorem establishes a lower bound for all learning algorithms, which we formalize as functions from data samples to binary classifiers.

the lower bound applies not only to learning linear classifiers.


Theorem 1 improves significantly over the convergence rate established for the pairwise conditional gradient variant in [16].

the number of iterations to reach an ✏ error in the analysis of [16] depends linearly on |V|!, where |V| is the number of vertices of P. Throughout this section we let h t denote the approximation error of Algorithm 3 on iteration t, for any t 1, i.e., h We start by proving that the iterates of Algorithm 3 are always feasible.


The flanker acts as clutter with respect to the target object.

we train our models to recognize even MNIST digits-i.e.


While matrix completion is a well studied problem [2] to recover missing measurements, the setting with Capture-C is slightly different.

the number of highly confident entries for the n ⇥ n adjacency matrix of rank r may be less than nr log n, which is suggested for matrix completion [2].


Note that when G is undirected (more precisely, as above, when G is a directed graph having for every pair of nodes i� j ∈ V either no arcs or length-two cycles), then mas(G) = α(G), otherwise mas(G) ≥ α(G).

when G is itself a directed acyclic graph, then mas(G) = |V |.


Our objective is to find the set of voxel time courses S defined as where S * denotes a subset of the entire collection of voxels time courses S extracted from a single fMRI scan.

we aim at finding spatially distributed voxel responses S that best explain the naturalistic stimuli, without making any prior assumptions about location and size of voxel subsets.


Disjunctions use the signal from the survey propagation messages without overcommitting to a particular variable assignment too early (as in decimation).

without loss of generality, if we are given marginals µ i (1) > µ i (0) and µ j (1) > µ j (0) for variables i and j, the new update adds the streamlining constraint x i _ x j to the problem instead of overcommitting by constraining i or j to its most likely state.


In this section, we provide some experimental results that confirm the accuracy of our theoretical predictions.

these results verify the predicted linear rates of convergence under the conditions of Corollaries 2 and 3.


The intuition of the proposed SBA algorithm is that if the graphon is smooth, neighboring crosssections of the graphon should be similar.

if two labels u i and u j are close i.e., |u i − u j | ≈ 0, then the difference between the row slices |w(u i , •) − w(u j , •)| and the column slices |w(•, u i ) − w(•, u j )| should also be small.


Our approach is to treat both the inputs and outputs of the GP as latent variables.

we introduce k Gaussian distributed pseudoinputs η = (η 1 , .


That is, the choice x t ∈ X defines a parameter of a mixed strategy from which a actual move ω(x t , s t ) is drawn; for instance, for additive zero-mean Gaussian noise, x t defines the center of the distribution from which x t + s t is drawn.

noise does not allow the adversary to play any desired mixed strategy.


Our theory enables us to account for several detailed aspects of retinal function.

the primate retina is dominated by four types of ganglion cells, ON midget and parasol cells and their OFF counterparts.


Here, we denote by a ≻ a ′ the event that a wins a ′ and by P (a ≻ a ′ ) the probability that a ≻ a ′ happens.

we assume that the feedback from the oracle follows the following two-valued random variable: where the probability P (a ≻ a ′ ) is determined by the oracle.


Most often the data is embedded in Euclidean space and the distances that we work with are Euclidean distances.

probably the most studied center based clustering problem is the Euclidean k-means problem.


Instead of taking the farthest point in the simplex, this technique is designed to shift the estimate of a topic to a high density region, where true topics are likely to be found.

given a (current) cone S ω (v), we re-position the cone by updating v := argmin v m∈Sω(v) wm 2 (1 − cos( wm , v)).


In this section, we validate our method for action recognition on two public datasets: Sports dataset [25] and UCF101 [20] dataset.

we consider three sets of attributes: humanlabeled attribute set (HLA set), data-driven attribute set (DDA set) and the set mixing both types of attributes (Mixed set).


As highlighted in [19], providing gap-free results was an open question for decades and is a more reliable goal for practical purposes.

they proved that the block Krylov method converges in time , where ε is the multiplicative approximation error.


Given a parameter , the adversarial frequency measures how often f fails to be (x * , )-robust.

if f has high adversarial frequency, then it fails to be (x * , )-robust for many inputs x * .


The solution fλ,m of the corresponding minimization problem can now be written as, where A † denotes the Moore-Penrose pseudoinverse of a matrix A, and  for any t > 0, where In practice, leverage scores are onerous to compute and approximations ( li (t)) n i=1 can be considered [16,17,24] .

in the following we are interested in suitable approximations defined as follows: Definition 1 (T -approximate leverage scores).


samples X n ∼ p n , which in turn are used to estimate f (p).

given property f , we would like to construct an estimator f : X * → R such that f (X n ) is as close to f (p) as possible.


One of the most important features of the graph Laplacian is that it is coordinate free.

it does not depend on any special coordinate system.


From this assumption it follows that information density is optimized near the channel capacity, where speakers maximize the rate of information transmission while minimizing the danger of a mistransmitted message (see also Aylett (2000); Aylett and Turk (2004); Genzel and Charniak (2002)).

uID is an optimal solution to the problem of rapid yet error-free communication in a noisy environment.


(30) Then by non-expansiveness of proximal operation (Lemma 2), we have Since for z ∈ T , ∥H t z∥ 2 ≥ √ m∥z∥ Ht , (31) leads to where last inequality follows from Lipschitz-continuity of ∇  In this section, we study the convergence behavior of Proximal Gradient method and Proximal Newton method on high-dimensional real data set with and without the CNSC condition.

two loss functions -logistic loss L(u, y)=log(1 + exp(−yu)) and ℓ 2 -hinge loss L(u, y)=max(1 − yu, 0) 2 -are used in (3) with ℓ 1 -regularization h(x) = λ∥x∥ 1 , where both losses are smooth but only logistic loss has strict convexity that implies the CNSC condition.


Note that once λ is computed ĉ is given by a minimizer of problem (2.3) for K = K(λ).

for square loss regularization this requires solving the equation (2.5) with In this section we present our experiments on optical character recognition.


This bound can be expressed on a scale that is independent of the temperature parameter γ; intuitively, ∆ max corresponds to a utility difference so slight that the user identifies the best item only with probability 0.56 under R L with temperature γ.

the maximum loss is so small that the user is unable able to identify the preferred item 44% of the time when asked to compare the two items in S. This derivation can be generalized to sets of any size k, yielding , where L W (•) is the Lambert W function.


The dominating factor of the bound on t becomes (5) Table 1 summarizes several interesting cases, and compares the iteration bound of ASDCA to the iteration bound of the vanilla SDCA algorithm (as analyzed in Shalev-Shwartz and Zhang [2013a]) As can be seen in the table, the ASDCA algorithm interpolates between SDCA and AGD.

aSDCA has the same bound as SDCA when m = 1 and the same bound as AGD when m = n. Recall that the cost of each iteration of AGD scales with n while the cost of each iteration of SDCA does not scale with n. The cost of each iteration of ASDCA scales with m. To compensate for the difference cost per iteration for different algorithms, we may also compare the complexity in terms of the number of examples processed (see Table 2).


A key tenet in any FDA trial is the use of randomization for the treatment assignment, which precisely protects against biases that might be introduced by physicians.

physicians may prescribe Drug A for their wealthier patients who have better nutrition than their less wealthy ones, when unknown to the doctors, the wealthy patients would recover without treatment.


We then also propose plane-wise metrics.

we generate the normals from the predicted depths using the NYU toolbox [24], and evaluate the degree variation (var.)


This bound has a few more parameters than Corollary 3.1.

the term , which is instantiated to m −1/2+1/p in the proof of Corollary 3.1, catches the mass of points discarded due to the outer bracket, as well as the resolution of the (inner) cover.


Accordingly, the update rule for ρ takes the form, To those familiar with advantage-actor-critic methods [21] (A2C and its asynchronous analogue A3C) PCL's update rules might appear to be similar.

advantage-actor-critic is an on-policy method that exploits the expected value function, to reduce the variance of policy gradient, in service of maximizing the expected reward.


It returns a mixed policy ψ such that µ(ψ) − µ E 2 ≤ as long as T and m are sufficiently large.

their algorithm seeks to "match" the expert's feature expectations.


This is an example of the avoidance of the curse of dimensionality in the manner mentioned above, by a method whose performance depends on the intrinsic dimensionality of the data instead of the full dimensionality of the feature space.

our method is practical in that it works with Euclidean distances on R N . we do not assume any knowledge of the quantities pertaining to the intrinsic geometry of the underlying submanifold such as its metric tensor, geodesic distances between its points, its volume form, etc.


In this paper we proposed a distributionally robust approach to mitigate the conservatism of the robust MDP framework and incorporate additional a-prior probabilistic information regarding the unknown parameters.

we considered the nested-set structured parameter uncertainty to model a-prior probabilistic information of the parameters.


When the distribution function µ f0 p q does not change abruptly with the expression of ε U n pf 0 q can be significantly simplified.

if for all P p0, c 0 s it holds that µ f0 p { log nq ě µ f0 p q{rlog ns Op1q , then ε U n pf 0 q can be upper bounded as It is also noted that if µ f0 p q has a polynomial behavior of µ f0 p qβ for some constant β ě 0, then Eq.


A predictor Y is derived from a random variable R and the protected attribute A if it is a possibly randomized function of the random variables (R, A) alone.

y is independent of X conditional on (R, A).


We approximate the Markov chain described in (7) by considering a priori that all alternatives have equal strength.


Moreover, the primal variables can be easily recovered from the dual variables by the KKT conditions.

)τ e . Subsequently, the decision function f (x) = w ϕ(x) + b is a linear combination of k(x i , x)'s defined on both the labeled and unlabeled examples, as in standard manifold regularization.


Intuitively, each cluster function corresponds to a partial marginalization of the factors contained in cluster v. Since each cluster function is defined over a subset of the variables in the original graph, we require some additional notation to represent these sets.

for a cluster v, let A(v) be the arguments of its cluster function, and let V(v) be the set of all arguments of its children, In a slight abuse of notation, we let A(v) be the arguments of the node v in the original graph, so that if v is a variable node A(v) = v and if v is a factor node A(v) = N (v).


However, more work needs to be done to explore the strengths and weakneses of these algorithms.

simple sequential importance resampling is known to break down when applied to datasets with many observations, although we are optimistic that methods for addressing this problem that have been developed for Dirichlet process mixture models (e.g., [5]) will also be applicable in this setting.


An admissible randomized strategy for this relaxation is given by (6).

the expected regret of BISTRO+ is upper bounded by for any L ≥ K. setting L = (KT / log(N )) ).


Nevertheless, we show how to bound the regret of TS in a very specific case of n × m rank-1 matrices, and we leave the generalization of these results for future work.

we analyze the regret of PTS in the setting of Gopalan et al.


Observation: Assume the marginal distribution P(Y = 0) and P(Y = 1) is fixed for any distribution P ∈ C. Then, the distribution in C with the minimum HGR correlation between X and Y is the distribution P obtained by solving (5).

ρ(X, Y ; P) ≤ ρ(X, Y ; P), ∀ P ∈ C, where ρ(X, Y ; P) denotes the HGR correlation coefficient under the probability distribution P. Based on the above observation, from now on, we call the classifier δ(•) in (7) as the "Rényi classifier".


Therefore, we structure the experiments to compare against this baseline.

we choose to implement learnets using similar network topologies for a fairer comparison.


5: end for Output: x K where z k = (k+2)y k /2 − (k/2)x k . Correspondingly, we can also obtain the following analysis tools for our convergence analysis using the proposed equations ( 4) and (5).

the following equations ( 7) and ( 8) can be viewed as the Riemannian space counterparts of (6).


However, the conditions there are much more stringent than the SE property in terms of the dependence on the problem dimensions.

their quadratic convergence can only be guaranteed in a much smaller neighborhood.


Our bounds require us to assume that all sites f i (x) are β m -strongly log-concave with slowlychanging log-function.

if we note The target distribution p(x) then inherits those properties from the sites.


Further suppose we have a simple linear scoring function defined by where I is the indicator function.

p i is the best match for query q i , but p 3 does not score highly for either query feature alone.


Tractability is not necessarily sufficient to obtain algorithms that work effectively in practice.

although the problem of large margin estimation can be formulated as a quadratic program (QP) in several cases of interest [2,19], and although this formulation exploits enough of the problem structure so as to achieve a polynomial representation in terms of the number of variables and constraints, off-the-shelf QP solvers scale poorly with problem and training sample size for these models.


In some cases we already know that certain larger sub-structures can be significant features for solving the target problem.

significant larger sub-structures, which the conventional methods cannot deal with efficiently, should have the possibility of further improving the performance.


Problem (5) decomposes into k independent problems.

the minimizers {Z * (t) j : j ∈ V w \B(t)} are determined as the solution of the (T w , B w )-problem.


In this work, we answer both questions in the affirmative, bridging the gap between its use in the non-stochastic and stochastic settings.

we propose a stochastic variant of the cubic-regularized Newton method.


It is also an acoustic cue that seems amenable to modeling by NMF.

we can imagine the basis functions in NMF as harmonic stacks of individual periodic sources (e.g., voices, instruments), which are superposed to give the magnitude spectrum of a mixed signal.


An output unit in the second layer is the maximum of d 2 branch units.

the output unit is the upper envelope of d 2 hyperplanes, and it represents a convex PWL function.


On the other hand, EH-Hash's hash functions are significantly more expensive to compute.

it requires O(d 2 ) time, whereas H-Hash requires only O(d).


See the appendix for sufficient conditions.

under this assumption, we have that the above setting for w t is in fact a minimizer of the primal objective, since (λ t 1 , .


An immediate corollary is that non-analyticity points (w.r.t ∆) of the MIs are the same in the coupled and original models.

defining ∆ Opt,coup := sup{∆ | lim L→∞ lim n→∞ I w,L (S; W)/(n(L+1)) is analytic in ]0, ∆[}, we have ∆ Opt,coup = ∆ Opt .


We can extend the preceding example to exhibit a dimension-dependent gap between the LP and the SDP for random weight matrices.

for a random network with m hidden nodes and input dimension d, with high probability, .


The new method outperforms all other methods, and in particular the methods without time delays.

the new algorithm is capable of approximating 97% of the trajectory data with only 3 sources, while PCA and ICA require more than 6 sources to achieve the same level of accuracy.


convex function f that satisfies P f +κ = P f • P κ for all gauge κ.


First, we ran a number of gradient-based and decision-based adversarial attacks against a baseline model (ResNet32) and tested the efficacy of the resulting adversarial images against the cache models (a similar procedure was recently used in [12]).

we considered the following four adversarial attacks.


This paper develops three causal structure learning algorithms, each of which discovers all dynamic causal graphs that explain the observed measurement data, perhaps given undersampling.

these algorithms all learn causal structure in a "rate-agnostic" manner: they do not assume any particular relation between the measurement and system timescales.


The updates M i,t−1 → M i,t and b i,t−1 → b i,t are performed at node i through vector xt both when i = i t (i.e., when node i is the one which the context vectors in C it refer to) and to a lesser extent when i = i t (i.e., when node i is not the one which the vectors in C it refer to).

this is because, as we said, the payoff a t received for node i t is somehow informative also for all other nodes i = i t . because we are assuming the underlying parameter vectors u i are close to each other, we should let the corresponding prototype vectors w i,t undergo similar updates, so as to also keep the w i,t close to each other over time.


Note that for the feed-forward networks with BN layers, we can obtain a reliable estimation of m b,W and m b,z .

we assume that W b is row homogeneous [Ba et al., 2016], i.e., they represent the same level of information and are roughly of similar magnitude, and define  17) and ( 18) of the fully connected layer with BN are computed as follows, We next evaluate the scale-amended SGD on training VGG nets [Simonyan and Zisserman, 2015] for image classification tasks with two datasets: CIFAR-10 [ Krizhevsky and Hinton, 2009] and CIFAR-100 [Krizhevsky and Hinton, 2009].


Each training game was played within 1m 1s Fischer time controls.

both players start with a minute on the clock, and gain an additional second every time they make a move.


These theoretical results suggest two simple yet efficient algorithms for the joint M -best diverse problem, which outperform competitors in terms of runtime and quality of results.

as we show in the paper, the new methods compute the exact M -best diverse labelings faster than a popular method of Batra et al., which in some sense only obtains approximate solutions.


In the resulting graph, v 4 is d-separated from v 1 implying that v 4 is independent of v 1 in the distribution, P (v 4 , v 2 , v 1 |do(v 3 )).


Given a solution Λ * to ( 6), the optimal model parameters W * for (3) can be recovered via The equivalence relation form of log-linear regression can be used to derive useful relaxations of EM variants for directed models.

by exploiting Theorem 1, we can now re-express the regularized form of the joint EM objective (1) strictly in terms of equivalence relations over the hidden variable values = min where h ranges over the hidden variables, and Note that ( 8) is an exact reformulation of the joint EM objective (7); no relaxation has yet been introduced.


The principle is simple: because random sampling does not change the clustering structure of data, a clustering method with high consistency should preserve the pairwise relationships of samples in different trials.

we test each clustering method with J (= 100) trials.


Disadvantageous-inequity-averse agents are better than A3C at maintaining cooperation via punishment in commons games (Figure 4).

a single disadvantageous-averse agent can fine defectors, generating a sustainable outcome.


Although DP requires the same number of messages updates as BP and MP, DP updates are more expensive because they require the computation of convolutions.

each variable-to-factor update rule (4) requires (N − 2)L convolutions, where N is the number of neighbors of the variable node and L is the number of states in the random variable.


If ⇥ is a convex bounded subset of the strictly positive orthant in R d which is invariant under permutations, then k • k ⇥ is a symmetric gauge function.

this readily applies to both the k-support norm and box-norm.


In contrast to most DP-based approaches, our construction is motivated by the intrinsic relation between Dirichlet processes and compound Poisson processes.

we discussed three operations: superposition, subsampling, and point transition, which produce DPs depending on others.


2 Note that r * S (x 0 ) can be equivalently written When S = R d , r * (x 0 ) := r * R d (x 0 ) is the adversarial (or worst-case) perturbation defined in [17], which corresponds to the (unconstrained) perturbation of minimal norm that changes the label of the datapoint x 0 .

r * (x 0 ) 2 corresponds to the minimal distance from x 0 to the classifier boundary.


We can calculate the variational objective up to a constant without involving the exact posterior p.z j x/.

we do not require calculating the normalizing constant of the posterior, which is typically intractable.


Affinities (or similarities) of all n data points are computed with respect to these m anchors in linear time O(dmn) where m n. The true affinity matrix A o ∈ R n×n is then approximated by using these affinities.

an anchor graph leverages a nonlinear data-to-anchor mapping (R ) /M , where δ j ∈ {1, 0} and δ j = 1 if and only if anchor u j is one of s m closest anchors of x in U according to some distance function D() (e.g., 2 distance), t > 0 is the bandwidth parameter, and M = m j=1 δ j exp(− ) leading to z(x) 1 = 1.


For each popped element s ∈ S, we find k such that 0 < k ≤ r − x(s) with f (kχs|x) kc(χs) ≥ θ with binary search.

if there is such k, we update x with x + kχ s . Finally, we push s again with the key f (χs|x) c(χs) if x(s) < r. The correctness of this technique is obvious because of the DR-submodularity of f . the key of each element s ∈ S in the queue is always at least f (χs|x) c(χs) , where x is the current vector.


In this paper, we consider this question from the optimization's perspective and give an answer: the Hessian of the local back-matching loss is responsible for the difficulty of training deep nets with BP.

we start from a penalized loss formulation, which takes the intermediate feature outputs as variables of the optimization and the penalty is to enforce the coordination (architecture) connection [Carreira-Perpinan and Wang, 2014].


Let w ∞ = w max = 1 for ease of presentation.

suppose ∆ the top-K items are identified as desired.


There exists an infinite dimensional separable metric space (X , ρ) and a realizable distribution μ over X × {0, 1} such that no k n -NN learner satisfying k n /n → 0 when n → ∞ is Bayes-consistent under μ.

this holds for any space and realizable distribution μ that satisfy the following condition: The set C of points labeled 1 by μ satisfies Since µ(C) > 0, Eq.


[18] obtains superb approximation guarantees 3 , its runtime is exponential in the representation length of domain elements.

it is designed for a discrete version of the problem, in which centers come from a finite set Y , and the runtime of their algorithm is at least linear in |Y |.


[2016] takes a different approach and leverages known polynomial constraints on the covariance matrix of the observed variables in a given Gaussian latent tree.

the postulated latent tree is tested with an aggregate statistic formed from estimates of the polynomial quantities involved.


The PDF P(On Iva) gives the probability of presence of the object class On given contextual information.

the PDF P{on Ive) evaluates the consistency of the object On with the context vc.


Our main objective in the following experiments is to demonstrate that our LIRPG-based algorithm can extend to a different class of domains and a different choice of baseline actor-critic architecture (namely, PPO instead of A2C).

we explore domains from the Mujoco continuous control benchmark [Duan et al., 2016a], and used the open-source implementation of the PPO  algorithm from OpenAI  as our baseline agent.


Any (possibly randomized) unbiased statistical algorithm that solves Z with probability " requires at least m calls to the Unbiased Oracle for < .

if ÷ AE 1/6, then any algorithm with success probability at least 2/3 requires at least min{¸/4, 1/48"} samples from the Unbiased Oracle.


Actions: The available actions of the agent consist of two groups, one for scaling the current window to a sub-window, and the other one for translating the current window locally.

the scaling group contains five actions, each corresponding to a certain sub-window with the size 0.55 times as the current window (see Fig.


Thus A * is always at least as good as A u .

in addition, since The difference between those two quantities is the weighted quadratic variation of the σ k around their weighted mean Σ w . it is the variance of the (σ k ) 1≤k≤K .


In the case of a Markov chain, learning with hidden state is exactly the problem of parameter estimation in hidden Markov models.

we must update the estimates of w, /tl and ur for each state, and aij for each state transition (out of the holding state), in order to maximize the likelihood of the sequence of observations The standard algorithm for hidden Markov model parameter estimation is the Baum-Welch method [10].


The PIP loss respects the unitary-invariance.

if where U is a unitary matrix, then the PIP loss between E 1 and E 2 is zero because In addition, the PIP loss serves as a metric of functionality dissimilarity.


Then the optimal action-value function Q * (s, a) is equal to R(s, a).

r(s, a) is the best similarity function.


Moreover, starting from this initial estimate, RAF converges linearly to the true solution x.

to reach any -relative solution accuracy (i.e., dist(z T , x) ≤ x ), it suffices to run at most T = O(log 1 / ) RAF iterations (cf.


Our assumptions essentially characterize the probability of pulling near-optimal arms.

given µ * ∈ [0, 1] as the best possible mean-reward and β ≥ 0 a parameter of the mean-reward distribution, the probability that a new arm is -optimal is of order β for small , i.e.


This paper thus proposes to study the problem of learning a two-layer overparameterized neural network using SGD for classification, on data with a more realistic structure.

the data in each class is a mixture of several components, and components from different classes are well separated in distance (but the components in each class can be close to each other).


In time-series settings [11][12][13], such as audio denoising and missing data imputation considered later in the paper, this means that the number of pseudo-datapoints must grow with the number of datapoints if restoration accuracy is to be maintained.

m must be scaled with N and so pseudo-datapoint schemes have not reduced the scaling of the computational complexity.


While no progress has been made on the general form of the problem, some progress has been made in interesting special cases.

if the bounded convex loss functions are also assumed to be Lipschitz, Flaxman et al.


Theorem 1 also ensures that, if sufficient memory is available, D2-CTW will eventually perform as well as CTW with optimal depth bound K = D. This follows from the fact that, for every node s at depth d s ≤ D in a CTW suffix tree, if β s t ≥ 1 for all t > τ , then the D2-CTW suffix tree will be at least as deep as d s at context s after some time t ≥ τ .

at some point, the D2-CTW model will contain the "useful" sub-tree of the optimal-depth context tree.


We propose a variant for video evaluation, which measures both visual quality and temporal consistency.

we use a pre-trained video recognition CNN as a feature extractor after removing the last few layers from the network.


It is known that there exists a unique cross-covariance operator Σ Y X : holds for all f ∈ H X and g ∈ H Y .

the cross covariance operator represents the covariance of f (X) and g(Y ) as a bilinear form of f and g. if Y is equal to X, the self-adjoint operator Σ XX is called the covariance operator.


Another way to specify a parametric graphical model is by assuming that each node-conditional distributions is a part of the exponential family [29].

the conditional distribution of x s given x −s is assumed to be where C is the base measure, D is the log-normalization constant and N (s) is the neighborhood a the node s. Similar to (2), the neighborhood of each node can be estimated by minimizing the negative log-likelihood with l 1 penalty on θ.


We see that the proportion of misses for the fastest two levels-1 and 2-of the cache for B = 1 increase significantly with the number of cores, while increasing the batch size to B = 10 substantially mitigates cache incoherency.

we maintain (near) linear increases in iteration speed with little degradation in solution quality (the gap f ( x) − f (x ⋆ ) output by each of the procedures with and without batching is identical to within 10 −3 ; cf.


More importantly, in blind image inpainting task, the proposed method provides solutions to some complex problems that have not been tackled before.

we can automatically remove complex patterns like superimposed text from an image, rather than simple patterns like pixels missing at random.


It is intuitive to think that separating male faces from female faces is a much more difficult task than separating faces from background, needing more detailed attention, so more resources should be allocated to face images than background images.

a good tradeoff for RNP is to prune the neural network more when dealing with background images and keep more convolutional channels when inputting a face image.


Interestingly, our lower bound for the Gaussian model also applies to such worst-case distributional shifts.

if the adversary is allowed to shift the mean θ by a vector in B ε ∞ , our proof sketched in Section C transfers to the distribution shift setting.


While performance was significantly improved, these state of the art methods may be restricted by the assumption that under the learned domain-invariant feature representations, the source classifier can be directly transferred to the target domain.

this assumption may not hold when the source classifier and target classifier cannot be shared.


3 We use standard notions of total variation and absolute deviation.

the total variation of a sequence {µi} n i=1 is defined as Alternative approaches.


We greatly expand this set of features, then use PSTD to synthesize a small set of high-quality combined features.

we add the entire 100-step state vector, the squares of the components of the state vector, and several additional nonlinear features, increasing the total number of features from 16 to 220.


Given a set of trajectories obtained following policy π θ , we can produce an estimate ∇ θ J µ (θ) of the policy gradient and we assume to be able to produce a vector = [ 1 , .

. . , m ] T , so that the i-th component of the approximation error is bounded at least with probability 1 − δ: Given the approximation error vector , we can adjust the bound in Corollary 5.1 to produce a new bound that holds at least with probability (1 − δ) m . to preserve the inequality sign, the estimated approximation error must be used to decrease the L2-norm of the policy gradient in the first term (the one that provides the positive contribution to the performance improvement) and to increase the L1-norm in the penalization term.


Prior research has suggested that larger values of λ lead to superior results.

an example of [1] and the approximation error bounds of [17,18], both of which are restricted to the case of a fixed policy, suggest that approximation error is amplified by a factor of 1/(1 − α) as λ is changed from 1 to 0.


R n . We introduce an empirical risk formulation that has favorable global geometry for gradient methods, as soon as m = O(kd 2 log n), under the model of a d-layer fully-connected neural network with random weights.

when suitable deterministic conditions on the generator and measurement matrix are met, we construct a descent direction for any point outside of a small neighborhood around the true k-dimensional latent code and a negative multiple thereof.


Second, the experiments clearly indicate that the results obtained by minimizing the domination loss are better than the other ranking losses, no matter what error is used for evaluation.

employing the domination loss yields significantly better results than using the disagreement loss which has been the commonly used decomposition method in categorization problems [7,10,6,4].


The rate limit implies an expectation of decisions.

the λ-based loss is minimized without explicitely estimating the loss of any decisions that are implied by the rate limit.


The class of functions F is a class of submodular functions by Lemma 10 (in Appendix C).

we now show that every function in F is efficiently learnable from samples drawn from any distribution D. we show that for any ✏, 2 (0, 1) the functions are (✏, ) PAC learnable with the absolute loss function (or any Lipschitz loss function) using poly(1/✏, 1/ , n) samples and running time.


It thus provides an invariant approach to measuring agent-agent and agent-environment interactions.

nash averaging encourages a maximally inclusive approach to evaluation: computational cost aside, the method should only benefit from including as many tasks and agents as possible.


However there is uncertainty on the output y to predict for a given x.

p (y|x) is diverse and Q(y|x) should be diverse as well.


These properties suggest that the hippocampus is better paralleled with a generative model than a replay buffer.

deep generative models such as deep Boltzmann machines [32] or a variational autoencoder [17] can generate high-dimensional samples that closely match observed inputs.


The Evaluation Module (E-Module) is used to determine whether a phrase is a complete caption.

given an input phrase P , the E-Module encodes it into a vector z e , using a two-level LSTM model as described above, and then evaluates the probability of P being a complete caption as It's worth noting that other properties could also be checked by the E-Module besides the completeness.


The moment-to-moment amplitude fluctuations of these local rhythms reflect variable functional states of the underlying neuronal cortical networks and can be used for brain-computer interfacing.

the pericentral µand β rythms are diminished, or even almost completely blocked, by movements of the somatotopically corresponding body part, independent of their active, passive or reflexive origin.


A priori, this isn't obvious even for modular functions.

note that indexing elements in order they are picked by the greedy algorithm doesn't suffice since there are many functions and we need to balance values amongst all.


The third algorithm, which we call H-RLS (Hierarchical Regularized Least Squares), is a simplified variant of APPROX-H-BAYES in which the thresholds τ i,t are set to zero.

we have g i,t (x t ) = {w i,t x t ≥ 0} where the weights w i,t are defined as in (5) and updated as in the APPROX-H-BAYES algorithm.


We begin demonstrating the power of asymmetry by considering an asymmetric variant of the above problem.

even if S is symmetric, we allow associating with each object x i two distinct binary codewords, u i ∈ {±1} k and v i ∈ {±1} k (we can think of this as having two arbitrary mappings u i = f (x i ) and v i = g(x i )), such that S ij = sign( u i , v j − θ).


We also assume the 0/1 labels can be flipped with probability as large as 1 2 .


The major challenge is that evaluating R(F ) requires performing inference.

if we define then we have that Since A(θ) contains a maximization, this is a saddle-point problem.


If the lower bound behaves roughly like W 2 (P U , P f −1 (V ) ), our simulation result also serves as an empirical evidence that W 2 (P U , P f −1 (V ) ) weighs more on high precision.

we generate 10000 uniformly distributed samples in a 10-dimensional unit 2 -ball.


Experimental results with both real and simulated users show that the proposed reinforcement learning framework achieves better retrieval performance than existing techniques.

we observe that feedback based on natural language is more effective than feedback based on predefined relative attributes by a large margin.


Dobra [6] showed how to construct a Markov basis for moves in a complete contingency table given a decomposable set of margins.

let A be decomposable and let n A be consistent with A = V , so that each variable is part of an observed margin.


Unlike many learning problems, there is no natural distribution over TPQs to draw from, and thus, active selection of TPQs is essential.

we want the learner to select TPQs for which the responses will be most useful toward the goal of learning the target policy.


Interpreting Z 0 = (X 0 , y 0 ) as the empty set and the associated Tikhonov matrix as C ,0 = I, all the results stated above can be extended to the case r = 0.

we define the algorithm as w h (Z 0 ) = argmin w2R d w h 2 = h and its transfer risk as E 0 .


It can be viewed as a decomposition procedure where solutions to sub-problems are found locally, and coordinated via constraints to find the global solution.

it is a form of augmented Lagrangian method which applies partial updates to the dual variables.


We also note that the CapProNet+X consistently outperforms the backbone counterparts with varying dimensions c of capsule subspaces.

with the WideResNet backbones, in most cases, the error rates are reduced with an increasing capsule dimension c on all datasets, where the smallest error rates often occur at c = 8.


We establish that for deep linear networks, there is a direct connection between the smoothness parameter of the training loss function and the largest singular value of the estimated linear function.

we show that if the gradient descent algorithm can converge to a solution with a large step size, the function estimated by the network must have small singular values, and hence, the estimated function must have a small Lipschitz constant.


Algorithm 1 Metropolis-within-Gibbs sampler for missing data imputation using a trained VAE Inputs: Observed data x obs , trained VAE (f θ , g γ ), number of iterations T Outputs: Markov chain of imputations xmiss 1 , ..., xmiss T . At each step of the chain, rather than generating codes according to the approximate posterior distribution, we may use this approximation as a proposal within a Metropolis-Hastings algorithm (Metropolis et al., 1953;Hastings, 1970), using the fact that we have access to the unnormalised posterior density of the latent codes.

at each step, we will generate a new code zt as a proposal using the approximate posterior Ψ(z|g γ (x obs , xmiss t−1 )).


The Tsallis entropy is in fact a generalization of the Shannon entropy, as one obtains the latter as a special case of the former asymptotically.

it is easy to prove the following uniform convergence: We emphasize again that one can easily show that Tsallis-smoothing bandit algorithm is indeed identical to INF using the appropriate parameter mapping, although our analysis is simpler due to the notion of differential consistency (Definition 2.2).


Our analysis also reveals a problem of this method for non-separable problems.

the formulation over-penalizes classification error.


The proximity operator of the 1 -norm is given by a simple soft-thresholding.

for the case of the TV semi-norm, we have J ∈ PSS x (T x ) with where I = supp(D * x).


Classical estimators from this field can be classified into two categories: either (i) they are computationally efficient but incur an error that scales polynomially with the dimension d, or (ii) they are provably robust (in the aforementioned sense) but are hard to compute.

essentially all known estimators in robust statistics (e.g., the Tukey median [Tuk75]) have been shown [JP78,Ber06,HM13] to be intractable in the high-dimensional setting.


These kernels can be described in terms of our notation as follows (see also Remark 3.3).


However, say we have a learning algorithm that assumes everything in the same cluster has the same label (so the hypothesis space H consists of all rules that do not split clusters).

then if the graph of which clusters are associated with which is an expander graph, then the distributions will be expanding with respect to H. given a labeled example x, the learning algorithm will generalize to x's entire cluster C i , then this will be propagated over to nodes in the associated clusters C j in X 2 , and so on.


Finite matric spaces are associated with a complexity measure which is very different from the Minkowski dimension (i.e., the covering/packing complexity).

we exhibit a phase transition between dimension d = 0 and d ≥ 1 in the rate of growth of the regret induced by the metric.


We have evaluated the hierarchical multi-class learning strategies on several object classes.

we used: UIUC multi-scale cars [22], GRAZ [4] cows and persons, Weizmann multi-scale horses (adapted by Shotton et al.


We now consider a setting where the graph can be time-varying.

we assume that for each time interval t = 1, 2, . . . , T , we observed a snapshot of the graph L (t) ∈ L n×n .


The operator is in control of the network, thus can adaptively sample the matrix of pair-wise latencies.

the operator can request full columns of the matrix by measuring one host to all others.


Second, we observed that the volunteers were rapidly exhausted from reading multiple articles in sequence.

we find the proposed benchmark, WebNav, as a good benchmark for machine intelligence but not for comparing it against human intelligence.


It is natural to restrict to nonlinear mappings that act radially, preserving the spherical symmetry.

one can show that the generating function of p(x wht ) is completely determined by its radial marginal distribution: p r (r) = r d−1 β f (−r 2 /2), where r = x wht , Γ(•) is the standard Gamma function, and β is the normalizing constant that ensures that the density integrates to one.


Following [3], there is a natural way to define "features" in a Reeb graph and measure their "importance".

given a function f : X → IR, imagine we plot its Reeb graph R f (X) such that the height of each point z ∈ R f (X) is the function value of all those points in X mapped to z.


(1), the resulting W is often the same as the leading eigenvectors of Q up to an r × r rotation.

smoothing by random walk itself can bring little improvement unless we impose extra constraints or regularization.


Global Voting: We determine the globally top-2k attributes by a majority voting among all locally selected attributes in the previous step.

we rank the attributes according to the number of local machines who select them, and choose the top-2k attributes from the ranked list.


We adopt the same performance measures used in ILSVRC10.

for every image, each tested algorithm will produce a list of 5 object categories in the descending order of confidence.


This corresponds perfectly with the the appropriate measure of accuracy for evaluating the classifier in a CAD system.

even if one of the candidates that refers to the underlying malignant structure (radiologist mark) is correctly highlighted to the radiologist, the malignant structure is detected; i.e.


The following proposition extends the result for the one-dimensional total variation [4,23] to all submodular functions and their Lovász extensions: Proposition 5 (Proximal problem for ℓ 1 -penalized problems) The unique minimizer of 1 2 w − z 2 2 + f (w) + λ w 1 may be obtained by soft-thresholding the minimizers of 1 2 w − z 2 2 + f (w).

the proximal operator for f + λ • 1 is equal to the composition of the proximal operator for f and the one for λ • 1 .


As previously mentioned, α-covers can be seen as real valued generalizations of dominating sets.

an α-cover is a dominating set for binary weight graphs and α = 1.


• Graphs with Random Sparsity Structures: We use the procedure mentioned in Example 1 in [11] to generate inverse covariance matrices with random non-zero patterns.

we first generate a sparse matrix U with nonzero elements equal to ±1, set Σ −1 to be U T U and then add a diagonal term to ensure Σ −1 is positive definite.


The RTN model based on AlexNet (Figure 1) outperforms all comparison methods on most transfer tasks.

rTN substantially improves the accuracy on hard transfer tasks, e.g.


This oracle policy takes as input a sequence of predicted labels ŷ<t , the input x, and the free label multiset with respect to its predictions, Y t = Y\ {ŷ <t }.

it outputs a distribution whose entire probability (1) is evenly distributed over all the items in the free label multiset Y t . Definition 2 (Oracle).


This entails that the multi-scale Bernstein's condition holds whenever the Bernstein's condition does, thus al-lows us to consider a larger class of problems.

our results are also valid with the Bernstein's condition.


We represented each image as a bag-of-features vector in a manner similar to [25].

we identified 1024 representative SURF features points [26] from 10 × 10 gray-scale patches chosen randomly from the training images; this partitions the space of image patches (represented with SURF features) into Voronoi cells.


From ( 10) and ( 13), the θ-th power of the Korobov kernel k α is given by If αθ ∈ N, this is exactly the kernel k αθ of the Korobov space W αθ Kor ([0, 1]) of lower order αθ.

w αθ Kor ([0, 1]) is nothing but the θ-power of W α Kor ([0, 1]).


We leverage over recent advancements in the stochastic optimization literature to design a backprop-agation procedure that does not have a learning rate at all, yet it is as simple as the vanilla SGD.

we reduce the SGD problem to the game of betting on a coin (Section 4).


In fact, the Last Step Minimax (LSM) principle is a general recipe for online prediction, which states that the algorithm should minimize the worst-case regret with respect to the next outcome [16].

it should act as the minimax algorithm given that the time horizon is one iteration ahead.


(Because a success probability can be improved efficiently using standard repeat-and-test techniques [19], we follow the common practice of excluding this success probability from our analysis.)

we are interested in computationally efficient learning algorithms which have running time poly(n, 1/γ, 1/ε).


The hope is that zeroing the return contributions from Explore-rollouts will encourage the RNN to account for the impact of casting a wider sampling distribution on the final meta-reward.

during Explore-rollouts the policy will take actions which may not lead to immediate rewards but rather to the RNN hidden weights that perform better system identification.


Without loss of generality, the intercept is set to zero by centering the response and standardizing the predictors as To reduce the complexity of the model and leverage the structural information in X m , we assume that the coefficient tensor W to be both low-rank and sparse.

we assume W can be decomposed via CP decomposition as r , where each rank-1 tensor W r is possibly sparse, or equivalently, the vectors in its representation w (n) r , n = 1, . . . , N , are possibly sparse, for all r = 1, . . . , R. Here we impose sparsity on the rank-1 components from CP decomposition -rather than on W itself [Chen et al., 2012, Tan et al., 2012.


In fact, this observation naturally generalizes to trades between three or more agents and the same convergence analysis applies.

our analysis also holds when bilateral trade with a fixed market maker is replaced by multilateral trade among arbitrarily overlapping subsets of agents.


From Equation 6, if we just use Q ||| instead of Q, we get This equation shows that the unknowns in matrix Q ||| can be found by exploiting the fact that R i is a truncated rotation matrix (as was done in [1]).

if Λ2i−1:2i denotes the two rows of matrix Λ at positions 2i − 1 and 2i, then we have where I 2×2 is an identity matrix, giving three indepedent constraints for each image i.


Note that this illustration holds just in the ideal case where the mixing matrix A is given according to such a hierarchical model.


From the analysis of previous work one concludes that, in general, it is preferable, both in terms of accuracy and interpretability, to adopt a fully functional approach to the problem.

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


The result is generally biased, however [7] have shown that this algorithm offers a very advantageous bias-variance trade-off in case of manifolds with noise.

we use the output Ŷ of PRINCIPALCURVES to find a subset of points that (1) lie in a high density region relative to most directions in R D and (2) are "in the middle" of their neighbors, or more formally, have neighborhoods of dimension at least d. this is a good heuristic to avoid "border effects", or other regions where the d-manifold assumption is violated.


The noisy data are described by the random variable X + W = Y ∼ ν, and , where Γ is the gamma function.

ρ2 (µ, ν) ≤ σ √ D. The magnitude of the bound, and hence the worst-case sensitivity of the estimation procedure to noise, is controlled by the noise variance, by the extrinsic dimension, and by the distortion exponent.


Note that this is just a very simple one-dimensional example, but the outlined approach works in any dimension and for any convex objective function, even if we just have access to stochastic subgradients [Orabona and Pal, 2016].

if the gradients are bounded in a range, the same reduction works using a continuous coin.


Empirically, this illustrates that a neural network can be decomposed as a collection of diverse sub-networks.

the error correlation in the tic-tac-toe data set is around 0.25, which reconciles our observation that the full network performed better than the best sub-network.


We assume that E[R t |H t , θ, A t ] = f θ (A t ).

the realized reward is the mean-reward value corrupted by zero-mean noise.


In many natural machine learning settings, one is not only faced with data that may be corrupted or deficient in some way (classification noise or other label errors, missing attributes, and so on), but with data that is not uniformly corrupted.

we might be presented with data of variable quality -perhaps some small amount of entirely "clean" data, another amount of slightly corrupted data, yet more that is significantly corrupted, and so on.


The goal of this section is to evaluate the merits of using SDDs as a target representation in tractable learning for complex queries.

we want to address the following questions: Q1 Does capturing mutual exclusivity allow LearnSDD to learn more accurate models than ACMN?


This remark allows us to work with small matrices (of at most d × d instead of r × r) and to avoid the problem of estimating d, the intrinsic dimension of the data manifold.

we evaluate the tangent subspace around each sampled point x i using weighted (local) Principal Component Analysis (wPCA) and then express g R r | TpM directly in the resulting lowdimensional subspace as the unit matrix I d . The tangent subspace also serves to define a local coordinate chart, which is passed as input to Algorithm 1 which computes H(x i ), G(x i ) in these coordinates.


Of course, in practice, we never perform Gibbs sampling on a variational distribution but instead on the true posterior distribution which is proportional to e f (x) (where we assumed that − log q v (x) is not part of f (x)).

at each Gibbs step we simulate a new value for some x i from the posterior conditional distribution that is proportional to e f (x \i ,xi) and where x \i are the fixed values for the remaining random variables.


Our starting point is a model which regresses a response variable y ∈ R to observed covariates (predictor variables) x ∈ R D by a non-linear functional relationship.

we consider a multi-linear structure to account for the interactions among the covariates in the model: (1) Here, w i for i = 0, . . . , D are bias and linear weights as in the standard linear regression model, J is the number of multi-way interactions where Z j , β j for j = 1, . . . , J represent the interactions, i.e., sets of indices of interacting covariates and the corresponding interaction weights, respectively.


The shapes dataset was used to benchmark performance on rotation, scaling and translation analogies.

we generated 48 × 48 images scaled to [0, 1] with four shapes, eight colors, four scales, five row and column positions, and 24 rotation angles.


It shows that sparse coding lets bases faraway from the encoded data have nonzero coefficients, while local coordinate coding allows only nearby bases to get nonzero coefficients.

sparse coding on this data does not ensure a good locality and thus fails to facilitate the nonlinear function learning.


Since these messages function as any other network activation, gradients can be passed back along the channel, allowing end-to-end backpropagation across agents.

the network, which we call a C-Net, outputs two distinct types of values, as shown in Figure 1(b), a) Q(•), the Q-values for the environment actions, which are fed to the action selector, and b) m a t , the real-valued vector message to other agents, which bypasses the action selector and is instead processed by the discretise/regularise unit (DRU(m a t )).


Note that solving this problem from observed data on users as we described is made challenging both by the fact that unobserved variables causally affect both community membership and sharing, creating spurious correlations, and because social network membership introduces dependence among users.

for realistic social networks, every user's activity potentially depends on every other user's activity (even if indirectly).


However, the performance of these approaches are only empirically evaluated, and there is no analysis that proves performance guarantees.

eM algorithms require an initial starting point which is typically randomly guessed.


6, our model learns meaningful latent representations.

for the discrete factor of variation, JointVAE is able to better separate the classes than other models.


As seen before, to evaluate the quality of a ranking function f R (x|Θ) is necessary to specify the nature of a cost function c(x, y|Θ).

we consider cost definitions corresponding to associate penalties whenever uncorrect decisions are made (e.g.


As expected, performance at layer one is the same as the baseB points.

if we learn eight layers of features, save the first layer of learned Gabor features and color blobs, reinitialize the whole network, and retrain it toward the same task, it does just as well.


The setting in which the data distribution is supported on a low dimensional manifold embedded in a high dimensional space has only been considered more recently.

kernel density estimators on manifolds have been described in [36], and their pointwise consistency, as well as convergence rates, have been studied in [25,23,18].


The negative log-likelihood (NLL) of the data under this model is given by D d=1 NLL d , where Similar to [10], we infer the parameters (θ, σ) and latent variables X by minimizing NLL using stochastic gradient-based optimization.

we update the parameters and latent variables with a randomly selected batch of datasets B as where t denotes training iteration, η is the learning rate, N n denotes the total number of observations for pipeline n, and N Bn denotes the number of observations for pipeline n in the batch (we use the convention ∂NLL d ∂xn = 0 when the entry y n,d is unobserved).


4 The PSDD is a complete and canonical representation of probability distributions.

pSDDs can represent any distribution, and there is a unique PSDD for that distribution (under some conditions).


Several results exist showing that generalization takes place only when such correlation exists (nofreelunch; luckiness), and many classic estimators of performance (eg the margin) can be understood as estimating this relation.

selecting a kernel in this class of systems amounts to the classic feature and model selection problems in machine learning.


While this demonstrates that LP tightness is necessary for the convergence and correctness of BP, it is unfortunately not sufficient in general.

bP may not work even when the corresponding LP relaxation to the MAP inference problem is tight.


In particular, if µ is the Lebesgue measure (i.e., a uniform density over the unit interval), all percentiles are equally weighted and the µ-robust policy will optimize the expected cumulative reward over the distribution P(M).


Within a statistical learning setting, we propose and study an iterative regularization algorithm for least squares defined by an incremental gradient method.

we show that, if all other parameters are fixed a priori, the number of passes over the data (epochs) acts as a regularization parameter, and prove strong universal consistency, i.e.


The LN model has two important features.

first, the spike trains of the cell are given by a conditionally (inhomogeneous) Poisson process given x; that is, there are no dynamics in this model beyond those induced by x and K. Second, equation (1) implies: the conditional probability of firing is constant along (hyper)planes in the input space.


We then show that it makes L2L applicable to RSNNs (section 5), similarly as for LSTM networks.

we show that meta-RL [6,7] produces new motor control capabilities of RSNNs (section 6).


Given a risk parameter δ ∈ (0, 1], we aim for a δ-correct algorithm, that satisfies P µ (µ ∈ H m) ≥ 1 − δ for all µ ∈ H. Our goal is to build δ-correct algorithms that use a small number of samples τ δ in order to reach a decision.

we want the sample complexity E µ [τ ] to be small.


If the data are i.i.d., then the (B, κ)-Bernstein condition is satisfied with κ = 1 and B = 2λmax µ , where λ max is the maximum eigenvalue of E [xx ⊺ ] and µ = E[yx], provided that µ > 0.

if x t is uniformly distributed on the sphere and y t = sign(⟨ū, x t ⟩) is the noiseless classification of x t according to the hyper-plane with normal vector ū, then B ≤ c √ d for some absolute constant c > 0.


Our results suggest that symbolic mismatch-based comparative approaches may have clinical utility in identifying high risk patients, and can provide information that is complementary to existing clinical risk variables.

we note that the hazard ratios we report are typically considered clinically meaningful.


Note that as long as p(x) > 0 implies q(x) > 0, any choice of q will yield consistent risk estimates because weighting factors account for the discrepancy between sampling and test distribution (Equation 3).

∆n,q is guaranteed to converge to ∆ as n grows large; any approximation employed to compute q * will only affect the number of test examples required to reach a certain level of estimation accuracy.


Unfortunately, none of the aforementioned techniques are truly lifted.

they do not exploit first-order structure to the fullest extent.


The purpose of this paper is to analyze how the computational capabilities of randomly connected recurrent neural networks in the domain of real-time processing and the type of dynamics induced by the underlying distribution of synaptic weights are related to each other.

we will show that for the types of neural networks considered in this paper (defined in Sec.


Each slice is then whitened [5], and finally we perform sparse k-means on each slice separately, k denotes an integer which is a parameter of our algorithm.

letting X ∈ R n×p denote a slice of data (n: number of data points and p: dimensionality of the slice) and C n,k denote the set of cluster assignment matrices: Warm-starting k-means with warm started NMF.


Intuitively, such a limit on representational precision should strongly influence the form of the optimal code.

it should be possible to increase the number of limited capacity units in a population to form a more precise representation of the sensory signal.


Fortunately, we can exploit the following key property of our convex activity shaping framework: the instantaneous average intensity only depends on Ψ(t) through matrix-vector product operations.

we start by using Theorem 3 to rewrite the multiplication of Ψ(t) and a vector v as We then get a tractable solution by first computing e (A−ωI)t v efficiently, subtracting v from it, and solving a sparse linear system of equations, (A − ωI)x = e (A−ωI)t v − v , efficiently.


The sample version of the EM algorithm-the method actually used in practice-is obtained by simply replacing the expectations in equations (3) and equation (4a) by the sample-based counterpart.


This popularity is expressed by the total number of occurrences for the feature k which is given by m k = n−1 i=1 z ik .

we draw z nk from N B(z nk ; m k , n n+1 ) which has a mean value equal to m k n . Once we have sampled from all represented features we need to consider a sample from the set of unrepresented features.


The approximate problem (14) is argmin-continuously differentiable irrespective of the chosen permutation σ sorting y θ if and only if any of the following equivalent conditions hold.

we can equivalently say that the optimum has to lie on the interior of the face.


To validate the predictive power of the Replica MAP Claim for finite dimensions, we performed numerical simulations where the components of x are a zero-mean Bernoulli-Gaussian process.


As discussed in the introduction, LFR has fundamental differences from the proposed framework.

lFR only considers binary-valued D, and consequently, we restrict D to be binary in the experiments presented here.


We next generated random grid networks with frustrated interactions.

each edge was given either an attractive factor or repulsive factor, at random each with probability 1 2 .


Our approximate reconstruction is accomplished using a modification of the reconstruction attack techniques of prior work.

we employ tools from the fingerprinting codes literature [3,22,6] but we output quantitative scores, rather than a hard in/out decision about what is in the sample.


Finally in Section 4, we show how the submodular Bregman can be used in applications in machine learning.

we show how the proximal framework of the submodular Bregman generalizes a number of mirror-descent style approximate submodular optimization algorithms.


3 When they are used for evaluating the risk (e.g., in cross-validation), is by default the zero-one loss, namely 01 (t, y) = (1 − sign(ty))/2; when used for training, 01 is replaced with a surrogate loss [27].

[15] showed that if satisfies a symmetric condition: we will have which can be minimized by separating X p and X u with ordinary cost-sensitive learning.