Moreover, as in standard Q-learning, our approach does not store old data.

our approach differs from other batch methods, which need to wait for all data to be received before starting computation, and require multiple passes over the data.


This definition is based on the orientation tuning of the input currents to the excitatory model cells in the orientation domain (0.6 < map OSI < 0.9).

if the sum of input currents is positive (negative) for all presented orientations, recurrent excitation (inhibition) is dominant, and the regime thus excitatory (EXC; respective inhibitory, INH).


Numerical evaluations of bias and variance of estimators discussed in the present section only rely on the computation of coefficients θ α,β defined by where α, β are non-negative integers and (Z, Z ) are bivariate normal (1) with r = ρ.

we have E ρ [ ρ lin ] = θ 1,1 , Var ρ ( ρ lin ) = (θ 2,2 − θ 2 1,1 )/k.


We introduce a new type of test, the e-test, and present the first nonlinear PSR that can be applied to a general class of dynamical systems.

in the first such result for PSRs we show that there exist controlled dynamical systems whose PSR representation is exponentially smaller than its POMDP representation.


Then M * (x) is a nonempty and compact interval for P X -almost all x ∈ X.

given a function f : X → R, Lemma 2.3 shows that for P X -almost all x ∈ X there exists a unique t * (x) is the element in M * (x) that has the smallest distance to f (x).


Following the method described in [11,12], we used AdaBoost during training to select a small set of SIFT features from among all the SIFT features computed for each sample in the training set.

each training image was represented by a set of SIFT features {F i,j , j = 1, ...n i }, where n i is the number of SIFT features in sample I i .


Both types consist of three individual grafts for each player: one to play each card with complete information.

table 3: Sampled expected winnings in Texas Hold'em of the row player against the column player in millibets per hand (mb/h).


In this subsection, we will discuss on this issue.

we propose a bootstrap procedure to iteratively refine word allocation based on the learned word embedding in the LightRNN model: (1) For cold start, randomly allocate the words into the table.


We employ a standard tracking algorithm (KLT point tracker [10]) to "lift" the visual observations to the velocity space.

for each video, we first run the tracking algorithm, and we obtain velocities by simply using the center locations of each of the tracked moving objects between frames.


However, there still remain several problems for future work.

it is unknown under which conditions a specific optimization procedure for (9) can produce an optimal solution that exactly restores the target matrix L 0 .


Second, although not obvious, from experimentation we show that phone streams different from the language being spoken provide complimentary information for speaker recognition.

accuracy improves with these additional systems.


In an nCRP model of personal choice each individual picks a tree path by walking from the root node down the tree, from node to node.

when situated at a particular parent node, the child node c i individual i chooses is modeled as a random variable that can be either an existing node or a new node: (i) the probability that c i is an existing child node k is proportional to the number of persons who already chose node k from the same parent, (ii) and a new node can be created and chosen with probability proportional to γ 0 > 0, which is the nCRP concentration parameter.


But if Y is multivariate, we may be also interested in how the structure of P (• | X) varies as a function of X.

let G(x) be the undirected graph corresponding to P (• | X = x).


Nonetheless, generative classifiers also have several advantages, among them straightforward EM methods for handling missing data, and often better performance when training set sizes are small.

it has been shown that a simple generative classifier (naive Bayes) outperforms its conditionally-trained, discriminative counterpart (logistic regression) when the amount of available labeled training data is small [11].


First we notice that not all of the algorithms are efficient in this setting.

algorithms that optimize the smooth dual (either soft-max or L 2 smoothing) need to enumerate factor configurations in order to compute updates, which is prohibitive for the global cardinality factor.


We now examine the relationship between learning to defer and rejection learning.

we will show that learning to defer is a generalization of rejection learning and argue why it is an important improvement over rejection learning for many machine learning applications.


An important drawback of MCMC approaches is that it might be extremely difficult to monitor their convergence and that they may require a very large number of samples before actually converging.

over 100, 000 sample paths were necessary to reach convergence in the case of the double-well system.


In this work, we focus on supervised learning techniques where systematic uncertainties can be associated to a data generation process that is not uniquely specified.

the lack of systematic uncertainties corresponds to the (rare) case that the process that generates training data is unique, fully specified, and an accurate representative of the real world data.


Different from perturbation-based attacks, we propose to synthesize unrestricted adversarial examples entirely from scratch using conditional generative models.

we first train an Auxiliary Classifier Generative Adversarial Network (AC-GAN) to model the class-conditional distribution over data samples.


We use advantage actor critic (A2C) to train the above reinforcement learning policy.

let θ = {W i } be the parameters in graph memory representation f (•;θ), and φ be the parameter used in π(a t |T (<t) ,G;φ), our objective is to maximize the expected policy reward: The baseline function b(T (<t) ,G;ψ) parameterized by ψ is used to estimate the expected return, so as to reduce the variance of policy gradient.


In this paper, we introduce TUCRL, the first algorithm able to perform efficient exploration-exploitation in any finite Markov Decision Process (MDP) without requiring any form of prior knowledge.

for any MDP with S C communicating states, A actions and Γ C ≤ S C possible communicating next states, we derive a O(D C √ Γ C S C AT ) regret bound, where D C is the diameter (i.e., the length of the longest shortest path between any two states) of the communicating part of the MDP.


For instance, consider a nonlinear system with additive noise of the form, where G is a tall matrix, and consequently rank(G) < dim(ξ t ).

the process noise covariance matrix is singular.


A network gains exponentially efficiency through the reuse of such maps of different filters of later layers.

the neural networks having more linear regions can represent functions of higher complexity and have strong expressive power.


If the image is modified by the addition of a distortion vector, x + αû, where û is a unit vector, and scalar α controls the amplitude of distortion, the model can be used to predict the threshold at which the distorted image can be reliably distinguished from the original image.

one can express a lower bound on the discrimination threshold in direction û for any observer or model that bases its judgments on r (Seriès et al.


We hypothesize that a simple fixed architecture of excitatory and/or inhibitory connections is sufficient to account for the cyclic trends in discrimination thresholds.

we propose that a recurrent connection profile whose strength varies as a function of (a) the similarity between preferred motion patterns and (b) the distance between receptive field centers, is computationally sufficient to recover the trends in GMP/COM performance (Fig.


Examples of functions that satisfy this assumption will appear later in this section.

the gradient, as a map from R n (equipped with ≤ K ) to R n (equipped with ≤ K * ), is order-reversing.When ψ is twice differentiable, assumption 1 is equivalent to w, ∇ 2 ψ(u)v ≤ 0, for all u, v, w ∈ K. For example, this is equivalent to Hessian being element-wise non-positive when K = R n + .


Assume that the loss function is 1-Lipschitz continuous, then the dependence of the corresponding generalization bound on the architecture for each N k,d k p,q,wid  We present a general framework for capacity control on WN-DNNs.

we provide a satisfying answer for the central question: we obtain the generalization bounds for L 1,∞ WN-DNNs that grows with depth by a square root term while getting the approximation error controlled.


The Kikuchi/Bethe approximation is exact if the a-clusters form a singly-connected structure.

exact inference is obtained when the a-clusters correspond to cliques in a junction tree.


While encouraging, these efforts focused exclusively on statistical issues, ignoring computation altogether.

the proposed algorithms exhaustively enumerate candidate value functions to eliminate the ones that violate Bellman equations, an approach that is computationally intractable for any function class of practical interest.


We evaluate our framework along a number of different axes pertaining to its ability to learn disentangled representations through the provision of partial graphical-model structures for the latents and weak supervision.

we evaluate its ability to (i) function as a classifier/regressor for particular latents under the given dataset, (ii) learn the generative model in a manner that preserves the semantics of the latents with respect to the data generated, and (iii) perform these tasks, in a flexible manner, for a variety of different models and data.


Many of the results come with more work than their counterparts in statistical learning theory.

instead of training sets we have to work with trees, making the results somewhat involved.


If we can recognize when this happens  and snap the right picture at each spatial location, then recovering the desired ground truth image would be simply a matter of stitching these correct observations together.

the question that we will be exploring in this paper is not where to look, but when!


The t w triples are chosen by first uniformly selecting a group g and then uniformly selecting three indices i, j, and k from group g and finally assigning a weight of w g . For the t a triples, the sampling procedure first selects an index i from group g i with a probability proportional to the weights of the group.

indices in group g are chosen proportional to w g . Two indices j and k are then selected uniformly at random from groups g j and g k other than g i .


By operating directly on Python source code, Tangent is able to achieve a separation of concerns that other AD libraries do not.

we achieve compositionality with tools in the Python ecosystem, such as debuggers, profilers and other compilers.


We define the perspective projection operator P which projects an α-degree field of view (FOV) from I s to W pixels on the the tangent plane n = (θ, φ).


(9) has an insightful interpretation: each factor template F learns a multiclass SVM given by parameters w F from factors f ∈ F , while each factor is augmented with messages λ f passed from all variables related to f . Despite the insightful interpretation, formulation (9) does not yield computational advantage directly.


Instead of treating it as a constant for all words we attempt to infer its values for different words and topics respectively.

we assume that (1c) is replaced by We refer to this setting as downstream conditioning, in analogy to the upstream conditioning of [14] (which dealt with topical side information over documents).


A fundamental departure of our approach from previous approaches is to consider WM as inherently probabilistic.

instead of proposing that task set is stored in an all-or-none fashion, we wish to allow for task set-as well as all cortical representations-to be treated as random variables.


Let AE ´ µ Õ ¼ be the set of players that can be directly affected by player 's behavior.

the transfer risk parameter Õ of player with respect to player is either zero, in which case the player has no direct effect on player 's behavior; or it is constant, in which case, the public safety 1 ´½ Ü µAE µ of player with respect to player ¾ AE ´ µ is the same as for any other player in AE ´ µ.


To make a fair comparison, all algorithms adopt the same experimental setup.

for all the compared algorithms, we set the penalty parameter C = 5, and employ the same Gaussian kernel with σ = 8.


Our construction instead relies on designing F and g such that when S is "large", then we can make use of the concentration result of Lemma 1 and when S is "small", functions in F and g are deterministically close to each other.

we introduce for H ⊆ N of size h: The value of the parameters α and h will be set later in the analysis.


Since DIAL learns to communicate in a continuous channel, our results offer an illuminating perspective on this topic.

figure 7 shows that, in the switch riddle, DIAL without noise in the communication channel learns centred activations.


The system is trained by reducing the error shown below (c and s are the cover and secret images respectively, and β is how to weigh their reconstruction errors): It is important to note where the errors are computed and the weights that each error affects, see Figure 3.

note that the error term ||c − c || does not apply to the weights of the reveal-network that receives the container image and extracts the secret image.


A relation can be based on the fact that for all µ, lim α↑1 (1 − α)J µ − λ µ ∞ = 0, as explained in Section 2.

based on this, we have lim for all policies µ and probability distributions π.


Unfortunately, robustness in this context comes with a cost: when minimizing a convex loss, even a single data point can dominate the result.

any (non-constant) convex loss function exhibits necessarily unbounded sensitivity to even a single outlier [17, §5.4.1].


Then, in order to build the DAG, we draw a directed edge from l ij of depth k − 1 to l ij of depth k if and only if p ij is a sub-path of p ij .

all ascendants of l ij consist of the compressed labels obtained from sub-paths of p ij of size < k. Similar to Weisfeiler-Lehman kernel, we assign a uniform weight between parents and children.


Next, we apply the KDR method to classification problems, for which many conventional methods of dimensionality reduction are not suitable.

sIR requires the dimensionality of the effective subspace to be less than the number of classes, because SIR uses the average of X in slices along the variable Y . CCA and PLS have a similar limitation on the dimensionality of the effective subspace.


Algorithm 1 PartitionFunction(T , D) (i) .

v (T ) (i) is the set of all vertices in the trellis containing the element x i and V (T ) (i) is the set of all vertices that do not contain x i .


At the same time, we need a criterion to describe convergence of sequences of bases.

we would want to know how close an iterative process is to obtaining a self-refining basis.


This is because according to (10), when the primal-dual parameters (w, α) converge to the optimal parameters (w * , α * ), we have (∇φ i (w) + λnα i ) → 0, which means that even if the learning rate η t stays bounded away from zero, the procedure can converge.

this is the same effect as SVRG, in the sense that the variance goes to zero asymptotically: as w → w * and α → α * , we have SDCA is also a variance reduction method for SGD, which is similar to SVRG.


This demonstrates a surprising degree of robustness to β.

γ β is close to Γ * if β is adjusted to be close to β * , and the choice of β = 1/2 always yields a 2-approximation to Γ * .


On the other hand, now consider the Fredholm kernel from Eqn 6 (or similarly from Eqn 7): k F (x e , z e ) = k(x e , u)p(u) • k H (u, v) • k(z e , v)p(v)dudv, and set the outer kernel k to be the Gaussian kernel, and the inner kernel k H to be the same as target kernel k target H . We can think of k F (x e , z e ) as an averaging of k H (u, v) over all possible pairs of data u, v, weighted by k(x e , u)p(u) and k(z e , v)p(v) respectively.


Its expectation E(h 1 (x)) on a test example x equals c 1 x 1 + c2 2 .

at the same time, the expectation E(h 1 (x k )) on a training example x k is different and equals (c we experience a prediction shift of h 1 .


If the updated values of these functions using newer M-step estimates lie in these intervals, we increase the number of samples collected in the next MCEM loop.

similar to [13], we start off with the following test function for HMC-EM (Alg.


Our approach significantly outperforms existing state-of-the-art object proposal methods on the challenging KITTI benchmark.

for 2K proposals our approach achieves a 25% higher recall than the state-of-the-art RGB-D method MCG-D [14].


To improve mixing rates, we first integrate out the Dirichlet variables Φ, by exploring the conjugacy between a Dirichlet prior and multinomial likelihood.

we can integrate out Φ and perform Gibbs sampling for the marginalized distribution: where C t k is the number of times topic k being assigned to the term t over the whole corpus; is a function defined with the Gamma function Γ(•).


Next, we study how the recovery error changes as the sparsity of W varies.

we applied MMV prox on the data sets of size d = 100, m = 400, and n = 10 with k (the number of nonzero rows of W) varying from 0.05d to 0.7d, and used W − W p 2 F /(dn) as the recovery quality measure.


In this paper, we leverage recent advances in generative modeling [19,20,21].

we train an Auxiliary Classifier Generative Adversarial Network (AC-GAN [20]) to model the set of legitimate images for each class.


Algorithm 1 and Theorem 1 extend readily to the Bayesian setting in which f ⇠ GP(0, k(x, x 0 )).

since the proof of Theorem 1 is based on confidence bounds, the only change required is selecting t to be that used for the Bayesian setting in [31].


Although the maximizer of a quadratic function depends continuously on the quadratic form, its dependence is weak and thus, both papers failed to obtain an informative bound for the excess risk of RFSVM in the classification problem.

such an approach requires RFSVM and KSVM to be compared under the same hyper-parameters.


Inspired by SDA's ability to learn noise specific features in denoising tasks, we argue that in unsupervised feature learning problems the type of noise used can also affect the performance.

instead of corrupting the input with arbitrarily chosen noise, more sophisticated corruption process that agrees to the true noise distribution in the data can improve the quality of the learned features.


We performed two further experiments to answer the following questions: • What happens if we continue training HCIAE-D-NP-ATT in an adversarial setting?

we continue training by maximizing the score of the ground truth answer a gt t and minimizing the score of the generated answer ât , effectively setting up an adversarial training regime L D = −L G . The resulting discriminator HCIAE-GAN1 has significant drop in performance, as can be seen in Table .


More formally, if the product is the graph G = G 1 × G 2 , then the vertex set of G is the Cartesian product of the vertex sets of G 1 and G 2 .

in other words, for any vertex v 1 in G 1 and any vertex v 2 in G 2 , there exists a vertex (v 1 , v 2 ) in G. Additionally, the edge set of G is such that, for any edge the set of hypercube graphs (or, identically, the set of Hamming cubes) can be derived using the Cartesian product operator.


This defines a probabilistic mapping to the topic variables, which can be seen as an economical representation encoding essential information of the pose.

from a image I new , we estimate the topic proportions θ(I new ) as:  The computation of this score requires only a dot product between low dimensional unit vectors θ, so our approach represent an efficient method for matching and clustering poses in large datasets.


It operates a frequency-wise comparison in the sense that, at every frame n, the spectral coefficient v in at frequency i is compared to its counterpart vin , and the results of the comparisons are summed over i.

a small displacement in the frequency support of one observation may disproportionally harm the divergence value.


The greedy scheme is to choose S i that maximizes ∇ Si F (δ t ) ∞ .

choose the set of coordinates that correspond to maximum gradient of the function F . Intuitively this corresponds to choosing the block that promises the maximal (local) decrease in objective.


Establishing convergence of Q-learning with function approximation is substantially more difficult than for a normal Q-table for a finite MDP, and there are a number of well-known counterexamples.

finite discretization may cause a loss of an MDP's Markov property [9].


In the current work we build on these advances by considering, not how search might operate on a pre-specified semantic representation, but rather how the representation itself can be learned from data (i.e., human-produced semantic fluency lists) given a specified model of the list-generation process.

we model search as a random walk on a set of states (e.g.


We compared the performance of the CRM algorithm to a simple unconstrained no-regret algorithm that treats both the true-positive and false positive probabilities similarly, but with different weight.

the reward at stage n of this algorithm is g n (w) = f n (a n ) I {b n = 1} − wf n (a n ) I {b n = −1} for some weight parameter  w ≥ 0.


Somewhat counter-intuitively, we show that despite being computationally tractable to optimize, submodular functions cannot be minimized from samples to within a desirable guarantee, even when these functions are learnable.

we show that there is no algorithm for minimizing a submodular function from polynomially-many samples drawn from any distribution that obtains an additive approximation of 1/2 o(1), even when the function is PAC-learnable.


Then the sequence of primal and dual variables {w t , α t } t≥0 of Quartz (Algorithm 1) satisfies: where θ is defined in (7).

if we fix ≤ P (w 0 ) − D(α 0 ), then for In order to put the above result into context, in the rest of this section we will specialize the above result to two special samplings: a serial sampling, and the τ -nice sampling.


On the other hand, if the coin lands Tails, it is considered unlucky and P 1 receives an EV of −$0.50 for action Sell.

(P 1 must on average pay $0.50 to get rid of the coin).


This is much easier to do for ⇣ m,j as there are much fewer of these: individually they have higher correlation with the backpropagated gradient @L SGVB D /@b m,j , so the covariance is easier to estimate.

measuring the effect of ⇣ m,j on @L SGVB D /@b m,j is easy as ⇣ m,j is the only random variable directly influencing this gradient via b m,j .


The complete data likelihood is then Inference is typically accomplished using Markov chain Monte Carlo (MCMC) sampling [9].

although the posterior for γ has a simple closed form, p(γ|{τ i }) ∝ Γ(N + a, 1 + b), sampling from f is more complicated.


Intuitively, this objective enforces that more similar points i and j should have a lower common ancestor in the tree because the weight w i,j is large and having a smaller least common ancestor ensures that |leaves(T [i ∨ j])| is smaller.

more similar points should be separated at lower levels of the hierarchical clustering.


We believe that our method can also be extended so that to model the case where the entries of the regression vector have a finite support, even if irrationally valued, and the entries of Y are only observed through their magnitude.

we expect that the method of the present paper applies to the phase retrieval problem at least in some of the cases and this is one of the current directions we are exploring.


In the case where g is strongly convex it is possible to obtain linear convergence rates that depend on the ratio γ = µ/L as opposed to the sublinear convergence rates discussed above.

we obtain the following convergence rate on the iterates of the basic proximal-gradient method: Proposition 3 (Basic proximal-gradient method -Strong convexity) Assume (H), that g is µstrongly convex, and that we iterate recursion (3) with y k = x k . Then, for all k 1, we have: A consequence of this proposition is that we obtain a linear rate of convergence even in the presence of errors, provided that e k and √ ε k decrease linearly to 0.


To address this problem, we extend the formulism in (10) by including the kernel learning mechanism.

we assume that a set of m kernel similarity matrices K 1 , K 2 , .


In this paper, we reverse the idea of training parameters of the network given fixed inputs to instead optimizing the inputs (i.e., actions) subject to fixed parameters (effectively the transition and reward parameterization assumed a priori known in planning).

that is, as shown in figure 2, given transition T (s t , a t ) and reward function R(s t , a t ), we want to optimize the input a t for all t to maximize the accumulated reward value V . we want to optimize all actions a = (a 1 , .


This example is an enhanced version of the fused lasso [12], with some graph structure exploited to improve feature selection in biostatistic applications [6].

given some graph whose nodes correspond to the feature variables, we let f ij (x) = |x i − x j | for every edge (i, j) ∈ E. For a general graph, the proximal map of the regularizer Similar as above, each f ij is 1-Lipschitz continuous w.r.t.


These theoretical results are growing and our goal is to show how they can be leveraged to show posterior concentration rates for deep learning.

we will build on properties of deep ReLU networks characterized recently by Schmidt-Hieber (2017).


More recently, however, it became clear that STDP can exhibit different dependencies on the temporal order of spikes.

it was found that the reversed temporal order (first post-then presynaptic spiking) could lead to LTP (and vice versa; RSTDP), depending on the location on the dendrite [9,10].


The dilemma faced by our analyst is how to reconcile the logically incoherent outputs of the two detectors.

how to ascribe probabilities in a way that is logically consistent, but still retains as much as possible the expert assessments of the detectors.


To leverage the additional information from trusted labels, we propose a new loss correction and empirically verify it on a number of vision and natural language datasets with label corruption.

we demonstrate recovery from extremely high levels of label noise, including the dire case when the untrusted data has a majority of its labels corrupted.


6 Discussion and future research The LoSST algorithm is suitable for locally stationary structures, but there are obviously limits.

it will perform poorly if the generating structure changes very rapidly, or if the datapoints are a random-order mixture from multiple structures.


For a p-dimensional MHP, the intensity function of the i-th dimension takes the following form: where the constant µ i is the base intensity of the i-th dimension, N j (t) counts the number of arrivals in the j-th dimension within [0, t], and f i,j (t) is the triggering function that embeds the underlying causal structure of the model.

one arrival in the j-th dimension at time τ will affect the intensity of the arrivals in the i-th dimension at time t by the amount f i,j (t − τ ) for t > τ .


Next, we conduct experiments for multilabel classification with missing labels.

we remove a fixed fraction of training labels randomly from each dataset considered.


Let T be a trellis over A proof of Theorem 1 can be found in the Appendix in Section E. By making a minor alteration to Algorithm 1, we are also able to compute the value of and find the clustering with maximal energy.

at each vertex in the trellis, v, store the clustering of D(v) with maximal energy (and its associated energy).


As pointed out in [14], the definition of user expectation is different in this case.

in our setting, each player selects arms based on their empirical means, whereas in the Bayesian setting, it is possible for a player to also consider posterior distributions of arms for decision making.


Since the approximative weights (11) can be evaluated sequentially, the idea is to start with p = 1 and then increase p until the weights have, in some sense, converged.

in our experimental work, we have used the following simple approach.


We know that, if the hypothesis of inequality (11) are satisfied, the linear loss based fairness is close to the EO.

these two quantities are closer when ˆ is small.


This paper develops upper and lower bounds on the influence measure in a network, more precisely, the expected number of nodes that a seed set can influence in the independent cascade model.

our bounds exploit nonbacktracking walks, Fortuin-Kasteleyn-Ginibre type inequalities, and are computed by message passing algorithms.


We have outlined a theory of elicitation complexity which we believe is the right notion of complexity for ERM, and provided techniques and results for upper and lower bounds.

we now have tight bounds for the large class of Bayes risks, including several applications of note such as spectral risk measures.


On the other hand, many researchers have found convolutional networks (ConvNets) [17] [18] are useful in extracting information from raw signals, ranging from computer vision applications to speech recognition and others.

time-delay networks used in the early days of deep learning research are essentially convolutional networks that model sequential data [1] [31].


We address this problem using Dirichlet processes [3], a tool from nonparametric Bayesian analysis for learning mixture models whose number of components is not fixed, but instead estimated from data.

we extend the recently proposed hierarchical Dirichlet process (HDP) [4,5] framework to allow more flexible sharing of mixture components between images.


Each CS decision tree is trained in an iterative way and, similar to the case of standard decision trees, a decision about whether to branch new nodes or produce a leaf is taken based on a subset of the training samples Z ⊆ D. However, in contrast to the standard setting, the learning process depends on the order in which nodes of the tree are grown because split functions depend on t (<m) which in turn is affected by the node ordering.

we impose an explicit ordering on the recursive calls of function g described in Equation ( 4).


Our first measure is based on the hypothesis that human observers have an "effective order parameter".

their performance on the target curve tracking task behaves like e −N KH where K H is an effective order parameter which difference from the true order parameter K might reflect a human bias towards straight lines or ecological shape priors.


Even though great care was taken in collecting the highest quality training data, there were a number of imperfections in the training data that could not be avoided: (a) The smallform-factor, low-cost cameras presented significant differences in their default settings.

the white balance of the two cameras were somewhat different; (b) To maximize image quality, the automatic gain control and automatic exposure were activated.


Unfortunately, the extension of this methodology to the more general tensor setting presents some difficulties.

we shall prove in this paper that the tensor trace norm is not a tight convex relaxation of the tensor rank.


0 through the convergent subsequence, where V (•) satisfies the following ODE: To solve ODE (7), we rotate the coordinate to decouple each dimension.

there exists an eigenvalue decomposition such that Note that, after the rotation, e 1 is the optimum corresponding to v 1 .


The symmetric formulation can be obtained by decoupling ρ and θ (r) in the bi-linear constraint ρ(r) θ(r) = θ.

we introduce θ(r) = θ (r) ρ(r) and perform the optimization w.r.t.


Our goal is to estimate model parameters θ 0 ∈ Θ given only access to sentences x ∼ P θ0 .

suppose we have an observation function φ(x) ∈ R m , which is the only lens through which an algorithm can view the data.


Unsupervised methods, on the other hand, do not require annotations but yield disentangled representations that are usually uninterpretable and dimension-wise uncontrollable.

the user has no control over the semantic encoded in each dimension of the obtained codes.


From the first observation above, we will now introduce a new definition of margin, m, which performs a maximization over a superset Several examples are NP-hard m for H (DAGs, trees or cardinality constrained sets), but poly-time m for H being a set of binary strings.

we can encode any DAG (in H) as a binary string (in H), but not all binary strings are DAGs.


In particular, under linear sparsity (k = αn for some constant α), the theorem shows that m ≍ 2α MAR • SNR n log n measurements are necessary for sparsity recovery.

similarly, if m/n is bounded above by a constant, then sparsity recovery will certainly fail unless when SNR • MAR is bounded, the sparsity ratio k/n must approach zero.


Previous work has used a Boltzmann weighted average operator [20,5].

this operator has been used by [5] to propose an iterative algorithm converging to the optimal maximum reward policy inspired by the work of [15,39].


The model is obviously incomplete, and requires testing in richer environments.

we have yet to explore how habits get created from actions as the maximal advantage goes to o. I am very grateful to Christian Balkenius, Bernard Balleine, Tony Dickinson, Sham Kakade, Emmet Spier and Angela Yu for discussions.


We focus on linear-functions of the form f (x) = sign(p) for a quantity p = w x, w ∈ R d , called the margin.

the algorithm maintains a set of K weight vectors.


This leads to a novel label tree learning technique that is more efficient and effective.

we eliminate the onevs-all training step while improving both efficiency and accuracy in testing.


A convex cone K is a set of points closed under linear combinations of positive coefficients.


Task Module Given a subtask graph G, the remaining time steps step t ∈ R, an eligibility vector e t and a completion vector x t , we compute a context embedding using recursive-reverse-recursive neural network (R3NN) [5] as follows: where [•] is a concatenation operator, b θ , t θ are the bottom-up and top-down encoding function, φ i bot,a , φ i top,a are the bottom-up and top-down embedding of i-th AND node respectively, and φ i bot,o , φ i top,o are the bottom-up and top-down embedding of i-th OR node respectively (see Appendix for the detail).

the w i,j + , Child i , and P arent i specifies the connections in the subtask graph G. w i,j + = 1 if j-th OR node and i-th AND node are connected without NOT operation, −1 if there is NOT connection and 0 if not connected, and Child i , P arent i represent a set of i-th node's children and parents respectively.


However, it should be noted that these degrees should be treated differently because only "relevant" images represent a "concept" in the user's mind, while "non-relevant" images may represent a number of other concepts different from user's interest.

while it is meaningful to treat the degree of relevance as a degree of membership to the class of relevant images, the same does not apply to the degree of non-relevance.


We use the mean of all the entries of the output as a hard threshold to round each entry.

if an entry is less than the threshold, it is rounded to 0 and to 1 otherwise.


In this paper we tackle the problem of obtaining a tight characterization of the sample complexity which a particular learning rule requires, in order to learn a particular source distribution.

we obtain a tight characterization of the sample complexity required for large (Euclidean) margin learning to obtain low error for a distribution D(X, Y ), for X ∈ R d , Y ∈ {±1}.


We next empirically study the dependence of sampling times on key factors that govern our theoretical bounds.

we run Markov chains on chain-structured Ising models on a partition matroid base and DPPs on a uniform matroid, and consider estimating marginal and conditional probabilities of a single variable.


The convergence rate for minimizing a smooth function, roughly speaking, scales only like 1/t.

this rate does not improve even when the function is also strongly convex.


An efficient approach to solving the constrained optimization problem (10) is the mirror descent algorithm (MDA) with Bregman divergence chosen to be generalized Kullback-Leibler divergence [2,18,21].

let f (✓ d ) denote the cost function in (10), then the MDA updates the MAP estimate of ✓ d iteratively according to: ✓ d,`d enotes the estimate of ✓ d,`a t the `-th iteration, T d,`d enotes the step-size of MDA, and (x, y) is the Bregman divergence chosen to be (x, y) = x T ln(x/y) 1 T x + 1 T y.


The fact that F [v] < F max can be seen either by simply noting that χ 2 [v] ≥ 0, or from the data processing inequality, which applies not only to Kullback-Leibler divergence, but also to Rényi divergences [12,13,1].

the variance in the firing rate explained by a given dimension F [v] cannot be greater than the overall variance in the firing rate F max .


However, we note that for the purpose of an associative memory, the recovery phase should be neurally feasible and computationally simple.

each node (or storage unit) should be able to recover the coordinated associated to it locally by applying simple computation on the information received from its neighboring nodes (potentially in an iterative manner).


One can verify that ∆ = |1 − α| < 10 −7 is likely in [6].

the required sample size could be O 10 14 .


neuromodulation, input statistics) most rules are sensitive to parameters.

recent work has focused on two different formulations of spike-timing-dependent plasticity rules.


What's more, TrailBlazer exploits MDPs with specific structure by exploring only a fraction of the whole search space when either κ or d is small.

we showed that if the set of near-optimal nodes have non-vanishing action-gaps, then the sample complexity is O(1/ε 2 ), which is the same rate as Monte-Carlo sampling.


The algorithms from [Tewari and Bartlett, 2007] can be used to compute an optimistic minimax policy.

for each (s, a) ∈ F , its transition function is chosen pessimistically from U(s, a).


The cumulative regret of an allocation policy A after time n is: where X * t is the reward that the algorithm would have received at time t if it had pulled the optimal arm i * .

r n is the cumulative loss due to the fact that the allocation policy does not always pull the optimal arm.


We introduce a new family of online learning algorithms based upon constraining the velocity flow over a distribution of weight vectors.

we show how to effectively herd a Gaussian weight vector distribution by trading off velocity constraints with a loss function.


The key difference from traditional empirical Bayes methods is that we are able to prove a regret bound for a BO method that uses estimated parameters to construct priors and posteriors.

we use frequentist concentration bounds to analyze Bayesian procedures, which is one way to certify empirical Bayes in statistics [49,13].


The first approach investigates the problem from a statistical perspective.

it assumes that the observations are generated by a (sparse) linear combination of the features, and investigates the asymptotic or probabilistic conditions required for Lasso to correctly recover the generative model.


We say that R is separately exchangeable if its distribution is invariant to separate permutations on its rows and columns.

for each n, m ≥ 1 and each pair of permutations π ∈ S n and σ ∈ S m , in MATLAB notation.


The functions β CQ and β SD are derived under local alternatives and asymptotic assumptions that are similar to the ones used here to obtain β RP .

all three functions can be obtained allowing p/n to tend to an arbitrary positive constant or infinity.


Define the following three MDPs in Φ obtained starting from φ and changing the kernels for (state, action) pairs in U 1 ∪ U 2 .

let (p, q) be some transition and reward kernels.


. . , z n ] and then learn regressors V in the second stage.

z is obtained by directly solving (1) but without the L 1 penalty term: where M = Z T Z.


From this observation, the estimation problem inherent to perception is linked to the definition of an adequate generative model.

the simplest generative model to describe visual motion is the luminance conservation equation.


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

when applying their algorithm to the Euclidean space, one must first discretize the space of possible centers, and account for the error introduced by this discretization.


3 The problem and our approach Formally, we have a random variable X that takes values in a set of combinatorial objects M. For example, X could be a random tree drawn from the collection M of all trees in some graph G. We can think of the members of M as being the valid configurations among all possible sets in 2 V , so that any configuration not in M should get a probability of zero.

we consider distributions over the configurations M of the general form where F : 2 V → R is the objective function and • is the Iverson bracket.


This goal is achieved by using parameterized 1-dimensional convolutional filters.

the kth of K filters for channel c is The frequency !


Now, in a real world scenario, it is likely that people decide rankings of candidates based on a few issues such as war, abortion, economy and gay marriage.

an individual will decide the ranking of the candidates based on the opinions of candidates on these issues.


This section demonstrates that the continuous problem (2) provides an exact relaxation of the combinatorial problem (1).

we provide an explicit formula that gives an exact correspondence between the global minimizers of the continuous problem and the global minimizers of the combinatorial problem.


In contrast, the operators we defined above have argument lists containing only those variables that are free parameters.

when the operator move(A, B) takes two arguments, A and B , it means that the other variables (such as C , the block under A) are derivable from the relations in the rule schema.


We show that our algorithm makes a conservative use of the privacy budget.

if the underlying non-private learner yields a classification error of at most α ∈ (0, 1), then our construction answers more queries, by at least a factor of 1/α in some cases, than what is implied by a straightforward application of the advanced composition theorem for differential privacy.


This regularization ranges from element-sparsity for variable selection to a low-rank matrix for dimension reduction [1,2,3,13,21].

for any ∈ N d , denote the -th row vector of P by P and P = ( .


More interesting results follow when only some forms of independence hold.

it has been shown that Dec-MDPs with both transition-and observation-independence, but not reward-independence, are NP-complete [8,7].


From Figure 1(a) we can see that TSDE outperforms all the three algorithms in randomly generated MDPs.

there is a significant gap between the regret of TSDE and that of UCRL2 and TSMDP.


We use key algorithmic components of SIEVE-STREAMING [Badanidiyuru et al., 2014], namely greedy thresholding and binary search, combined with a novel analysis to prove a constant factor approximation for -weakly submodular functions (defined in Section 3).


Additionally, distribution-specific learners and learners with query access to the function have been studied [12,7].

membership queries tend to make agnostic learning easier.


Identify the eigenvector v that has the largest inverse participation ratio among the q eigenvectors in U .


The difference is that in this case, the task hierarchy encodes extra information that cannot be deduced just from the models.

the task hierarchy tells the agent that good policies consist of gold/wood collection moves followed by deposit moves.


We see that Zell consistently outperforms the other algorithms across the five data sets.

the performance of Zell is encouraging on the FRGC data set which has the largest numbers of clusters and samples.


A is entry-wise Gaussian and the feasibility is achieved via increasing data dimension.

we then minimize we discretize the ODE (11) for p = 2 with integrators of different orders, i.e., s ∈ {1, 2, 4} and compare them against GD and NAD.


While models usually assume each single trial is independent of the others, trials are actually dependent when the decision maker has control over trial termination.

the decision maker has a motivation to terminate each trial quickly to get the reward, and proceed to the next one.


Some of these include: • Multivariate Regression with Covariance Estimation (MRCE-ℓ 2 ): With the task inverse covariance matrix Σ −1 = I K and the bias term set to zero, our model results in the ℓ 2 regularized weights variant of the MRCE model [17] which would be equivalent to minimizing the following objective: • Multitask Relationship Learning for Regression (MTRL): With the output inverse covariance matrix Ω −1 = I K and the sparsity constraint on Σ −1 dropped, our model results in the regression version of the multitask relationship learning model proposed in [22].

the corresponding objective function would be: In [22], the − log |Σ −1 | term is dropped since the authors solve their cost function in of Σ and this term is concave in Σ.


We scale up the algorithm to terabytes of real-world SS and BT data that contains hundreds of millions of users and hundreds of thousands of features, by leveraging the scalability characteristics of the algorithm and the inherent structure of the problem including data sparsity and locality.

we demonstrate two somewhat orthogonal philosophies of scaling algorithms to large-scale problems, through the SS and BT implementations, respectively.


We first study the effectiveness of the algorithm, measured by the EV value of its output after the last sample, and compare it to the nuclear norm based online RPCA (OR-PCA) algorithm [8].

we vary the intrinsic dimension d from 0.02p to 0.5p, with a step size 0.04p, and the corruption fraction ρ from 0.02 to 0.5, with a step size 0.04.


Therefore, the value function of any policy is a linear combination of the expected discounted frequency (count) of encountering each state-action feature.

the value function of the expert's policy is approximated by a linear combination of the empirical averages of the features, estimated from the demonstration (the trajectories).


Given access to a vector u = m k=1 α k (u)A k + P A ⊥ u (where P A ⊥ is the projection onto the orthogonal complements of the range of A), some recovered columns A 1 , .

. . , A r , and corresponding rows of A † , we may zero out the components of u corresponding to the recovered columns of A. Let- u is orthogonal (in the •, • C space) to the previously recovered columns of A.


As a proof of concept we present a BCI classifier that is robust to changes in the level of parietal α-activity.

the EEG decoding still works when there are lapses in vigilance.


However, as reported in previous studies [15,19], it is very important to take into consideration the relationships between different bits in HFL.

to learn compact hash codes, we should coordinate the learning of hash functions for different bits.


From the view of learning with data dependent hypothesis spaces, theoretical analysis of GNKR is established to illustrate its generalization bound and learning rate.

the fast learning rate arbitrarily close to O(m −1 ) is obtained under mild conditions, where m is the size of subsampled set.


This leads us to a Bayesian nonparametric approach for constructing sparsity-inducing penalties.

we have illustrated the construction of the LOG and EXP penalties.


. . , Λ K }, our goal of graph regularization parameter selection is to choose one Λ ∈ G n , such that the true graph E is contained in E( Λ) with high probability.

we want to "overselect" instead of "underselect".


The neural network is composed of primary filters, while the strength of the network will be fully disclosed through secondary filters in computation.

we develop versatile filters in both spatial and channel dimensions.


. . , p k ) ∈ k i=1 W i be any vector with p l = 0 for l = j.

for (x, y) ∈ D we define we encode the data into the subspace Z, and select any point p in W , then decode the concatenated vector.


This assumption originates from the fact that the convergence of MP-and FW-type algorithms fundamentally depends on an alignment assumption of the search direction returned by the LMO (i.e., z t in Algorithm 1) and the gradient of the objective at the current iteration (see third premise in [39]).

for Algorithm 1, the LMO is assumed to select a descent direction, i.e., ∇f (x t ), z t < 0, so that the resulting weight (i.e., γ for Variant 0) is always positive.


The resulting state of applying a sequence of complete actions π = a 1 , ..., a n in s with respects to D i is defined as: A planning problem with incomplete domain D is P = D, I, G where I ⊆ F is the set of propositions that are true in the initial state (and all the remaining are false), and G is the set of goal propositions.

an action sequence π is considered a valid plan for P if π solves the problem in at least one completion of D . Given that D can be exponentially large in terms of possible preconditions and effects, validity is too weak to guarantee on the quality of the plan.


We consider a pseudo-Euclidean inner product defined as follows: Let C = ADA T where D is a diagonal matrix with non-zero diagonal entries, and define •, • C by u, v C = u T C † v. When D contains negative entries, this is not a proper inner product since C is not positive definite.

gives that the columns of A are orthogonal in this space.


To limit the contribution of the top gate we force its main weight to be equal to +1 but its bias to be arbitrary (this is necessary also to avoid cases where the input to the top threshold gate is equal to 0).

if the lower gate as weights w i and activation S = N i=0 w i x i , then the final output is given by O = sign(sign S + b) in the pure threshold gate case, and by O = sign(ReLU (S) + b) in the ReLU case.


We need to see if − a n , 1 ∈ W . Note that W is the convex set formed by starting with the graph of x → e ηx on the domain [−1, 1], including the line segment connecting this curve's endpoints (−1, e −η ) to (1, e ηx ), and including all of the points below this line segment but above the aforementioned graph.

w is precisely the set It remains to check that 1 is sandwiched between the lower and upper bounds at x = − a n . Clearly the lower bound holds.


In this work, we restrict our attention to closeness measures that arise from norms.

we seek a distance matrix M so that, where • is a norm, W is a symmetric non-negative weight matrix, and ' ' denotes the elementwise (Hadamard) product of two matrices.


When f (A, w) ⌧ f ⇤ , CombCascade can distinguish A from A ⇤ without learning the expected weights of all items in A.

combCascade acts implicitly on the prefixes of suboptimal solutions, and we choose them in our analysis such that the probability of observing all items in the prefixes is "close" to f ⇤ , and the gaps are "close" to those of the original solutions.


While it is still common practice in both sparse coding and manifold learning to collect data in an orderless fashion, other work has used time-series data to learn spatiotemporal representations [57,41,30] or to disentangle form and motion [6,9,13].

the combination of topography and temporal coherence in [30] provides a strong motivation for this work.


It is surprising because the intermediate posteriors have something to say about the data, even if they condition on hyperparameter configurations that do not maximize the marginal likelihood.

when we harbour uncertainty about η, should we be more Bayesian?


To address this challenge, we approximate 4 using a simple penalty method.

given ρ > 0, we arrive at the next formulation 5 by introducing two auxiliary variables Z 1 and Z 2 with some penalty function, such as the squared Frobenius norm.


We define the queue-regret as the difference in expected queue lengths for the two policies.

the regret is given by: Here (t) has the interpretation of the traditional MAB regret with caveat that rewards are accumulated only if there is a job that can benefit from this reward.


For subset selection under additive noise, POSS using By comparing Theorem 1 with 3, we find that the approximation bounds of POSS and the greedy algorithm under multiplicative noise are nearly the same.

for the submodular case (where γ X,k = 1 for any X and k), they are exactly the same.


Updating α with fixed β can be achieved by computing Updating β can be done via the Newton-Raphson method β ← β − l (β) l (β) , where The likelihood ratio distance is related to several existing methods.

we show that under the exponential family distribution, it leads to several widely used distance measures.


Our study suggests that the filter properties of STDP originating from the dynamics of pre-and postsynaptic activity contributions are in fact exploited for learning in the brain.

shifts in baseline rates, as well as the frequency and the respective phases of pre-and postsynaptic rate modulations induced by theta oscillations could be tuned to match the values that make STDP most susceptible for synaptic modifications.


Previously information theoretical approaches to neural coding have been used to quantify the amount of information that a neuron or a neural network is able to encode or transmit [6][7][8].

algorithms based on the maximization of the mutual information between the output and the input of a network, also called infomax principle [9], have been used to detect the principal (or independent) components of the input signal, or to reduce the redundancy [10][11][12].


However, we use different types of basis functions to analyze varying degrees of adaptation.

we study whether data-dependent basis functions improve over data-agnostic basis functions.


Neurons however, also possess a variety of potassium currents that are able to limit these large excursions in voltage.

recent observations show that A-type potassium currents are abundant in dendrites of a variety of neurons [7] .


Hand and Voroninski [8] further establish that a fully connected generative network with weights following Gaussian distribution can be inverted given only compressive linear observations of its last layer.

they show that under mild technical conditions Problem (3) has a favorable global geometry, in the sense that there are no stationary points outside of neighborhoods around the desired solution and its negative multiple with high probability.


2 can be rewritten as L π = p 0 r 0 + c/r 1 , where r i is the mean firing rate when the inputs are generated from f i , and p 0 is the fraction of time when f 0 is applicable (as opposed to f 1 ).

if the rate of change is slow compared to neuronal firing rates, then optimal processing amounts to minimizing the "spontaneous" firing rate during f 0 and maximizing the "stimulus-evoked" firing rate during f 1 .


For that purpose, define a matrix Φ(s, s) = 1 s=θ(s) Algorithm 2: Solve (3.1) in Line 15 of Algorithm 1 Input: z s , q s -sorted such that z s is non-decreasing, indexed as 1 . . . n Output: ξ s,a -optimal solution of (3.1) // Determine the threshold 5 do where s ∈ S, s ∈ S, and 1 represents the indicator function.

each column corresponds to a single aggregate state with each row entry being either 1 or 0 depending on whether the original state belongs into the aggregate state.


Here we start our experiment with triple pole-balancing using exactly the same configuration, and then we let KBSF refine its model M by incorporating more sample transitions through update rules (3) and ( 4).

we used Algorithm 2 with a 0.3-greedy policy, t m = t v = 10 6 , and n = 10 7 .


The corresponding sub-counts n vjk can be viewed as the result of the contribution of underlying biological process k to the expression of gene v in sample j of domain d. The probability parameter p (d) j , which depends on the sample index, accounts for the potential effect of varying sequencing depth of sample j in domain d. More precisely, the expected expression of gene v in sample j and domain d is , and hence the term kj ) can be viewed as the true abundance of gene v in domain d, after adjusting for the sequencing depth variation across samples.

it comprises of contributions from both domain-dependent and globally shared latent factors, where the amount of contribution of each latent factor can automatically be learned for the sample relevance across domains.


We define U (I) to be the maximum payoff reachable from a history in I, and L(I) to be the minimum.

u (I) = max z∈Z,h∈I:h z u P (I) (z) and L(I) = min z∈Z,h∈I:h z u P (I) (z).


A special case of MDP under which PSRL has been recently extensively studied is MDP with state resetting, either explicitly or implicitly.

in [Osband et al., 2013;Osband and Van Roy, 2014] the considered MDPs are assumed to have fixed-length episodes, and at the end of each episode the MDP's state is reset according to a fixed state distribution.


This is sensible since, by the design of a MSR, trader i's belief-dependent action influences the action of each of (rational) traders i + 1, i + 2, . . . so that the action of each of these successors, in turn, has a role to play in determining the market impact of trader i's belief.

if τ j = τ > 0 ∀j ≥ 1, then the aggregation weights satisfy the inequalities α n j+1 / α LMSR assigns progessively higher weights to traders arriving later in the market's lifetime when they all exhibit identical constant risk aversion.


By treating the perturbations as stochastic we may obtain a more realistic estimate (though no longer a strict bound) on the resulting error.

let us describe the error functions log e ts (x s ) for each x s as a random variable with mean zero and variance σ 2 ts .


Moreover, the last propositions show that we indeed can estimate the correct hull of the sparsity pattern if the sufficient condition is satisfied.

if we can make the groups such that the between-group correlation is as small as possible, we can ensure correct hull selection.


Since the rank function is neither differentiable nor convex, its relaxation such as Schatten-p norm has been used in respective optimization formulations.

we consider the smoothed nuclear norm (i.e., Schatten-1 norm) minimization [18,20] that corresponds to ] d×r is a given matrix with missing entries, Ω indicates the positions of known entries and λ is a weight parameter and ε > 0 is a smoothing parameter.


In this section, we focus on the development of theory of ERM for risk minimization.

we learn a model w by solving the following ERM problem corresponding to (1): where z 1 , .


Again, our proposed methods surpasses the performances of the existing methods.

the convergence of Hybrid was significantly fast and stable.


Corollary 1 For the location family where each ζ j is a normal distribution with mean zero and with fixed variance, or Gumbel distribution with mean zeros and fixed shape parameter, l( θ; D) is concave.

the log-likelihood function for P-L is concave.


However, if the algorithm would choose a different center depending on the specific value of α used from the interval [α , α h ], then we fork the algorithm, making one copy for each possible center.

we partition [α , α h ] into a finite number of sub-intervals such that the next center is constant on each interval.


The quality of the generated answers of our mQA model on this dataset is evaluated by human judges through a Turing Test.

we mix the answers provided by humans and our model.


However, all the works above focus only on the empirical problem, and can all be seen as instances of SAA (ERM) approach to the stochastic optimization (learning) problem (1).

the analyses in these works bounds suboptimality on the training objective, not the population objective (1).


In this work, we develop a procedure that can take arbitrary probabilistic graphical models, represented as factor graphs over discrete random variables, and compile them into chemical reaction networks that implement inference.

we show that marginalization based on sum-product message passing can be implemented in terms of reactions between chemical species whose concentrations represent probabilities.


Here, we present a constructive proof of Theorem 1.

we will describe an algorithm to determine the function values from Q which will be the original f as long as properties P1 and P2 are satisfied.


The driving force behind such results are the isoperimetric inequalities which can be extended from uniform to general logconcave distributions.

computing the volume of a convex body can be seen as a special case of integration of a logconcave function, and there has been a number of major results on mixing time for sampling from logconcave distributions [17,18].


. . , A k } denote the partition induced by T . Let j(A) denote the depth of A and note that λ(A) = 2 −j(A) where λ is the Lebesgue measure on R d . Define T to be the collection of all DDTs and A to be the collection of all cells corresponding to nodes Let M be a dyadic integer, that is, M = 2 L for some nonnegative integer L. Define T M to be the collection of all DDTs such that no terminal cell has a sidelength smaller than 2 −L .

no coordinate is split more than L times when traversing a path from the root to a leaf.


Thirdly, they can both explain Win-Stay strategy.

under the maximization assumption, choosing the option with maximum value in RL model, the rewarded option (Win) will reinforce this choice (i.e.


We show that the use of a probabilistic model over link graphs allows us to represent and exploit interesting subgraph patterns in the link graph.

we have found two types of patterns that seem to be beneficial in several places.


D can be asymmetric, where d ij = d ji for some pairs of data points.

how well data point i represents data point j can be different from how well j represents i.


In addition, we assume the existence of a set of k A equality and k B inequality constraints on the random variables, that is, A(x) = a, where A : D → R k A , a ∈ R k A and B(x) ≤ b, where Both equality and inequality constraints restrict the possible combinations of values the random variables X can assume.

we set f (x) = 0 whenever any of the constraints are violated and constrain the domain of integration, denoted D, for the normalization constant correspondingly.


Therefore, to compute all splits for a given depth t, we introduce a data structure to record the most recent splits at depth t − 1.

we create a mapping between each feature value and the node i at depth t that it is currently assigned to.


Thus, for a fair comparison, these hyperparameters are specified using cross-validation on a subset of data.

we further partition the training set into 80% for learning and 20% for validation.


For example, when X's are frequent but Y is presented, individuals are predisposed toward producing the X response, and this predisposition must be overcome by the perceptual evidence from the Y. Jones and Braver (2001) also performed an fMRI study of this task and found that anterior cingulate cortex (ACC) becomes activated in situations involving response conflict.

when one stimulus occurs infrequently relative to the other, event-related fMRI response in the ACC is greater for the low frequency stimulus.


In fact, SGD is actually overall better, but as b increases, its performance degrades more quickly, eventually performing worse than AG.

even in the least favorable scenario for AG (high L(w ) and small , see the tables in Sec.


We evaluate the proposed distance learning technique by semi-supervised clustering.

we first learn a distance function from the given pairwise constraints and then apply the learned distance function to data clustering.


In our problem, this translates to permitting the generating process to have an unrestricted number of latent components, but requiring that only a small number of them contribute to the composition of the histogram represented by any data instance.

the latent components must be learned such that the mixture weights with which they are combined to generate any data have low entropy -a set with low entropy implies that only a few mixture weight terms are significant.


Our notion of margin ρ in (ρ, ∆, ε)-separated instances is analogous to the margin separation notion used by the above results on spectral clustering [15,6].

we require a margin of ρ = Ω(∆/ε 2 ) where ∆ is our scale parameter, with no extra √ k factor.


Other models may have not allowed conclusions about reliability estimates in the brain.

[13] introduced an alternative model that also leads to equivalence with the drift diffusion model, but explains differences in behaviour by different mean measurements and their representations in the generative model.


In this section, we review some landmark methods that are closely related to our work.

we focus on neural-network-based methods.


To describe people's internal and external foraging behavior, various random walk models have been proposed.

internal foraging has been modeled as sampling: in order to gather relevant information for making a decision, people draw samples from a mental representation using random-walk algorithms such as Markov chain Monte Carlo (MCMC).


In electrophysiology, the most popular connectivity methods are variations on the autoregressive (AR) framework [5].

granger causality (GC) and related methods, such as partial directed coherence and directed transfer function, have been successfully applied to many kinds of neuroscientific data [6,7].


As far as we know, this has remained an unanswered issue, with no suggested alternatives.

all normalization methods, including BN, use an L 2 normalization (variance computation) to control the activation scale for each layer.


But from a probabilistic standpoint it seems contrived.

it produces a very confident posterior covariance, which does not capture global error.


One way to deal with the first problem is to search until we find a solution which covers an acceptable percentage, p, of the training data.

the search oracle is modified to be |ρ ≥ n * m * p. The second problem is addressed in the next section.


These advantages help make Advise robust to infrequent and inconsistent feedback, and fair well with an inaccurate estimate of C. A primary direction for future work is to investigate how to estimate Ĉ during learning.

a static model of C may be insufficient for learning from real humans.


However, for upstream supervised topic models, the discriminant function for prediction involves an intractable computation of posterior distributions, which makes the max-margin training more delicate.

we present a joint max-margin and max-likelihood estimation method for learning upstream scene understanding models.


Under mild conditions, we show that the proposed algorithm has policy improvement bound by theoretical analysis.

the policy π is uniformly as good as, or better than the behavior policy π.


However, this can be computed by finding the parameters of an OU process that matches q 0 in the moment matching Kullback-Leibler divergence.

if q * ({x t }) minimises D[q 0 ({x t })||q * ({x t })], then the parameters of q * are given by where m bw t and V bw t are computed by the backward Kalman-Bucy filtering equations.


Thus, we need to minimize (f 1 , g) + Ω(f 1 , g) with respect to f 1 and g. Since both f 1 and g are gradient boosted trees, we naturally adopt a stage-wise approximation for the objective.

we define an impurity function which on the one hand approximates the negative gradient of (f 1 , g) with the squared loss, and on the other hand penalizes the initial acquisition of features by their cost c α .


The connections through the inference network are somewhat analogous to our lateral connections.

there are inference paths from observed inputs to reconstructed inputs that do not go all the way up to the highest layers.


In this paper we propose intuitively reasonable sufficient conditions under which active learning achieves lower label complexity than that of passive learning.

we focus on the A 2 algorithm [BAL06] which works in the agnostic setting.


Note that neither of these steps depends explicitly on the observed values of the neural signal u t or the decoded output xt .

co-adaptation can be simulated without ever actually generating the stochastic process of intention, encoding and decoding.


Temporal frequencies were chosen so that the set of cells at each spatial frequency had the same pattern of velocity tunings.

the sets tuned to low (0.0625 cycles/pixel), medium (0.125 cycles/pixel), and high (0.25 cycles/pixel) spatial frequencies had velocity tunings of 0.25, 0.5, 1.0, 2.0, and 4.0 pixels per time frame.


The results achieved by deep neural networks for prediction tasks have been impressive in domains where data is structured and available in large amounts.

convolutional neural networks (CNNs) [14] have shown to model well the local appearance of natural images at multiple scales, while also representing images with some invariance through pooling operations.


Let S u,s denote the set of vertices which are at distance s from vertex u in the graph defined by edge set E 1 .

i ∈ S u,s if the shortest path in G 1 = ([n], E 1 ) from u to i has a length of s. Let T u denote a breadth-first tree in G 1 rooted at vertex u.


The controller's location-choice actions simply receive a reward equal to the priority map value at the glimpse location, which encourages the controller to choose locations according to the priority map.

for locations (x, y) 1 , ..., (x, y) k sampled from the controller, define R iloc = P (x,y)i .


To overcome this limitation, we present a novel hybrid model, EigenNet, that uses the eigenstructures of data to guide variable selection.

it integrates a sparse conditional classification model with a generative model capturing variable correlations in a principled Bayesian framework.


We consider a representation for which we expect the weight matrix to be dense.

we only consider the value of the player's hand and of the dealer's hand and we generate features as the Cartesian product of these two discrete variables plus a feature indicating whether the hand is soft, for a total of 280 features.


Since n 1 n 2 50 for all patient sets, our collection of twosample problems is genuinely high-dimensional.

we have 14 × ( 6 2 + 4 2 + 6 2 ) = 504 problems under H 0 and 14 × (6 • 4 + 6 • 4 + 6 • 6) = 1176 problems under H 1 -assuming that every gene set was differentially expressed between two sets of patients with different cancers, and that no gene set was differentially expressed between two sets of patients with the same cancer type.


We defer all the details to Appendix D for saving space.

we introduce a simple variational inference method to approximate the optimal posterior, which turns out to perform well in practice.


When used to guide decision-making, linear regression analysis typically treats estimation of regression coefficients separately from their use to make decisions.

estimation is carried out via ordinary least squares (OLS) without consideration of the decision objective.


Nevertheless, other representations, for example representing the parameters of a mean-field approximation to the true posterior [5,9,10], would also be possible and similarly informative about uncertainty.

we use Gibbs sampling, as it allows for neurally plausible recall dynamics [7].


The present work can thus be seen as exposing a close connection between statistical estimation and numerical optimization of convex functions.

our results imply that the local minimax complexity can be viewed as a computational analogue of Fisher information in classical statistical estimation.


(2) There is no mechanism for distinguishing between certain low-frequency image components and a bias field.

the method may mistake signal for noise in certain cases when removal of the true signal reduces the entropy of the brightness distriibution.


We term this method as learning with target priors (LTP).

lTP learning seeks parameter θ that maximizes the log likelihood of f θ (X) on a uncorresponded training set with regards to p(y).


Given the output q of Policy 1, we obtain an approximate Q-value for each (continuous) state-action pair (x, a) 2 Z via the nearest-neighbor average operation, i.e., Q T h (x, a) = ( NN q) (x, a); here the superscript T emphasizes that the algorithm is run for T time steps with a sample size of T . As a main result of this paper, we obtain finite-sample analysis of NNQL policy.

we find that the NNQL policy converges to an "-accurate estimate of the optimal Q ⇤ with time T that has polynomial dependence on the model parameters.


Each neuron in the optimal network will seek to reduce this loss function by firing a spike.

the i th neuron will spike whenever L(no spike in i) > L(spike in i).


If we break down problem (1) into element-wise product, it becomes: where A ij:ab corresponds to the edge similarity between edge Here H ∈ {0, 1} 2n×n 2 is a selection matrix over the elements of x sufficing assignment constraints according to (1).

we relax x into continuous domain and let f prod (x ia , x jb ) = x ia x jb : We generalize problem (3) by replacing f prod with f δ : where f δ is a 2D quasi-delta function in the continuous domain (f δ (x, y) = 1 if x = 1 and y = 1, and f δ = 0 otherwise).


The resulting classifier defines a nonlinear separating boundary in the original feature space.

equations ( 1) and (2) imply that the classification function f K (x) is directly proportional to the signed distance from the input point to the separating boundary computed in the higher dimensional space defined by the mapping Φ K . the function output depends only on the projection of vector Φ K (x) onto w and completely ignores the component of Φ K (x) that is perpendicular to w. This suggests that in order to create a displacement of Φ K (x) that corresponds to the differences between the two classes, one should change the vector's projection onto w while keeping its perpendicular component the same.


We used posterior predictive checking to assess the extent to which our model is a "good fit" for the NHC email network [16,17].

we defined four network statistics (i.e., four discrepancy functions) that summarize meaningful aspects of the NHC network: generalized graph transitivity, the dyad intensity distribution, the vertex degree distribution, and the geodesic distance distribution.


In fact, we could make the constant arbitrary small for large enough T . In our experiments we focused on pipelined optimization.

we used two different training sets that were based on e-mails: the TREC dataset [5], consisting of 75,419 e-mail messages, and a proprietary (significantly harder) dataset of which we took 100,000 e-mails.


This theoretical study opens up several avenues for further inquiry.

the tightness of our envelope for early times, less than O( M r ), remains an open question, and we are currently pursuing several conjectures.


5 of how our algorithm can also be used to solve other practical problem which can be cast under the Marginal-MAP framework.

the Minimum Bayes Risk [9] problem for decomposable loss-functions can be readily solved under this framework.


One useful result from their analysis is a derivation of the optimal schedule along a given path.

the value of F(γ) using the optimal schedule is given by (γ) 2 /2, where is the Riemannian path length defined by Intuitively, the optimal schedule allocates more distributions to regions where p β changes quickly.


Let us assume that each image of the database has been represented by a number of low-level features, and that a (dis)similarity measure has been defined so that the proximity between pairs of images represents some kind of "conceptual" similarity.

the chosen feature space and similarity metric is meaningful at least for a restricted number of users.


More formally, a tensor X ∈ R I1×•••×I M is a multidimensional array with elements that can each be indexed by a vector of positive integers.

every element X i1•••i M ∈ R is uniquely addressed by a vector (i 1 , • • • , i M ) such that 1 ≤ i m ≤ I m for all m. Each of the M dimensions of X is called a mode and represents a particular component of the data.


The expectation is over the random selection of the target feature.

can we find clusters that match as many unobserved features as possible?


We extend the above strategy by explicitly representing the norm of VQ residuals, learning a PQ codebook only on the unit-normalized rotated VQ residuals, while separately scalar quantizing the residual norm scales.

multiscale quantization employs the following steps: (1) vector quantization of the dataset; (2) learned rotation of the vector quantization residuals; (3) reparameterization of the rotated residuals into direction and scale components; (4) product quantization of the direction component; (5) scalar quantization of the scale component.


A reason for this is that in the general case, likelihood calculation has exponential complexity in the number of discrete elements, limiting the usefulness of the models.

these methods are impractical for likelihood-based estimation of information-theoretic quantities which requires many likelihood evaluations.


For a given dictionary Φ a separate network must be trained via SGD, to which we add a unique extra dimension of randomness via an online stochastic data-generation strategy.

to create samples in each mini-batch, we first generate a vector x * with random support pattern and nonzero amplitudes.


The main advantage of the patch-based approach is that object-detection can be reduced to a binary classification problem.

we compute P (O c i = 1|v c i ) for each class c and patch i (ranging over location and scale), where O c i = 1 if patch i contains (part of) an instance of class c, and O c i = 0 otherwise; v c i is the feature vector (to be described below) for patch i computed for class c. To detect an object, we slide our detector across the image pyramid and classify all the patches at each location and scale (20% increments of size and every other pixel in location).


Each subproblem is equivalent to computing a conjugate norm and can be solved in closed form.

the matrix Z is updated at each iteration by Z = max{0, x1 T + 1y T + Dzg T ⇧ 0 }.


Computational complexity: the per-iteration computational complexity of Algorithm 1 scales linearly with the number of known entries Y Ω , denoted by |Ω|.

the per-iteration computational cost depends on the following ingredients.


Concretely, let f (i) j represent the j th feature value (rows) for the i th example (columns), where f . Our method simply involves first normalizing the feature distribution matrix by rows, then by columns and finally summing up the absolute value of all entries.

we first normalize each feature to be equally active by dividing each feature by its 2norm across all examples: fj = f j / f j 2 .


Therefore, the data center can achieve an arbitrary fidelity D(R), where D(•) denotes the distortion rate function of {X(t)}.

the data center recovers almost complete information if R exceeds the entropy rate of {X(t)}.


11  We developed an RL-agent (M-Walk) that learns to walk over a graph towards a desired target node for given input query and source nodes.

we proposed a novel neural architecture that encodes the state into a vector representation, and maps it to Q-values and a policy.


An advantage of using TS∞ is that aleatoric and epistemic uncertainties are separable [Depeweg et al., 2018].

aleatoric state variance is the average variance of particles of same bootstrap, whilst epistemic state variance is the variance of the average of particles of same bootstrap indexes.


The model parameters, θ, are specified as By setting W 3 and W 4 to be zero matrices, the TSBN can be viewed as a Hidden Markov Model [1] with an exponentially large state space, that has a compact parameterization of the transition and the emission probabilities.

each hidden state in the HMM is represented as a one-hot length-J vector, while in the TSBN, the hidden states can be any length-J binary vector.


Figure 3 shows that when using fixed code length K, as the Super-Bit depth N gets larger (1 < N  min(d, K)), the MSE of SBLSH gets smaller, and the gap between SBLSH and SRP-LSH gets larger.

when N = K, over 30% MSE reduction can be observed on all the datasets.


Differential privacy: First, we spell out how the differential privacy based approach from [7] can be applied to this more general setting.

a simple corollary of results in [7] is that for a dataset consisting of i.i.d.


Specifically, we optimized each one of the 30 tasks after warm-starting the optimization with 10 observations drawn uniformly at random from each of the remaining 29 tasks.

we have for each task N t = 10 and warm-starting is therefore based on 290 evaluations.


Recently, several promising segmentation algorithms have been proposed based on nonparametric Bayesian methods [10,11,12].

an approach which couples Pitman-Yor mixture models [13] via thresholded Gaussian processes [14] has lead to very promising initial results [10], and provides a baseline for our later experiments.


= We assume η is α-Hölder for α ∈ (0, 1], i.e., Next, we illustrate some of the key conditions verified by dyadic trees which standard results build on.

we want the diameters of nodes of T 0 to decrease relatively fast from the root down.


Nevertheless, the ORN transient as an interval indeed offers unique features to expedite the classification.

the novelty gained due to observing consecutive samples during the transient is on average greater than the informational gain obtained during the steady-state.


By studying the ODE, we establish the global convergence of Async-MSGD, and the rate of convergence will be established later.

we consider a continuous-time interpolation V ⌘,⌧ (t) of the solution trajectory of the algorithm: For t 0, set V ⌘,⌧ (t) = v ⌘,⌧ k on the time interval [k⌘, k⌘ + ⌘).


We represent this with an observation matrix N Ω of size n × m, whose (i, j)-th entry is the noisy observation if (i, j) ∈ Ω and 0 otherwise.

the social graph on n users is generated as per the stochastic block model (SBM) [21].


For each i ∈ I, let us collect the cliques that i participates in: define sets We show that u can be feasible if and only if u Ji + u Ki ≥ 1 for all i ∈ I, where u S denotes a product.

u can be feasible if and only if, for each i, (⇒) If u j ≤ x Pj x Qj for all j ∈ V, then having u Ji + u Ki ≥ 1 is necessary: if both u Ji = 0 and u Ki = 0 for any i it would mean there exists j ∈ J i and k ∈ K i for which x Pj = 1 and x Q k = 0, contradicting any unique assignment to x i .


All methods require a regularization parameter, which obviously determines the outcome.

tiWnet and TiMT use the same, constant parameter throughout all 100 distance matrices and obtain the final W via annealing.


Modeling catching as a continuous partially observable Markov decision process and employing stochastic optimal control theory, we discover that the four main heuristics described in the literature are optimal solutions if the catcher has sufficient time to continuously visually track the ball.

by varying model parameters such as noise, time to ground contact, and perceptual latency, we show that different strategies arise under different circumstances.


Further, our converse results in Section 2.3 provides a fundamental limit on the achievable error rates under differential privacy.

with data collected from an ε-locally differentially privatization mechanism, one cannot achieve an asymptotic type II error smaller than 2(e ε + 1) D kl (P 0 ||P 1 ) , whenever ε ≤ ε * , where ε * is dictated by Theorem 2.3.


Here we show that transformation invariance can be handled in our framework via representers in RKHS.

gradients with respect to the transformations are bounded linear functionals.


We will assume that S i ∩ S j = ∅ for all i, j and that ∪ i S i = {1, . . . , N }.

block coordinate minimization algorithms work as follows: at each iteration, first set δ t+1 = δ t . Next choose a block S i and set: where we use F i (δ Si ; δ t ) to denote the function F restricted to the variables δ Si and where all other variables are set to their value in δ t . at each iteration we fully optimize only over the variables δ Si while fixing all other variables.


In addition, we expect that the data points in each cluster be associated with the representatives in that cluster only.

definition 1 Given dissimilarities {d ij } i,j=1,...,N between N data points, we say that the data partitions into n clusters {C i } n i=1 according to the dissimilarities, if for any data point j in any C j , the largest dissimilarity to other data points in C j is strictly smaller than the smallest dissimilarity to the data points in any C i different from C j , i.e., max i ∈Cj the data partitions into clusters {C i } n i=1 , when the interclass dissimilarity is smaller than the intraclass dissimilarity.


One can note that the proposed method outperforms all tested methods in most cases.

ours improves the AUROC of LID from 82.2% to 95.8% when we detect CW samples using ResNet trained on the CIFAR-10 dataset.


From the second order Taylor expansion with Lagrangian remainder and the assumption that is twice differentiable, it follows that there exists a constant C( θ * , D) depending only on the local minimizer θ and the dataset D such that the following inequality holds for every sample in the dataset and every ε ∈ [0, ε 0 ), Summing the above inequality over all samples in the dataset and recalling that Ln ( θ * +∆) ≥ Ln ( θ * ) holds for all ε ∈ [0, ε 0 ), we obtain Finally, we complete the proof by induction.

for the base hypothesis where p = 0, we can take the limit on the both sides of the above inequality as ε → 0, using the property that δ a can be either positive or negative and thus establish the base case where p = 0.


y NEW is obtained by a simple linear projection of its sparse coefficients.

if ↵ NEW is the K-sparse code of x NEW , then y NEW = Y ↵ NEW .


For the overlap case (γ < 0), supervised learners are always limited by the error incurred due to averaging across decision sets (n −1/d ).

for the collection of distributions with γ < −γ 0 , a faster rate of error convergence is attained by SSL compared to SL, provided m ≫ n 2d .


Therefore, GDMM-subFMO achieves orders-of-magnitude speedup over other methods.

when running on ChineseOCR and EUR-Lex, each iteration of SSG, GDMM, BCFW and Soft-BCFW take over 10 3 seconds, while it only takes a few seconds in GDMM-subFMO.


We demonstrate that we can learn high quality decision trees for the half-cheetah problem instance in the MuJoCo benchmark.

we used a neural network oracle trained using PPO [28] to extract a regression tree controller.


our work derives PAC-Bayes bounds for hypothesis stable Hilbert space valued algorithms.


However, the resulting algorithm cannot be justified by existing analyses of Random Features, since the Representer theorem does not hold for the ℓ 1 -regularized problem [15,16].

since the prediction cannot be expressed as a linear combination of kernel evaluations, a small error in approximating the kernel function cannot correspondingly guarantee a small prediction error.


First, we believe that our model's unusual compatibility with larger, less manicured data sets will prove to be a major strength moving forward.

the skip-gram language model we constructed included only a modestly sized vocabulary, and was exposed only to the text of a single online encyclopedia; we believe that the gains available to models with larger vocabularies and trained on vastly larger text corpora will be significant, and easily outstrip methods which rely on manually constructed semantic hierarchies (e.g.


A property that we often desire from our prior distributions is that they have the ability to become arbitrarily strong.

under the interpretation of a Bayesian prior as "inferences from previously-seen data", we would like to be able to imagine an arbitrary amount of such data and construct a highly-informative prior when appropriate.


A is a symmetric re-scaled zero-filled matrix of observed entries.

here p ∈ (0, 1) is a parameter that controls the probability of observing a particular entry in A, corresponding to a sample complexity of O(n 2 p).


+ is strictly proper if and only if it is c-calibrated for all c ∈ ∆n .

a continuous strictly proper loss is c mid -calibrated.


A later datum stays at node with probability (N +1)/(N +N ≺• +α(| |)+1), where N is the number of previous data that stopped at , and N ≺• is the number of previous data that came down this path of the tree but did not stop at , i.e., a sum over all descendants: N ≺• = ≺ N . If a datum descends to but does not stop then it chooses which child to descend to according to a Chinese restaurant process where the previous customers are only those data who have also descended to this point.

if it has reached node but will not stay there, it descends to existing child i with probability (N i +N i≺• )/(N ≺• +γ) and instantiates a new child with probability γ/(N ≺• +γ).


This means that the optimal policy for the robust MDP studied in this paper can be found quite efficiently.

the robust MDP in Theorem 5 might not seem to allow an efficient optimization without the knowledge of the relation to the corresponding risk-sensitive MDP, whose optimal policy is readily available with dynamic programming.


We also propose softmax losses accordingly under such representation framework.

the proposed softmax losses supervise network learning by also taking the SphereConv activations from the last layer instead of inner products.


Of course, any problem can be embedded into the complete graph by setting to zero a subset of the {θ st } parameters.

(for a graph G = (V, E), we simply set θ st = 0 for all pairs (s, t) / ∈ E.) We first focus on the marginal polytope MARG(K n ) ≡ MARG(K n ; φ) of valid dual variables {µ s , µ st }, as defined in Eqn.


These results indicate the model is functioning correctly, but raise issues for future work.

the considerable improvement achieved by reducing the timescales indicates that careful modeling of the calcium transient is essential, and that faster calcium indicators have the potential to dramatically improve the ultimate accuracy of weight inferences.


• Out-of-sample performance guarantees: Adopting a distributionally robust framework allows us to invoke results from the measure concentration literature to derive finite-sample probabilistic guarantees.

we establish out-of-sample performance guarantees for the classifiers obtained from the proposed distributionally robust optimization model.


We set the source classifier f S as the outputs of the residual block to make it better trainable from the source-labeled data by deep residual learning [8].

if we set f T as the outputs of the residual block, then we may be unable to learn it successfully as we do not have target labeled data and thus standard back-propagation will not work.


Ideally, we would want to be able to read out both the present input and the past inputs, such that x t−n ≈ D n r t , where n is an elementary time step, and D n are appropriately chosen readouts.

we will in the following assume that there is a matrix M such that D n M = D n+1 for all n. the input history should be accessible via xt−n = D n r = D 0 M n r t . Then the cost function we would like to minimize is a straightforward generalization of Eq.


Moreover, since we are mostly interested in the properties of the metric space, we focus on embeddings based purely on the Poincaré distance and on models with comparable expressivity.

we compare the Poincaré distance as defined in Equation ( 1) to the following two distance functions: Euclidean In all cases, we include the Euclidean distance d(u, v) = u − v 2 .


Instead of adding a asymmetric Gaussian matrix, the algorithm can add the a symmetric matrix with i.i.d.

gaussian entries N . for 1  i  j  d, the variable N ij is an independent Gaussian random variable with variance 2 .


We define the combined error map as Obj Err(Obj) * Obj where * represents pointwise multiplication.

we restrict the error map for each object to the object itself, and then sum the results.


Analysis In this paragraph, we borrow the regret from online learning to measure the performance of FESL-c.

we give a loss bound as follows which shows that the performance will be improved with assistance of the old feature space.


For this purpose, we assume that each node i in the suffix tree has a hidden probability q i of stopping at node i when following a path from the root of the tree to add a customer.

(1 − q i ) is the "penetration probability" when descending an infinite depth suffix tree from its root (Figure 2).


Structured learning has been successfully applied to many complex real world problems such as word/sequence alignment [22,24], graph matching [6], static analysis of binary executables [14] and segmentation [3].

we attempt to find the decision boundary that maximizes the margin between the correct association z * n and the closest runner-up solution.


We consider three baselines: (i) Individual Corruption: for each corruption type i 2 [m], we construct an oracle that trains a neural network using the training data perturbed by corruption i, and then returns the trained network weights as ✓ t , for every t = 1, . . . , T . This gives m baselines, one for each corruption type; (ii) Even Split: this baseline alternates between training with different corruption types between iterations.

call the previous m baseline oracles O 1 , ..., O m . Then this new baseline oracle will produce ✓ t with O i+1 , where i ⌘ t mod m, for every t = 1, ..., T ; (iii) Uniform Distribution: This more advanced baseline runs the robust optimization scheme with the Hybrid Method (see Appendix), but without the distribution updates.


Similar ideas have been proposed in the past.

[SG05] also trained a parametric student model to approximate a Monte Carlo teacher.


They take the simple Gaussian form: Finally, the binary label y ∈ {0, 1} is used to detect the coincidence of the variables z, z in the latent space.

y follows a Bernoulli distribution with mean value: Eq.


One useful application of this theorem is in matching graphs with Euclidean affinities, since Euclidean distances are conditionally negative definite of order 1 (Wendland, 2004).

the affinities are Euclidean distances of points in Euclidean spaces of arbitrary dimensions, where This class contains, besides Euclidean graphs, also affinities made out of distances that can be isometrically embedded in Euclidean spaces such as diffusion distances (Coifman and Lafon, 2006), distances induced by deep learning embeddings (e.g.


To analyze Problems 1 -3, we introduce the concept of a curvenormalized polymatroid 1 .

we define the κ f -curve-normalized version of f as If κ f = 0, then we set f κ ≡ 0.


The bound of equation ( 3) is tight if and only if there exists a configuration y @ kD that for each & (' v v Q£ ) achieves the maximum defining £ ge £ ¦ 2 .


To derive the posterior (2), we first reformulate the likelihood as a multivariate Gaussian distribution over z.

after taking out constant terms with respect to z from the likelihood, it immediately becomes proportional to the canonical form Gaussian over z with ν = BΣ −1 y and Λ = BΣ −1 B , which is equivalent to the standard form Gaussian with mean Λ −1 ν and covariance Λ −1 .


The first result is interesting in its own right since it corresponds to translating optimization guarantees into statistical results.

we derive a relation the excess risk of the FALKON algorithm f λ,M,t from Alg.


Thus, statistical feedback is a difference in potential outcomes.

an additive feedback model In order to get a handle on feedback as defined above, we assume that feedback enters the model additively: , where f is a feedback function, and y (t) i is the prediction published at time t. we assume that the predictions made by our model at time t + 1 are the sum of the prediction the model would have made if there were no feedback, plus a feedback term that only depends on the previous prediction made by the model.


The parameter k plays the same role as the parameter 1 − δ in standard dropout, which controls the number of features to be dropped.

the expected total number of the kept features using multinomial dropout is k and that using standard dropout is d(1 − δ).


Thus we set the horizontal x-coordinate of the camera in the world to zero and allow ϑ to be the only variable extrinsic parameter.

the position of the camera is constrained to a circular arc on the y, z-plane (Figure 2).


This solves (3) for s = kgk 1 /kgk q and satisfies E[kL q (g)k 2 2 ] = kgk 1 kgk q .

for q = 2 we get 1-bit QSGD while for q = 1, we get TernGrad.


The advantage of MMD over other nonparametric distance measures is discussed in [30,15,16,31].

mMD statistic defines a metric, and whenever MMD is large, the samples are "likely" from different distributions.


Based on prior work (see [7,8]), the large sample behavior of d ∞ is well understood.

d ∞ (T p h , T p0 ) converges to the supremum of an appropriate Gaussian process, on the basis of which we can construct confidence intervals for the d ∞ metric.


These networks demonstrate that existing, powerful neural network learning algorithms can learn representations that perform complex relational binding of information.

these networks had to bind together object identity, location, and relationship information to answer a number of questions about input displays containing two objects.


(t, b) is more frequent than (t, 1−b) in S . By the above discussion, we will have that v(t * ) = v * .

the protocol identifies a set of T domain elements, containing all of the heavy-hitters.