We will keep track of the weighted cumulative votes of the first i weak learners for the sample x t by , where α i t is the weight of W L i , l i t is its prediction and e j is the j th standard basis vector.

for the optimal algorithm, we assume that α i t = 1, ∀i, t. the booster makes the final decision by simple majority votes.


As has been pointed out in [4], the shape of unimodal tuning functions can strongly influence the coding accuracy of population codes of angular variables.

box-like tuning functions can be superior to cosine tuning functions.


We stress on the fact that boundaries in the Z-space can be very different from the boundaries in the image space.

as g is in general non-linear, f might be a highly non-linear function of the input space, while z → (f • g)(z) is a linear function in z.


Moreover, in the semi-stochastic gradient v t , there are also two reference gradients and we denote them by g (0) t = rF (e x) and g Back to our One-epoch-SNVRG, we can define similar reference points and reference gradients as that in the special case of SVRG.

for t = 0, . . . , t }, l = 0, . . . , K, which is set to be x (l) t = x t l with index t l defined as Specially, note that we have x (0) t = x 0 and x (K) t = x t for all t = 0, . . . , Q K l=1 T l 1.


Minimizing the exponential loss or the log loss on real data often fails to produce finite parameters.

this happens when for some feature U V U V 9 BA DC FE HG 4 § U V 9 BA DC qv E 19 BA TG FG % for all E and A with v w 9 fA hG $ (3) or U V 9 BA DC FE HG 4 § U V 9 BA DC qv E 19 BA TG FG & for all E and A with v w 9 fA hG $ Hc This is especially harmful since often the features for which (3) holds are the most important for the purpose of discrimination.


These different similarity functions are controlled by the choice of three items: G r , F r and the parameter d representing the degree of the tensor product operator.

we will focus on the case G r = I and on Λ d (A, B) as a representative of the non-linear family.


Furthermore, conditioning information, in the form of macroeconomic variables, can be of importance, but exhibit the cumbersome property of being released periodically, with explanatory power that varies across the forecasting horizon.

when making a very long-horizon forecast, the model should not incorporate conditioning information in the same way as when making a short-or medium-term forecast.


In this section, we analyze the computational complexity of solving the optimization problem (2) and prove that the problem is NP-hard.

we proceed by showing that the following sub-problem of (2): for a fixed π ∈ Π R , is NP-hard.


Since in our network the complex cell receptive field size in the image is only 50% greater than that for the simple cells, this implies translation invariance: over the time (or space) that a simple cell of the correct orientation gives a strong but transitory response, the complex cells provides a strong continuous response.

its response is invariant to the phase that determines the profile of the simple cell response.


An appropriate choice of the bounds ensures that the algorithm always improves the objective values for Problems 1 and 2.

choosing ft as a modular upper bound of f tight at X t , or ĝt as a modular lower bound of g tight at X t , or both, ensures that the objective value of Problems 1 and 2 always improves at every iteration as long as the corresponding surrogate problem can be solved exactly.


More precisely, the corollary follows directly from successive application of that single expert relative loss bound, and then our Fixed-share relative loss bound (Theorem 1): ≤ log(m) + (T − 1) min j=1,...,m We start by finding the smallest discrete set of switching-rate parameters so that any additional regret due to discretization does not exceed (T − 1)δ, for some threshold δ.


This implies that, if the initial state x[0] is randomly drawn from a continuous distribution, then almost surely x [k] does not converge to the origin, yet |x[k]| converges to 2 /4.

with probability 1, the state x[k] does not converge to a fixed point, such as a local optimum or a saddle point, even though the estimation error converges to a finite non-optimal value.


For a continuous Y , or unless there is a mass probability at V α , we have The equivalence between the robust MDP with the objective (2) and the risk-sensitive MDP with the objective (3) can be shown by the use of the following alternative definition of CTE: where Q is the set of probability mass functions, q, whose support is a subset of the probability mass function, q 0 , of Y such that q(y) ≤ q 0 (y)/α for every y in the support of q 0 .

let C π i be the cost incurred at the i-th epoch with policy π so that Then, by the recursive definition of ICTE and the translation invariance 1 of CTE, it can be shown that where the second equality follows from (7).


The complete implementation is available in the tensor2tensor library on github.

the layout is given by: mesh_shape = [("all", n)] computation_layout = [ ("vocab", "all"), ("d_ff", "all"), ("heads", "all")] the dimensions representing the vocabulary size, the size of the feed-forward hidden layer, and the number of attention heads are each split across all processors.


From this perspective, Gold's theorem [8] asserts that languages with infinite recursion, such as most human languages, are impossible to learn from positive evidence alone.

linguistic exceptions, such as the restrictions on verb alternations mentioned above, are cited as being impossible to learn empirically.


samples f ⇠ D (as in our setting), as well as the ability to compute exact gradients rF (v) in a potentially expensive operation.

the basic intuition is to use an exact gradient rF (v) at some "anchor point" v 2 W as a kind of "hint" for what the exact gradient is at nearby points w. SVRG leverages the theorem that rf (w) rf (v) + rF (v) is an unbiased estimate of rF (w) with variance approximately bounded by L(F (v) F (w ? ))


Notice that, in sharp contrast to the nearest neighbors-based methods, which first fix the number of neighbors or the neighborhood radius and then compute the weights between points in each neighborhood, we do the two steps at the same time.

the optimization programs ( 4) and ( 5) automatically choose a few neighbors of the given data point, which approximately span a low-dimensional affine subspace at that point.


We also make a restriction on the features, analogous to the usual Markovian assumption made in CRFs, and assume that every component g k of g is a function only of x, s j , and the label y j−1 associated with the preceding segment s j−1 .

we assume that every g k (j, x, s) can be rewritten as g k (j, x, s) = g k (y j , y j−1 , x, t j , u j ) (2) for an appropriately defined g k . In the rest of the paper, we will drop the g notation and use g for both versions of the segment-level feature functions.


We begin with a general but relatively naive lower bound that depends only on the geometric structure of the parameter space, as captured by its metric entropy.

given a subset Θ ⊂ R d , we say {θ 1 , .


To circumvent this issue, practitioners have resorted to asynchronous implementations, which emphasize parallel efficiency by using potentially stale stochastic gradients for computation.

each worker in asynchronous implementations can process a mini-batch of data independently of the others, as follows: (1) The worker fetches from the parameter servers the most up-to-date parameters of the model needed to process the current mini-batch; (2) It then computes gradients of the loss with respect to these parameters; (3) Finally, these gradients are sent back to the parameter servers, which then updates the model accordingly.


Here we address this problem by providing a theoretical connection between spike-triggered covariance analysis and nonlinear subunit models.

we show that a "convolutional" decomposition of a spike-triggered average (STA) and covariance (STC) matrix provides an asymptotically efficient estimator for class of quadratic subunit models.


The tree policy target is the average tree policy of the MCTS at the root.

we try to match the network output to the distribution over actions given by n(s, a)/n(s) where s is the position we are scoring (so n(s) = 10, 000 in our experiments).


We revisit the classical problem of optimal experimental design (OED) under a new mathematical model grounded in a geometric motivation.

we introduce models based on elementary symmetric polynomials; these polynomials capture "partial volumes" and offer a graded interpolation between the widely used A-optimal design and D-optimal design models, obtaining each of them as special cases.


One traditional way to combine these features is through multiple kernel learning.

we take the average of individual kernels constructed based on individual features, and train a one-vs-all SVM using the joint average kernel.


For all M ≥ 1, any external field φ, and any M -cover 3.

for all choices of the external field φ and any 2-cover (H; if all of the edge potentials are strictly positive, and the graphical model has even one negative cycle, then there exists an external field φ and a 2-cover (H; φ H , ψ H ) of (G; φ, ψ) such that Z(G; φ, ψ) 2 < Z(H; φ H , ψ H ).


If this submodular Hamming distance is used for k-means clustering, then the mean-finding step becomes an instance of the SHmin problem.

if cluster j contains documents C j , then its mean takes exactly the following SH-min form: Structured prediction: Structured support vector machines (SVMs) typically rely on Hamming distance to compare candidate structures to the true one.


Note passive learning here is also subject to oracle epiphany.

the labels y t are produced by the same oracle epiphany model, some of them can be ⊥ initially.


We show that taking into account the interaction between the identified components leads to significant improvements in the few-shot generalization.

we show that a non-trivial interaction between the similarity metric and the cost function can be exploited to improve the performance of a given similarity metric via scaling.


To mimic the procedure, we propose to combine retrieval and generation for automatic report generation.

we first compile an off-the-shelf template database T that consists of a set of sentences that occur frequently in the training corpus.


We followed [12] in simulating sensor failure by introducing some missing ranges for these signals.

we have a missing range of [0.6, 1.2] for the Bramblemet tide height sensor and [1.5, 2.1] for the Cambermet.


Our theoretical result suggests a cutting-plane approach to the MWM problem, where BP is used as the LP solver.

we examine the BP solution to identify odd-sized cycle constraints -"cuts" -to add to the MWM-LP relaxation; then construct a new GM using our graphical transformation, run BP and repeat.


The goal of this work is to learn a function h : P → Y given a set of example pairs {(P i , y i )} m i=1 , where P i ∈ P and y i ∈ Y.

we consider a supervised setting in which input training examples are probability distributions.


The tapering estimator Ωk , defined in (7), of the precision matrix Ω p×p with p > n 1 2α+1 satisfies: , and a universal constant C > 0.

the estimator Ω = Ωk with k = n 1 2α+1 satisfies: Given the result in Equation ( 10), it is easy to show that setting k = n 1 2α+1 yields the optimal rate by balancing the size of the inside-taper and outside-taper terms, which gives Equation (11).


Note that γ i is chosen to minimize q − (q i , γ) , and also that q − (q i , γ) = (q, γ−1 .

γ * i is a warping that aligns q i to q.


An optimal stopping rule is to stop the first time the expected cost of continuing exceeds that of stopping, and to choose δ = 0 or δ = 1 to minimize the probability of error given the accumulated evidence (see [10]).

we define the continuation region at time t by Although we have obtained an expression for the optimal policy in terms of Q(t, p) and Q(t, p), computing Q(t, p) is difficult in general.


We believe this drawback is offset by the strong empirical performance across many domains of deep neural networks in general, and VAEs in particular.

we propose to parametrize the causal graph of Figure 1 as a latent variable model with neural net functions connecting the variables of interest.


For example, from the perspective of a teacher and a student, we came up with a question that simply providing the teacher's knowledge directly without any explanation can be somewhat insufficient for teaching the student.

when teaching a child, the teacher should not use his/her own term because the child cannot understand it.


(C1) Conditional Independence: We assume that distribution of non-zero entries of x * is conditionally independent and identically distributed.


As our new results will now show, there is a limit to this sort of complexity reduction: other relatively obvious combinations of independence relationships do not bear the same fruit.

we show the NP-completeness result to be specific to fully transition-and observation-independent problems.


To tune the parameters µ and λ, we use 5-fold cross validation (CV) separately for each experiment.

we randomly divide Ω X and Ω Y into five disjoint subsets each.


Interactive POMDPs (I-POMDPs) [7] generalize POMDPs to multi-agent settings by replacing POMDP belief spaces with interactive belief systems.

an I-POMDP augments the plain beliefs about the physical states in POMDP by including models of other agents.


Toward this end, we present an effective and efficient integration of the two existing approaches for geometric invariance, i.e., STN-based feature extraction networks [5,19] and geometric matching networks [30,31], that includes a novel weakly-supervised loss function tailored to our objective.

the final geometric field is recursively estimated by deforming the activations of feature extraction networks according to the intermediate output of the geometric matching networks, in contrast to existing approaches based on geometric matching which consider only fixed, upright versions of features [30,31].


In this paper, we consider the more general setting of regularized Empirical Risk Minimization (ERM) when data cannot fit into memory.

we generalize the existing block minimization framework based on strong duality and Augmented Lagrangian technique to achieve global convergence for general convex ERM.


Suppose that there exists a constant M such that p ∞ ≤ M and p ∞ ≤ M . Suppose also that the density difference f = p − p is a member of Besov space with regularity α.

f ∈ B α 2,∞ where B α 2,∞ is the Besov space with regularity α, and where α denotes the largest integer less than or equal to α and ω r,L2(R d ) is the r-th modulus of smoothness (see [10] for the definitions).


In a minor abuse of notation, we write L(θ, β) = L(θ, q(•; β)).


However, we note that static analysis is not sufficient to capture all dependencies required in the factor graph to accurately model the process of generating training labels.

static analysis can (i) pick up spurious dependencies among HFs that are not truly dependent on each other, or (ii) miss key dependencies among HFs that exist due to dependencies among the DSPs in the HFs.


These weight change values are expressed in the chip as subthreshold currents.

the signal that triggers positive weight updates is represented by an I UP current, and the signal that triggers weight decreases if represented by the I DN current.


Note that computing the membership probability of any element x i in any cluster C j , as is done in mixture models, is ill-suited for our goal.

this computation assumes a fixed clustering whereas our work focuses on computations performed with respect to the distribution over all possible clusterings.


In a small neighborhood dθ around θ, we have: Since Z n+1 is completely random and H is smooth, the above shows that on Θ\{θ * 1 , .

. . , θ * K } Z n+1 is simply a Poisson process with rate measure α Γ(1+c)Γ(n+c+σ) Γ(n+1+c)Γ(c+σ) H. it will have Poisson(α Γ(1+c)Γ(n+c+σ) Γ(n+1+c)Γ(c+σ) ) new atoms, each independently and identically distributed according to H. In the IBP metaphor, this corresponds to customer n+1 trying new dishes, with each dish associated with a new draw from H. The resulting Indian buffet process is as described in the introduction.


The lengthscales are set to 14.5 m and the prior standard deviation of heights is 10 m. We assume a noise standard deviation of 0.075 m. Since the safety feature of each state transition is a linear combination of heights, the GP model of the heights induces a GP model over the differences of heights, which we use to classify whether state transitions fulfill the safety constraint.

the safety depends on the direction of travel, that is, going downhill is possible, while going uphill might be unsafe.


On the contrary, this condition turns out to be equivalent to boostability.

according to our criterion, it is neither too weak nor too strong as a weak-learning condition.


In [30], the authors propose a mutual-exclusivity loss function that forces the set of predictions for a multiclass dataset to be mutually-exclusive.

it forces the classifier's prediction to be close to one only for one class and zero for the others.


The weight vector p t is the probability distribution over the n base classifiers that maximizes the entropy, subject to the constraint that p t correctly classifies all patterns seen in previous trials with a given margin γ.

it maximizes the entropy of p t subject to the constraints that p t • x s ≥ 1/2 + γ whenever y s = 1 for s < t, and p t • x s ≤ 1/2 − γ whenever y s = 0 for s < t. We show that, if there is a weighting p * , determined with benefit of hindsight, that achieves margin γ on all trials, then this on-line maximum entropy procedure makes at most ln n 2γ 2 mistakes.


In our active learning setting, our goal is to output a boundary g that is close to g * while making as few interactive queries to the labeler as possible.

we want to find an algorithm with low query complexity Λ(ǫ, δ, A, L, g * ), which is defined as the minimum number of queries that Algorithm A, acting on samples with ground truth g * , should make to a labeler L to ensure that the output classifier We now introduce three conditions on the response of the labeler with increasing strictness.


Similarly to (26), the robust optimization problem for a given user j is where g j (G) = min mi2G f j (m i ).

for the user with feature vector u j , our goal is to find the group of movies to recommend such that the entire collection of movies is robust with respect to the user's preferences.


Note also that as ρ becomes small, the variance v ρ becomes large, and likewise, confidence intervals based on Φ ρ ( F ; c) become wider.

there is a trade-off between the width of the confidence interval and the size of the bound (10).


To analyze lim m→∞ P(x (k) I ≤ 0), we use a standard probabilistic analysis technique known as 'density evolution' in coding theory or 'recursive distributional equations' in probabilistic combinatorics [9,13].


To make progress on the computational question we exploit a key observation: for any loss function, its corresponding clipped loss can be indirectly expressed by an auxiliary optimization of a smooth objective (if the original loss function itself was smooth).

given a loss ℓ(y, x ⊤ θ) define the corresponding ρ-relaxed loss to be for 0 ≤ ρ ≤ 1; see Figure 1.


Running (A, B)-PROD with an appropriately tuned instance of A = EXP3 (see [3]) and B = UCB (see [2]), with the parameter setting suggested in Corollary 1 simultaneously guarantees against any assignment of the loss sequence.

if the losses are generated in an i.i.d.


Assuming communication constraints are severe, we examine the extent to which alternative processing rules can avoid a loss in (MAP or MPM) performance.

given a directed graphical model with binary-valued hidden variables and real-valued noisy observations, we assume each node may broadcast only to its children a single binary-valued message.


Additionally, instead of combining all the base kernels in the forest F , we preselect a subset of them based on the taxonomy structure.

from each taxonomy, we select base kernels that correspond to the nodes on the path from the root to the leaf node class.


In all cases, the non-extreme columns are conic combinations of two of the selected columns, i.e., each column in H has at most two non-zero values.

the non-extreme columns are conic combinations of the two extreme columns that "sandwich" them in the matrix.


Despite this trend to rely on linear classifiers and overcomplete feature representations, sparse coding is still a flat model, and efforts have been made to add flexibility to the features.

deep Coding Networks [17] proposed an extension where a higher order Taylor approximation of the non-linear classification function is used, which shows improvements over coding that uses one layer.


Note that π (j) can be made arbitrarily complex.

even if π (j) is linear, the adaptive mechanism D (j) a is quadratic in x, and as such escapes the bottleneck.


From Lemma 2 we immediately get This upper bound can be computed for Z = ab ∈ A k,q for different norms.

for Ω(Z ), we have ab 1 ≤ √ kq and Ω k,q (ab ) = ab * = 1 which lead to the corollary: is an atom, the expected errors of the oracle estimators ẐOracle Ω k,q , ẐOracle 1 and ẐOracle * using respectively the (k, q)-trace norm, the 1 norm and the trace norm are upper bounded as follows: When the smallest entry in absolute value of a or b is close to 0, then the expected error is smaller for ẐOracle 1 , reaching σ 2 log(m 1 m 2 ) on e 1 e 1 while not changing for the two other norms.


Lemma 5 is proved by explicitly constructing a classifier (tester) p ψ from any sequence of low regret.

for any assortment sequence tS t u T t"1 , we construct p ψ as p ψ " 0 if 1 T ř T t"1 Ir1 P S t , 2 R S t s ě 1{2 and p ψ " 1 otherwise.


Under some mild conditions, we show that this two-stage procedure achieves minimax rates of convergence up to a logarithmic factor, even after only one iteration of EM.

given any δ ∈ (0, 1), we provide the bounds on the number of workers and the number of items so that our method can correctly estimate labels for all items with probability at least 1 − δ.


The encoder employs a Tree-LSTM [29] to compute embeddings for both the entire source tree and each of its sub-tree.

consider a node N with the value t s in its one-hot encoding representation, and it has two children N L and N R , which are its left child and right child respectively.


We represent this scene map y(n) in terms of a set of coefficients {w i (n)} K i=1 at each location n, corresponding to various spatial derivatives.

these coefficients are related to the scene map y(n) through convolution with a bank of derivative filters {k i } K i=1 , i.e., For our task, we define {k i } to be a set of 2D derivative-of-Gaussian filters with standard deviations 2 s pixels, for scales s = {1, 2, 3}.


In this setting, we assume that we have bundled purchase history data from n users.

there are two categories of interest with d 1 and d 2 alternatives in each category respectively.


A distribution with few modes would lead to more stable batches and, for example, the representation of a given sample would not vary along different batches.

it is desirable that a sample, if repeated twice in multiple batches, would produce the same network activations postnormalization.


In this paper, we propose to explicitly account for compression while training the initial deep network.

we introduce a regularizer that encourages the parameter matrix of each layer to have low rank in the training loss, and rely on a stochastic proximal gradient descent strategy to optimize the network parameters.


When each Player i chooses a mixed strategy x i independently, the payoff function extends naturally by player can change her mixed strategy for achieving a higher payoff.

for any Player i, let x −i denote the mixed strategies chosen by all players other than Player i, then for any mixed strategy A NE is said to be fully mixed if no probability density in any of the x i is zero.


To illustrate this, let P (k) denote a sequence of partitions.

then (9) shows that lim k→∞ an arbitrarily ill-balanced partition leads to arbitrarily poor values of its Product Cut.


Let � T = R, and let ψ : {1, 2, 3} × R→R + be defined as (see Figure 3) We will show there are 3 points in R ψ satisfying the conditions of Theorem 7.

consider t1 = 1, t2 = 2, and t3 = 3, giving z 1 = ψ( t1 ) = (0, 1, 2) � , z 2 = ψ( t2 ) = (1, 0, 1) � , and T here is a convex set and ψ : � T →R 3 is a convex function.


hinge loss) case, the number of points may be reduced to poly(ε −1 ) by elementary methods.

it is in easy task to prove that if we omit all but a random subset of log(ε −1 )/ε 2 data points, the hinge loss corresponding to the obtained separator w.r.t the full set of points will be O(ε).


A fundamental aspect of our approach is that we can train stochastic processes that need not exactly obey detailed balance, yielding access to a larger and potentially more powerful space of models.

this enables us to relax the weight symmetry constraint of undirected graphical models corresponding to neural networks, yielding a more brain like iterative computation characteristic of asymmetric biological neural circuits.


in [10] proposed a self-tuning spectral clustering algorithm that computes a different local kernel width for each data point x i .

the local kernel width for each x i is computed as the distance of x i to its kth nearest neighbor.


Proving socalled "direct-sum" results has become a standard tool, namely the fact that the amount of resources required for solving d copies of a problem (with different inputs) in parallel is equal to d times the amount required for one copy.

there is no saving from solving many copies of the same problem in batch and the trivial solution of solving each of them separately is optimal.


For a function f : W → R, we can then consider the following diagram - Since both ρ and I are bijective, it is easy to see that f not only defines a function g : X → R by g := f • I • ρ, but conversely, every function g : X → R has such a representation and this representation is unique.

there is a one-to-one relationship between the functions X → R and the functions W → R. Let us now assume that we have a kernel k W on W with RKHS H W and canonical feature map [31,Theorem 4.21] shows that the RKHS H X of k X is given by Since, for f ∈ H W , the reproducing property of Let us now assume that X is compact and that k W is one of the universal kernels considered in Theorem 2.1 or the Gaussian RBF kernel (4).


Recall that we assumed above that ka i k 2 = 1 for all a i .

we have the following lemma about b g. In order to ensure that this estimator is sparse, we fix some sparsity budget s. we require P i p i = s. This is a sparsity on average constraint.


In this work we present an approach to estimation and learning of generative probabilistic models in a distributed context where certain sensor data can be missing.

we show how traditional centralized models, such as probabilistic PCA and missing-data PPCA, can be learned when the data is distributed across a network of sensors.


The conditional distribution of Y given X = x is obtained by standard Gaussian formulas.

the conditional covariance matrix of Kernel Smoothing Estimators.


Our first experiments stress-test our proposed algorithm in an ideal case for manual batching.

we train a model on a bi-directional LSTM sequence labeler [12,23], on synthetic data where every sequence to be labeled is the same length (40).


Then A x is the intensity profile of pixel x (We address pixels with a single number because the images are represented in a scan-line vector form).

a x is an N -dimensional vector (where N is the number of images) that holds the intensity values of pixel x in each image in the ensemble.


We found that this small amount of weight decay was important for the model to learn.

weight decay here is not merely a regularizer: it reduces the model's training error.


This argument is also supported in a recent study by [16], where the hitting and commute times are shown to be dominated by the inverse of degrees of vertices.

they do not take into account the graph structure at all.


Valid molecules can still be unrealistic.

it is possible to generate valid molecules that are very sterically strained such that it is unlikely that they will be stable at ordinary conditions.


Then we fix the modeling agent i's strategy level to be 2 and try to observe the changes of j's samples which represent different possible models or strategy levels.

we start from equal number of samples that representing j as level-1 I-POMDP, level-0 POMDP, and frequency based agents, and then gradually learn that the majority of samples converge or become close to the ground truth: j is a level-1 I-POMDP.


In order to evaluate the behaviour of AICO-T we applied it to a reaching task with varying starttarget distance.

for a fixed start point we considered a series of targets lying equally spaced along a line in task space.


We use simple 1D chain CRFs with first order Markov dependency between name tags.

we have clique potentials joining adjacent labels (y i , y i+1 ), but which are independent of the text itself, and clique potentials joining words and labels (x i , y i ).


However, the motivation of hashing problem is fundamentally different from that of the search problem considered in this paper.

the goal of learning to hash is to embed data into compact binary codes so that the hamming distance between two codes reflects their original similarity.


For any non-empty subset S of [m] and any k × m matrix M we refer by M S to the k by |S| sub-matrix of M that contains only the columns of M corresponding to S. Also define the scalar product u, v ≡ 1 m m i=1 u i v i for u, v ∈ R m . Finally, the subgradient of a convex function f : R m → R at point x ∈ R m is denoted by ∂f (x).

remember that the subgradient of the ℓ 1 norm, x → x 1 is given by (3.1) One main challenge in the proof of Theorem 2 lies in the fact that the function x → C A,y (x) is not -in general-strictly convex.


We would like to employ this idea to indicate high similarity of Image-B to Image-A, and lower similarity of Image-C to Image-A.

regions in one signal (the "query" signal) which can be composed using large contiguous chunks of data from the other signal (the "reference" signal) are considered to have high local similarity.


. . , ω(A T )) , which are both T × d matrices, we can write this model in a matrix form: where This assumes that the noise is driven by time-series dynamics (a martingale increment), where each coordinates are independent (meaning that features are independently corrupted by noise), with a sub-gaussian tail and variance uniformly bounded by a constant σ 2 .

no independence assumption between the N t is required here.


The optimization landscape of (4) has been well studied in existing literature.

we impose the following assumption on Assumption 1.


With the intent of deriving stronger approximation guarantees, we begin this section by introducing a 'nicer' measure of the quality of approximation afforded by Φ.

instead of measuring J * − Φr * in the L ∞ norm as we did for our previous bounds, we will use a weighted max norm defined according to: J ∞,1/ψ max x∈X |J(x)|/ψ(x), where ψ : X → [1, ∞) is a given weighting function.


There are many existing work on ways to approximate D * (X n ).

a new randomized algorithm based on threshold accepting is developed in [12].


The other benchmark model comes from Goldberger and Ben-Reuven [10], which unifies the two steps of the first benchmark model into a joint fashion.

the noise transition matrix (y → ỹ) modeled by a nonlinear softmax layer connects the classifier (x → y) with noisy labels ỹ for the end-to-end training.


Proposition 1 states that for any optimal solution D of the supervised objective, there exists an optimal solution D * of the (K + 1)-class objective such that D and D * share the same generalization error.

using the (K + 1)-class objective does not prevent the model from experiencing any arbitrarily high generalization error that it could suffer from under the supervised objective.


This Lemma can be obtained with a slight modification of results in [18].

no matter how poorly structured a particular dictionary is with regard to a given sparsity profile, there will always be sparse coefficients we are guaranteed to recover (provided we utilize a convergent algorithm).


The simplest assumes that the network structure, after the introduction of the hidden variable, is fixed.

we assume that the "true" structure of the network is indeed the result of applying our transformation to the input network (which was produced by the first stage of learning).


Going from the penalization of supports to the penalization of level sets introduces some complexity and for simplicity in this section, we only consider the analysis in the context of orthogonal design matrices, which is often referred to as the denoising problem, and in the context of level set estimation already leads to interesting results.

we study the unique global minimum ŵ of the proximal problem in Eq.


To demonstrate the performances of different methods on recovering the overall transcriptional regulatory network, we randomly selected 10 TFs in the benchmark network and plotted the subnetwork in Figure 2 which has 70 edges with nonzero regulatory potential scores.

the gray edges form the benchmark network, the red edges are those identified correctly and the green edges are those incorrectly inferred by each method.


However, it should be noted that for short horizons the hierarchy is not quite the same, and the growth rate is not logarithmic; this question is raised in .

on short horizons the Best-Arm Identification procedure performs very well with respect to the others, and starts to be beaten (even by the gap-aware strategies) only when T ∆ 2 is much larger that 10.


LSPI applies also with minor modifications to Markov games [6].

at each iteration, LSPI evaluates the current policy using the stored samples and keeps the learned weights to represent implicitly the improved minimax policy for the next iteration by solving the linear program above.


We propose a highly efficient method to directly compute the tensor contractions without forming the input tensor explicitly.

given the sketch of a tensor, each tensor contraction can be computed in O(n + b log b) operations, regardless of order of the source and destination tensors.


A computationally more efficient alternative is nonconvex optimization.

we reparameterize the m ⇥ n matrix variable M in the optimization problem as UV > with U 2 R m⇥k and V 2 R n⇥k , and optimize over U and V . Such a reparametrization automatically enforces the low rank structure and leads to low computational cost per iteration.


As discussed in the introduction, we wish to compare our objective value to that obtained by linear predictors.

we will use the notation to denote the expected loss of a linear predictor parameterized by the vector w. By Assumption 1, this function is convex and twice-differentiable.


Spectral graph theory provides insight into the topological structure encoded by the SR.

we showed that eigenvectors of the SR can be used to discover a hierarchical decomposition of the environment for use in hierarchical RL.


[9] analyzed all the transformations on similarities that preserve existence of efficient LSH search.

they showed that if S is a similarity for which there exists an LSH family, then there also exists an LSH family for any similarity which is a probability generating function (PGF) transformation on S. PGF transformation on S is defined as P GF (S) = ∑ ∞ i=1 p i S i , where S ∈ [0, 1] and Similar theorem can also be shown in the case of 3-way resemblance.


Variational sparse GPR can alternatively be formulated to approximate the posterior of the latent function by a full GP parameterized by the inducing points and the statistics of inducing functions [1,26].

titsias [26] proposes to use is the conditional probability in the full GP.


As it turns out, the fact that the feedback is only partially censored makes a stochastic bisection search possible with minimal assumptions on φ(•, C).

we only need that φ(•, C) be Lipschitz, while the technique developed in [20] for a dynamic pricing problem requires φ(•, C) to be bi-Lipschitz.


Finally, by taking advantage of Nesterov-style acceleration [Nesterov, 2007], the worst-case convergence rate of the algorithm after k steps can be improved from The procedure (8) can also be used to solve the additive (6) and sparse additive (7) extensions via a block coordinate descent algorithm.

given a set of estimates d j (j = 1, . . . , p) we can fix all but one of the vectors d j and optimize over the non-fixed vector, by solving for some vector r j ∈ R n . For additive waveMesh (λ 2 = 0), this reduces to the univariate problem which can be solved via the algorithm (8).


W p = b (since all constraints are linear, we abuse notation and consider B, W to be matrices).

the hyperplanes corresponding to the constraints in W intersect in p. Now, for all directions d ∈ R n such that there exist active constraints W i , W j with W i d < 0 and W j d > 0, the line segment through p induced by d must necessarily be 0.


We assume a recent generalization of the Dawid-Skene model introduced in [22] to model the responses, which captures the heterogeneity in the tasks as well as the workers.

each new arriving worker is parametrized by a quality parameter p j ∈ [0, 1] (for the j-th arriving worker), which is i.i.d.


More generally, we can assume that λ is some function of variable i.

we assume λ = a/i + b + ci where a, b and c are non-negative numbers.


Ultimately, we want to find the nodes that have the greatest effect on Risk(P (X )), so we must condition Risk(P (X )) on the beliefs at each node.

if we learned that the true marginal probabilities of node x were π x , what effect would that have on our current risk, or rather, what is Risk(P (X )|P (x) = π x )?


Two-stage Estimator: the workers' parameters are first estimated using the control items, and are then used to predict the target items.

scoring workers: Predicting target items: where we use the maximum likelihood estimator as a general procedure for estimating parameters.


We now assume that the functions v sj (x j ) are uniformly bounded from a Sobolev space of order two.

let {ψ jk ∶ k = 0, 1, . . .} denote a uniformly bounded, orthonormal basis with respect to L 2 [0, 1], and assume that v sj ∈ H j where Then the family of random variables Z (V,w) ∶= √ n ⋅ w ⊺ (Σ(V ) − Σn (V ))w is sub-Gaussian and sample continuous.


Finally, by processing matches with a 4D convolutional network we establish a strong locality prior on the relationships between the matches.

by design, the network will determine the quality of a match by examining only the information in a local 2D neighbourhood in each of the two images.


. . , U d ) T follows a uniform distribution in S d−1 .

when Eξ 2 exists, Θ j (Cov(X)) = Θ j (K).


Schur complement deflation is unique in that it preserves orthogonality in all subsequent rounds.

if a vector v is orthogonal to A t−1 for any t, then This further implies the following proposition.


A graph is generated from an inhomogeneous random graph model if the vertex set contains N nodes and all edges are independent.

for any two nodes i, j ∈ V , i connects to j with some probability p ij and this event is independent of the formation of any other edges.


This is the first work we know of in this line of research of stochastic adaptive algorithms for statistical learning, hence many questions are still open.

it is not clear if high probability bounds can be obtained, as the empirical results hint, without additional hypothesis.


Consider a smoothed scenario, with the z-variable of the adversarial move (z, y) perturbed by an additive uniform noise σ = Unif[−γ/2, γ/2] for some γ ≥ 0.

the actual move revealed to the player at time t is (z t + s t , y t ), with s t ∼ σ.


For a complete theory, the following questions are worth asking: • Under what conditions, the global solution with non-convex penalty is statistically better than the (one-stage) convex relaxation solution?

when does it lead to better prediction accuracy or generalization error?


Let D π t be the distribution on state-action pairs at time t under policy π.

a sample (s, a) is drawn from D π t by first drawing s 1 ∼ α(•), then following policy π for time steps 1 through t, which generates a trajectory (s 1 , a 1 , .


We make similar observations for ECG and Sine.

different positional bias leads to visibly different Recall T values.


q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Airline passengers Gym memberships  Results are shown in Figure 5b, demonstrating that participants converged to intuitively plausible patterns.

for both the volcano and the airline passenger data, participants converged to compositions resembling those found in previous analyses [6].


1: for all (i, j) ∈ U t do 3: 4: end for for all (i, j) / ∈ U t do 7: end for 10: x t ← X (µ t , K t ) 11: end for Consensus propagation is a distributed protocol because computations at each node require only information that is locally available.

the messages F ij (K t−1 ) and G ij (K t−1 ) transmitted from node i to node j depend only on {µ t−1 ui , K t−1 ui |u ∈ N (i)}, which node i has stored in memory.


In this paper, we study these questions by considering the competitiveness of a simple residual network (composed of an arbitrarily deep, nonlinear residual unit and a linear output layer) with respect to linear predictors (or equivalently, 1-layer networks).

we consider the optimization problem associated with training such a residual network, which is in general non-convex and can have a complicated structure.


Imagine that ( ¥ 0) ¢¡ £ ¥ ¢¦ §£ ¢ 21 and ( ¥ 3) ¢¡ " £ ¢¦ 4 £ ¢ 4 1 have been stored as episodic patterns.

their EP representations are stored in the hippocampus and are available for recall and replay.


Therefore, the overall computational complexity of the approach is O(MN log N + N 2 Kd), which can be prohibitive for large N and M. One natural way to reduce the complexity of the graph traversal part of Isomap is to not run Dijkstra's algorithm N times.

instead of generating the entire N × N matrix of dissimilarities, generate an interesting subset of n rows, n << N. There are a family of MDS algorithms, here called Rectangular Dijkstra (RD) MDS algorithms.


Let f be the formula in M from which f was derived foreach logical variable in f that was substituted by the j-th value in ∆iR to yield f do w = w × |πj| where πj is the j-th cell of π return M The key advantage using Algorithm Partition-Ground is that the lifted algorithm (LMAP) will have much smaller space complexity than the one using Algorithm Constrained-Ground.

unlike the latter, which yields O(n|U |) ground atoms (assuming each predicate has only one argument in U ) where n is the number of constants in the domain of U , the former generates only O(k|U |) ground atoms, where k << n. The following example illustrates how algorithm partition-ground constructs a new MLN.


This result holds regardless of the underlying distribution.

unlike algorithms such as [12], we do not require independence of the binary response variables (labels).


( 3), we can simplify it to where h p η (h|X T θ)h using the property of GB-RBM that the elements of h are independent given X T θ.

assume g = h p η (h|X T θ)h, then g i = σ(W i X T θ), where W i is the ith row of W and σ is the logistic function σ(x) = 1/(1 + exp (−x)).


A node-perturbation in a GGM can be equivalently represented through a perturbation of the entries of a row and column of the corresponding precision matrix (Figure 1).

we can detect a single node perturbation by looking for a row and a corresponding column of Θ 1 − Θ 2 that has nonzero elements.


AltMinPhase has similar statistical complexity to that of PhaseLift and PhaseCut while being much more efficient computationally.

for accuracy ǫ, we only need to solve each least squares problem only up to accuracy O (ǫ).


We emphasize that this is a very general definition of analog computation in recurrent neural networks.

we do not specify any definite output node but allow the output to occur at any node.


Despite these advances, the work possesses a number of limitations.

our closed world assumption is problematic, although we believe the extension to open worlds is relatively straightforward.


Strings: Efficient computation of string kernels using suffix trees was described in [9].

it was observed that expansions of the form m i=1 α i k(x i , x) can be evaluated in linear time in the length of x, provided some preprocessing for the coefficients α and observations x i is performed.


In this paper, we propose that salience may be defined based on a strategy for maximum information sampling.

shannon's self-information measure [9], -log(p(x )), applied to the joint likelihood of statistics in a local neighborhood decribed by w, provides an appropriate transformation between probability and the degree of infom1ation inherent in the local statistics.


The situation is even worse in the presence of noise where active learning appears to be particularly hard.

prior to this work, there were no known efficient active algorithms for concept classes of super-constant VC-dimension that are provably robust to random and independent noise while giving improvements over the passive case.


Secondly, while we also consider more structured distributions (monotone, k-histograms), the techniques developed in the study of distributed parameter estimation do not apply to our problems.

those results reduce to the problem of learning a high-dimensional vector (say, where each coordinate parametrizes a spherical Gaussian distribution), where the value at each coordinate is independent of the others.


If the reward function R is bounded, with |R t (x, a)|  R for all (x, a, t), then this also implies a bound on regret over episodes experienced prior to time T , defined by Regret(T ) = ).


Our main contribution is the KRONDPP model along with efficient algorithms for sampling from it and learning a Kronecker factored kernel.

inspired by the algorithm of [25], we develop KRK-PICARD (Kronecker-Kernel Picard), a block-coordinate ascent procedure that generates a sequence of Kronecker factored estimates of the DPP kernel while ensuring monotonic progress on its (difficult, nonconvex) objective function.


[6] defined a scoring matrix for the c-class classification problem.

it is such It follows from this definition that ΘΘ = Π −1 − 1 n 1 c 1 c . In the literature [15], the authors presented a specific example for Θ = (θ 1 , .


Here we extend this idea to Markov games and show that it promotes cooperation in several types of sequential social dilemma, via a profitable interaction with policy learnability.

we find that inequity aversion improves temporal credit assignment for the important class of intertemporal social dilemmas.


3 Problem Formulation We use bold letters to denote vectors.

x ∈ R p denotes an authentic sample without corruption, e ∈ R p is for the noise, and z ∈ R p is for the corrupted observation z = x + e. Here p denotes the ambient dimension of the observed samples.


For global differential privacy, we can generalize the definition of staircase mechanisms to hold for all neighboring database queries x, x (or equivalently within some sensitivity), and recover all known existing optimal mechanisms.

the geometric mechanism shown to be optimal in [17], and the mechanisms shown to be optimal in [14,16] (also called staircase mechanisms) are special cases of the staircase mechanisms defined above.


RAI is a constraint-based structure learning algorithm.

it learns a structure by performing independence tests between pairs of variables conditioned on a set of variables (CI-tests).


We derive here new generalization bounds, based on Rademacher Complexity theory, for model selection and error estimation of linear (kernel) classifiers, which exploit the availability of unlabeled samples.

two results are obtained: the first one shows that, using the unlabeled samples, the confidence term of the conventional bound can be reduced by a factor of three; the second one shows that the unlabeled samples can be used to obtain much tighter bounds, by building localized versions of the hypothesis class containing the optimal classifier.


The eigengap in Assumption 3.1 implies that the optimal solution is identifiable.

the optimal solution U ⇤ is unique up to rotation.


At each iteration, SDCA chooses i at random according to p t , and updates the α i corresponding to the loss φ i to increase D. This scheme has been used for particular losses before, and was analyzed in [6] obtaining linear rates for general smooth losses, uniform sampling and l 2 regularization, and recently generalized in [10] to other regularizers and general sampling distributions.

[10] show improved bounds and performance by statically adapting to the global smoothness properties of losses; using a distribution iterations to obtain an expected duality gap of at most ε.


For benign problem instances, the benefit of using the taxonomy is the vastly improved dependence on the number of arms.

without a taxonomy or any other structure, regret of any algorithm for stochastic MAB scales linearly in the number of (near-optimal) arms, for a sufficiently large t. let N δ be the number of arms x such that δ 2 < ∆(x) ≤ δ.


This result is a corollary of the specific results for each class, which is proved in the appendix.

a more complete statement for unimodality, log-concavity and monotone-hazard rate, with precise dependence on both n and " is given in Theorems 7, 8 and 9 respectively.


One of the key issues preventing neural networks from being visualizable and understandable is that they assume the variable relationships in data are extremely high-dimensional and complex [23].

each hidden neuron takes as input all nodes from the previous layer and creates a high-order interaction between these nodes.


In order to achieve this type of structured sparsity at multiple levels of the hierarchy, we utilize an overlapping-group-lasso penalty recently proposed in [21] for genetic association mapping problem, where the goal is to identify a small number of SNPs (inputs) out of millions of SNPs that influence phenotypes (outputs) such as gene expression measurements.

given the tree hierarchy T = (V, E) over image categories, each node v ∈ V of tree T is associated with group G v , composed of all leaf nodes in the subtree rooted at v, as illustrated in Figure 1(b).


If we restrict our attention to the case of linear reward functions we can develop an efficient algorithm to compute an approximate best response.

we consider the case where the reward for a state (s, θ) is defined as a linear combination of state features for some feature function φ : R(s, a H , a R ; θ) = φ(s) θ.


The kernel embeddings can also be generalized to conditional densities p(X|z) [9] µ Given this embedding, the conditional expectation of a function f ∈ F can be computed as E X|z [f (X)] = f, µ X|z F . Unlike the ordinary embeddings, an embedding of conditional distribution is not a single element in the RKHS, but will instead sweep out a family of points in the RKHS, each indexed by a fixed value z of the conditioning variable Z.

it is only by fixing Z to a particular value z, that we will be able to obtain a single RKHS element, µ X|z ∈ F. conditional embedding is an operator, denoted as C X|Z , which can take as input an z and output an embedding, i.e., µ X|z = C X|Z φ(z).


Since the columns are 1 -normalized, the columns of H have non-negative entries and sum to one.

all columns of XD −1 are in the convex hull of the columns indexed by K. The problem of determining K is reduced to finding the extreme points of a convex hull.


We write OPT S (Y ) to denote the lowest possible cost of k centers from Y .

oPT S (Y ) = min C⊆Y, |C|=k {cost S (C)}.


In the following Theorem, we show that this kernel function plays an important role in characterizing optimizers of optimization (6).

we show that the objective function of the neural network optimization has a term where this kernel function is applied (component-wise) to the inner product matrix among vectors u 1 ,...,u r . Theorem 1 The loss function (3) for a matched PNN can be written as where ψ(.)


By making multiple predictions, a system can improve coverage of lower density areas of the solution space, without sacrificing performance on the majority of examples.

by optimizing for the oracle loss, a multiple-prediction learner can respond to ambiguity much like a human does, by making multiple guesses that capture multi-modal beliefs.


While the literature on stochastic optimization is vast, we emphasize that an optimization guarantee does not necessarily imply a bound on regret.

we directly build on an optimization method that has been developed by Nemirovski and Yudin [14,Chapter 9].


In all our experiments, we use a decreasing step size for SGLD as suggested by [15].

we use t = a(b + t) −γ , where the parameters a, b and γ are chosen for each dataset to give the best performance of the algorithm on that particular dataset.


Then For any i, j, with i ∈ I l and j ∈ S l \ I l , we must have y i = y j .

also, for any i, j with y i = y j and i, j ∈ S l , either i ∈ I l or j ∈ I l . there is a one-to-one correspondence between 1) edges i, j for which i, j ∈ S l and either i ∈ I l or j ∈ I l and 2) edges i, j for which i, j ∈ S l and y i = y j .


In the setting considered in this paper, the covariance function K θ is composed of a view kernel that models covariances between views, and an object kernel that models covariances between objects.

the covariance between sample n (with corresponding feature vectors x pn and w qn ) and sample m (with corresponding feature vectors x pm and w qm ) is given by the factorized form (Bonilla et al., 2007;Rakitsch et al., 2013): Observed versus unobserved feature vectors Our model can be used when either one, or both of the view/sample feature vectors are unobserved.


As the proposed depth-wise separable channel-wise convolution can directly replace the depth-wise separable convolution, we attempt to apply the replacement here.

the same special depth-wise convolution is employed, but is followed by a channel-wise convolution with a kernel size of d c whose number of output channels is equal to the number of classes.


The restricted isometry property [7,6] (or the related restricted eigenvalue [2] or compatibility conditions [19]) have been used to establish guarantees on the estimation and model selection errors of the Lasso or similar approaches.

bickel, Ritov and Tsybakov [2] show that, under such conditions, with high probability, The same conditions can be used to prove model-selection guarantees.


In nonparametric Bayesian regression, we specify prior probability distributions over the whole (infinitely dimensional) space of functions f i (t).

in the GP regression framework this distribution is chosen to be a zero-mean GP.


. . , j(k, N )) ∈ Z N , which represents the how outdated each of the blocks are.


As the main contribution of this work we establish a close relationship between diversity with submodular energies and the parametric submodular minimization.

the joint M -best diverse labelings can be obtained by running a non-parametric submodular minimization (in the special case -max-flow) solver for M different values of γ in parallel, for certain diversity measures.


We implemented the steps of this pipeline with several optimizations.

we did not fully compute the approximate distributions L and S: rather, we lazily computed their entries that were requested by Algorithm 1.


Here, we consider a delayed-memory XOR task, which performs the exclusive-or (XOR) operation on the input history stored over extended duration.

the network receives binary pulse signals, + or −, through an input channel and a go-cue through another channel.


We find that in every experiment, for the training set sizes considered, the normalized hybrid algorithm with R = 2 has test error that is either the lowest or very near the lowest among all the algorithms.

it almost always outperforms the  In this section, we give a distribution free uniform convergence bound for our algorithm.


An important distinction is that we require strong convexity of the function F (w) with respect to the norm w . This is rather different than requiring the loss function z → (z, y) be strongly convex on the reals.

the loss of a linear predictor, w → ( w, x , y) can never be strongly convex in a multi-dimensional space, even if is strongly convex, since it is flat in directions orthogonal to x.


[9] who cleverly observed that x ∈ D m := DIS(A m ) can be efficiently determined using a single call to an ERM oracle.

to apply their method, we use the oracle to find 4 h = arg min{err(h, Zm−1 ) ) is at most γ∆ m−1 .


Inspired by the success of accelerated full gradient methods (e.g., [12,1,22]), several recent work applied Nesterov's acceleration schemes to speed up randomized coordinate descent methods.

nesterov [13] developed an accelerated randomized coordinate gradient method for minimizing unconstrained smooth convex functions, which corresponds to the case of Ψ(x) ≡ 0 in (1).


One simple internal model that subjects may have about the nature of the stimulus sequence in a 2-alternative forced choice (2AFC) task is that the statistical contingencies in the task remain fixed throughout the experiment.

they may believe that the experiment is designed such that there is a fixed probability γ, throughout the experiment, of encountering a repetition (x t = 1) on any given trial t (thus probability 1−γ of seeing an alternation x t = 0).


To further verify this, we compare our L2 norm verification signal on all the sample pairs with those only constrain either the positive or negative sample pairs, denoted as L2+ and L2-, respectively.

the L2+ only decreases the distances between DeepID2 features of the same identity, while L2-only increases the distances between DeepID2 features of different identities if they are smaller than the margin.


Generally, the sparsity of a DNN model can be considered in two aspects: the sparsity of connections among neurons and the sparsity of neuron activations.

the sparsity of activations also include that of middle-layer activations and inputs, which can be treated as a special case.


Instead, we allow it to be a projection of a latent structure (Pearl, 2009).

we assume the presence of additional hidden variables Q that are not in G inv but induce dependency (for example, "interactive forks" (Pearl, 2009)) among H. For clarity, we omit these variables from the graph and use bi-directional edges to represent the dependency induced by them.


Our main convergence results are also readily generalized to stochastic target policies by replacing the sub-sampling process described in Section 2 with a sample-weighting process.

instead of including or excluding transitions depending upon whether the action taken matches a deterministic policy, we include all transitions but give each a weight.


In this paper, we show that the latter is superior and describe an algorithm that can do so, outperforming convolutional methods.

we present tiled convolutional networks (Tiled CNNs), which use a novel weight-tying scheme ("tiling") that simultaneously enjoys the benefit of significantly reducing the number of learnable parameters while giving the algorithm flexibility to learn other invariances.


Unlike the generic network-based imitation learning, the MPC policy can exploit its inherent structure.

because the network contains a well-defined notion of the dynamics and cost, it is able to learn with much lower sample complexity that a typical network.


In this section, we consider how to compose the nonlinear mappings induced by kernel functions.

we show how to derive new kernel functions which compute the inner product after successive applications of the nonlinear mapping Φ(•).


to find the correct setting of γ 1 , it suffices to perform a binary search over the possible values of γ 1 in the interval (−vol(G), λ 2 (G)) until the correlation constraint is satisfied, that is, until (s T D G x) 2 is sufficiently close to κ 2 1 , see [8,14].


Note that if q = p/2, the regularization term on α j in Problem (2) becomes the standard p-norm.

if {p, q} = {2, 1} in Problem (2) as used in the methods of [15] and [1], the 2 -norm regularizer is applied to α j .


To achieve this, we implemented a version of adaptive hyperparameter boosting, in which rounds of optimization are interleaved with boosting and hierarchical stacking.

suppose that N ∈ N and S is a screening stimulus set.


To learn a metric which approximates the ideal geometric setup described above, we introduce, for each training point, a conditional distribution over other points (as in [5]).

for each Ü we define a conditional distribution over points such that If all points in the same class were mapped to a single point and infinitely far from points in different classes, we would have the ideal "bi-level" distribution: Furthermore, under very mild conditions, any set of points which achieves the above distribution must have the desired geometry.


Other works have studied the risk landscapes of neural network optimizations for more complex structures under various model assumptions [9,10,11,12,13,14,15,16,17,18,19].

references [13,14,15] consider a two-layer neural network with Gaussian inputs under a matched (realizable) model where the output is generated from a network with planted weights.


Such implicit bias, which can also be viewed as a form of regularization, can play an important role in learning.

implicit regularization has been shown to play a crucial role in training deep models [14,13,18,11]: deep models often generalize well even when trained purely by minimizing the training error without any explicit regularization, and when there are more parameters than samples and the optimization problem is underdetermined.


the size of label set of the child that the example follows, relative to its parent's size.

given w and P , the ambiguity for an example x is Note that A ∈ [0, 1].


from a Gaussian with unknown mean and unit variance.

let Θ = R, Y = R, and P (y ≤T |θ) be the product Π T t=1 N (y t ; θ, 1).


Analytic differentiation of optimization Finally, our work relates methodologically to a recent trend of incorporating more structured layers within deep networks.

recent work has looked incorporating quadratic programs [Amos and Kolter, 2017], combinatorial optimization [Djolonga and Krause, 2017], computing equilibria in games [Ling et al., 2018], or dynamic programming [Mensch and Blondel, 2018].


In this paper, we propose a novel processing technique to learn code semantics, and apply it to a variety of program analysis tasks.

we stipulate that a robust distributional hypothesis of code applies to both human-and machine-generated programs.


IHDP data [16] is an experimental dataset collected by the Infant Health and Development Program.

a randomized experiment was conducted, where intensive high-quality care were provided to the low-birth-weight and premature infants.


We therefore learn to ensemble by a gating component as: where g i is the importance score of the i-th branch's logits z i , and z e is the logits of the ONE teacher.

we denote the original branch as i = 0 for indexing convenience.


It is important to note that the pruning strength is an upper bound on the approximation error.

the optimal values are guaranteed to lie within the ranges of the approximate ranged value function.


The trades that achieve this optimal drop in risk are called efficient: Our following key result shows that efficient trades have remarkable structure: once the state r and subset S is specified, there is a unique efficient trade, up to cash transfers.

the surplus is removed from the position vectors and then redistributed as cash to the traders; the choice of trade is merely in how this redistribution takes place.


Second, when we have more sample points close to the boundary, the variance of the parameters S N is lower.

when we emphasize samples with high prediction variances, the uncertainty of parameters tends to be reduced, akin to the variance reduction strategy in active learning [30].


Hardt and Roth [30] showed a lower bound of Ω( √ kn + √ km) on additive error by showing a reduction to the linear reconstruction attack [18].

any algorithm that outputs a low rank matrix with additive error o( √ kn + √ km) cannot be differentially private!


These provide a richer picture of the step-size control performed by the line search.

they show that the line search chooses step sizes that follow a nontrivial dynamic over time.


On the other hand, OCCA is blind to and hence worst at the encoding error.

its encoding error is even worse than that of the baseline PBR.


That is, any continuous Markov process (with no jumps) with the desired invariant distribution can be cast within our framework, including HMC, Riemann manifold HMC, SGLD, SGHMC, their recent variants, and any future developments in this area.

our method provides a unifying framework of past algorithms, as well as a practical tool for devising new samplers and testing the correctness of proposed samplers.


To be able to exploit this, we denote M = { mi } M i=1 as the set of possible values for illuminant chromaticity m, and construct it from a training set.

we quantize 1 the chromaticity vectors { mt } T t=1 of the illuminants in the training set, and let M be the set of unique chromaticity values.


However, these methods cannot be directly applied in kernel space and have assumptions about the underlying distributions of the data.

as noted by the authors, spectral hashing assumes a uniform distribution over the data, a potentially restrictive assumption in some cases.


Our motivation for studying SIMs under such a constraint can ultimately be traced to the vast sufficient dimension reduction (SDR) literature.

we would like to point out [22] as a source of inspiration.


The transformation of the individual loss reduces penalties of all examples, and in particular benefits correctly classified data.

if such examples are "far enough" from the decision boundary, like in the 0-1 loss, their penalty becomes zero.


We note that F t i is submodular and that if F t (S) ≥ 1 then F t i (S) ≥ 1.

in this sense F t i is an easier objective than F t . Also, for any v, the gain of F t i at ∅ is the normalized gain of F t at S i−1 .


. . , the m th according to π k−m+1 and then starts again.

formally, this can be written as It is straightforward to show that the value v π k,m of this periodic non-stationary policy π k,m is the unique fixed point of the following operator: Finally, it will be convenient to introduce the following discounted kernel: for any pair of values v and v , it can easily be seen that T 3 Tightness of the performance bound of Theorem 1 The bound of Theorem 1 is tight for API in the sense that there exists an MDP [4,Example 6.4] for which the bound is reached.


The next result shows that, when the number of voters tends to infinity (and the weight of each voter tends to zero), the variance of W Q will tend to 0 provided that the average of the covariance of the risks of all pairs of distinct voters is ≤ 0.

the variance will always tend to 0 if the risk of the voters are pairwise independent.


The model selection problem (7) may now be solved using, e.g., Quasi-Newton methods such as BFGS which only require function value and gradient at a hyperparameter setting.

reaching the minimizer of f too closely is not important.


For p = 2, [20] has shown that mass transport can be formulated as a fluid mechanics problem.

the Wasserstein distance can be formulated as a distance arising from a Riemannian metric on the space of probability densities.


This paper presents a probabilistic-graphical model that can be used to infer characteristics of instantaneous brain activity by jointly analyzing spatial and temporal dependencies observed in electroencephalograms (EEG).

we describe a factor-graph-based model with customized factor-functions defined based on domain knowledge, to infer pathologic brain activity with the goal of identifying seizure-generating brain regions in epilepsy patients.


Our idea is to develop the analogue of a Newton iteration-rather than gradient descent-for the purpose of sampling distributions more efficiently.

we design an algorithm where each map T k is now computed as the perturbation of the identity function along the direction that minimizes a certain local quadratic approximation of J.


Therefore, we focus on sets of size ≤ m/ 3 (at most 1/ 3 elements for each function) to find an initial set such that the remaining elements have marginal value ≤ 3 V i for f i , for every i.

one can try all possible initial sets of this size (i.e.


We prove our new method converges globally and linearly at a faster rate compared with the method in [8].

the main contributions of this paper can be summarized as follows: (i) We show that the existing ADMM in [8] exhibits a slow and fluctuating "tail convergence", and provide a theoretical understanding of why this phenomenon occurs.


The main feature of proposed algorithms is summarized as follows: Our algorithms achieve a sample complexity of O(1/ε) with much better dependency on the problem dimension than the ones in existing work.

if the smallest magnitude of the non-zero components of the optimal solution is not too small, the rate can be boosted to near the minimax optimal sample complexity of full information algorithms.


(5) satisfies d( Θ, Θ 0 ) ≤ with probability larger than 1 − δ.

this is achieved by taking λ = 6 log(4q/δ)/(nα 2 (1 − ρ)) .


. . , y m } ⊆ Y two sets of observations between which we would like to find a correspondence.

we would like to find some permutation π ∈ Π m on m terms, that is m×m and π1 m = 1 m and π such that the pairs Z(π) := (x i , y π(i) ) for 1 ≤ i ≤ m correspond to dependent random variables.


For local constant approximation, the error term α x−x ′ is the first order in x−x ′ ; for local linear approximation, the error term β x−x ′ 2 is the second order in x − x ′ ; for local quadratic approximation, the error term ν x − x ′ 3 is the third order in x − x ′ .

if f (x) is smooth with relatively small α, β, ν, the error term becomes smaller (locally when x − x ′ is small) if we use a higher order approximation.


The ergodic decomposition theorem for stationary processes (see, e.g., 7) states that any stationary process can be expressed as a mixture of stationary ergodic processes.

a stationary process ρ can be envisaged as first selecting a stationary ergodic distribution according to some measure W ρ over the set of all such distributions, and then using this ergodic distribution to generate the sequence.


In this paper we propose to use relational modeling [17,16,27] in order to quantify functional coherent groups of resting state networks.

we investigate how this line of modeling can be used to discriminate patients with multiple sclerosis from healthy individuals.


Described this way, it is clear that the Plackett-Luce model is basically a partial size-biased permutation of the atoms in G [10], and the existing machinery of random measures and exchangeable random partitions [11] can be brought to bear on our problem.

in Section 2 we will use a gamma process as the prior over the atomic measure G. This is a completely random measure [12] with gamma marginals, such that the corresponding normalized probability measure is a Dirichlet process.


For this, we recall a closely related result shown by de Farias and Van Roy [1] for the ALP.

de Farias and Van Roy [1] showed that given an appropriate weighting (or in their context, 'Lyapunov') function ψ, one may solve an ALP, with ψ in the span of the basis functions Φ; the solution to such an ALP then satisfies: provided β(ψ) ≤ 1/α.


Throughout the execution, we use the inline comments in order to prescribe a feasible (but not necessarily optimal) assignment of the data points to (a subset of k of) the current candidate centers.

we maintain an array ASSIGN, where we write ASSIGN[j] = u (for some center u in our current set of candidate centers) to denote that all of the points in the optimal cluster S * j are assigned to the candidate center u.


These ideas have been exploited also for deriving novel safe policy-search approaches [15,16,17,18,19] that have obtained significant empirical results.

policy-gradient methods are among the most commonly used RL techniques to solve complex high-dimensional tasks.


We will also show that the capsule projection can be viewed as normalizing the multiple columns of the weight matrix simultaneously to form an orthogonal basis, which makes it more effective in incorporating novel components of input features to update capsule representations.

the capsule projection can be viewed as a multi-dimensional weight normalization in capsule subspaces, where the conventional weight normalization is simply a special case of the capsule projection onto 1D lines.


As we will show, for many samplings of interest small enough parameter v can be obtained in time required to read the data {A i }.


Then with probability at least 1 − δ, C is (σ, ǫ) consistent.

the clusters containing A[X] and A ′ [X] are internally connected and mutually disconnected in C(r) for r defined by where ρ is now slightly modified (in constants), i.e., ρ := min σ 7 , ǫτ 72d , τ 24 .


In this setting, we would like to predict the total mass of galaxy clusters, using the dispersion of velocities in the direction along our line of sight.

we will use the 'ML1' dataset, as obtained from the authors of [16,17], who constructed a catalog of massive halos from the MultiDark mdpl simulation [9].


We observe that the full tree2tree model significantly outperforms the other alternatives.

on JC-BL, the full tree2tree's program accuracy is 30 points higher than the tree2tree model without parent attention feeding.


Corollary 4 is consistent with known beamforming results.

it is known that A † is optimal (in terms of minimum mean squared error) for underdetermined ICA [19, section 3B].


Repeating the experiment over other instances of (1) obtained by solving (10) with randomly selected x, z, we generally obtain graphs that look like Figure 2.

sGD usually escapes spurious local minima even when they are engineered to exist.


This has been shown for linear mappings [1,11], as well as for non-linear ones [7,14,20].

[20] proposed to encourage the shared-private factorization to be nonredundant while simultaneously discovering the dimensionality of the latent space.


Assumptions 3 and 4 are the key technical conditions under which we can generalize the analyses in [2,3,4] to the general GLM case.

we can show that the statements of the following propositions hold, which show that the random vectors X following the GLM graphical model in (10) are suitably well-behaved: Proposition 3.


Viewing E = A -A as a perturbation to the "ideal" A that results in A = A+E, we ask: When can we expect the resulting rows of Y to cluster similarly to the rows of Y?

when will the eigenvectors of L, which we now view as a perturbed version of L, be "close" to those of L?


Moreover, alternative choices of u (v) can help define new algorithms.

we will refer to the algorithm based on the surrogate loss defined by u (v) = ue v as StructBoost, in reference to the exponential loss used in AdaBoost.


We use the standard accuracy measure to compare various methods.

for a given algorithm, we compute predicted label y j for each sequential data point X test j , 1 ≤ j ≤ m in the test set and then compute accuracy as 1 m m i=1 I[ y i == y i ].


Here, 𝐸 𝑖 and 𝐸 𝑖+1 are the system energies at solution configuration 𝑖 and 𝑖 + 1 respectively; 𝐾 is the Boltzmann constant; and 𝑇 is the system temperature.

a step will be taken when a lower energy is found.


7 For a generic kernel a0, the shift-coherence is bounded by | a0, sτ [a0] | ≈ 1/ √ k for any shift τ .

8 under a Bernoulli-Gaussian model, for each j, E[z 2 j ] = θ i =0 a0, si[a0] 2 .


In the same spirit, we investigate the classification calibration property of the AT k loss.

we first obtain the population form of the AT k objective using the infinite limit of (2) We then consider the optimization problem where the infimum is taken over all measurable function f : X → R. We say the AT k (aggregate) loss is classification calibrated with respect to p if f * has the same sign as the Bayes rule f c . The following theorem establishes such conditions.


The decomposition of relations over latent factors allows a significant reduction of the number of parameters and is motivated both by computational and statistical reasons.

our approach is quite scalable both with respect to the number of relations and to the data samples.


We also generate mini-batches in which the three extrinsic scene variables are held fixed but all other properties of the face change.

these batches consist of many different faces under the same viewing conditions and pose.


Since [∆ − ∆ * ] + is a monotonically increasing function of ∆, its upper bound is derived directly from the upper bound of ∆.

the upper bound of ∆ is given by and the upper bound of the surrogate loss with a fixed ∆ * is given by When the original space is fixed, the problem is not straightforward since the surrogate loss becomes a V-shaped function with ρ > 1.


However, for such multiclass problems, a complete theoretical understanding of boosting is lacking.

we do not know the "correct" way to define the requirements on the weak classifiers, nor has the notion of optimal boosting been explored in the multiclass setting.


If the regularization constant c is chosen so that then, there exists a sample size m such that (10) and (11) are satisfied for all m > m and n = ↵m.

if ↵ = 2 we can satisfy (10) and (11)  Proof.


Here we show that several characteristic phenomena of multistability derive naturally from applying standard MCMC inference to Markov random fields (MRFs) -high dimensional, loosely coupled graphical models with spatial structure characteristic of many low-level and mid-level vision problems.

we capture the classic findings of Gamma-distributed mode-switching times in bistable perception; the biasing effects of contextual stimuli; the situations in which fused (rather than bistable) percepts occur, and the propagation of perceptual switches in traveling waves across the visual field.


On the prediction of out-of-sample travel times, ML outperformed both a random model and the constant model used for the initialization (a(x, s, γ) = α(x) with α(x) obtained from (19) with uniform p path ).

when all journeys in the test dataset are considered, ML outperforms the baseline method with a 24% improvement.


Beyond imitation learning, we further study our SYNPO for reinforcement learning (RL) under the same transfer learning setting.

we use PPO [20] to fine-tune the three top performing algorithms on GRIDWORLD.


Our final result shows that a decision rule that does not directly use z as an input variable or for determining thresholds will have lower accuracy than the optimal rule that uses this information.

we show that DLPs are suboptimal for trading off accuracy and impact parity.


But we view as our primary contributions the proposed algorithms and theoretical results, which strengthen in important ways previous results on LP approaches and unify certain ideas in the approximate DP literature.

highlights of our contributions include: 1.


Indeed, we can recover the coupling layer from the autoregressive transformation of MAF by setting µ i = α i = 0 for i ≤ d and making µ i and α i functions of only x 1:d for i > d (for IAF we need to make µ i and α i functions of u 1:d instead for i > d).

both MAF and IAF can be seen as more flexible (but different) generalizations of Real NVP, where each element is individually scaled and shifted as a function of all previous elements.


Bias and MSE In [17], the bias and MSE of the standard SG-MCMC algorithms with a Kth order integrator were analyzed, where the order of an integrator reflects how accurately an SG-MCMC algorithm approximates the corresponding continuous diffusion.

if evolving x t with a numerical integrator using discrete time increment h induces an error bounded by O(h K ), the integrator is called a Kth order integrator, e.g., the popular Euler method used in SGLD [3] is a 1st-order integrator.


First we note that the objective K is shifted by a constant if we multiply all variances by a factor σ 2 i → ασ 2 i , which leaves its minima invariant.

the objective is only sensitive to changing ratios between eigenvalues.


Our first observation is that the collision probability can be well approximated using the χ 2 similarity: Figure 4 shows this approximation is better than 1 π cos −1 (ρ 1 ).

in sparse data, the approximation is very accurate (except when ρ χ 2 is close to 1), while the bound 1 π cos −1 (ρ 1 ) is not sharp (and the curve is not smooth in ρ 1 ).


Inspired by recent model distillation methods in [20,29,34], we use a smoothed underlying distance matrix to solve the "vanishing gradient" problem when updating B and Λ.

the 9) is replaced by a Sinkhorn distance with the smoothed underlying distance, i.e., y S (B(R), λ m (A); D τ θ ), where (•) τ , 0 < τ < 1, is an element-wise power function of a matrix.


While the existence of very small adversarial perturbations seems counter-intuitive in this low-dimensional illustrative example (i.e., r in and r unc can be large for some choices of f ), we show in the next sections that this is the case in high dimensions.

if for all i, P(C i ) ≤ 1 2 (the classes are not too unbalanced), we have To see the dependence on the number of classes more explicitly, consider the setting where the classes are equiprobable, i.e., P(C i ) = 1 K for all i, K ≥ 5, then This theorem is a consequence of the Gaussian isoperimetric inequality first proved in [33] and [34].


In this paper we give necessary and sufficient conditions under which kernels of dot product type k(x, y) = k(x . y) satisfy Mercer's condition and thus may be used in Support Vector Machines (SVM), Regularization Networks (RN) or Gaussian Processes (GP).

we show that if the kernel is analytic (i.e.


Note that we do not consider privacy beyond ratings.

we omit any subsequent release of information due to users exploring items recommended to them.


OvA classifiers are usually implemented using a winner-take-all (WTA) strategy that associates a real-valued function with each class in order to determine class membership.

an example belongs to the class which assigns it the highest value (i.e., the "winner") among all classes.


In this paper we propose to extend the sparse coding framework to learn motion dictionaries.

we cast the problem of learning spatio-temporal primitives as a tensor factorization prob-lem and introduce tensor group norms over the primitives that encourage sparsity in order to learn the number of elements in the dictionary.


While our model is flexible enough to deal with different sets of unlabeled examples in the support set and the query set, for a task T ∼ P (T ) we only use a single unlabeled set of examples U T to follow the same training scheme in [Ren et al., 2018], for a better comparison with our baseline models.

for MetaGAN with MAML, we set S u T = S s T and Q u T = U T . For MetaGAN with RN, we set S u T = ∅ and Q u T = U T in loss eq.


Finally, we extend our results to general stochastic games.

lthe agents can choose the equilibrium selected by a fixed shared algorithm.


However we now give an explicit construction for the minimal method of choosing the weak hypothesis for which the number of iterations of greedy covering and InfoBoost grow linearly in k and the number of iterations of AdaBoost with Bias is quadratic in k. For any dimension N we define an example set which is the rows of the following (N + 1) × N dimensional matrix x: All entries on the main diagonal and above are +1 and the remaining entries −1.


We show that this changes the likelihood function in such a way that the Bayesian estimator model can account for reported perceptual behavior.

we compare the model's predictions to human motion discrimination data and demonstrate that the model accounts for the commonly observed perceptual adaptation effects of repulsion and enhanced discriminability.


Larger variation indicates stronger relations.

we consider a neuron f l j (p) a candidate (of witness) if its variation is larger than the median of all the observed variations in the layer.


However, such an equivalence does not extend to the stationary points of the two problems.

it is possible that a stationary point of eq.


Hartigan (1981) expanded on this idea and formalized the notions of high-density clustering, of the cluster tree and of consistency and fractional consistency of clustering algorithms.

hartigan (1981) showed that single linkage clustering is consistent when D = 1 but is only fractionally consistent when D > 1.


Their experimental evidence fell somewhat short in the last part, where "exponentation through active membrane conductances" should invert logarithmic encoding.

the authors observed that "In 7 out of 10 neurons, a third-order power law best described the data" (sixth-order in one animal).


unchanged in any world state) but unknown.

the applicability of the action in a state s ⊆ F depends on possible preconditions p that are realized (i.e.


Observe that this model has an interesting property in that it allows us to share statistical strength across word representations of different languages.

we show in our experiments how we can improve cross-lingual classification performance between English and German when a large amount of parallel data exists between English and French and only a small amount of parallel data exists between English and German.


NLRN uses the fewest parameters but outperforms all the competitors.

nLRN benefits from inherent parameter sharing and uses only less than 1/10 parameters of RED.


In this paper, we explore how learning can be applied to the discovery of mathematical identities.

we propose methods for finding computationally efficient versions of a given target expression.


It was recently shown that tuning properties of neurons in monkey motor cortex are adapted selectively in order to compensate for an erroneous interpretation of their activity.

it was shown that the tuning curves of those neurons whose preferred directions had been misinterpreted changed more than those of other neurons.


Every feature in the encoder is r-sparse with non-zero loadings on the same set of r original dimensions; and, you cannot do better with a row-sparsity of r. Further, the batch algorithm does not distinguish between the k-factors.

there is no top component, second component, and so on.


We can combine these two variables into one and use it to adjust q(w) and q(α) jointly.

we introduce the variable c: Setting c = c 0 = 1, we perform the regular updates over q(τ ), q(w) and q(α).


as the problem of discovering the mapping from image I 1 to image I 2 and searching for an image I 4 that has the same relation to image I 3 as I 1 to I 2 .

we find a function T (parametrized by θ) that maps each pair of images (I 1 , I 2 ) to a vector x 12 = T (X 1 , X 2 ; θ).


We use this change of variables to define a new function f , parameterized by θ, that is equivalent to the original function f : The gradient and the Hessian of this new function f are (by the chain rule): A gradient descent iteration θt = θt−1 − η∇ f ( θ) for the transformed function corresponds to for the original parameter θ.

by left-multiplying the original gradient with a positive definite matrix D −1 , we effectively apply gradient descent to the problem after a change of variables θ = D 1 2 θ.


While Theorem 1 shows that there are situations where at least some of the approval winners may not be MLE best alternatives, it does not paint the complete picture.

in both profiles used as counterexamples in the proof of Theorem 1, it holds that every MLE best alternative is an approval winner.


In case of dense connections, however, the number of parameters explodes with a factor 7 18 4 S /S.

roughly a factor 265 for 32×32 input, and 11,327 for 256×256 input.


Our main contribution is a novel loss-estimation technique that enabled us to prove strong regret bounds under various partial-feedback schemes.

our results largely improve on the best known results for the sleeping bandit problem [13], which suffers large losses from both from an initial exploration phase and from explicit exploration rounds in the main phase.


In this paper, assuming p(p+1) > m, we show that if the search space of (SDP) is compact and if the search space of (P) is a regularly defined smooth manifold, then, for almost all cost matrices C, if Y satisfies first-and second-order necessary optimality conditions for (P), then Y is a global optimum of (P) and, since p(p+1) 2 ≥ m, X = Y Y is a global optimum of (SDP).

first-and second-order necessary optimality conditions for (P) are also sufficient for global optimality-an unusual theoretical guarantee in non-convex optimization.


Definition 2 For any function f with n inputs, we define the sensitivity S(f ) as the maximum, over all inputs, of the difference in the value of f when one input of f is changed.


Finally, we run experiments on both simulation and real-world data and show our method outperforms the standard approaches to this problem.

we use data from a large-scale RCT measuring the effect of small classrooms and teacher's aids [WJB + 90, Kru99] to obtain ground-truth estimates of causal effects, which we then try and reproduce from a confounded observational study.


An ideal SSL algorithm is effective even with very little labeled data and benefits from additional unlabeled data.

we recommend combining SVHN with SVHN-Extra to test performance in the large-unlabeled-data regime.


In this situation, discrimination control and utility are still guaranteed for samples drawn from q D,X,Y that are transformed using p Ŷ , X|X,Y,D , where the latter is obtained by solving (6) with p D,X,Y .

denoting by q Ŷ |D and q X, Ŷ the corresponding distributions for Ŷ , X and D when q D,X,Y is transformed using p Ŷ , X|X,Y,D , we for n sufficiently large (the distortion control constraints (4) only depend on p Ŷ , X|X,Y,D ).


Experimental results show that the proposed scheme can outperform previous spatial-domain counterparts by a large margin on several datasets, without or with negligible accuracy loss.

the proposed algorithm can acquire accuracy gain for the ResNet on CIFAR-10, while achieving an impressive compression and acceleration of the network.


Let split k ∈ {1, . . . , p} be the index of the feature that the root node of the k th tree split on and suppose that that split reduces the impurity of the label by r k . Then DStump assigns feature j the score  Choosing d: We pick the number of features to use in our local linear model via a greedy forward selection procedure that relies on the fact that features can be sorted by their s j scores.

we evaluate the predictive accuracy of MAPLE on a held out validation set for d = 1, . . . , p and choose the value of d that gives the best validation accuracy.


The goal is to have agents that do not adapt to interruptions so that, should the interruptions stop, the policy they learn would be optimal.

agents should learn the dynamics of the environment without learning the interruption pattern.


First, it highlights that the fact that the architecture "collects" information from all different edges of the graph, in an invariant fashion via the α, φ functions.

the functions φ (after summation) aggregate all the features around a given node, and then α (after summation) can collect them.


The model (Figure 5) is based on the following observation: an auto-association network fed with a sequence of composite images in which some fragment/location combinations are more likely than others develops a nonuniform spatial distribution of reconstruction errors.

smaller errors appear in those locations where the image fragments recur.


5) Human intervention should be minimized.


We show that causal queries of the type P (y|do(x)) can be recovered even when the associated probabilistic relations such as P (y, x) and P (y|x) are not recoverable.

causal effects may be recoverable even when Y is not separable from its missingness mechanism.


We compute the optimal policy using Bellman's dynamic programming principle (Bellman, 1952).

we iteratively compute the expected cost of continue and stop as a function of the belief state b t (these are the Q-factors for continue and stop, , then the optimal policy chooses to continue; otherwise, it chooses to stop; therefore, the belief state is partitioned by the decision policy into a continuation region and a stopping region (details omitted due to lack of space).


Significant progress has already been achieved on Conjecture 1.

the conjecture is set for k = 2, with the achievability part proved in [15,16] and [17], and the impossibility part in [10].


The Reeb graph R f (X) provides a simple yet meaningful abstraction of the input domain X w.r.t function f . Computation in discrete setting.

assume the input domain is modeled by a simplicial complex K. a k-dimensional simplex σ is simply the convex combination of k + 1 independent points {v 0 , .


While our upcoming guarantees are all concerned with the cone membership test itself and not the specific implementation, these variations provide better algorithmic options and are more robust to noise.

we choose to use a variation (in the box below) that is a bounded feasible linear program for our implementation of the cone membership test.


This is not a contradiction because the norm on dual representations • z,r is in fact just a pseudo-norm, not a proper norm.

the relationship between H and q is many to one, and several matrices can correspond to the same q.


This approach towards SFM was revitalized in 2006 when Fujishige and Isotani [6,7] announced encouraging computational results regarding the minimum norm point algorithm.

this algorithm significantly out-performed all known provably polynomial time algorithms.


In this paper, we study the gradient descent algorithm as a discrete-time dynamical system during training deep neural networks, and we show the relationship between the step size of the algorithm and the solutions that can be obtained with this algorithm.


This results in asynchronous updates, in which the activity of a neuron x i changes stochastically as a function of its input cue xi , the activity of all other neurons, x \i , and neighbouring synapses, W i,• and W •,i .

the Gibbs sampler results in a sigmoid transfer function, with the total current to the neuron given by the log-odds ratio: with the terms I in/out rec defining the evidence from the incoming and outgoing synapses of neuron i, and the constants a and b determined by the prior over patterns and the noise model.


Let n (i) be the length of document x (i) .

for any k ∈ {0, 1}, we have: where 1{•} is the indicator function (1{True} = 1, 1{False} = 0), and we have applied Laplace (add-one) smoothing in obtaining the estimates of the word probabilities.


We note that although tree-structured graphs provide a relatively restrictive class of models, junction trees [14] can be used to extend some of our results to more general graphs.

we can still support changes to the parameters of the distribution (evidence and conditional relationships), although changes to the underlying graph structure become more difficult.


Remarkably, our theory reveals how all these detailed retinal properties arise as a natural consequence of the statistical structure of natural movies and realistic energy constraints.

our theory simultaneously accounts for: (1) why it is beneficial to have these multiple cell types in the first place, (2) why the four properties of cell density, spatial RF-size, temporal filtering speed, and contrast sensitivity co-vary the way they do across midget and parasol types, and (3) quantitatively explains the variation in midget to parasol density ratios over retinal eccentricities.


This problem is feasible in both λ j s and t and hence we can drop the minimization over individual constraints to obtain an equivalent problem: min λj ∈∆n j ∀j,t t subject to nj k=1 λ jk ≤ t. One can now eliminate t by reintroducing the max j and interchange the min λj ∈∆n j ∀j with other variables to obtain: Now one can derive the standard dual of (3) wrt.

to the variables w jk , b, ξ i alone, leading to: where α, γ are Lagrange multipliers, The following points regarding (4) must to be noted: • (4) is equivalent to the well-known SVM [18] formulation with kernel 1 γ * j is the weight given to the j th component and λ * jk is weight given to the k th kernel of the j th component.


Previous approaches make the unrealistic assumption that all provided transformation operations preserve class labels for all data points.

for label mapping function y, any sequence of TF indices τ 1 , ..., τ L , and all data points x.


[11], which is later reused in all of the follow-up work, is that g t can be estimated by randomly perturbing the point x t . on round t, the algorithm chooses the point instead of the original point x t , where > 0 is a parameter that controls the magnitude of the perturbation, A t is a positive definite d ⇥ d matrix, and u t is drawn from the uniform distribution on the unit sphere.


In this approach, randomness comes from two sources, from the training data S and the random set T (w, x).

in Theorem 1, randomness only stems from the training set S. Now we need to produce generalization results that hold for all the sets T (w, x), and for all possible proposal distributions R(w, x).


Our exposition is inevitably woven around these three intimately related subjects.

we show that the way in which we generalize PCA is exactly analogous to the manner in which regression is generalized by GLM's.