First, if the focus of the analysis lies in the interaction between populations, then pCCA provides a more parsimonious description, as it focuses exclusively on the covariance between populations.

fA is unable to distinguish within-population covariance from between-population covariance.


To remove the correlation effect, it is possible to resort to feature decorrelation techniques.

existing techniques penalize high correlation explicitly; they either introduce a substantial computational overhead [10], or yield marginal improvements [11].


Observe from the figure that after 200 samples are revealed, the performance of OR-PCA is already satisfactory (E.V.> 0.8).

for GRASTA, it needs about 400 samples to achieve the same performance.


Previously, we considered the function of this neuromodulatory system on a time scale of minutes and longer, and suggested that it signals global uncertainty arising from gross changes in environmental contingencies.

norepinephrine is also known to be activated phasically by familiar stimuli in welllearned tasks.


Model selection of these methods is possible by cross-validation, which is a significant advantage over KMM.

logReg and KLIEP are computationally rather expensive since non-linear optimization problems have to be solved.


This can be expressed without creating new timestamps.

the "No Timestamps" model cannot fully represent a model of the tail's movement (which requires remembering what the head did when it was in the tail's position).


The focus of these works lies on deriving theoretical generalization bounds for supervised learning.

this work aims at developing practical procedures for model assessment, which are also applicable in semi-and unsupervised settings.


In these applications there is a natural place where one might divide data into episodes: for type 1 diabetes treatment, one might treat each day as an independent episode, and for digital marketing, one might treat each user interaction as an independent episode.

each day is not actually independent in diabetes treatment-a person's blood sugar in the morning depends on their blood sugar at the end of the previous day.


(2017) have previously argued that humans are bad at shaping reward.

our results show that demonstrations can provide consistent exploration guidance.


The only work in combinatorial bandits we are aware of that does not require the above assumption on the expected reward is [15], which is based on a general Thompson sampling framework.

they assume that the joint distribution of base arm outcomes is from a known parametric family within known likelihood function and only the parameters are unknown.


On the one hand, the raw micro-policy π raw aims for short-term optimality.

the macro-policy π macro can attend via φ to sequences of actions that lead to a macro-goal and bias the agent towards good long-term behavior.


The derivation of heavy-tailed process classification (HPC) is similar to that of standard multiclass GPC with Laplace approximation in Rasmussen and Williams [12].

due to the nonlinear transformations involved, some nice properties of their derivation are lost.


In [8] it is observed that EP is not efficient enough to be practical while VB suffers from a large bias.

collapsed Gibbs sampling also has its own problems: one needs to assess convergence of the Markov chain and to have some idea of mixing times to estimate the number of samples to collect, and to identify coherent topics across multiple samples.


In addition ICA was studied by another recently introduced MI estimator [16].

the derivative of the estimators that are based on order statistics can hardly be computed and therefore the optimization of such numerical criteria can not be based on gradient techniques.


proposed a robust LDA algorithm to deal with data uncertainty in classification applications by formulating the problem as a convex problem.

in many applications, it is not easy to obtain the information about data uncertainty.


Eigenvalue problems associated to a symmetric and positive semi-definite matrix are quite abundant in machine learning and statistics.

considering the eigenproblem from a variational point of view using Courant-Fischer-theory, the objective is a ratio of quadratic functions, which is quite restrictive from a modeling perspective.


When α ≥ 1.5, this upper bound is fairly sharp.

when α ≤ 1, the bound is not tight, especially for small α.


Additionally, we performed some experiments on the Netflix data set.

we cannot compare to any of the other methods on that data set as to the best of our knowledge, COFI RANK is the first collaborative ranking algorithm to be applied to this data set, supposedly because of its large size.


In the nonlinear case, the data are first mapped into a high-dimensional feature space where a linear decision boundary is computed.

to extend directly the previous analysis to the nonlinear case, one would need to compute the matrix C, in feature space, and the new kernel function due to the high dimension of the feature space, it is impossible to do it directly.


This baseline system requires significant 3D a priori knowledge, including   the intrinsic and extrinsic parameters of the camera, and a rough initial segmentation of each test image.

our approach has to infer these aspects of the scene from the image data.


This means that the attention map has not regularized the classifier well as it has not fully identified the target regions.

along the tracking process, the reciprocative learning scheme helps the attention map cover the whole target region and thus strengthens the instance awareness.


[3] also aims at computing the number of certain clusters in the solution space.

sP counts only the number of clusters with a "typical size", and would show non-zero values in Figure 2 only for average vertex degrees between 4.42 and 4.7.


Our formulation is also reminiscent of a Latent-SSVM with the indicator flags {ρ i,m | m ∈ [M ]} taking a role similar to latent variables.


Neural responses are typically studied by averaging noisy spiking activity across multiple experimental trials to obtain firing rates that vary smoothly over time.

particularly in cognitive tasks (such as motor planning or decision making) where the neural responses are more a reflection of internal processing rather than external stimulus drive, the timecourse of the neural responses may differ on nominally identical trials.


Most standard approaches to preventing catastrophic forgetting discussed in section 2 are either not applicable to a variational context, or do not scale well due to memory requirements.

thanks to learning a generative model of the observed environments, we can prevent catastrophic forgetting by periodically hallucinating (i.e.


Discussion We note reweighted strategy [26] was also used in solving problems like zero-norm of the parameters of linear models.

it cannot be directly used to solve "exclusive group LASSO" problem proposed in this paper, and cannot handle arbitrary structures on feature space.


This is not surprising, as for the SME we have to halve the sample size in order to construct a valid test.

recall that the PhD test has an inflated Type I error for large noises, which means that its results should be considered with caution in practice.


For each of these trends the flP model outperforms the causal power model, with overall R 2 values of 0.95 and 0.35 respectively.

it is important to note that the responses of the human subjects in columns 8-10 (contingencies {l.00 , 0.50} , {0 .80 , 0.40} , {0 .40, O.OO}) are not quite consistent with the predictions of flP : they show a slight V-shaped non-linearity, with P( C -+E) judged to be smaller for 0.80 , 0.40 than for either of the extreme cases.


Although they result in efficient description and indexing in many cases they are limited to the matching accuracy of the original input space.

our approach learns a nonlinear feature mapping that is specifically optimized to result in highly accurate descriptor matching.


From (7) we see that λ n appears in the conditional posterior p(f |y, K, λ, γ) as first order terms, thus we can write where n and γ (i) are the estimates of f n and γ at the i-th iteration, and u n . From (7) and (8) we can obtain the EM updates: In the ECM setting, learning the parameters of the covariance function is not as straightforward as in MCMC.

we can borrow from the GP literature [7] and use the fact that we can marginalize f while conditioning on λ and γ: Note that K is a function of X and θ. Estimation of θ is done by maximizing log Z(y, X, λ, γ, θ).


Despite a different origin, the resulting algorithm, Expectation BackPropagation (EBP), is very similar to BP in form and efficiency.

it has several additional advantages: (1) Training is parameter-free, given initial conditions (prior) and the MNN architecture.


We find our online Laplace approximation to maintain higher test accuracy throughout training than placing a quadratic penalty around the MAP parameters of every task, in particular when using a simple diagonal approximation to the Hessian.

the main difference between the methods lies in using a Kronecker factored approximation of the curvature over a diagonal one.


Technically, BMC leverages the regularized Bayesian inference (RegBayes) principle [15], which has shown promise on supervised learning tasks, such as classification [16,17], link prediction [18], and matrix factorisation [19], where max-margin constraints are introduced to improve the discriminative power of a Bayesian model.

little exploration has been devoted to the unsupervised setting, due in part to the absence of true labels that makes it technically challenging to enforce max-margin constraints.


( 6) does include a sum over distinct pairs of units, as well as a sum over coupled triplets of units, such triplets are excluded by the bipartite structure of the RBM.

coupled quadruplets do contribute to the fourth-order term and therefore fourth-and higher-order approximations require much more expensive computations [21], though it is possible to utilize adaptive procedures [19].


Generally, Algorithm 2 does not ensure its convergence.

the following theorem shows that running Algorithm 2 with sufficiently large batch sizes will not increase the objective values too Algorithm 3: much.


By reinterpreting this function as a (artificial) marginal likelihood, it is easy to see that it can also be maximized using an EM-type algorithm in the spirit of [5].

the observation that we are dealing with a trans-dimensional distribution enables us to go beyond EM.


To do this, we fit a Gaussian copula process by using a type of Warped Gaussian Process [15].

our methodology varies substantially from Snelson et al.


LDA models are arguably among the most successful recent learning algorithms for analyzing count data such as text.

they can take days to learn for large corpora, and thus, distributed learning would be particularly useful for this type of model.


Note that a constant ratio approximation algorithm implies a linear regret in our model.

these works consider a more involved problem from a buyer's valuation perspective.


So one way to test for linear dependence would be to use the Gaussian copula family.

the likelihood of copula-based models relies on the CDF which has no closed form solution for the Gaussian family.


For strong attractive potentials on the 3 x 3 grid, the TRP /BP approximation in panel (a) is very poor, as reflected by relatively loose bounds on the actual marginals.

the Kikuchi approximation in (b) is excellent, as revealed by the tightness of the bounds.


We can therefore expect to obtain "richer" metrics γ K than when being restricted to RKHSs (see Examples 1-3).

one disadvantage of the RKBS framework is that γ K (P, Q) cannot be computed in a closed form unlike γ k (see Section 4.3).


The time-honored solution is to use wavelets.

in the absence of a natural dilation operator, defining wavelets on a discrete space is not trivial.


8 Interestingly, this sub-function is convex, as well as the constraint (which is now linear), resulting in a unique optimum, as in Bayesian networks.

even when θ is a feasible point, the unique optima of these sub-functions may not be feasible when combined.


Due to the open nature of recommendation systems and their reliance on user-specified judgments for building profiles, various forms of attacks are possible and have been discussed, such as the random attack and random product push/nuke attack [4,5].

these attacks are not formally analyzed and cannot be optimized according to specific collaborative filtering algorithms.


MOHEX is a highly optimised algorithm, utilising a complex, hand-made theorem-proving algorithm which calculates provably suboptimal moves, to be pruned from search, and an improved rollout policy; it also optionally uses a specialised end-game solver, particularly powerful for small board sizes.

our algorithm learns tabula rasa, without game-specific knowledge beside the rules of the game.


However, they considered a bandit setting, where at each iteration any arm can be played.

we consider exploring an MDP, which introduces restrictions in terms of reachability that have not been considered in Bayesian optimization before.


Noting that the only non-vanishing steps are PQ, QR and QP, we obtain We note that the number a 0 accounts for the agents who contribute to both steps in the attractor, and thus can complete the attractor dynamics alone in the extremely diverse limit.

the number a cum arises from the £rst step PQ arriving at Q.


We train the student network using 2 loss: We will define a potential function g, and show that if g is small, the gradient points to partially correct direction and we get closer to W * after every SGD step.

g could be large and thus gradient might point to the reverse direction.


Convergence Rate Previous works have shown the convergence of distributed SGD with unbiased stochastic quantization functions [5,46].

no convergence rate is given in [46], and the convergence analysis on non-convex optimization in [5] shows that the average `2 norm of the gradients has a constant variance blowup term.


As discussed, a key difficulty of our setting that prevents us from directly applying well studied algorithms for OLO, is that essentially all standard algorithms require to exactly solve (or up to arbitrarily small error) some linear/convex optimization problem over the convexification of the feasible set CH(K).

not only that our approximation oracle O K (•) cannot perform exact minimization, even for ↵ = 1 it is applicable only with inputs in R d + , and hence cannot optimize in all directions.


This makes it hard to decrease (or increase in FNS) the threshold φ for the MS-distance enough to filter off many data.

on such high dimensions, both the linear search and the cover tree algorithm also show poor performance.


In this case, the transition would be of the form In the standard GP regression setting, the inputs and targets are fully observed and f can be learned using exact Bayesian inference [13].

this is not the case in GP-Vol, where the unknown {v t } form part of the inputs and all the targets.


Once Z is computed, Algorithm 1 can be run for the remaining time steps.

it needs to be modified slightly to take into account the budget consumed during the first T 0 rounds.


We also show how this convex optimization can be modified for the design of smoothing function specifically for the sequential algorithm.


Extensions exist to perform approximate inference with other likelihood functions [2,3,4,5].

the assumptions of the Gaussian process are not the only set of reasonable assumptions, and are not always appropriate.


I.e., s(0, 0) = s(1, 1) = 1, s(n, 0) = 0 for n > 0 and s(n, m) = 0 for m > n. Other entries can be computed as s(n + 1, m) = s(n, m − 1) + ns(n, m).

for large l (k) u. , it is often more efficient to sample ρ k,i by simulating the table assignments of the Chinese restaurant according to Eq.


Thus, the pi x are sufficient for predicting the most likely class for each example.

as shown by Hastie and Tibshirani, they are not accurate probability estimates because they tend to underestimate the differences between the pi x values.


A reader familiar with the multi-armed bandit literature would expect an adaptive sampling algorithm to have a large advantage over uniform sampling when there is a large diversity in the means of H 1 since larger means can be distinguished from µ 0 with fewer samples.

one should note that to declare all of H 1 as discoveries, one must sample every arm in H 0 at least as many times as the most sampled arm in H 1 , otherwise they are statistically indistinguishable.


This approximation should work well if high scoring vectors lie close to a low-dimensional affine space; the approximation can be improved by taking more principal components.

the approximation will work poorly if the cell vectors have a "blobby" distribution, which appears to be the case here.


The improvement with respect to the baseline is lower (≈ 2% gain at a precision of 90%) and we find similar developments of the curves.

this comes somewhat expectedly as the training data does not properly reflect the occlusions we actually want to model.


A standard measure of an optimization algorithm is how fast the algorithm converges to an optimal solution.

finding the global optimal solution to a generally non-convex problem is intractable [13] and is even a NP-hard problem [10].


In Line 6, G denotes the space of functions over which risk minimization is performed in the classifier.

we will show (in Theorem 1) that the variational distance between the distribution of one of the samples in U 0 2 and f CI (x, y, z) is very small for large n. the samples in U 0 2 are not exactly i.i.d but close to i.i.d.


Letting m r (z gives the third and fourth k-statistics [15].

we are interested in estimating the gradients (for Algorithm 2) and Hessians (for Algorithm 1) of the cumulants rather than the cumulants themselves.


We notice that the perplexity strictly decreases when we increase the decoder size.

for practical reasons, we do not increase the decoder size beyond 4096.


Since the correlation matrix M ij is symmetric, it has up to n(n + 1)/2 independent elements.

the unknown mixing matrix C has potentially n 2 elements (for n sources and n detectors).


In most of our experiments the size parameter is a twodimensional vector including the covariance of an isotropic Gaussian and a scaling factor.

in general s i can represent an arbitrary parametric distribution: for instance, in the supplement we We show 2D-to-1D projection for illustration purposes, but in practice we perform 3D-to-2D projection.


In this paper, we propose the use of an established formalism from mathematical psychology, additive clustering, as a means of automatically constructing binary representations for objects using only pairwise similarity data.

existing methods for the unsupervised learning of additive clustering models do not scale well to large problems.


This changes the saliency from a bottom-up identification of locations where center and surround differ, to a topdown identification of locations containing the target in the center and background in the surround.

the procedure of [22] (based on feature ranking) is not biologically plausible.


Both PLS and RMH are effective dimensionality reduction methods with comparable performance.

the components selected in PLS are, in general, more difficult to interpret because they involve whole trajectories.


responses of additional neurons) will very probably grow IB(K).

the original concepts will be embedded as a substructure in the larger lattice, with their ordering relationships preserved.


It follows that the SD is invariant to coordinate permutations and depends only on the sparsity level, i.e., the number of nonzero elements.

this result is in general not true for a nonorthogonal sparse basis, indicating that the precise location of the spikes (sparsity pattern) and not just their number has an effect on the SD.


Combining Professor Forcing with the PixelRNN would be an interesting area for future research.

the PixelRNN parallelizes computation in the teacher forcing network in a way that doesn't work in the sampling network.


Estimation in continuous parameter spaces in some cases allows for closed-form solutions (such as in least-squares regression), or if not we can resort to computing gradients (for smooth continuous functions) or subgradients (for non-smooth continuous functions) which give us a generic tool for finding a local optimum of an arbitrary continuous function (global optimum if the continuous function is convex).

unless P=NP there is no general tool to efficiently optimize discrete functions.


For the first part, cutting-plane methods solving multiple intermediate LPs for the minimum weight matching problem have been discussed by several authors over the past decades [17,18,19,20] and a provably polynomial-time scheme was recently suggested [21].

lPs in [21] were quite complex to solve by BP.


Taking traffic signs as an example, an adversary performing perturbation-based attacks can draw graffiti [4] or place stickers [18] on an existing stop sign in order to exploit a classifier.

the attacker might want to go beyond this and replace the original stop sign with a new one that was specifically manufactured to be adversarial.


A number of inverse-graphics inspired methods have recently been proposed in the literature [15].

most such methods rely on hand-crafted rendering engines.


As can be seen in Figure 1 and Figure 2, parallelization of the BCD algorithm results in more efficient algorithm.

the computational gain does not grow linearly with the number of processors.


A natural modeling framework for capturing such effects is structured prediction, which optimizes over complex labels, while modeling within-label interactions.

it is unclear what principles should guide the design of a structured prediction model that utilizes the power of deep learning components.


Much of current research on computation of solution concepts has focused on complete-information games, in which the game being played is common knowledge among the players.

in many multi-agent situations, players are uncertain about the game being played.


We say that f is weakly Lipschitz with respect to if for all x, y ∈ X , Note that weak Lipschitzness is satisfied whenever f is 1-Lipschitz, i.e., for all x, y ∈ X , one has |f (x) − f (y)| ≤ (x, y).

weak Lipschitzness implies local (one-sided) 1-Lipschitzness at any maxima.


As we introduced in Section 1, some DL methods [4,5,7] learn a shared dictionary for all classes and a classifier on the coding coefficients simultaneously, while some DL methods [8,9,10] learn a structured dictionary to promote discrimination between classes.

they all employ 0 or 1 -norm sparsity regularizer on the coding coefficients, making the training stage and the consequent testing stage inefficient.


In this case the output y w (x) is not a differentiable function of w. As one smoothly changes w the output y w (x) jumps discontinuously between discrete output values.

so one cannot write when the input space X is continuous the gradient ∇ w E [L(y, y w (x))] can exist even when the output space Y is discrete.


In the case of Bayesian optimization, the objective is to locate the global maximum of an expensive-to-evaluate function.

we are interested in the entire set of points that define a contour of the function.


The authors employed model averaging to explain the results [3] and the data were found to have a very high likelihood under their model.

they do not analyse variability in the subjects' responses and this aspect of behavior as a function of spatial disparity is not readily obvious in their published data.


For a matrix with IJ rows in total, e.g., B A, in general, the calculation of statistical leverage score is prohibitively expensive.

due to the special structure of the KRP B A, the upper bound of statistical leverage score, which is sufficient to obtain the same guarantee by using slightly more samples, can be efficiently estimated, as shown in Theorem 3.2.


Text inputs are represented by the mean of all hidden states.

in this text representation matrix for both approaches, the embeddings are completely independent, without leveraging the semantic relatedness indicated from the graph.


(This choice was best in [4] and was also made in [3,12].)

we did not investigate any careful tuning of scaling factors.


The single-class alignment shown in this figure coerces the feature in region A for class 1 to be inappropriately collapsed in time, and the overall width of the main broad peak in class 2 to be inappropriately narrowed.

our multi-class model handles these features correctly.


In the previous example, the algorithm had access to both the complete path demonstrated by the teacher, and the same input data (overhead image) the teacher used while generating the path.

in this example not only is the input data different (since the teacher generally controls the robot from behind and to the side using their own prior knowledge of the environment and highly capable vision system), but we face the additional challenge of estimating the path planned by the teacher at a particular time step from the vehicle motion we observe in future time steps, when the teacher is using additional data.


Intuitively, X ρ can be thought of as the region where the distribution is concentrated, that is ρ(X ρ ) = 1.

there are many different sets having this property.


The optimism principle built on concentration inequalities automatically strikes a balance between exploitation and exploration, eliminating the need to manually adjust the learning rate or the exploration ratio.

when importing analysis from MDPs to SGs, we face the challenge caused by the opponent's uncontrollability and non-stationarity.


A novel study of k-NN by Samworth, [11], derives a closed form expression for the optimal weight vector, and extracts the optimal number of neighbors.

this result is only optimal under several restrictive assumptions, and only holds for the asymptotic regime where n !


If there is a potential for X in a possibly rare regime in X to be fully correlated with Y such that Y = f (X), then the hypercontractivity coefficient is maximum: s(X; Y ) = 1.

just as HGR correlation is not the only one satisfying Rényi's original axioms, the hypercontractivity coefficient is not the only one satisfying our axioms.


Explanations that do not depend on model parameters or training data might still depend on the model architecture and thus provide some useful information about the prior incorporated in the model architecture.

in this case, the explanation method should only be used for tasks where we believe that knowledge of the model architecture on its own is sufficient for giving useful explanations.


(2009) characterized the distributions that can be realized by a RBM with k parameters as residing within a manifold inside the entire space of distributions on {0, 1} n whose dimension depends on k. For sub-exponential k this implies the existence of distributions which cannot be represented.

this kind of result gives us no indication of what these hard-torepresent distributions might look like, leaving the possibility that they might all be structureless or otherwise uninteresting.


The theorem of the previous section gave a bound on the expected regret.

a much stronger claim can be made that essentially the same regret bound holds with very high probability (exponential tail).


In fact, since a continuous time white-noise process contains power at all frequencies and infinite power in total, most of the white-noise content will be undeterminable, as it is suppressed by the filter (or filtered out).

for the same reason, these components do not affect prediction of f (t).


For node-compatible graphs, there is no baseline, because it is a new task.

we will compare the results of using regularization versus not and show the effectiveness of the proposed method.


The three large elements of the msc vector for different values of λ correctly reflect the fact that the sphere has dimension two.

note that for very large values of λ the performance of the embedding degrades since we put more emphasis on the sparsity of the solution.


To make the prediction more accurate, incorporating the player-interaction effects are demanding.

people are also interested in the cooperation effects between players.


Earlier papers [9,11] assume that the ideal similarities are constant within a block in which case the assumption is trivially satisfied by the definition of the ideal matrix.

more generally this assumption is necessary to show that the entries of the eigenvector are safely bounded away from zero.


This can yield a posterior mean reward function with an optimal policy again inconsistent with the data.

the MAP does not involve an objective function that is integrated over the reward function space; it is simply a point that maximizes the posterior probability.


We use the same local decoding procedure that is standard in dual decomposition [31].

evaluating the objective Q(x B ) involves a potentially difficult sum over x A , making it hard to score each decoding.


The cube in the middle can be viewed in multiple ways.

the views on the left and right require a very specific viewing angle.


Therefore, (a) may become a dominant factor to be minimized during the optimization process and as a trade-off, (b) will be easily compromised, which makes it difficult to learn a good association between data modalities.

the MinVI objective focuses on modelling the conditional distributions (Equation ( 4)), which is arguably easier to optimize.


The problem remains NP-hard, even if the optimal policies have a compact piecewiselinear representation using α-vectors.

we can also show that given a suitable set of points that "cover" R * (b 0 ) well, a good approximate solution can be computed in polynomial time.


The standard CCD method follows the standard cyclic order (1, 2, . . . , n) as described in Section 2.

we can construct a CCD method that follows an alternative deterministic order by considering a permutation ⇡ of {1, 2, . . . , n}, and choosing the coordinates according to the order (⇡(1), ⇡(2), .


If the primal update is solved exactly which makes sufficient progress, the dual update should take small step, e.g., sADMM.

if the primal update takes small progress by adding proximal terms, the dual update can take full gradient step, e.g.


Knowing the observations and policies of other agents is not a particularly restrictive assumption; if our goal is to train agents to exhibit complex communicative behaviour in simulation, this information is often available to all agents.

we can relax this assumption if necessary by learning the policies of other agents from observations -we describe a method of doing this in Section 4.2.


A second advantage is that (6) requires samples from a smaller (and perhaps better-behaved) probability space.

the Rao-Blackwellized estimate ( 6) is more expensive to compute than (5) because we must compute conditional expectations.


Ideally, we prefer to derive new algorithms that find the posterior distribution ρ that minimizes the PAC-Bayesian bound on the expected loss.

we can also use PAC-Bayesian bounds in order to estimate the expected loss of posterior distributions ρ that were found by other algorithms, such as empirical risk minimization, regularized empirical risk minimization, Bayesian posteriors, and so forth.


Our optimal teaching set (0, 2, 8) has Teaching Impedance T I = 2.65.

the set (1, 3, 6) has T I = 4.51 and the previous set (317, 965, 1933) has T I = 956.25 due to its size.


In the experiments above, only two types of data are used: labeled training examples and unlabeled test examples.

an additional source is the noisy labels from the Tiny Image dataset (the keyword used to query the image search engine).


The associated RIP constant δ < 1 is not too much worse than than the figure quoted in Theorem 6.

spurious local minima must cease to exist once δ < 1/5 according to Theorem 2.


It is shown in the above subsection that by enforcing a carefully designed optimality error and greedily train the networks, we can obtain a theoretically convergent solver for nonconvex optimization.

it is indeed challenging to utilize strict mathematical models to exactly formulate the complex data distributions in real-world applications.


From (6) this leads to a parameter posterior infinitely peaked at zero for many weights Wm with the consequence that J.tMP correspondingly comprises very few non-zero elements.

the learning procedure in [6] relied upon heuristic re-estimation equations for the hyperparameters, the behaviour of which was not well characterised.


The algorithms in [2,11] run in poly(n/ ) time, as they involve solving a linear program of poly(n/ ) size.

the running time of our algorithm is sublinear in n. All the testing algorithms in this paper follow from a simple and general technique that may be of broader interest.


Such algorithms are well adapted to stochastic and adversarial domains, due to their sampling nature and the convergence guarantee to min-max values.

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


Then for > 0 with probability at least 1 the buyer loses a surplus of at least Bounding the number of lies: Although we argued in the previous lemma that lies during the learning phase cause the surplus-maximizing buyer to lose surplus relative to the truthful buyer, those lies may result in lower prices offered during the exploit phase, and thus the overall effect of lying may be beneficial to the buyer.

we show that there is a limit on how large that benefit can be, and thus we have the following high-probability bound on the number of learning phase lies.


Explicit correspondences between features have an advantage over a BoW as information loss in the vector quantization can be avoided.

the drawback with this approach is that the identification of corresponding points with minimum distortion is also computationally expensive.


marginalization is not tractable for all models, such as the Ising model, sigmoid belief networks, and (untruncated) Bayesian nonparametric models.


Our first concern is usually with the accuracy of the classifier.

in some applications, the size of the support is equally important.


DDP provides a feedback controller which is important in coping with unknown distur-bances and modeling errors.

as shown in equation ( 2), the feedback controller is indexed by time, and development of a time independent feedback controller is a future goal.


1(a), every neuron in this model also receives a full set of outputs from the earlier network layer or input data as in a standard FNN.

in this case, the connection maps at the dendrite branch level are sparse.


This inspires the work of kernel online learning [5,6] that uses a linear model in the feature space to capture the nonlinearity of input data.

the kernel online learning approach suffers from the so-called curse of kernelization [7], that is, the model size linearly grows with the data size accumulated over time.


It was shown that these problems are alleviated in DiscoGAN [24], CycleGAN [26] and ALICE [32] via additional 1 , 2 or adversarial losses.

these methods lack of explicit probabilistic modeling of observations, thus could not directly evaluate the likelihood of given data samples.


We shall consider an array of K × K Hawkes processes, where K is the number of clusters in a partition drawn from a CRP restricted to the individuals V . As in the IRM, the CRP allows the number of processes to grow in an unconstrained manner as the number of individuals in the graph grows.

unlike the IRM, these Hawkes processes will be pairwise-dependent: the Hawkes process governing events from cluster p to cluster q, will depend upon the Hawkes process governing events from cluster q to cluster p. Let N pq be the counting measure of the (p, q)th Hawkes process.


A choice of the learning rate 0: < k according to Theorem 1 yields a convergent iteration.

this might not be the best choice with respect to asymptotic convergence rate.


Finally, Hyvärinen proposed Score Matching Networks [11], which are based on the somewhat surprising observation that one can model unknown derivatives of the function without actual access to its values -all that is needed is a sampling based strategy and specific penalty.

such an estimator has a high variance [28], thus it is not really useful when true derivatives are given.


The analysis of such data is far simpler than our setting as electrophysiological measurements of the sub-threshold membrane potential of a post-synaptic neuron can enable highly accurate detection of strong and weak incoming connections.

we are concerned with the more challenging setting of noisy calcium imaging measurements of suprathreshold post-synaptic spiking activity.


Strictly speaking, it is not a valid comparison because the experimental setups of VOC06 competition and ours are totally different.

we illustrate them as references to show how closely our approach can reach the best supervised methods in VOC 06 for the localization.


The classification task is easy if one knows that the concept is graspability.

the challenge of inferring the concept is extremely difficult because there are many dimensions along which these objects vary and any one-or more-could be the classification dimension(s).


Of the four methods compared here, only ABCD is able to model heteroscedasticity, explaining why it is the only method with a substantially different set of significant p-values.

the procedure is still potentially failing to capture structure on four of the datasets.


Then all the threads are synchronized to perform the actual update to the variables.

the method can potentially diverge; indeed, this is mentioned in [23] about the potential divergence when the number of threads is large.


However, their approach does not tackle the problem of 3D scene understanding and 3D object detection.

we propose a generative model which jointly reasons about these two tasks.


Such an oracle could be realized through convex optimization methods and non-parametric representations; see Appendix D.2.

due to practical computational barriers, we opt for parametric realizations of the oracle, i.e neural networks.


In this term, the dependency in S is slightly worse (Kwidth(S) instead of K).

this extra regret is limited to arms belonging to an "-approximation of the Pareto front, i.e.


[5] and Hein and Audriushchenko [14] theoretically show that the sensitivity to adversarial examples can be controlled by the Lipschitz constant of DNNs and propose Parseval training and cross-Lipschitz regularization, respectively.

these methods usually require approximations, making them less effective to defend very strong and advanced adversarial attacks.


In Section 4.2, we will show that optimal decision-theoretic predictions (i.e., that maximize expected gain) can be made in several settings by a simple sorting procedure, and this will be our primary way of using the learned model.

here, we consider the task of producing a distribution over labels y, given θ(x).


The minimization of the upper bound (3) amounts to the equivalent problem 3 min where c k ∈ R n represents the unknown true gradient.

that is, with respect to the bounds k , u k , we can write 3 we derived the optimal solution for a fixed c k ∈ C k . this is not sufficient to find the optimal solution for an arbitrary c k ∈ C k . Just computing the optimal solution for an arbitrary (but fixed) c k ∈ C k is unlikely to yield a good solution.


Since our goal is to show the performance of our algorithm, we assume we can "submit" our solution x for unlimited times, which is unreal in a real contest like KDD-Cup.

even with very few iterations, our algorithm does converge fast to a reasonable small RMSE, as shown in Figure 3.


All of these previous nonparametric relational models employed MCMC learning algorithms.

the conditionally conjugate structure of our HDPR model allows us to easily develop a stochastic variational inference algorithm [17,2,9].


A key assumption made by matrix factorization-and completion-based collaborative filtering algorithms is that the underlying rating matrix is of low-rank since only a few factors typically contribute to an individual's form utility [5].

a user's demand is not only driven by form utility, but is the combined effect of both form utility and time utility.


[19] extended their work by feeding in the per-pixel predicted labels using a CNN classifier to the next stage of the same CNN classifier.

their propagation structure is not adaptive to the image content and only propagating label information did not improve much over the prior work.


Principal component analysis (PCA) is widely used for data pre-processing, data compression and dimensionality reduction.

pCA suffers from the fact that each principal component is a linear combination of all the original variables.


The genes known to be frequently mutated across GBM subtypes are somewhat enriched out of those that appear to be perturbed according to the PNJGL estimates (3 out of 10 mutated genes were detected by PNJGL; 37 out of 258 total genes were detected by PNJGL; hypergeometric p-value = 0.1594).

fGL detects every gene as having node perturbation (Figure 4(a)).


Another well-studied thread in this space involves leveraging knowledge graphs to improve natural language question answering (QA) [4,5,47].

the focus of these QA approaches is understanding natural language, whereas we focus on queries that are in logical form.


Note that this result not only holds for GPs, but for any predictor whose output is a linear combination of the training outputs.

most of the physical constraints that rule non-rigid motion are more complex than simple linear functions.


A natural choice of greedy heuristic might be a bound on the total error rate.

in many cases the final objective function is not an appropriate choice for the greedy heuristic in greedy algorithms.


In order for those methods to perform as well as they did with the smaller version of Frogger, the value for B[s, a] needs to be set higher and decayed more slowly by manually finding new values for b and d. Thus, like r h , the optimal values to b and d are dependent on both the domain and the quality of feedback.

the estimated feedback consistency, Ĉ, used by Advise only depends on the true feedback consistency, C. For comparison, we next show how sensitive Advise is to a suboptimal estimate of C. Interactions with a real human will mean that in most cases Advise will not have an exact estimate, Ĉ, of the true feedback consistency, C. It is presumably possible to identify a value for Ĉ that is close to the true value.


Conventionally, the binary classification approach has been employed, in which whether or not text belongs to a category is judged by the binary classifier for every category.

our approach can simultaneously detect multiple categories of text using PMMs.


We expect batch-mode active learning methods to use more label requests than their fully-sequential cousins.

if the cost to obtain a batch of labels is sublinear in the size of the batch, then we may sometimes expect the total cost used by a batch-mode learning method to be significantly less than the analogous fully-sequential algorithms, which request labels individually.


For example, the humpback whale drops the kernels belonging to the whole Behavior taxonomy block, and gives the strongest weight to "hairless", and "habitat".

by failing to select some of the upper-level nodes, it focuses only on the most confusing fine-grained problems.


This implies that determining when the algorithm should stop and which new component to add is hard.

a significant amount of research has been carried out on sparse PCA recently, and we thus propose to leverage some of the recently proposed relaxations and heuristics to solve this rank 1 sparse PCA problem (see [8,20] and references therein).


Thus the dataset is fixed and we typically resort to techniques such as model selection where each model corresponds to one histogram (see Arlot [2007] for an extensive study of such).

we here ask a very different question, that is how to optimally sample in an online setting in order to minimize the approximation error of some histogram.


One can achieve linear convergence under (7) or (8).

the weakly PL condition [6] may be considered as a generalization of (weak) convexity.


Moreover, with the appropriately chosen dimension, a 32 dimensional µ 2 reaches an even lower EER at 1.34%.

the negative results of using µ 1 also validate the success in disentangling utterance and segment level attributes.


Many modern machine learning techniques require large amounts of training data to reach their full potential.

annotated data is hard and expensive to obtain, notably in specialized domains where only experts whose time is scarce and precious can provide reliable labels.


Therefore, we expected the improvement from learning the tree to be marginal.

the improvement over the baseline was significant, showing that transferring information between related classes helped.


. . , y n . Conditioned on all other variables being false, each variable is equally likely to be true or false.

the x variables and the y variables contradict each other.


How to measure them properly is unclear in the literature (Section 3.2).

numerous experiments have suggested that there exists a tradeoff between the two when dimensionality reduction happens [34], yet this tradeoff still remains a conceptual mystery in theory.


Traditionally, in the non-private literature, the goal is to minimize the multiplicative error γ, with the current state-of-the-art (non-private) algorithm achieving multiplicative error of γ = 6.357 (with no additive error) [2].

in spite of the long line of works on private k-means [8,28,14,18,27,33,31,32,17,6,29,21], all of the existing polynomial time private algorithms for the problem obtained only a super constant multiplicative error.


Since there are exponentially many indicator terms, they resort to constructing approximations to the influence function, for which a strong technical condition needs to be satisfied; this condition need not however hold in most settings.


There are now several large scale deployments of differential privacy used to collect statistical information about users.

these deployments periodically recollect the data and recompute the statistics using algorithms designed for a single use.


Our analysis assumes the least squares loss.

it is possible to handle more general loss functions with a more complicated derivation.


In this paper, we study the rate of ∆ as a function of : We identify a function T : R + → R + such that when ∆ = O(T ( )), we can give an algorithm that outputs a point x such that f (x) ≤ min x f (x) + within time poly d, 1 over a natural class of well-conditioned convex bodies.

when ∆ = Ω(T ( )) 1 , we also prove an information theoretic lower bound that any algorithm outputs such x must use super polynomial number of evaluations of f . Our result can be summarized as the following two theorems: Theorem (Algorithmic upper bound, informal).


Using a KL-bound for the update of the search distribution is a common approach in the stochastic search.

such information theoretic bounds could so far only be approximately applied either by using Taylor-expansions of the KL-divergence resulting in natural evolutionary strategies (NES) [2,11], or sample-based approximations, resulting in the relative entropy policy search (REPS) [9] algorithm.


Though DiscoGAN is an unsupervised learning method, it achieves impressive results.

with supervision provided by 10% paired data, ∆-GAN generally generates more accurate edge details of the shoes.


To see this, consider the case where the starting kernel K is a poor fit to the data.

in this case, a large initial step size η will probably be accepted; even though such a step will likely result in the truncation of many eigenvalues at 0, the resulting matrix will still be an improvement over the poor initial K. with many zero eigenvalues, the new K will be near-diagonal, and, unfortunately, Equation (5) dictates that if the current K is diagonal, then its gradient is as well.


This results in a looser bound in (15) for large α. as a function of the conditional entropy H(G|X n ) and the size of the set of all graphs with p nodes.

a naïve application of Fano's inequality results in a trivial lower bound as the set of all graphs, which can be realized by G ER (p, c/p) is "too large".


As a result, the lack of 3D structure inevitably limits some practical applications of these generative models.

we present an image synthesis method powered by a disentangled 3D representation.It allows a user to change the viewpoint easily, as well as to edit the object's shape or texture independently.


An important effort to show that fitting the training data exactly can under certain conditions be theoretically justified is the margins theory for boosting [37] and other margin-based methods [6,24,28,29,34].

this theory lacks explanatory power for the performance of classifiers that perfectly fit noisy labels, when it is known that no margin is present in the data [12,51].


Unlike regression or classification, unsupervised graph learning lacks at present an error function, so it seems difficult to apply the bias-variance framework here.

we have conducted a wide range of empirical tests to understand the properties of the ensemble MSTs, and to compare them to the other graph construction methods, in terms of the error in the geodesic distances (if known a priori).


The issue of diverging marginals and time-inhomogeneity has not been previously recognized as a practical problem.

the increasing interest in probabilistic models that contain large numbers of deterministic constraints (see, e.g.


This approach can find an embedding with the least distortion ↵.

solving semidefinite programming problem is much more costly than computing Bourgain embeddings.


The condition ∆ L < ∞ is crucial for Theorem 3 and cannot be removed.

in practice, statistical models of interest do not necessarily satisfy such boundedness conditions.


[22] used EM to learn high-dimensional linear dynamical models form multiple observations, an approach which they called 'stitching'.

their model assumed highdimensional dynamics, and is therefore limited to small population sizes (N ≈ 100).


The data center collects separate samples from all the L sensors and uses them to recover the original sequence.

since {X(t)} ∞ t=1 is not the only pressing matter which the data center must consider, the combined data rate R at which the sensors can communicate with it is strictly limited.


For Dec 8 , in Figure 2, we can see that, when considering the original architecture with 512 neurons per layer, our approach only yields a small reduction in parameter numbers with minimal gain in performance.

when we increase the initial number of neurons in each layer, the benefits of our approach become more significant.


Since r ≤ ǫτ /72d, by Lemma 7 kµ by assumption, this ball contains at least k sample points, and hence X i is kept.

if X i ∈ S σ−r , then the set B M (X i , r) contains mass at most Thus by Lemma 6 B M (X i , r) contains fewer than k sample points, and hence X i is removed.


Biophysical models of neuronal dynamics are of central importance for understanding the mechanisms by which neural circuits process information and control behaviour.

identifying which models of neural dynamics can (or cannot) reproduce electrophysiological or imaging measurements of neural activity has been a major challenge [1].


The work of [6] also offers a weak guarantee wherein they are only able to tolerate a 1/ √ d fraction of corruptions.

[6] allow corruptions in covariates as well.


Such property assures the local search algorithms to converge to global minima [GHJY15,LSJR16,NP06,SQW15].

establishing it for concrete problems can be challenging.


Due to additive property of Polya-Gamma distribution, y ∼ PG(n, ρ) if x i ∼ PG(1, ρ) and y = n i=1 x i .

this sampler can be slow when n is large.


For test points, the learnt regression function is applied in the reduced space and then standard recovery algorithms from CS literature are used to obtain sparse predicted labels [8,9].

in this method, learning of the decision functions is independent of the sparse recovery and hence in practice, it requires several measurements to match accuracy of the standard baseline methods such as 1-vs-all SVM.


One can note that all the quantities entering in the right-hand side of ( 6) are known, so that it is not hard to choose µ and γ in such a way that they satisfy the conditions of Theorem 2.1.

in practice, this theoretical choice may be too conservative in which case it could be a better idea to rely on cross validation.


We have observed that when the generating model is misspecified WGANTPP outperforms the other methods without leveraging the a priori knowledge of the parametric form.

when the exact parametric form of data is known and when it is utilized to learn the parameters, MLE with this full knowledge performs better.


This makes the applicability of this approach for large real life problems rather limited.

if we would know a subspace (spanned by the d columns of a matrix V ∈ ℜ (nt+nw)×d ) in which (or close to which) the label vector lies, we can restrict the feasible region for Γ, leading to a much more efficient algorithm.


Since the N sample points are independent, the consistency result holds.

if the center point is the center of mass, and it converges to a bad point, then such a test will not be consistent.


Although our sample-based approach is intuitively straightforward, its implementation has two drawbacks: (1) One has to be cautious of samples that are close together, because they lead to unbounded gradients; and (2) all samples must be considered when computing the force on a data point, which makes the approach computation intensive.

in [10,7] approximations are proposed that reduce the computational complexity of the approach.


3, we can observe that the efficiency is improved by about one magnitude on average compared to that of the solver without screening.

when λ/λ max < 0.5, strong rule can not identify any inactive features and thus the running time is almost the same as that of the solver without screening.


Recent progress has focused mainly on continuous approximation and convex relaxation of the hard cardinality constraint.

we consider an alternative discrete spectral formulation based on variational eigenvalue bounds and provide an effective greedy strategy as well as provably optimal solutions using branch-and-bound search.


a larger choice for q, affects the guaranteed convergence rate as ρ ∼ q/n.

as long as one step of a q-memorization algorithm is as expensive as q steps of a 1-memorization algorithm, this insight does not lead to practical improvements per se.


The values are larger here: R − = 0.147, 0.27, 0.118 for small2-4 respectively, showing the importance of the meta-strategy.

these are still significant reductions in JPC: 19.5%, 36.5%, 59.9%.


We think this might be due to the fact that ODIN relies on the perturbation of the network classifier output, which becomes less accurate as the number of classes grows from 10 to 100.

gPND does not use class label information and copes much better with the additional complexity induced by the increased number of classes.


Among these, 1 corresponds to the case where every node is assigned by ψ to C 1 , and, similarly, for 0, to C 2 .

1 Ci are the indicators of the clusters C i and hence correspond to the ground truth community assignment.


Specifically, Q(s) and the divergence between Q(s o ) and P (s o ) are parameterized with two neural networks, i.e., one generator and one discriminator, and then play an adversarial game [12].

different from the original GANs, our model has one extra discriminator (called as reconstructor), since the first term in Eq.


G-MLE predominantly produces 'safe' and less informative answers, such as 'Yes' and or 'I can't tell'.

our proposed model G-DIS does so less frequently, and often generates more diverse yet informative responses.


From Section 2.4, assume r(•) is three times continuously differentiable on [0, 1], at the boundary, the asymptotic order of the bias is , which is smaller than the interior points.

the asymptotic order of the variance is O p {(3σ 2 e (n + 1) 2 )/(k(i)(k(i) + 1)(2k(i) + 1)} and attains O p (n 2 ), as i is close to either 2 or n − 1.


However, since the new user is not willing to label his/her accelerometer data due to troublesomeness, no labeled sample is available for the new user.

unlabeled samples for the new user and labeled data obtained from existing users are available.


Kalai and Servedio [7] showed that, if data is corrupted with noise at a rate η, it is possible to boost the accuracy of any noise-tolerant weak learner arbitrarily close to 1 − η, and they showed that it is impossible to boost beyond 1 − η.

we show that, in the presence of noise at a rate arbitrarily close to 1/2, the AUC can be boosted arbitrarily close to 1.


As we expect that the number of required hidden units grows with the dimension of the datasets we have limited ourselves in the experiments to low-dimensional datasets.

these bounds are likely not to be tight, so that there might be room for improvement in terms of dependency on the number of hidden units.


The Kalman filtering framework seems to work well in these example domains.

there are some cases where we may need to apply more sophisticated techniques.


The instance-based approach is used because it is a baseline method for category representation.

more advanced approaches like Bayesian approach can be easily adapted.


If the robot succeeds at achieving the goal configuration when the visual statistics of the pair of initial and goal image is similar to before and after image in the training set, then this would not be a convincing demonstration of generalization.

if the robot is able to displace objects into goal positions that are much farther apart as compared to position of objects before and after a single poke then it might suggest that our model has not simply overfit but has learnt something about the underlying physics of how objects move when poked.


The clustering step of our method groups the training data into different bins based on their therapy sequences.

the complete treatment history is not necessarily available for all patients in our clinical data set.


Some learning algorithms are generic, e.g., working on arbitrary unstructured vectors in d , such as ordinary SVMs, decision trees, neural networks, and boosting applied to generic learning algorithms.

other learning algorithms successfully exploit the specific topology of their input, e.g., SIFT-based machine vision [10], convolutional neural networks [6,7], time-delay neural networks [5,16].


In our application, the x n are one-dimensional time indices.

since the model can be generalized to multi-dimensional inputs, we do not restrict ourselves to the one-dimensional case.


Moreover, we also find that recalls of the single optimal search path RL become stable with a few steps and hardly increase with the increasing of steps.

the recalls of Tree-RL keep increasing as the levels of the search tree increase.


our posterior distribution p( 9) represents our "optimal" beliefs about the different possible values of 9*, given our prior knowledge and the evidence.


3 One can drop this condition by considering slightly larger e N (t) and smaller e NK (t).

we keep the condition as it simplifies e N (t) and e NK (t).


We showed that neither forward greedy nor backward greedy algorithms are adequate by themselves.

through a novel combination of the two ideas, we showed that an adaptive forward-back greedy algorithm, referred to as FoBa, can effectively solve the problem under reasonable conditions.


Indeed, Gibbs sampling is the standard inference algorithm used in nonparametric Bayesian methods, most of which are based on the Dirichlet process [3,4].

recent work has suggested that sequential Monte Carlo methods such as particle filtering can provide an efficient alternative to Gibbs sampling in Dirichlet process mixture models [5,6].


Since its early beginning [24,34], the PAC-Bayesian theory claims to provide "PAC guarantees to Bayesian algorithms" (McAllester [24]).

despite the amount of work dedicated to this statistical learning theory-many authors improved the initial results [8,21,25,30,35] and/or generalized them for various machine learning setups [4,12,15,20,28,31,32,33]-it is mostly used as a frequentist method.


It is clear which features should be learned when all of the line segments occur independently and when the line segments in each object always occur together (the line segments and the objects respectively).

in the intermediate cases of non-perfect co-occurence, what should be learned?


They are also be an important part of the early design process, helping us explore rough ideas and solutions in an informal environment.

despite their ubiquity, there is still a large gap between how people naturally interact with sketches and how computers can interpret them today.


In contrast, second-order methods which have access to the Hessian of f can exploit curvature to more effectively escape saddles and arrive at local minima.

constructing the full Hessian can be prohibitively expensive in high-dimensions.


To be explicit, if is defined by an edge set © £ ¦ ¤£ © , then e £ n must have zeros in all elements corresponding to edges not in © £ ¦ .

given an edge ¥ belonging to two trees H and §¦ , the quantity e ©£ n H can be different than e £ n ¦ $ .


The cross-polytope LSH functions were originally proposed in [15].

the analysis in that paper was mostly experimental.


Binarization typically provides a significant speed up by packing bits into 64-bit integers, allowing a CPU or GPU to perform a single xnor operation in lieu of 64 floating-point multiplications.

heterogeneous tensors are essentially composed of sparse arrays of bits.


In general, this limiting Gaussian process will be infinite-dimensional.

if we choose the exponential nonlinearity (f (.)


Central to our contribution are recent developments of scalable methods for general OT that leverage the idea of entropic regularization [Cut13, BCC + 15, GCPB16].

the apparent practical efficacy of these approaches came without theoretical guarantees.


Another key feature of cortical circuits is the detailed and tight balance of excitation and inhibition, which has received growing support both at the experimental (Froemke et al., 2007;Xue et al., 2014;Froemke, 2015) and theoretical level (van Vreeswijk and Sompolinsky, 1996;Brunel, 2000;Vogels and Abbott, 2009;Hennequin et al., 2014Hennequin et al., , 2017.

the computational processes that are facilitated by these architectures and dynamics are still elusive.


In the rest of this paper we will focus on a model with a GP prior over the transition function and a parametric likelihood.

our variational formulation can also be applied to the double GP case (see supplementary material).


For instance, setting a small κ in such cases will likely lead to detrimental under-estimation.

setting high κ values decreases the estimator robustness ( this makes the loss function approach the 2 loss).


Max-norm regularizer has been extensively studied in the last decade as it promotes an effective low rank estimation of the underlying data.

maxnorm regularized problems are typically formulated and solved in a batch manner, which prevents it from processing big data due to possible memory bottleneck.


Thus, our approach is based on Gibbs sampling.

there are two constraints that significanlty complicate sampling.


When a short training procedure is required, optimizing the basis could be too costly.

if the focus lies on fast inference, we argue to optimize the basis to obtain a compact representation.


The rapid pace of innovation we have seen in the last several years has been enabled largely by tools that have automated the error-prone aspects of engineering, such as writing code that computes gradients.

our contention is that operation batching is increasingly becoming another aspect of model coding that is error prone and amenable to automation.


For the case of monotone link function, [38] analyze a nonconvex least squares estimator under the assumption that the covariate is sub-Gaussian.

the success of their estimator hinges on the knowledge of the link function.


Finally, we have yet to account for neurophysiological data within this framework, and constraints at the neural implementation level.

versions of our computations are oft suggested for intra-areal and feedback cortical circuits; and smoothness principles form a key part of the association field connection scheme in Li's 24 dynamical model of contour integration in V1.


For example, orientation selective neurons in superficial layers in ferret visual cortex receive more than 50% of their cortico-cortical excitatory inputs from neurons with orientation preference which is less than 30 o apart.

this input structure is rather broad in terms of stimulus-specificity: cortico-cortical connections between neurons tuned to dissimilar stimulus orientation also exist [4].


In principle, one could alternatively use a feature vector representation using these frequencies directly, as in (a).

applying the log transformation yields a feature space with fewer isolated points in R 2 , as in (b).


Since the eigenvalues λ i ( Σ) are observable, they may be used to estimate ν and guide the selection of /n = n −θ .

from a practical point of view, we found it easier to select via cross-validation in numerical experiments, rather than via an estimate of ν.


In general, we will have more than two clusters.

notice that any point x in B r/ √ m (F m θ0,i,j ) (for some i, j) is much closer to F θ0,i,j than to any other cluster boundary.


In [10], the focus is on non-monotone functions and online problems on the positive semidefinite cone.

the focus of this paper is on monotone functions on the positive orthant.


We also list the published result of [22] in the table.

it is important to remember that it is not directly comparable to other numbers in Table 1(a), since [22] uses different image features and different subsets of the dataset unspecified in the paper.


For instance, indoor spaces, pictures containing people, particularly if their face is visible, close up views on objects, animals, are more memorable than buildings, pictures of natural landscapes, and natural surfaces in general (like mountains, grass, field).

to date, there is no work which has attempted to predict which local information from an image is memorable or forgettable, in an automatic manner.


Convergence analysis of this algorithm can be established using the idea summarized above; see details in appendix.

in partice, instead of the quadratic upper bound (which has a theoretical garantee easier to derive), one may also consider minimizing an approximation to the Taylor expansion, which would be closer to a Newton type method.


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

as the number of observed features K increases, model bias decreases and λ approaches 0, placing increasing weight on OLS.


The focus of this paper is solving inference problems, not generating samples, so we do not investigate the sampling properties of MP-DBMs extensively.

it is interesting to note that an MP-DBM can be viewed as a collection of dependency networks [9] with shared parameters.


As in the latent dynamical system, the resulting model enforces a "bottleneck", whereby predictions of y t based on y 1 . . . y t−1 must be carried by the low-dimensional x t . State prediction in the RLM is related to the Kalman filter [7] and we show in the next section a formal equivalence between the likelihoods of the RLM and the latent dynamical model when observation noise is Gaussian distributed.

spiking data is not well modelled as Gaussian, and the generalisation of our approach to Poisson noise leads to a departure from the latent dynamical approach.


[4], where pair-wise queries are used for clustering.


Undoubtedly, CMS has lowest update time for the summary because it involves only calculating the bucket through hashing and then incrementing the respective counter.

its performance for "top-K problem" is highly thwarted by the overhead of maintaining probable top-K words summary.


The agents can win, if they both know the same correct tiling and interpret the sequence of observations as the position in the grid they are asked to put a tile on.

if both agents know different tilings or the tiling they share is not correct, then at least one trajectory will end in a bad state and has reward −(2 2k+2 ).


As explained in [1,2], Thompson Sampling with uniform prior for Bernoulli rewards can be slightly adapted to deal with bounded rewards.

there is no notion of asymptotic optimality for this non-parametric family of rewards.


The second way is to simply collect more observations of transitions.

our results in this paper indicate that the effectiveness of this approach diminishes quickly as the sample size grows, and thus is recommended only when observations are cheap to obtain.


We note that while the top methods from the literature also achieve accuracies above 92% and compression factors above 20×, they employ a considerable amount of hand-designed steps, such as pruning, retraining, various types of weight clustering, special encoding of the sparse weight matrices into an index-difference based format and then finally use entropy coding.

we directly minimize the entropy of the weights in the training, obtaining a highly compressible representation using standard entropy coding.


Template matching approaches have also been applied to deformable 2D objects, where very efficient solutions exist [9,11].

these methods do not extend easily to the case of 3D surfaces.


In DPI and CBMPI, we managed to obtain results similar to CE, only after multiplying the per iteration budget B used in the D-T experiments by 10.

cBMPI and CE use the same number of samples, 150, 000, 000, when they converge after 2 and 60 iterations, respectively (see Figure 5).


Given a distribution p over permutations, the matrix M k of k'th order marginals is where S j1...j k i1...i k is the two-sided coset µ j1... Clearly, these matrices satisfy a number of linear equations, and therefore are redundant.

it can be shown that for for some appropriate basis transformation matrix T k , i.e., the Fourier matrices { p(λ)} λ : λi=n−k capture exactly the "pure k'th order effects" in the distribution p. In the spectral analysis of rankings, as advocated, e.g., in [7], there is a lot of emphasis on projecting data to this space, Marg k , but using an FFT this takes around O(n 2 n!)


Practitioners have developed a wide range of manual adjustments to overcome these limitations, from adjusting the electrode position to isolate a single neuron, to manually performing the clustering for spike identification.

previous studies have shown that there is great variability in manual sorting results [17], and that human choices for cluster parameters are often suboptimal [18].


It showed that not only is joint training necessary for optimal performance, but also that standard SIFT still outperforms many modern baselines.

their approach still relies on SIFT keypoints for training, and as a result it can not learn where SIFT itself fails.


Recent achievements have shown that alternating online and offline optimization is a successful choice for efficient trajectory reuse.

deciding when to stop optimizing and collect new trajectories is non-trivial, as it requires to account for the variance of the objective function estimate.


We found that with scale and object rotation variability suppressed (but with speed and position variability retained), the sensor was able to nearly perfectly identify the objects.

with all sources of variability present, the SVM was just above chance in its performance, while combinations of variability are more challenging for the sensor than others (details can be found in supplementary information).


The overall structure of these moves could be easily converted to an MCMC or simulated annealing algorithm.

in our experiments this search outperformed simple MCMC methods in terms of speed and accuracy.


The most related work is probably [18], which investigates contextual combinatorial bandit with submodular reward.

it considers a special submodular reward function that depends on a linear combination of individual rewards and assumes a constant set of arms which is very different from the volatile arms in our work.


We further demonstrated how this computed reward can be used by different learning mechanisms to form appropriate Pavlovian or instrumental associations.

it should be noted that concerning food (and other ingestible) outcomes, the basic form of the drive-reduction definition of reward (equation 2) has four interrelated problems, some of them used traditionally to criticize the drive reduction theory: (1) Post-digestive nutritive feedback of food, defined by the drive reduction equation, might occur hours after ingestion.


Their approach was later signficantly extended [Ruttor et al., 2013, Ruttor and Opper, 2010, Vrettas et al., 2015.

similiar to [Lyons et al., 2012] they study parameter estimation in stochastic dynamical systems while our work focuses on deterministic systems.


Recent advances in graph embedding and visualization have focused on undirected graphs, for which the graph Laplacian properties make the analysis particularly elegant [1,2].

there is an important number of graph data, such as social networks, alignment scores between biological sequences, and citation data, which are naturally asymmetric.


The figure shows that the three methods give essentially identical performance, and that after 3 examples, all three methods have a regret per start state of about 2 units, which is less than the cost of a single -5 penalty.

the three formulations differ in their ease of training and in the information they require.


It is very difficult to compare a separation method with other CASA techniques, because their approaches are so different in many ways that an optimal tuning of their parameters would be beyond the scope of this paper.

we compared our method with Wiener filtering [4], that provides optimal masking filters in the frequency domain if true spectrogram is given.


Much work has focused on extending existing knowledge bases using patterns or classifiers applied to large text corpora.

not all common knowledge that is obvious to people is expressed in text [5,6,2,7].


One of the most popular geometries for PDs is the Wasserstein metric [Villani, 2003, Peyre and.

it is well-known that the Wasserstein distance is not negative definite [Reininghaus et al., 2015] (Appendix A).


However, the state-of-the-art alternating based algorithms (such as ANLS and HALS) for nonsymmetric NMF utilize the splitting property of (1) and thus can not be used for (2).

first-order method such as projected gradient descent (PGD) for solving (2) suffers from very slow convergence.


(2015) state that cross-entropy loss does not have margin-maximization properties, and add terms to the cross-entropy loss to encourage large margin solutions.

these terms encourage margins only at the output layer of a deep neural network.


So one way to recover C is to directly invert equation (11), which proves the decodability.

directly computing this inverse using the classical inversion algorithm might be expensive in more general cases.


[11] also analyze the slack re-scaling approach and theoretically show that using random structures one can obtain a tighter upper bound of the Gibbs decoder distortion.

these works do not take into account latent variables.


The theoretical RL literature o ers a variety of provably-e cient approaches to deep exploration [9].

most of these are designed for Markov decision processes (MDPs) with small finite state spaces, while others require solving computationally intractable planning tasks [8].


To apply Theorem 1, we need to learn distribution in C and find a q that is O(" 2)-close in 2 -distance to p. For the class of monotone distributions, we are able to efficiently obtain such a q, which immediately implies sample-optimal learning algorithms for this class.

for some classes, we may not be able to learn a q with such strong guarantees, and we must consider modifications to our base testing algorithm.


The second term dominates when the agent is not confident about its reward estimates, which promotes exploration.

when all the confidence terms are small, the algorithm exploits the best arm(s) [2].


We have The regret of the game with early stopping can be written as R T = T t=1 t s=1 e s (y t − ŷt ) 2 − ∆ * t . One might notice that δ * t = 0 for the choice y t = θ * t−1 x t , where θ * t−1 is the ordinary least squares solution on data through time t − 1, and the regret always increases.

this choice of y t may violate the label constraints, in particular, for B t = 1 and x t ∈ R increasing.


This illustrates a basic tension in the resulting bounds: in its primal form 2 (combination of trees), TRW is inefficient: it maintains a weight and O(|V |) parameters for each tree, and a large number of trees may be required to obtain a tight bound; this uses memory and makes optimization slow.

the dual, or free energy, form uses only O(|E|) parameters (the TRW messages) to optimize over the set of all possible spanning trees -but, the resulting optimization is only guaranteed to be a bound at convergence, 3 making it difficult to use in an anytime fashion.


With such definitions, the optimization problem may be interpreted as an upper-bound on the classification risk defined from ℓ FN and ℓ FP .

the machinery of Section 3.2 unveils a major problem: the SVM decision function provided by sign(f (x i ) + b) is not consistent with the probabilistic interpretation of SVM scores.


We have analyzed this case in Appendix C. To see how the probability of error must scale, we have used a generalized version of Fano's inequality (e.g., [23]).

when the number of queries is greater than zero, plus when queries can be adaptive, any such standard technique fails.


This again means that the overall algorithm fails to be a compression scheme as side information is required.

we can use the same approach described for KMP to bound the expected fit of the projections from the two views.


Focusing on Hawkes processes, the feature-based mixture models in [5,17,47] combine Hawkes processes with Dirichlet processes [2,36].

these methods aim at modeling clusters of events or topics hidden in event sequences (i.e., sub-sequence clustering), which cannot learn clusters of event sequences.


It may result in some accuracy loss when simply replacing the floating gradients with ternary gradients while keeping other hyper-parameters unchanged.

we are able to train large-scale DNNs by TernGrad successfully after making some or all of the following changes: (1) decreasing dropout ratio to keep more neurons; (2) using smaller weight decay; and (3) disabling ternarizing in the last classification layer.


The former is guaranteed to converge to an optimal policy, however, the size of the controller often grows intractably.

the latter restricts its search to controllers of a bounded size, but may get trapped in a local optimum.


Linear signal in 3d  optimal over both the bigger and smaller classes.

as soon as we enter the 3d world, it is no longer clear whether TV denoising can adapt to the smaller, inscribed Sobolev ball, whose minimax rate is faster, n −2/5 versus n −1/3 (ignoring log factors).


Many practical problems modeled by multi-agent MDPs use centralized policies to achieve fairness, e.g., network bandwidth allocation by telecommunication companies, traffic congestion control, public service allocation, and, more generally, fair resource allocation under uncertainty.

we can derive decentralized policies for individual agents from a maximin fairness policy π * by marginalizing it over the actions of all other agents.


[3] showed that if ρ(M −1 Hog N Hog ) < 1, the Markov chain admits N (µ, Σ) as stationary distribution where The above approach can be generalized to blocks of larger sizes.

beyond the block size, the Hogwild sampler does not have any tunable parameter allowing us to modify its incorrect stationary distribution.


The most expensive part of our algorithm involves computing similarities for all pairs of vertices in order to determine the set of nearest neighbors.

it would be possible to use approximate nearest neighbor techniques to greatly reduce the computation such that approximate nearest neighbor sets could be computed with significantly fewer than n 2 pairwise comparisons.


We observe that subgradient descent is again very slow to converge, and coordinate descent is also rather slow here (in fact, it is not even guaranteed to reach the optimum).

our FW algorithm converges orders of magnitude faster and finds a high quality solution (for runtime comparison see Appendix G).


The computational complexity of VDMGP is O(N M 2 K +N DK), just as that of SPGP-DR, which is much smaller than the O(N 3 +N 2 D) required by a full GP.

since the computation of the variational bound of VDMGP involves more steps than the computation of the evidence of SPGP-DR, VDMGP is slower than SPGP-DR.


Via delimited continuations, we get an expressive and define-by-run framework, similar to PyTorch.

tensorFlow and other define-then-run frameworks benefit from separating graph construction and graph execution into two stages, so that graph transformations/optimizations can be performed to target hardware-specific code (i.e., GPUs or TPUs).


1, provides a strong argument in support of adopting the sharp Sinkhorn over its regularized version.

while the gradient of the regularized Sinkhorn can be easily computed (see [8] or Sec.


It can be seen that, though the sub-model on each subset is mis-specified, DECO performs as if the full dataset were used as its performance is on par with lasso-full.


These networks use homogeneous binarization: all parameters in each layer or (more commonly) the whole model have the same low bitwidth (e.g., 2 bits).

modern hardware allows efficient designs where each arithmetic instruction can have a custom bitwidth, motivating heterogeneous binarization, where every parameter in the network may have a different bitwidth.


Importantly, the optimal batch-size depends on where we are in the learning process: far away from convergence we only need a rough idea of where to move which requires very few data-cases.

the closer we get to the true parameter value, the more resolution we need.


[4]) where the image is represented using a grid-structured MRF that is based on local connections between neighboring nodes.

when the graph is not sparse, and is not based on local grid connections, loopy BP almost always fails to converge.


Hence, the model sees only few examples of non-zero targets in a single epoch.

both RNN m and EncDec converge relatively faster than RNN b and do obviously not suffer from the r2f ordering.


Sampling θ x will, nevertheless, be harder since the covariance function hyper-parameters enter the expression in a non-trivial way.

we note that once the state trajectory is fixed, we are left with a problem analogous to Gaussian process regression where x 0:T −1 are the inputs, x 1:T are the outputs and Q is the likelihood covariance matrix.


One can show that if k(•, •) is a kernel then the function ϕ(•) and the feature space H exist; furthermore k(x, y) = ϕ(x), ϕ(y) [18].

directly performing linear PCA in the feature space might not be feasible because the feature space typically has very high dimensionality (including infinity).


Ontologies and knowledge bases such as WordNet [1], Yago [2] or the Google Knowledge Graph are extremely useful resources for query expansion [3], coreference resolution [4], question answering (Siri), information retrieval or providing structured knowledge to users.

they suffer from incompleteness and a lack of reasoning capability.


Thus, a common approach is to train object detectors from labeled images, which are widely available.

as seen in Figure 1, the domain of images and videos is quite different, as it is often the case that images of objects are taken in controlled settings that differ greatly from where they appear in real-world situations, as seen in video.


It is possible to rewrite this model so that the f factors do not have exponents, and therefore would not in gates.

this will necessarily change the approximation.


One such estimation is maximum mean discrepancy (MMD).

mMD suffers from its sensitive kernel bandwidth hyper-parameter, weak gradients, and large mini-batch size when used as a training objective.


Attention-based models, such as [2,25], offer some degree of interpretability and have been shown to also improve performance on downstream tasks.

soft attention, the most prominent attention model, often does not learn to single out human-interpretable inputs.


However, as more data points are available, the effect of the advice diminishes.

the advice refining methods are able to generalize much better with few data points, and eventually converge to a better solution.


In practice, this term only affects the computation of y(n) in the initial iterations when the value of β is small, and in later iterations is dominated by the values of w i (n).

we find that adding this regularization allows us to increase the value of β faster, and therefore converge in fewer iterations.


In fact, edge detectors can also produce pictures that highlight features which coincide with what appears to be relevant to a model's class prediction.

here the human observer is at risk of confirmation bias when interpreting the highlighted edges as an explanation of the class prediction.


Support Vector Machines [4] are based on this definition of margin, which we refer to as Sample-Margin.

an alternative definition, Hypothesis Margin, exists.


This greedy partitioning method parallels the classical algorithms for classification and regression that have been used in statistical learning for decades.

the strength of the procedures given in Definitions 1 and 2 is that they lend themselves to a theoretical analysis under relatively weak assumptions, as we show in the following section.


samples from the training distribution, this difference is small, and can in fact aid generalization.

minibatch-wise normalization may have significant drawbacks: For small minibatches, the estimates of the mean and variance become less accurate.


That is, 0 k is contained in the most recent play of a Nash equilibrium inside 6 .

if the two conditions above are not met, then agent ¢ chooses its best-response action in the same way as AP.


The terms P (R H ∩ H R (x A )) and P (R H ∩ H R (x A )) allows us to quantify the progress made on determining region validity.

naively computing these terms would require computing all hypotheses and assigning them to correct subregions, thus requiring a runtime of O 2 T . for the special case of Bernoulli tests, we can reduce this to O (T ) as we can see from the expression We can further reduce this to O (1) when iteratively updated (see supplementary for derivations).


Incorrect values for this parameter can result in failure to converge, not just slower convergence.

our algorithm automatically adapts to the true value of L with no tuning.


This quantity characterizes the energy costs of coding with a set of neurons.

the assessment of coding efficiency, as used by Treves and Rolls [13], is based upon the assumption that different stimuli activate small, distinct subsets of cells.


As shown in Remark 2, the values of ( 4) and ( 6) may be not equal.

with the condition of the asymptotic breakdown point and asymptotic onto-breakdown point of E 1 being the same, we can finally state our desired clean result.


Compared to explicit feedback (e.g., 1-5 star ratings), implicit feedback is more abundant and accessible in real-world applications.

the missing data of implicit feedback also brings two challenges.


Therefore it is a type of conditional HMM.

unlike the IOHMM and our proposed model, the input is unobserved, making learning completely unsupervised.


In the past decade, significant progress has been made toward the methodological development [13,8,30,22,7,14,28,19,27] as well as theoretical understanding [12,20,1,24,21,4,6,3,2,18,15] of sparse PCA.

all the above studies focused on estimating the leading eigenvector u 1 .


into a continuous function or a probability measure after it is sampled.

uncountable product space constructions are subject to all the subtleties of stochastic process theory, many of which do not occur in countable dimensions.


As the branch moves beyond the image boundary, the layered model interpolates its motion using long-range correlation with the branches in the center.

the single-layered approach incorrectly interpolates from local background regions.


Deep learning models can represent the data together with the model in fewer bits than a naive encoding, despite their many parameters.

we were surprised to observe that variational inference, though explicitly designed to minimize such codelengths, provides very poor such values compared to a simple incremental coding scheme.


In domain adaptation, the source and target data come from two different distributions.

if the source and target domains are reasonably close, we can employ a similar form of regularization using unlabeled data.


An alternative to GARCH are stochastic volatility models [3].

there is no evidence that SV models have better predictive performance than GARCH [4,5,6].


The linear UCB algorithm we adapt in this work is a canonical approach to the linear bandit problem, using the principle of "optimism in the face of uncertainty."

recent work [24] shows that all such "optimistic" algorithms are sub-optimal, and instead proposes adapting to the decision set in a particular way by solving an intricate optimization problem.


Exact nearest neighbour algorithms organize the appearance descriptors into trees which can be efficiently searched [17].

these methods work well only for descriptors of small dimensionality n (typically n < 20), and their speedup vanishes for larger n (e.g.


(As we have seen, variance is not full rank.)

the mean and second moment satisfy both by Lemma 1.


It plays the same role as the state of a deterministic finite-state automaton.

the recurrent network enjoys a continuous and infinite state space (a high-dimensional Euclidean space), as well as a learned transition function.


Some notable studies have been carried out regarding the behavior of these methods for infinitely large sample size (von Luxburg et al., 2008;Rohe et al., 2011), which provide sufficient confidence to practitioners about the effectiveness of these methods.

recent developments in computer vision have led to a plethora of applications, where the model deals with multi-way affinity relations and can be posed as uniform hypergraphs.


Recognition is then posed as a constraint satisfaction problem, as in [4,5], or as an inference problem on a graphical model, as in [1,16,17,20].

in many real-world sketches, it is difficult to extract these primitives reliably.


6 As expected, the firing probability of this unit is close to one for one of the interpretations and close to zero for the other, especially in the initial time period after a perceptual switch.

as the neuron's firing rate and synaptic activity deviate from their low average levels, the synaptic efficacy changes as shown in the plot.


The number of classes could be as large as the number of rounds, which is Θ(log(k)), and their algorithm uses roughly m ,δ samples from each class.

algorithm R1 uses only m ,δ samples across all classes and saves a factor of Θ(log(k)) in the sample complexity.


Traditionally, for complicated inpainting tasks, an inpainting mask that tells the algorithm which pixels correspond to noise and require inpainting is supplied a priori.

in various situations this is time consuming or sometimes even impossible.


The usual maximum Shannon entropy criterion, which selects the next example x maximizing x]] where D is the previously observed labeled examples, can be thought of as a greedy heuristic for building a policy π maximizing H(π).

it is still unknown whether this greedy criterion has any theoretical guarantee, except for the non-adaptive case.


Much of the seminal work in CS [9,10,11] has focused on sufficient conditions on A such that (3) is guaranteed to perfectly recover the true signal, so that ŝ = s 0 .

many large random measurement matrices A which violate sufficient conditions proven in the literature still nevertheless typically yield perfect signal recovery.


Previous work learning sparse codes of image sequences has shown that it is possible to recover local, direction-selective components (akin to translating Gabors) [11].

this type of model does not capture the abstract property of motion because each unit is bound to a specific orientation, spatial-frequency and location within the image-i.e., it still suffers from the aperture problem.


Conceptually, the log-likelihood objective implicitly trades off between model error in different regions of the input/output space, but does so in a manner largely opaque to the modeler, and may ultimately not employ the correct tradeoffs for a given task.

there is an alternative approach to solving (1) that we describe as the model-free "black-box" policy optimization approach.


The MDS problem endeavors to find an ensemble of points in a prescribed metric space (usually a Euclidean space) such that the distances between these points are close to the set of input distances.

the MN problem does not seek to find an embedding.


On one hand, different conclusions on the consistency of pairwise ranking methods have been obtained in our work and in [11].

we have shown that there exists an connection between the rank-differentiable property and the low-noise setting (see Figure 1).


Otherwise CLSR may be an interesting alternative although this method does not make any assumption about the sparsity of f + and its goal is not to recover a possible sparse f + but only to make good predictions.

in some sense our method finds a sparse solution in the fact that the regression function g L lies in a space G M of small dimension M N and can thus be expressed using only M coefficients.


In principle, past inputs could be read out from the instantaneous activity of the network.

the theoretical principles underlying the ability of recurrent networks to store temporal sequences in their transient dynamics are poorly understood.


A natural hope is that Theorem 2.1 extends to a non-empty C since adding more cycles can help to reduce the integrality gap of C-LP.


We consider the full eigenfunctions ofO(a, b, aI, bl ) below.

the case that Piepenbrock & Obermayer (1999) studied of a flat arbor function (u A = 00) turns out to be special, admitting two equilibrium solutions, one flat, one with topography, whose stability depends on (3.


It is widely assumed that prefrontal cortex (PFC) plays a role in maintaining and updating working memory.

relatively little is known about how PFC develops useful working memory representations for a new task.


It is tempting to posit that conv(F) = C holds more generally in the Bethe case.

as the following example shows, settings arise where conv(F) ( C. Details are contained in Appendix C.2.


This tends to favor frequent n-grams [11,23], leading to issues such as incorrect semantic coverage, and lack of diversity.

our proposed paradigm proceeds in a bottom-up manner, by representing the input image with a set of noun-phrases, and then constructs captions according to a recursive composition procedure.


Our code, including an implementation of various literature baselines, is publicly available at https: //github.com/amitz25/PCCoder Automatic program synthesis has been repeatedly identified as a key goal of AI research.

despite decades of interest, this goal is still largely unrealized.


AS gives a very intuitive account of population sparseness, and is invariant to both multiplicative and additive changes in firing rate.

since it discards most of the information about the shape of the distribution, it is a less sensitive measure than TR.


An ensemble of two FS-LSTM models (1.20 BPC) outperforms cmix (1.23 BPC) [24], the current best text compression algorithm on enwik8 [27].


If G 2 > H, we would like to use as large a learning rate as possible in order to maximize reward.

if G 2 < H, the algorithm will obtain negative reward, and the best it can do is to cut its losses by setting η as small as possible.


Consider any m < M . The ζ (m) conditions on f (m) constrain the value of f (M ) -the confidence band β is large, it means that we have not constrained f (m) sufficiently well at x t and should query at the m th fidelity.

querying indefinitely in the same region to reduce β t−1 in that region will not help us much as the ζ (m) elongation caps off how much we can learn about f (M ) from f (m) ; i.e.


Players discover ways to collect unlimited lives or health, and by taking advantage of these exploits, they can easily complete an otherwise difficult game.

in the process of doing so, they may have forfeited the opportunity to learn the skill required to master the game as intended by the game designer.


In the preceding example, the Vapnik-Chervonenkis (VC) dimension, which characterizes the sample complexity of supervised learning, is 1.

the eluder-dimension, which will we define below, is n. To highlight conceptual differences between the eluder dimension and the VC dimension, we will now define VC dimension in a way analogous to how will define eluder dimension.


With K 1 members in each group, each representative has competence µ(K 1 ) < µ(K 2 ).