Existing first-order knowledge compilation turns out not to admit an intuitive completeness result.

we generalized the existing Independent Partial Grounding compilation rule to the domain recursion rule.


is the mean of n independent Bernoulli(0.5) random variables (since the rewards are chosen randomly), and has expected value 0.5.

by the Hoeffding inequality, taking a union bound over all policies π ∈ Π M , we have (3) Here, the probability is over the random MDP M . But since L outputs a policy in Π M , the chance of L outputting a policy π with U M (π) ≥ 1 − 2ǫ is bounded by the chance that there exists such a policy in Π M . ) By setting the right hand side to 1/4 and solving for m, we see that so long as we have that P (U M (π) ≥ 1 − 2ǫ) < 1/4.


Finally, we notice that ILP solves easily problem instances with, say, 60 nodes and vertex cover bound 8; see the results for carpo10000 data set.

in practice ILP scales up to significantly larger data sets and vertex cover number bounds than the combinatorial algorithm of Section 3.1.


When Π e is provided by an oracle (active learning), a query cost c > 0 is taken into account in (8), by subtracting c from u 1 .

the probability of exploration is reduced each time the agent makes a query to the oracle (i.e., y k t = 1).


Starting from the definition of resting state as functional coherent groups we search for functional units of the brain that communicate with other parts of the brain in a coherent manner.

what define functional units are the way in which they interact with the remaining parts of the network.


We also know from the proof of Theorem 2 that lim k→∞ lim h→∞ lim i→∞ λ i k,h = λ * ∞ .

there exist k 0 , h 0 , i 0 such that |λ i k0,h0 − λ * ∞ | < δ for any i > i 0 .


As shown in [2], the practical training of GANs is conducted over neural networks with parameters lying in a compact space.

we consider the following two compact parameter sets as taken in [5,8,22,24], In this paper, we study the minimax estimation problem defined as follows.


In Async-HDP, a processor can add new topics to its collection during the inference step.

when two processors communicate, the number of topics on each processor might be different.


The model is illustrated in Figure 1 jk ] of matrix Θ k . This way linking tendency from node i to node j is reflected based on their membership to group k. We then determine the overall link probability p (t) ij by combining the link affinities via a logistic function g(•) 1 .

where ǫ t is a density parameter that reflects the varying link density of network over time.


Eliminating x i in (2.3) by the representation above yields j .

x ij can be selected as a neighbor of x i if the orthogonal projection θ (i) j is small and (2.4) Assume all the x ij satisfy the above inequality, then we should approximately have We will use (2.5) as a criterion for adaptive neighbor selection, starting with a K-NN at each sample point x i with a large enough initial K and deleting points one by one until (2.5) holds.


of the target signal, it can be seen that the solution ŵ(t) of deterministic equation corresponding to (14) (which is of the same form as (10) with the obvious sign changes) converges to an eigenvector of the covariance matrix C 0 .

for β = 0 we expect the learning rule (14) to perform PCA for small learning rates α.


Let the conditional label density on C k be denoted by p k (Y |X = x).

a labeled training point (X, Y ) is obtained as follows.


The primary motivation in considering probably conditionally concave energies is that they enjoy (with high probability) the same properties as the conditionally concave energies, i.e., (i)-(iii).

locally minimizing probably conditionally concave energies over F can be done also with the Frank-Wolfe algorithm, with guarantees (in probability) on the feasibility of both the optimization result and the solution set of this energy.


For each of these, we solve the b-matching problem to find the most likely assignment of words to topics that satisfies the cardinality constraints.

the total running time is O((NT ) ⌧ (N + ⌧ ) 3 ).


we might consider a relaxation of the Gaussian LDS model in which these matrices are taken to be stationary from the outset, and are parametrised independently so that they are no longer constrained to take on the "correct" values as computed for Kalman inference.


Such dynamic behaviour is a primary reason why individual IP addresses tend to be such poor indicators of future malicious traffic [15,26].

we cannot assume that the data comes from a fixed distribution over the IP address space; the algorithm needs to adapt to dynamic nature of the malicious activity, and track these changes accurately and quickly.


[11, Lemma 1] The shrinkage operator S ν (•) is non-expansive, i.e., for any M and Proof.

for any given entry index (i, j), there are four cases: • Case 3: The sequence {M k } generated by the projected gradient descent iterations in Algorithm 1 with 0 < τ < min(2, 2 ρ ) converges to M * , which is an optimal solution of (2).


On the other hand, the parallels to the to the standard compressed sensing setup are limited because the available information is highly constrained.

the existing approaches of compressed sensing cannot be applied to the problem.


Under the following conditions, the rounded (re-optimized) estimator maintains bounded response: (i) If either y 1 remains bounded, or κ(x Note that the latter condition causes any convex loss to demonstrate unbounded response (see proof of Theorem 5 in Appendix B).

the approximate estimator is strictly more robust (in terms of bounded response) than regularized empirical risk minimization with a convex loss .


We thus conclude that OMP requires an identical scaling in the number of measurements to lasso.

at least for sparsity pattern recovery from measurements with large random Gaussian measurement matrices and high SNR, there is no additional performance improvement with the more complex lasso method over OMP.


All of these models assume that objects are already mapped onto locations in a perceptual space or conceptual hierarchy.

they are not able to make predictions about genuinely novel stimuli.


However, the goal of R is to rank the synthetic sentence lower than human-written sentences.

this can be treated as G and R play a minimax game with the objective function L: where θ and φ are the variable parameters in G and R, respectively.


Hence, we can write Ai5; = -(yfi5"";(I _ ,p)q,)T yfi5"";(J -,P)q,.


Thus, the objective function for multi-class clustering in (3) can be rewritten as: Cz,z(q) Dz(q) .

instead of minimizing J n mc , we can maximize J ′ n mc .


For instance, one could combine the analysis of LFP or calcium imaging data with that of spike trains.

with slight modifications to the model architecture, an analogous inference approach to the one presented here is applicable in a variety of contexts, including manifold learning, combinations of modalities, and beyond.


Finally, our experiments follow the standard online RL setup: after a model is updated based on the data observed so far, it is tested on newly generated episodes.

the numbers reported in Table 1 are not evaluated on the training dataset, so they truthfully reflect the actual average reward of the learned models.


By Little's theorem, the expected wait time is proportional to the expected number of packets waiting in each time slot [10]).

we want a policy that minimizes the expected value of The complexity of this problem is high.


They are in different scales: λ i (K) ≤ 1 (resulting from the fact that κ(x, y) ≤ 1 for all x and y), and λ i (A/ρ(n)) ≈ λ i (K/n) ≈ λ i (K) if n and ρ(n) are sufficiently large.

λ d (K) are independent of n for a fixed d and should be treated as Θ (1).


Convolutional neural networks (CNN) [21] are the gold-standard approach to problems such as image classification [20,35,9], object detection [32] and image segmentation [3].

there is a significant interest in improved CNN architectures.


The top-1 test error-rate of the final ensemble was 23.31.

in just 5 epochs we could improve the accuracy of the model by 0.56 using FGE.


The matrix is constructed such that columns near 64 have far more variability in the data -this is then responsible for additional "rank-like" structure.

we would intuitively expect the NMF algorithms to select additional columns closer to the end of the matrix.


Given K i g , K max g and K i H , K max H , and 0 < δ, δ g , δ H < 1, we define At any x k ∈ M, suppose that the sampling is done uniformly at random to generate S g and S H . Then, we have From Theorem 4.1, it can be easily seen that Assumption 4 follows with the same probability with It should be emphasized that if we use the typical condition (5) instead of Assumption 4, we obtain, e.g., for the sub-sampled Hessian H k . Considering that ∥η k ∥ goes to nearly zero as the iterations proceed, this obtained bound indicates that |S H | increases accordingly.

the size of the sub-sampled Hessian needs to be increased towards the convergence.


The Cramér-Rao inequality gives us a better understanding of the FIM.

assuming that η is an unbiased estimator, the variance-covariance matrix of η satisfies: the FIM is considered to be the inverse of the variance of an estimator, making it natural to use the matrix as a distance metric between the parameters.


( 6), the first term is the Kullback-Leibler (KL) distance from the trial distribution to the Bayes posterior, and the second term is a constant.

minimizing the free energy (6) amounts to finding a distribution closest to the Bayes posterior in the sense of the KL distance.


Knowing that and by the assumption of ||w|| 1 = 1, after simple calculations, we find that ρ ≤ h = ρ/||w|| 2 ≤ √ M ρ.

by minimizing d ∞ (0, P ) = ρ, (and ||w|| 1 = 1), h will be bounded and the volume of the simplex considered, as well.


One can show that the only function (up to a multiplicative constant) that can serve as penalty without breaking the property of scale equivariance is the logarithmic function.

we introduce an additional tuning parameter µ > 0 and look for minimizing the criterion If we make the change of variables φ 1 = Mβ/σ, φ 2 = Nβ/σ and ρ = 1/σ, we get a convex function for which the first-order conditions [20] take the form Thus, any minimizer of (2) should satisfy these conditions.


Observe that the expected loss function (3) is, generally speaking, nonconvex in the margins ∆ i,t (consider, for instance the logistic case g(∆) = 1 e ∆ +1 ).

we cannot directly minimize this expected loss.


On the contrary, the deterministic error bound does not suffer from such a kind of problem.

it provides a natural criterion for selective labeling.


First, notice that (by negation) the feasibility constraints in (1) can also be defined as: In other words, a solution w is feasible iff there exist no pair of samples from the two classes such that f w ( ) orders them incorrectly.

second, we will make the constraints in (1) more stringent: instead of requiring that equation (1) be satisfied for each possible pair (x + ∈ A + , x − ∈ A − ) in the training set, we will  Next, notice that all the linear inequality and equality constraints on (λ + , λ − ) may be conveniently grouped together as Bλ ≤ b, where, ) our constraints on w can be written as: Where the second equivalent form of the constraints was obtained by negation (as before), and the third equivalent form results from our third key insight: the application of Farka's theorem of alternatives [9].


Since the training dataset is more complicated, we thought the number of blocks (depth) in the network has much more impact on the classification performance because deeper and stronger networks will better learn the boundaries between classes.

the experiments on CIFAR-100 were designed to observe the changes depending on the depths of networks.


Put abstractly, adaptation involves short or long term changes to aspects of the statistics of the environment experienced by a system.

accounts of neural plasticity driven by such statistics, even if originally conceived as accounts of developmental (or perhaps representational) plasticity, 19 are automatically candidate models for the course and function of adaptation.


Both the encoder and decoder cells also use the information that is propagated horizontally.

at every iteration t, an encoder cell in the l-th layer receives three inputs: 1) the output of the encoder cell from the level below e l 1 (t), 2) the output d l (t 1) of the decoder cell from the same level from the previous iteration, 3) the encoder state s l (t 1) from the same level from the previous iteration.


In Alice's case, the situations where she tested Rob's learning behavior did not contain lava.

the lava-avoiding reward would have produced the same behavior as Alice's designed reward function in the (lava-free) environments that Alice considered.


( 20) tells us that when β is sufficiently small, The term in the denominator, b (0), is relatively easy to compute, and we show in Appendix C (in the supplementary material) that it is given by The key result of the perturbative analysis is that when the true distribution is out of the model class, the normalized bias can be increased by a term proportional to b (0) 1/2 .

the size of b (0) is crucial for telling us how big the bias really is.


Simplified equation When the tuning curves of the sensory cells are uniformly distributed Gaussians (e.g., spatial receptive fields), namely λ k (x) = λ max exp(−(x−k∆x) 2 /2σ 2 ), it can be shown [13] that for small enough ∆x, and a large number of sensory cells, Therefore the matrix Λ has no effect on the solution of (4), except for an exponential attenuation that is applied to all the cells simultaneously.

in cases where the number of sensory cells is large, Λ can be omitted from (4).


Note that whereas (J2 was previously described as a human-specified parameter, the analysis also suggests a particularly simple way of choosing it automatically: For the right (J2, Theorem 2 predicts that the rows of Y will form k "tight" clusters on the surface of the k-sphere.

we simply search over (J2 , and pick the value that, after clustering Y 's rows, gives the tightest (smallest distortion) clusters.


For example, assuming that and , this means that is more strongly connected to than .

by changing from small to large, we can obtain an order for the strength of connection between pairs of brain regions.


These are obtained by fitting the above function to the actual behavior of individuals in another PGG study [16].

this function is a simulation of real individuals' behavior in a PGG.


Recall that as ELIRL learns new tasks, the shared knowledge in L continually changes.

the modeled reward functions for all tasks continue to be refined automatically over time, without retraining on the tasks.


It only requires O(log(1/δ)) power iterations to compute a top SVD solution of accuracy δ.

for a sufficiently small δ > 0, the overall per-iteration complexity of the distributed method at the tth iteration is reduced to O(|Ξ| + max{n, p} log(1/δ)) at the central server, and O(|Ω k |(max{n, p} log(1/δ) + q)) at the kth worker.


In such a setting, the difference between the optimal and a near optimal policy may not be computed accurately.

it is useful to find all actions that are close to optimal so that the real optimal action is not missed.


However, the construction presented here is generic.

it is possible to derive similar results for radial basis function units, models of spiking neurons, and other unit types that are known to yield networks with bounded VC dimension.


One can show that the Haldane prior is equivalent to putting a flat prior on the log-odds log[q j /(1 − q j )]; also, it has the largest variance (and hence is "most uninformative") among all the possible distributions of q j .

although appearing to be extremely dichotomous, it is well known in Bayesian statistics as an uninformative prior of binomial distributions.


Note that our current dataset only has one property label per image (either for 'attribute' or 'action').

a negative analogy for one property may be positive for the other.


The optimization problem defined in Equation 5 is very likely to overfit if we cannot precisely estimate the hyperparameters, which automatically control the generalization capacity of the MRMA model.

for instance, σ U and σ V will control the regularization of U and V . it is more desirable to estimate the parameters and hyperparameters simultaneously during model training.


We show that the gradient field of a linear function has to be a parallel vector field (or parallel field in short).

we propose a novel approach called Parallel Field Regularization (PFR) to simultaneously find the function and its gradient field, while requiring the gradient field to be as parallel as possible.


Evidently θ * = 0, and by an asymptotic expansion we have that E[θ 1 ] = Ω(n − 1 2 ) (see the long version of our paper [22,Appendix D] for this asymptotic result).

the bias of θ 1 is Ω(n − 1 2 ), and the AVGM algorithm using N = mn observations must suffer mean squared error E[(θ 1 − θ * ) 2 ] = Ω(n −1 ).


Our goal is to construct a mapping f : V w2v → V rnn parameterized by a matrix W such that v = Wv for v ∈ V w2v and v ∈ V rnn .

inspired by [15], which learned linear mappings between translation word spaces, we solve an un-regularized L2 linear regression loss for the matrix W. any word from V w2v can now be mapped into V rnn for encoding sentences.


Observe that using Cauchy-Schwarz inequality, where σ max (A) denotes the largest singular value of A.

we have where where the last line inequality follows from (7).


A constant rebalanced portfolio (CRP) is an investment strategy which rebalances the wealth in every iteration to keep a fixed distribution.

for a CRP x ∈ ∆ n , the change in wealth is t (r t • x).


Then, the column space of P contains an approximately optimal solution for A, since AS − P N T 2 F and AS is a column pcp for A.

it follows from Lemma A.1 that with probability at least 98/100, there exists a good solution for A in the column space of P . Since we cannot compute this explicitly, we set up the following regression problem: min X A − P X 2 F . Again, we sketch the regression problem above by sampling columns of A according to the leverage scores of P . We can then compute X A = argmin X E A − E P X 2 F with probability at least 99/100, where E is a leverage score sketching matrix with poly k rows.


A second key limitation is that the LCMS method does not allow for dependencies between multiple biomarkers, measurement noise, or latent variables (such as unobserved confounders of the biomarker-outcome associations).

for instance, without allowance for noise in the biomarker measurements, non-zero conditional mutual information I(g i , y|x j ) will be interpreted as evidence of pleiotropy or reverse causation even when the relation between the underlying biomarker and outcome is causal.


For the algorithms considered in this paper, it will be more natural to consider reward-maximization rather than loss-minimization.

we consider online linear optimization where the goal is to maximize cumulative reward given adversarially selected linear reward functions f t (x) = g t • x.


This means choosing sufficiently small or sufficiently large σ (depending on P and Q) makes γ k σ (P, Q) arbitrarily small.

σ has to be chosen appropriately in applications to effectively distinguish between P and Q.


In addition, their approach is not based on learning parameters jointly, but estimates the covariance in each observation-subset separately, and then aligns these estimates post-hoc.

while this approach can be very effective and is important for theoretical analysis, it can perform sub-optimally when data is noisy.


In this paper, we first explain that, if the class prior in the unlabeled dataset is known, PU classification can be reduced to the problem of cost-sensitive classification [5] between positive and unlabeled data.

in principle, the PU classification problem can be solved by a standard cost-sensitive classifier such as the weighted support vector machine [6].


The approximation (3) can be computed using only matrix-vector multiplications, vector-vector inner-products and vector-vector additions O(M n) times each.

the time-complexity becomes O(M n A mv + M nd) = O(M n A mv ).


Additionally, this means that p(S) ≥ 2/3, so again with high probability, at least half of our m samples will lie in S. These first m /2 samples from S will be independent samples from (p|S), which is uniform, and therefore with 99% probability will pass the uniformity tester.

in this case, our algorithm will return "YES" with probability at least 9/10.


Then T j,j,j = Θ(v 3 1,j ) + E j,j,j , and one can show T j, * , * ) if and only if T j,j,j = ω(1/n), and therefore one would notice this by reading T j,j,j .

there can only be o(n 2/3 ) slices j for which T j, * , * for each of them we can afford to take O(n 2 ) 2 -samples and still have an O(n 2+2/3 ) = o(n 3 ) sublinear running time.


Markov Random Field (MRF) based exponential family of distribution allows for representing distributions in an intuitive parametric form.

it has been successful for modeling in many applications Specifically, consider an exponential family on n random variables X = (X 1 , .


Given a set A, we define SampleSubset(A,p) as the process of applying a Bernoulli trial with success probability p to each element of A and returning the subset of elements for which the trial was successful.

sampling ϕ global and ϕ eventual is accomplished by performing SampleSubset(τ , p G ) and SampleSubset(Ω, p E ).


Due to the wide diversity of PNs and LNs in terms of their time scales β and gains K, the performance of the network is highly sensitive to the agreement between the outcome of the generative process and the choice of parameters g E and g I .

it is not possible to give a one-sizefits-all value for these.


Note that learning is not only based on teacher spikes but also on non teacher-induced firing.

strongly correlated groups of inputs tend to cause autonomous (i.e., not teacher-induced) firing of the neuron which results in weight increases for all weights within the corresponding group of synapses according to well-known results for STDP [8,5].


Given the rapid advances in data collection technologies for biological systems, we expect that complex, high-dimensional, and feature rich data from complex dynamic biological processes, such as cancer progression, immune responses, and developmental processes, will continue to grow.

we believe our new method is a timely contribution that can narrow the gap between imminent methodological needs and the available data and offer deeper understanding of the mechanisms and processes underlying biological networks.


thus, we have By the assumption, we have for any U, which implies that (α 1 , • • • , α d , U * ) is a saddle point of the min-max problem in Eq.


In particular, for many applications, standard LP solvers perform poorly due to the large number of variables and constraints [33].

significant research effort has been put into designing efficient solvers that exploit the special structure of the MAP inference problem.


Still, finding the optimal sequence of subtasks by reflecting the long-term dependencies between subtasks and the context of observation is computationally intractable.

we found that it is extremely challenging to learn a good policy when it's trained from scratch.


Moreover, whereas the SMC estimate is invariant to permutations of the values on the axis, BMC makes use of the smoothness of the function.

a point in a sparse region is far more informative about the shape of the function for BMC than points in already densely sampled areas.


We split the words into a randomly sampled 9000 for training and 1000 for testing, and we used the standard test set to report results.

neither the words nor the sentences used during test time had been seen during training.


ImageNet consists of variable-resolution images, while our system requires a constant input dimensionality.

we down-sampled the images to a fixed resolution of 256 × 256.


We only need to model a subset of objects in the scene to provide enough constraints for volumetric reasoning.

we adopt a coarse 3D model of objects in the scene and model each object-volume as cuboids.


Naive extension of the previous DR methods in deep multiagent RL setting [16] is infeasible for large domains.

we develop novel approximation schemes that can compute DRs in the collective case even with thousands of agents.


For all t = 1, . . . , r, in turn, if the i-th feature of A is selected, then the next diagonal entry of D is set to 1/ √ rp i .

by using the notation of this paragraph, Algorithm 1 outputs the matrix Ã = ASD ∈ R n×r .


In this paper, following [5,30], we use the hinge loss: to measure the misclassification cost.The expected and empirical risks associated with the hinge loss are defined respectively as: and In theory, the excess misclassification error R(sgn(f )) − R(f c ) can be bounded by the excess convex risk E(f ) − E(f c ) [29,1,7].

the classification algorithm usually is constructed under structural risk minimization [22] associated with E z (f ).


We can then consider a quadratic regularizer of the form cV n /2kwk 2 to define the regularized empirical risk R n (w) := L n (w) + (cV n /2)kwk 2 and the corresponding optimal argument Since the regularization in ( 4) is of order V n and (3) holds, the difference between R n (w ⇤ n ) and L(w ⇤ ) is also of order V n -this may be not as immediate as it seems; see [22].

we can say that a variable w n satisfying R n (w n ) R n (w ⇤ n )  V n solves the ERM problem to within its statistical accuracy.


Although we can design tree-encoders and tree-decoders to handle trees with arbitrary number of branches, we observe that encoder and decoder for binary trees can be more effective.

the first step is to convert both the source tree and the target tree into a binary tree.


After mapping the two algorithms onto neural dynamics, we find that both can infer correct odors in less than 100 ms.

at the behavioral level, the two algorithms make very similar predictions.


Neural receptive fields, on the other hand, span a broad range of spatial scales, and exhibit distributions of spatial phase and other parameters unmatched by ICA and SC results [13,14].

as models of early visual processing, these models fail to predict accurately either the individual or the population properties of cortical visual neurons.


In our early experiments, we failed to obtain good performance by only using SRU or QRNN for recurrent layers.

we needed to try many different RNN structures, thus each training should not take much time.


On the other hand, when E i,j is positive, (Σ + E) −1 x * i will tend to be more negative and zeroed out by the threshold function.

e j,i will not be more negative at next iteration.


Moreover, such a treatment limits the negotiation to only two solutions.

even if there exists a better convention in which one agent compromise a little but the other is better off greatly, it will not be chosen.


The seller can compute an offline estimate of both this projection and by collecting a dataset of context vectors before starting to offer prices to the buyer.

in view of Proposition 1 and the strong convexity assumption, we see the learning phase of the LEAP algorithm is conducting a stochastic gradient descent to minimize the -strongly convex function F , where at each time step we update w t+1 = ⇧ W (w t 1 t gt ) and gt = g t is an unbiased estimate of the gradient.


One major reason for this is the wide "reality gap" between simulators and the real world.

there has concurrently been a push in the robotics community to collect real-world physical interaction data [4,5,6,7,8,9,10,11] in multiple robotics labs.


In practice, there may be fewer nodes since one need only to allocate space for contexts found in the data at least once, but their total number may still grow linearly with the length of the input string.

for problems such as partially observable RL, in which the amount of input data is unbounded, or for large |Σ| and K, the memory used by CTW can quickly become unreasonable.


Our guarantee of improvement holds, under well-specification, without requiring effect identification due to hidden confounding.

theorem 1 exactly captures the appeal of our approach.


Let N (F, r) be the size of the smallest -net over F with respect to the infinity norm at scale r = random landmark pairs then we have the following with probability greater than (1 − δ) This result tells us that in a large enough landmarked space, we shall, for each function f ∈ F, recover close to the best classifier possible for that transfer function.

if we iterate over the set of transfer functions (or use some gradient-descent based optimization routine), we are bound to select a transfer function that is capable of giving a classifier that is close to the best.


The computation of class-specific reconstruction error y − D * k P * k y 2 only has a complexity of O(mp).

the total complexity of our model to classify a query sample is O(Kmp).


A system of minimal complexity was built [8,9] and then the question was asked of what additional features should be added to this minimal system to enable a behavior that is missing in the previous design.

the two approaches start from different philosophical grounds, but will, hopefully, converge on similar solutions.


Therefore, these intervals are disjoint and sorted.

given a random number r, determining if r ∈ ∪ D i=1 [M i , M i + x i ] only needs binary search over d ranges.


The value of the variational distance depends on the response to the TPQ, which is unobserved at query selection time.

the agent computes the expected variational distance, Where P (y|ξ i , ξ j , D) = P (y|ξ i , ξ j , θ * )P (θ * |D)dθ * is the predictive distribution and is straightforwardly estimated using a set of posterior samples.


Our approach has some strong similarities with that of the second work noted above; however, the authors there do not establish or disprove stability, and instead note the presence of zero eigenvalues (which we will treat in some depth) as a motivation for their alternative optimization method.

we feel the works as a whole are quite complementary, and signify the growing interest in GAN optimization issues.


An alternative approach is to develop an adaptive method that contains a number of trackers that track different features or parts of a target object [3].

even though each tracker may fail under certain circumstances, it is unlikely all of them fail at the same time.


Further, to infer the full kdimensional subspace A, we need to consider at least m X ≥ k centers.

for each center ξ i , we define a set of m Φ directions and arrange them in a total of m Φ matrices We can now define the following linear system: where the linear operator A is defined as A(X) i = tr(Φ T i X), the curvature errors have been accumulated in e and the noise has been put in the vector z which is distributed as Given the structure of the problem, we can make use of several low-rank recovery algorithms.


Moreover, we also ask for a smooth algorithm, meaning one for which A and B in (6) are smooth.


Let S 1 denote a sum of these Bernoulli random variables.

a positive instance is incorrectly labeled if S 1 ≤0, and a negative instance is misclassified if S 1 >0.


In contrast, when simply perturbing an existing image, past work has been able to assume that the true label does not change if the perturbation is small.

during evaluation, to test whether the images are legitimate and belong to class y source , we use crowd-sourcing on Amazon Mechanical Turk (MTurk).


Note that the minimal separation rate n 2m 4m+1 is minimax optimal according to (Ingster [1993]).

d n,T ⇤ is optimal when ⌘ T ⇤ ⇣ n 4m/(4m+1) .


Moreover, we can observe that for k(x t , x i , α i ) j to have a significant value, two conditions must be satisfied: (a) α ij should have a large value, and (b) f T i f t should have a large value.

we interpret the pre-activation value Φ(x t , Θ) j as a weighted sum for the feature similarity f T i f t with the weight α ij .


Hence, to characterize the threshold for a small portion of the tail of this distribution, we must draw a very large number of samples just so that the estimate converges.

if we want an α = 0.01 threshold from the Null sample maximum distribution, we require many thousands of permutation samples -each requires randomizing the labels and recalculating all test statistics, a very computationally expensive procedure when v is large.


3, we also expose the simple independence between the subproblems because they are simple summations.

we can easily parallelize both optimization problems upto min(n, p) with little overhead and simple changes to the code-in our MATLAB implementation, we only changed a for loop to a parfor loop.


Note that, by Kirchoff's laws, r G (i, j) ≤ d G (i, j) and, so, R G ≤ D G . In this section we describe our method to compute the pseudoinverse of a tree.

let us begin by noting that the effective resistance is a better measure of connectivity than the geodesic distance, as for example if there are k edge disjoint paths of geodesic distance d between two vertices, then the effective resistance is no more than d k . the more paths, the closer the vertices.


The downside of RTRL is that for a fully connected RNN with n units the matrices H t and G t have size n × n and n × n 2 , respectively.

computing Equation 3 takes O(n 4 ) operations and requires O(n 3 ) storage, which makes RTRL impractical for large networks.


Under Assumptions 1 and 2, the bias of the optimal policy is bounded.

it makes sense to define the bias span H(M ) of a continuous state MDP M satisfying Assumptions 1 and 2 to be H(M ) := sup s λ π * (s) − inf s λ π * (s).


Define ϕ(x) = 1 p x p p , and note that (∇ϕ(x) Since a ijk has only three nonzero entries, we see that e only needs to be updated in three components.

in Algorithm 3.1 we may replace ( ) by an appropriate numerical computation of the parameter µ ijk and replace ( ) by the computation of the new value of e. Further details are available in [5].


While previous approaches to latent topic modeling assumed a fixed order such as unigrams or bigrams, the order is generally not fixed and unknown to us.

we used a Gibbs sampler for the Markov chain LDA [14] and augmented it by sampling Markov orders at the same time.


Denote by T 0j the overlapping part of T j corresponding to I 0 = I 1 ∩ I 2 as in the proof of Theorem 2.1.

we consider the overlapping parts Or equivalently, if we take an orthonormal basis G of the null space of [e, T T 01 ], −[e, T T 02 ] and partition j [e, T T j ] T , j = 1, 2.


But this leads to a complicated density dynamics, because the dynamics for v(t) consist of a continuous (differential equation ( 1)) and a discrete part (spike generation).

here we concentrate on an alternative description that allows a compact formulation of the desired I&F density dynamics.


For any µ ∈ ∆ p , let w µ ∈ H denote the mixture weight vector obtained from a sample of size pm by combining the p weight vectors w k , k ∈ [1, p], each returned by conditional maximum entropy when trained on the sample S k of size m. Then, for any δ > 0, with probability at least 1−δ, the following inequality holds: For the uniform mixture µ 0 = (1/p, .

. . , 1/p), the bound becomes Proof  by McDiarmid's inequality, which proves the first statement and the uniform mixture case since µ 0 = 1/ √ p. Theorems 2 and 3 help us compare the mixture weight w pm obtained by training on a sample of size pm versus the mixture weight vector w µ 0 .


then S t is defined from ws , where s is the largest number such that t s ≤ t. S t does not change for any t ∈ [t s , t s+1 − 1].


We have: Therefore, Since Tr is independent of the basis, choosing an appropriate basis we can write: (6) is true because ρ 0 (σ) = 1, ∀σ ∈ S n , and f ℓ1 = 1. f is obtained by truncating the Inverse Fourier Transform expression to the first two terms.

from (2), it follows that: Using the fact that ρ 0 (σ) = 1 ∀σ ∈ S n , fρ0 = 1, and plugging (6) into (7) gives the result of the proposition.


Having been provided with a TPQ the expert gives a response y indicating, which trajectory is preferred.

each TPQ results in a training data tuple (ξ i , ξ j , y).


Accordingly, to (locally) maximize η(θ), θ is incrementally updated with The direction of the NG is defined using a Riemannian metric.

an appropriate choice of the Riemannian metric for the task is required.


assumption over arbitrary parameter subsets, because doing so will cause incorrect estimates due to incompatibility of sub-results from different workers (e.g., imagine trivially parallelizing a long, simplex-constrained vector across multiple workers -independent updates will break the simplex constraint).

existing dataparallel schemes and frameworks, that cannot support sophisticated constraint and/or consistency satisfiability mechanisms across workers, are not easily adapted to model-parallel programs.


While there have been many recent works on modeling information diffusion [2,6,7,8] and network evolution [9,10,11], most of them treat these two stochastic processes independently and separately, ignoring the influence one may have on the other over time.

to better understand information diffusion and network evolution, there is an urgent need for joint probabilistic models of the two processes, which are largely inexistent to date.


Further, it uses a parametric approach, rank-m Singular Value Decomposition (SVD), and stochastic gradient descent for optimization.

dHA has a suitable time complexity for large datasets, and DHA does not require the training data when it computes the functional alignment for a new subject.


Optimizing the reward augmented maximum likelihood (RML) objective, L RML (θ; τ ), is straightforward if one can draw unbiased samples from q(y | y * ; τ ).

we can express the gradient of L RML in terms of an expectation over samples from q(y | y * ; τ ), ) to estimate ∇ θ L RML (θ; τ ) given a mini-batch of examples for SGD, one draws y samples given mini-batch y * 's and then optimizes log-likelihood on such samples by following the mean At a temperature τ = 0, this reduces to always sampling y * , hence ML training with no sampling.


In the reparameterized spike and slab prior, each pair of variables { w qm , s qm } is strongly correlated since their product is the underlying variable that interacts with the data.

a sensible approximation must treat each pair { w qm , s qm } as a unit so that { w qm , s qm } are placed in the same factor of the variational distribution.


When coupling increases, the dependence among the Z i t increases and tr(L z + ηI) −1 (and therefore α) decreases.

increased correlation among observational noise variables implies decreased regularization.


This procedure can be done regardless of the dimension.

it suffices to build a d-dimensional grid indicator function, which generalizes the notion of increasing trapezoid function into high dimensional space (Definition B.3 in the appendix).


In this paper we suggest a novel encoding for such higher level representations based on the analogy between CPs and differential equations: both formulations suggest a change of state given the current state of the system, and both usually encode a desired goal in form of an attractor state.

instead of shaping the attractor landscape of a policy tediously from scratch by traditional methods of reinforcement learning, we suggest to start out with a differential equation that already encodes a rough form of an attractor landscape and to only adapt this landscape to become more suitable to the current movement goal.


they do not contain any information on the object rotation) and the aspect assignments may also be suboptimal (see previous section).

we introduce latent variables h = (δp) that encode an adjustment to the ground-truth pose parameters y = (p).


Its variance is given by: where G is the ( ) covariance matrix of the noise, S is a vector formed by averaging each column of G , V is the average of all the elements of G and Q is the time-average of the mean ¦ .

9 depends only on the first and second moments of the response distribution; substitution of data-derived estimates of these moments into (2) yields a standard error bar for the estimator.


D-TS + : Implement the same operations as D-TS, except for the selection of the first candidate (Line 11 in Algorithm 1) is replaced by the following two steps: a (1) ← arg min R( 1) i ; D-TS + only changes the tie-breaking criterion in selecting the first candidate.

the regret bound of D-TS directly applies to D-TS + : Corollary 1.


( 14), and the second term is upper-bounded by 1 εn Σ Y X − Σ (n) Y X .


Indeed, if we set k = 2 in the dual functional (18) we would be able to conclude that the two dual functionals are identical (by a suitable change of variables) .

the role of the constant C complies with the findings of [10] by controlling the tradeoff between the number of margin errors and support vectors and the size of the margins: 2/ N ::; C ::; 2 such that when C = 2 a single margin error is allowed (otherwise a duality gap would occur) and when C = 2/ N all vectors are allowed to become margin errors and support vectors (see [11] for a detailed discussion on this point) .


Now we focus on the case where |S| = k. Define a neighbor S (i,j) of S as That is, the neighbor S (i,j) is given by replacing one of the elements of S with that of V \ S. Note that I S (i,j) − I S = e j − e i for any neighbor S (i,j) of S. Let S (i * ,j * ) be a neighbor that maximizes f (S (i,j) ) among all neighbors of S. Since a subset S of size ) and S * ← S (i * ,j * ) .

in either case it holds that γ ≥ f (S (i * ,j * ) ).


Let F be a self-concordant barrier of K with a parameter ν (see [20]).

for x, y ∈ K, ρ(x, y) is the distance in the Riemannian metric whose metric tensor is the Hessian of F . the metric tensor on the tangent space at x assigns to a vector v the length and to a pair of vectors v, w, the inner product v, w x := D 2 F (x)[v, w].


(Thus, peculiarly, it is negatively unbounded, an issue we discuss in §5.3.)


We apply our main result to non-private data-dependent Gaussian priors with a fixed covariance matrix.

we choose only the mean w 0 2 R p privately in a data-dependent way.


Similarly, we have Therefore, by Chebyshev's inequality, with 99% probability we have that .

assuming these hold, we have that if C/C is sufficiently large, if p is uniform, we accept, and if p is -far from uniform, then by Lemma 2.8, we reject.


Since the number of experiments required is at most logarithmic in the maximum clique size, none of our simulations would ever require more than five experiments to recover the causal graph and nearly always required just one or two.

the expected clique size (and hence number of experiments) required appears even smaller than the constant 13 produced in Theorem 4.1.


The model is constructed according to the principle of separation of time scales: processes that operate in the 100-1000 ms range are modeled as accurately as possible, those that vary on a much shorter time scale are assumed to instantaneously achieve their steady-state values, and those that occur over longer time scales, such as slow inactivation, are assumed constant.

the model does not incorporate currents which inactivate on a short time scale, and cannot provide a good description of rapid events such as the transitions between up and down states or the generation of action potentials.


In contrast, the S α similarities will propagate with similar speed.

the final joint similarity P * will have I m as the most similar to I 1 .


4 Contrary to the usual GP viewpoint [1], σn is not a hyperparameter in our method since the prior P θ does not depend on σn.

σn does also not contribute to the "penalty term" ln |Θ|.


Note that the new documents will likely have sentences that have not been seen before in the training samples.

the kernel matrix L needs to be reparameterized in order to generalize to unseen documents.


For example, in a seminal feature weighting method called Relief [7,8], the fixed TNs determined based on the uniform weights (i.e., the Euclidean distance) are used throughout the learning process.

the TN-weight consistency is Left: (a) The Euclidean feature space with w 1 = w 2 = 1/2.


However, subtractive normalization cannot prohibit w from running away when the error signal is turned off.

we additionally assume multiplicative normalization term η 3 w to limit the magnitude of w [4, p. 290, 314].


The ratio therefore is computed as a function of the Poissons P (n = 0|λ) when the spike count n is zero.

the Poisson expectation has to be multiplied by the time-length of the interval; call ∆t k = t k+1 − t k the length of the interval I k . Assuming that the neurons i = 1, . . . , N are independent we obtain: during the time-intervals where no action potential is observed, the diffusion drifts linearly with a slope equal to the sum over all neurons of the difference between the expected firing rate with stimulus 1 and the expected firing rate with stimulus 2.


Since L τ (θ) replaces the definition of Σ j with ωI ≽ Σ j , L τ (θ) is a looser bound than the one used by Algorithm 2.

performing θ τ +1 = arg max θ∈Λ L τ (θ) makes less progress than a step of Algorithm 1.


The variance of each task's reward distribution is normalized by dividing the advantage by the standard deviation of the reward: refer to A as the normalized advantage.

the gradient update to the parameters of the policy θ is the product of the advantage and expected derivative of the log probability of sampling an action: normalizing the advantage may also be seen as adapting the learning rate for each task.


Moreover, convergence rates for empirical processes on Markov chain sequences are also often given in terms of mixing coefficients that can ultimately be bounded in terms of π ⋆ and γ ⋆ (as we will show in the proof of our first result).

valid confidence intervals for π ⋆ and γ ⋆ can be used to make these rates fully observable.


In studies of visual cortex, modes could span neurons, time and stimuli [27].

given that tensors in the wild are almost never cubic, nor symmetric, to bridge the gap between theory and experiment, we go beyond prior work to derive and analyze Bayes optimal AMP algorithms for arbitrarily shaped high order and low rank tensor decomposition with different priors for different tensor modes, reflecting their different measurement types.


Rather, we only need a ranking of the best few candidates.

instead of evaluating the change in the objective exactly, we use an efficient approximation based on a small number, z, of training points which provide information about the residual between the current low-rank covariance matrix (based on inducing points) and the full covariance matrix.


Even though a maximum M of 699 would have been possible, P (M > 23|{ z i }) < 0.001.

we can accelerate the averaging process for quantities of interest (e.g.


Agglomerative heuristics have several appealing features: they are easy to implement, easy to tune, and their running time is r Opn 2 polylognq on a dataset of size n. Standard divisive heuristics based on graph partitioning or clustering methods (like for example the bisection k-means or the recursive sparsest-cut approaches) often involve solving or approximating NP-hard problems.

1 it is natural to ask how good the solution output by an agglomerative method is compared to the solution output by a top-down method.


Crucially, with a single sample, it is not possible to get an unbiased estimate of the gradient of the objective ⇢(u T x, v T y).

we consider a first-order oracle that provides inexact estimates of the gradient with a norm bound on the additive noise, and focus on inexact proximal gradient descent algorithms for CCA.


Next, since L i,t−1 is not assured anymore to be an upper bound on g t,i , we do not have any guarantee that the wealth Reward t,i is non-negative.

we enforce the positivity of the reward in line 8 of Algorithm 2.


Therefore, the worse case complexity of the second order term is less than ninoverlap exp(c).

we see that the second order method has the same exponential complexity as the first order method, but with a different polynomial prefactor.


Compared to ECW-RMED, D-TS + achieves much lower regret in the Condorcet case, lower or comparable regret in the non-Condorcet case, and much more robustness in terms of regret STD and delayed feedback.

the simplicity, good performance, and robustness of D-TS and D-TS + make them good algorithms in practice.


These can be computed efficiently if conjugate priors are available for the parameters.


Moreover, the events ( 4) and ( 5) will hold with probability ≥ 1 − ǫ if n ≥ (1/2δ 2 ) log(N 2 /ǫ).

any N -point subset of R D can be embedded, with high probability, into the binary cube of dimension O(log N ) in a similarity-preserving way.


In addition, the function | is convex and continuous as a positively weighted finite sum of convex and continuous functions.

the function f i = g i − h i is a DC function.


However, in order to perform proper uncertainty modeling, the model also needs to learn variances that reflect the nature of the observations.

it should generate more evidence when it is more sure of the outcome.


The green lines are the standard deviations around the ML value, and the red represent three times that.

while BDI is related to the effective reward size β, it is not related to the learning rate ǫ.


Thus, by the principle of induction, (Case I) There is no transition starting from s | w|+1 in this case, thus V | w|+1 = 0.

j=1 p | w|,j V j which is the same expression as the first case, and therefore j=i D j also holds for this case.


Here, we introduce an auxiliary function, By Corollary 2, we have that J (T) ≤ Q(T; T ), J (T ) = Q(T , T ), and Q(T, T ) has the same first-order derivative as J (T) at T . Because l and h are quadratic and convex, Q is quadratic and convex as well.

we can apply the optimization method in Section 3.2 to minimize J .


In most relevant contexts, α is close to 1; a representative value might be 0.99.


In some sense, this family of algorithms is "too conservative".

to achieve a more favorable regret guarantee, an entirely different algorithmic idea must be introduced.


By increasing the size of |L| (i.e., the number of lines), we can approximate every f ∈ F by some f ∈ F L,G arbitrarily closely.

characterizing the landscape of the loss function over PNNs can help us to understand the landscape of the unconstrained loss function.


On the left, where G is the tonic, input pool G, output pool 1, and the routing tier are in phase with one another (black clocks), while input pool C and output pool 6 are in phase with one another (grey clocks).

the raw note input, G, activates the invariant output 1, which corresponds to the perceived tonic invariant representation (heavy solid arrows).


However, when all cells (N = 7) were used in the analysis, I * 1 /I becomes only about 60%.

correlation seems to be much more important for decoding when population activities are considered than when only two cells are considered.


Furthermore, if we have certain prior knowledge about the interdependence between features, such as linkage disequilibrium between SNPs, we could easily incorporate them into our model.

our model provides an elegant framework for integrating complex data generation processes and domain knowledge in sparse learning.


Intuitively, if SNPs are located on these regions, they are more likely to be true eQTLs compared to those on regions without such annotations since they are related to genes or gene regulations.

it would be desirable to penalize regression coefficients less corresponding to SNPs having significant annotations such as TFBS in a regularized regression model.


Let l i denote a feed-forward neural network with sigmoid output parametrized by θ that maps the state s and action a to the Bernoulli's parameter of s i |s, a.

the probability of the next state s is finally given by: We emphasize that there is not necessarily any correspondence between the action vector a and the real actions that generated the observation pairs in the data.


This distance function needs not to depend on the kernel function and can be evaluated easily.

the implicitness of the feature space is avoided and optimization is possible.


Specifically, the number of mistakes at step k is [ i,k − max k <k i,k ] + and the associated loss is G k . Our main claim follows from combining the upper bounds in Equations 18 and 19.

algorithm OASM mimics the greedy policy π g . we decided to prove Theorem 1 based on counting how many times the policies π t and π g choose a different item.


We shall endow M k with the metric and define its covering number N ε M k as the minimal cardinality of a set The bound depends on an upper bound on base losses denoted B; this should be viewed as fixing a scale for the losses and is standard where losses are not naturally bounded (e.g., classical bounds on SVM kernel regression [19] use bounded kernels).

we have the following generalization result, whose proof can be found in Appendix D of the supplementary material.


Moreover, it is well known that the tail of any probability distribution generated by a WFA is sub-exponential.

though we do not require D Σ to be generated by a WFA, we do need its distribution over lengths to have a tail behavior similar to that of a distribution generated by a WFA.


It is difficult for a complete end-to-end framework to do this because it has to generate both correct poses and detailed appearance simultaneously.

we adopt a divide-and-conquer strategy, dividing the problem into two stages which focus on learning global human body structure Our contribution is three-fold.


In general, this conclusion also holds for cases when one applies general first-order optimization algorithms, such as AGD, on finite sums, as this typically results in a linear dependence on n. Secondly, if a given optimization algorithm obtains an iteration complexity of an order of n + L β w (0) − w * 2 / α for solving smooth and convex finite sum problems with a stochastic oracle, then n + L β−α n 2(α−1) = Ω(n 2 ).

β = α and α ≥ 2, indicating that an iteration complexity of an order of n + L w (0) − w * 2 / , as obtained by, e.g., SVRG++, is not attainable with a stochastic oracle.


A traditional frequentist's approach would use counters A Bayesian approach would model p(y 0 |y i ) with an (initially flat) Beta-distribution, and use the counters a i and b i to update this belief [3], leading to the same MAP estimate ŵi .

in both approaches a new example with y 0 = 1 and y i = 1 leads to the update where N i := a i + b i is the number of previously processed examples with .


Our methods alternate between x and y to break the non-separability of the augmented term ⇢ 2 kAx + By ck 2 .

at each iteration k, given ẑk := (x k , ŷk ) 2 dom(F ), ˆ k 2 R n , ⇢ k > 0, and k 0, we define the x-subproblem as If k > 0, then ( 7) is well-defined and has unique solution.


As we have seen, these steps are reduced to the ordinary EP steps if t = 1.

the proposed method can be regarded as an extention of the original EP to the t-exponential family.


denoting by ∆ t the difference in two consecutive dual objective values, D(λ 1 , .


And for computing histogram, we fix binwidth size and set the number of bins such that its range covers all {R i } M 1 elements of M number of graphs.

we had only one parameter, binwidth size, chosen from the set {0.001, 0.0001, 0.00001}.


We then performed the same test using the second held-out image.

for each iteration of the crossvalidation, two separate comparisons were made.


A very small update of w essentially keeps it at its randomly initialized value.

a very large update on the hand essentially re-randomizes w. we conjecture that FM2 causes the stochasticity of parameter updates to outweigh the effect of the training loss.


It is important that both Q functions learn from separate sets of experiences, but to select an action to perform one can use both value functions.

this algorithm is not less data-efficient than Q-learning.


Note that φ i is thus independent of x test , but not of x tr , y tr .

posterior inference over φ i must take into account both the evidence (training set) and the prior imposed by p(θ) and p(φ i |θ).


In this case, γ = 2λ( β − µβ) where β = Gβ is the Gaussian smoothing of β .

a component receives a high compensation weight γ k = 2λ( βk − µβ k ) if some neighboring spatial locations have already been selected (i.e., made "active") by the composite classifier.


Our bounds hold uniformly in that they hold for all hypotheses (functions from features to labels) at the same time.

unlike many bounds on standard generalization error, our bounds do not depend in any way on the richness of the hypothesis class under consideration measured in terms of, for instance, its VC dimension, or the margin of the selected hypothesis on the training sample, or any other property of the mechanism with which the hypothesis is chosen.


By our assumption on u and v, we have g(x, y) = u(x) − v(y) − (x − y) T ∇v(y) is continuous in x and y.

by Lemma 6, the assumption of non-emptiness of A cccp (x) for any x ∈ Ω ensures that A cccp is closed on Ω and so satisfies condition (3) in Theorem 2.


As negative test images we take all remaining images from the other categories.

we have the total of 86 training images, and the total of 169 test images.


z and , it satisfies the requirements for a standardization function.

plugging this function into (6), we have computing the implicit gradient requires only differentiating the CDF.


The major problem of this classification task is the limited number of samples -given the large number of genes.

feature selection is a prerequisite for good generalization [6,16].


Coupling the timing and the primitive model during learning has the effect of trying to learn primitives from data that are usually in the same place.

using the full model the inferred spikes are more compactly clustered at the Gaussian components due to the prior imposed from the timing model (the thick black lines correspond to Equation ( 4)).


One justification for using subspaces comes from the perspective that channel classification can be performed with inner products along one-dimensional subspaces.

the projection removes channel specific directions from the parameter space.


The reduced cost of a missing variable gives the maximum possible change in objective value per unit increase in that variable's value (when it is included in the formulation).

if all missing variables have non-negative reduced cost, then the current Restricted MLP cannot be improved and its optimal solution yields an optimal solution of the MLP.


For example, the relationship "on top of" does not hold in between two ground segments, i.e., a ground segment cannot be "on top of" another ground segment.

using an associative Markov network is very restrictive for our problem.


( 4), one can notice that our joint feature map is linear in Y .

if we take a loss that is linear in the first argument of , for instance the Hamming loss, the lossaugmented decoding is the maximization of a linear function over the spaces Y(X) that we can solve efficiently using dynamic programming algorithms (see Sec.


These conclusions also apply to the bandit framework when the algorithm is run with a positive exploration factor ε > 0.

by choosing a sufficiently small exploration factor, the end state of the EW algorithm in potential games with bandit feedback is arbitrarily close to a Nash equilibrium.


This requirement does not pose difficulty for analysis if µ X is continuous on X, since all queries will be unique with probability one.

our oracle epiphany model is parameterized by (β, K, U 1 , .


The proposed algorithm is in the tradition of algorithms like UCRL2 [11] in that it implements the "optimism in the face of uncertainty" maxim, here combined with state aggregation.

the algorithm does not need a generative model or access to "resets:" learning is done online, that is, in a single continual session of interactions between the environment and the learning policy.


This means that the attention scores in each feature channel may be far less than 1, and far less than most of the normal gate values in other gates, given the large spatial size of the input feature maps.

the attention mechanism needs to be modified to match the range of the sigmoid function in the gates.


The problem is solvable information-theoretically if there exists an algorithm that solves it, and efficiently if the algorithm runs in polynomial-time in n. Note that exact recovery in SBM(n, p, W ) requires the graph not to have vertices of degree 0 in multiple communities with high probability.

for exact recovery, we focus on W = ln(n)Q/n where Q is fixed.


Notice that the list L contains exactly 14 literals in all, one from each mistake.

also, we know from Lemma 3 that each of the "underlying groups" contributes at most one rule in L. the method of choosing features implicitly divides the class reptile into at least two groups, which appear to correspond to land reptiles and aquatic reptiles, and divides the somewhat nebulous class other into two groups, exemplified by worm and lobster.


Some of the modes may have full ranks if there is no sharing of information along them.

both the latent trace norm and the overlapped trace norm would suffer either from the heterogeneous multilinear rank or the heterogeneous dimensions in this context.


Each path uses one of four edges between every two adjacent vertical slices.

each edge is used by 1 4 of the classes, inducing a 1 4 K-vs-3 4 K subproblem.


Compared to variants of the Cascade model, the PBM is challenging due to the censoring induced by the examination variables: the learning algorithm observes actual clicks but non-clicks are always ambiguous.

combining observations made at different positions becomes a non-trivial statistical task.


We assume that this stochastic process is stationary and ergodic, and that p t is a conditional probability distribution, which for some (unknown) constant integer D with 0 < D ≤ t has the form: (1) and Σ is finite, there is a finite number of length-D strings on which the evolution of our stochastic process can be conditioned.

the stochastic process can also be represented by a time-invariant function Let s be a string of symbols from alphabet Σ with length |s| and elements [s] i=∈{1,...,|s|} .


Even if we knew in advance that we would receive only M non-adaptive queries, we would still need ⌦(log M/⌧ 2 ) samples to answer all of them accurately with high probability.

our price for non-adaptive queries is optimal up to constant factors.


Following [26], 0s are replaced with 0.5 before converting the data to compositional form.

we treat BMI as the response and the nutrient intake together with genera composition as predictors.


Intuitively, maybe noisy labels are not such a big problem at the beginning of the learning process.

we can initialize w 0 by running the vanilla perceptron for several iterations, and only then switch to our algorithm.


Finally, both GPFA and GLDS at p = 5 outperformed all GLMs, both for using AUC and mean-squared-error.

all four of our latent variable models provided superior fits to the dataset than GLMs.


We therefore approximate the solution for y u by minimizing L(D l , D u , y u , x s ), which leads to the following evaluation function for query selection: For computational simplicity, for the rest of this work, we choose a quadratic loss function, i.e., ℓ(y, y) = (y − y) 2 /2 1 .

it is straightforward to show where L = (K + λI) −1 and K = [κ(x i , x j )] n×n is the kernel matrix of size n × n. the evaluation function L(D l , D u , x s ) is simplified as Our goal is to efficiently compute the above quantity for each unlabeled instance.


Every convex, lower-semicontinuous function f has a convex conjugate function f * , also known as Fenchel conjugate [15].

this function is defined as The function f * is again convex and lower-semicontinuous and the pair (f, f * ) is dual to another in the sense that f * * = f . we can also represent f as Nguyen et al.


On the other hand, this objective is clearly impossible when the means and covariances are extremely close.

algorithms for parameter estimation of mixtures necessarily require some "separability" assumptions on the target parameters.


It turns out in our construction, the only second-order stationary points in the path are the final local minima.

we can also strengthen Theorem 4.1 to provide a negative result for approximating �-second-order stationary points as well.


Firstly, the posterior maintains the exact model, conditioned on U l . Secondly, we assume that the posterior distribution of {U l } L l=1 is factorized between layers (and dimension, but we suppress this from the notation).

our posterior takes the simple factorized form Thirdly, and to complete specification of the posterior, we take q(U l ) to be a Gaussian with mean m l and variance S l . A similar posterior was used in Hensman and Lawrence (2014) and , but each of these works contained additional terms for the noisy corruptions at each layer.


This is because we have to pay the cost every time we want to observe the state of an item, and the next item selected would depend on the previous states.

the adaptive setting in our paper is more difficult than the non-adaptive setting considered in previous works [10,12,13].


However, it is not always advisable to fabricate a new chip to test a hypothesis regarding a particular neural computation, and software models of spiking neural networks cannot typically execute or communicate with external devices in real-time.

we have designed specialized hardware that implements a reconfigurable array of spiking neurons for rapid prototyping of large-scale neural networks.


Note that multi-stream parallel processing executes multiple sequences concurrently, while multi-time step parallel processing, which is adopted in AM RNN, computes multiple output samples at a time.

conventional RNN, such as LSTM or GRU models, can be used for LM design.


To be more concrete, consider that the covariance matrix Σ is the identity matrix for which σ min (Σ) = σ max (Σ) = 1.

now suppose that q ≥ 2, which gives an upper bound in order to fulfill the xmin -condition in Prop.


This algorithm does not scale well, in fact it is exponential in the total number of weights in the network and the bits per weight.

we propose a randomized training algorithm which searches each node's weight vector independently.


While batch algorithms update parameters sweeping through all data, the incremental algorithm updates parameters for each data one at a time, which results in more parameter updates than batch algorithms.

the incremental algorithm sometimes converge faster than batch algorithms.


In our case all likelihood and prior factors are log-concave.

the posterior is unimodal and a Gaussian approximation is well suited.


However, the true global parameters θ are recovered inexactly due to noise : θ → θ and state indexing ambiguity via a permutation mapping P: θ → θP .

the parameters θP = ( ŌP , ĀP , νP ) obtained from the learning algorithm are in the following form: where P is the permutation matrix corresponding to the permutation mapping P. The presence of the permutation is a fundamental nuisance for MHMM learning since it causes parameter mixing between the individual HMMs.


This privacy requirement is strong enough that non-trivial functions will be computed only with some error.

there is a great need for understanding the fundamental tradeoff between privacy and accuracy, and for designing privatization mechanisms and communication protocols that achieve the optimal tradeoffs.


the topmost layer failed to learn a generative model of the toy shapes.

the top-most layer or pair of layers is necessary for generating complex hallucinations, as is the case with the higher associative visual areas (although processing in the cortical hierarchy is of course much more complicated than in our model).


Intuitively, the problem of estimating such capped densities is easier than full density estimation, but harder than partial density estimation.

if non-convexity is not an issue, it may be preferable to simply use c α -strictly proper losses.


(Beyond 50 topics, we find that TAM's perplexity stays about the same, and then begins to increase again once Z ≥ 75.)

in addition to scaling to more factors, f-LDA is more predictive than simpler multi-dimensional models.


This guarantees that with high probability the true MDP M is in M k . In Step 5, extended value iteration (see below) is used to choose a nearoptimal policy πk on an optimistic MDP Mk ∈ M k . This policy πk is executed throughout episode k (Step 6).

episode k ends when a state s is visited in which the action a = πk (s) induced by the current policy has been chosen in episode k equally often as before episode k. the total number of occurrences of any state-action pair is at most doubled during an episode.


Theorem 2 states that when in the absence of "gradient noise" (i.e., when ] remains finite.

necessarily { ∇V (x k ) } → 0 and the estimates approach a stationary point.


After taking the limit on the both sides of the inequality as ε → 0, we can prove that Equation (8) holds for p = k + 1.

by induction, we can prove that Equation ( 8) holds for any non-negative integer p. The proof of Lemma 2 follows directly from the results in reference (Zhang et al., 2012).


This is described by the following hypothesis class: It is known that the Rademacher complexity of H can be upper-bounded in terms of the trace of the combined kernel.

most existing algorithms for learning kernels [1,4,6] add the following constraint to restrict H: (4) As we saw in the previous section, however, the tail sum of the eigenvalues of the kernel, rather than its trace, determines the local Rademacher complexity.


Thus, Q(t, 0) becomes Similarly, on the event p t = 1, we have that P{θ = 1} = 1 and the policy attaining the infimum in ( 3) is τ * = t+1, δ * = 1.


(Without such a smoothness condition -for example, if some input x could definitively decide between some given θ 0 and θ 1 -then no such asymptotic independence statement can hold, since no more than one sample from such an x would be necessary.)

we can hope that we should be able to asymptotically approximate this optimal experiment by sampling in an i.i.d.


However, we are not sure whether C best can still keep ε-approximation property for C > Ct .

in line 3, we approximately solve the optimization problem (1) at C = Ct and obtain an approximate solution ŵ Ct .


., 0] T . The remaining n − N eigenvalues are zero.

b has total rank N and is positive semidefinite.


For K ≥ 3, h(V ) is a polynomial in V of even order K − 1, whose highest order term has a positive coefficient 1/(K − 1)!.

as V → ±∞, the function h(V ) goes to positive infinity and it thus has a global minimum at some value Ṽ ∈ R. At the global minimum, its derivative vanishes, .


We show that using a three-way connection, instead of the full connection between adjoining rows/columns, is sufficient for learning a dense affinity matrix and requires much fewer output channels of a deep CNN.

instead of using designed features and kernel tricks, our network outputs all entities of the affinity matrix in a data-driven manner.


Consider a k-partition P of V with k > 1, and compare P with a trivial partition {V } of V . Then, the number of clusters has increased by k − 1 and the clustering cost has increased by f [P] + c, where c is a constant.

it is natural to define an average cost of P by f [P]/(|P| − 1).


( 14), we find that the smallest error is obtained when Matrix A corresponds to the Hessian of the merit function for α = 1: A = H (α=1) .

among the various optimization strategies based on Rényi divergences, Kullback-Leibler divergence (α = 1) has the smallest asymptotic errors.


Even team Markov games (where the agents have common interests) can have multiple Nash equilibria, only some of which are optimal (that is, maximize sum of the agents' discounted payoffs).

learning in this setting is highly nontrivial.


In this setting the regression coefficients B and predictors X i are matrix valued, and response y j has mean h[trace(X Here the vec operator stacks the columns of its matrix argument.

the algorithm immediately applies if we replace B by vec(B) and X 1 , .


Since capsules of the same type model the same entity at different positions, we can safely assume that capsules of the same type that are close to each other should have similar poses and activations.

using the same transformation matrix on each capsule within a local receptive field would result in similar votes.


Let A 2 R n⇥m and n  k  m. Then the multiaffine polynomial is real stable.


Denoting the parts of Alice's behavior observable by Bob as x, then H(g | x) is the post-observation entropy in Bob's beliefs, and optimizing it is equivalent to optimizing I(g; x) = H(g) − H(g | x), since the pre-observation entropy H(g) is not dependent on Alice's behavior.

if Bob receives reward r when identifying the right goal, and 0 otherwise, then his expected reward is (1 − p err ) r. in this simplified setup, optimizing information is directly related to optimizing reward.


this approach does not use information about past query responses when selecting TPQs.


As an example, for the task of sentiment classification, y η is the predicted sentiment label of the phrase given by the subtree rooted at η.

during supervised learning, initial external errors are incurred on y, and backpropagated from the root, toward leaves [14].


With multiple bits of information at each of the parties, the existence of a differentially private protocol with a fixed accuracy for any non-trivial functionality implies the existence of a protocol with the same level of privacy and same level of accuracy for a specific functionality that only depends on one bit of each of the parties (as in [22]).

if we can obtain lower bounds on accuracy for functionalities involving only a single bit at each of the parties, we obtain lower bounds on accuracy for all non-trivial general functionalities.


Formally, this sequential test optimizes a cost function linear in the decision time and type I/II errors averaged over many trials [4].

under broad assumptions about the decision process, the optimal behaviour is simply to stop gathering data after reaching a threshold independent of the data history and collection time.


But in order to have lower upper bounds for both objective regret and the clipped long-term constraint T t=1 [g i (x t )] + compared with Proposition 3 in next section, we need to use time-varying stepsize as the one used in [9].

we modify the update rule of x t , λ t to have time-varying stepsize as below: If we replace the update rule in Algorithm 1 with Eq.


The distribution of the random variables in S is, in general, unknown, but a sample is available.

the problem is cast in terms of time series.


Unfortunately, the multi-view formulation of subspace learning does not have an obvious convex form, and current work has resorted to local training methods based on alternating descent minimization (or approximating intractable integrals).

there is no guarantee of recovering a globally optimal subspace.


To solve (4) suppose the problem has been solved to within its statistical accuracy for a set S m ⇢ S n with m = n/↵ samples where ↵ > 1.

we have found a variable w m for which Our goal is to update w m using the Newton step in a way that the updated variable w n estimates w ⇤ n with accuracy V n . To do so compute the gradient of the risk as well as the Hessian H n of R n evaluated at w m and update w m with the Newton step of the regularized risk R n to compute Note that the stepsize of the Newton update in ( 7) is 1, which avoids line search algorithms requiring extra computation.


We were particularly interested in the predictions that the two models made about the grammaticality of verb V4 in sentence structure S2, since this combination of verb and sentence structure never occurs in the data.

a generative learner receives implicit negative evidence that S2 is not grammatical for V4, while a discriminative learner receives no information.


Similarly, if we made this decision after n + q samples our conditional expected loss would be min x∈A μ(n+q) 1 (x).

we define the d-KG factor for a given set of q candidate points z (1:q) as d-KG(z where E n [•] is the expectation taken with respect to the posterior distribution after n evaluations, and the distribution of μ(n+q) 1 (•) under this posterior marginalizes over the observations y(z (1:q) ), ∇y(z (1:q) ) = y(z (i) ), ∇y(z (i) ) : i = 1, . . . , q upon which it depends.


Attention is known to improve learning performance by reducing the effective number of network parameters during learning.

the second element in our network is an attention module that outputs a vector of (attention modulated) values ψ(s).


If for some B j and B j+1 the first condition is satisfied with equality, we can merge the two sets so that the new coarser partition Π will also be optimal.

in the remaining of this section we will assume that the sets B j are chosen maximally.


[9,11]), due to their simplicity, under nonparametric assumptions on E[Y |X].

we adopt such standard assumptions here to illustrate the rates achievable by h Q * λ , following the more general statement of Theorem 1.


By decreasing the beam width from 128 to 32, the computation time can be reduced to less than half while the difference in WER is only about 0.02%.

the word piece model is more advantageous for real-time speech recognition.


Therefore, Finally, for any b ∈ ∆(∼ Γn ) and s ∈ S with b(s) = 1, and any other state ŝ with b(ŝ) = 1, it must be the case that s ∼ Γn ŝ and T n (0)(s, ŝ) = 0.

γn is the optimal value function for the MDP model over ∆(∼ n ).


If the iteration converges for singular A the fixed point depends on the initial value wO and is no longer unique.

for consistent RL algorithms we require that the Bellman error of all fixed points be the same.


The uncertainty parameters ξ on both sides of (2) are not necessarily the same.

any optimization procedure derived based on this test is more conservative than the problem in (2).


Second, we bound the minimum discrimination error in terms of the Jensen-Shannon information.

in order to obtain a relationship between I JS and Fisher Information, we use an expression already derived in [7], where p(r|θ + ∆θ) is expanded up to second order in ∆θ, which yields: Fisher Information provides a good approximation of the Jensen-Shannon Information for sufficiently small ∆θ.


Instead of solving this optimization problem directly, we equivalently solve the penalized objective function maximize where λ i ≥ 0 is a complexity parameter which has a one-to-one correspondence with the tuning parameter δ i .

in the Bayesian point of view, the constraint terms could be viewed as the logarithm of the prior distributions over the states S. the objective can be viewed as a log posterior distribution over S. Now the Viterbi algorithm is not applicable directly since at any time t, the state S it depends on all the states at all time steps, because of the regularization terms which are non-Markovian inherently.


The algorithm runs in two MapReduce rounds and the only information communicated is C i for each i if the data is already assigned to machines.


For example, if A i is random, finding A −1/2 i would be infeasible for large N . Additionally, the constraint R T i A i R i = I would lack any intuition.

we restrict A i = αI + βΦ T i Φ i with α > 0 and β ≥ 0.


Optimization is an integral part of most machine learning systems and most numerical optimization schemes rely on the computation of derivatives.

frameworks for computing derivatives are an active area of machine learning research.


Given a max set size of q, LMIP requires running Queyranne's algorithm [23] (complexity O(q 3 )) to minimize min X⊂A I(X, V \ X|F ) over every |A| ≤ q.

its overall time complexity is O(n q × q 3 ).


Another interesting aspect of the above result is that the convergence rate is linear.

to achieve a relative error of the total number of iterations is on the order of O(log(1 )).


In the finite sample setting, we seek a NDE that converges to f tar asymptotically.

we construct a weighted KDE where the kernel weights are lower in low density regions and higher in high density regions.


This is because (W − αββ T )y can be evaluated as the sum of Wy and αβ(β T y), each requiring O(N ) operations.

by enforcing the sparsity, the overall time complexity of SRcut is O(KM N ).


The variational distribution over the inducing point values u can be used to approximate a posterior process over the basis coefficients w i [Hensman et al., 2013].

given a representation x i , we have that where m k is the approximate posterior mean of the k th column of U and S kk is its covariance.


Since g j is a function of h j , it suffices to check whether the minimal square ℓ 2 -norm of g j is less than 1.

we solve the minimization problem of g j 2 2 w.r.t h j , which gives the optimal h j as, where is less than 1, then β j is zero and no update is needed; otherwise, we continue to the next step of checking whether β k j =0, ∀k, as follows.


The key intuition is that we stack L GNN modules and learn to assign nodes to clusters at layer l in an end-to-end fashion, using embeddings generated from a GNN at layer l − 1.

we are using GNNs to both extract node embeddings that are useful for graph classification, as well to extract node embeddings that are useful for hierarchical pooling.


Note that the present bound is sensitive to a shift of the features.

as a preprocessing step the features of the training patterns should be set to zero mean and the features of the test set shifted accordingly.


In the top two quadrants the robot is in the air, and in the bottom two quadrants the robot is on the ground.

the horizontal axis is a discontinuity of the robot dynamics, and trajectory segments end and often begin at the discontinuity.


Moreover, H(ρ 1 ) = ρ 2 and H(ρ 2 ) = ρ 1 , i.e., {ρ 1 , ρ 2 } is a periodic orbit.

all but a measure zero set S of initial conditions converge to the limit cycle {ρ 1 , ρ 2 }.


Therefore, we can train the LSVM by an iterative algorithm that alternates between inferring h on positive examples and optimizing the model parameter w. Dual form with fixed h on positive examples : Due to its nature of non-convexity, it is not straightforward to derive the dual form for the general LSVM.

as a starting point, we first consider a simpler scenario assuming h is fixed (or observed) on the positive training examples.


If we independently solve the Play subgame, we have no way of knowing that P 1 's value for Sell is lower than the blueprint suggested, so we may still assume there is a gift of 0.5 from the Heads state based on the blueprint.

in order to guarantee a theoretical result on exploitability that is as strong as possible, we use in our theory and experiments a lower bound on what gifts could be after subgame solving was applied to all other subgames.


First note that we can rewrite the convolution operator K using DFT matrices as, where B 2 R d⇥d is the DFT matrix and (•) H denotes conjugate transpose operation.

note that f  Hence, we can rewrite (16) in the frequency domain as, Since B is invertible, the uniqueness of the original C decomposition is equivalent to the uniqueness of e C decomposition.


Representational generality of our architecture Our work aims to extend existing models in behavioral game theory via deep learning, not to propose an orthogonal approach.

we must demonstrate that our representation is rich enough to capture models and features that have proven important in that literature.


Therefore, for consistent RL algorithms we require that the Bellman error of all fixed points be the same.

the quality of the obtained solution to the policy evaluation problem is independent of the initial value.


Nonetheless, the problem is still very sparse with ⇢=8.6e-3 and =4.2e-3.


Note that unlike in (3), the regularization is quite important here (i.e., the case of estimators of the form n i=1 β i k(x i , •)) without which the the linear system is under determined.

we propose the following class of estimators: where g λ (•) is a filter function and λ is referred to as a shrinkage parameter.


Because the problem ( 6) is nonconvex, an arbitrary or random initialization is unlikely to produce a global minimizer.

2 good initializations are critical for the proposed ADM algorithm to succeed.


Related to this issue is the question of estimating accurately the 2 error θ − θ 0 2 2 .

the above oracle bounds have the form θ − θ 0 2 2 ≤ C kλ 2 , with k = θ 0 0 the number of nonzero entries in θ 0 , as long as λ ≥ cσ √ log p. minimizing the bound does not yield a recipe for setting λ.


only the position of a bird's beak, there could be several very different plausible poses that satisfy the constraint.

a simple approach such as training a sparse autoencoder over keypoints would not suffice.


An increase in either K or p max increases the total cost of the algorithm.

the algorithm originally presented above does not perform well for small bandwidths.


Node Y~r~l is the output node of N;(/1), and node Y~~~l is also the output node of Nt.

the entire network has 3l + 6 nodes that are linear or product units and 3l nodes that compute functions h, fl' or f12.


The above naturally suggests an approximation based on a limited number of Fourier coefficients with respect to a certain subset of irreducible representations.

we will show that, indeed, the information matrix Q corresponds to the Fourier coefficient with respect to the first-order representation of the symmetric group S n . it yields a natural approximation.


A key obstacle of achieving this is that, while the conditional posteriors of U and V are Gaussians, neither their marginal and joint posteriors belong to well behaved classes (e.g., conjugate posteriors, or having closed forms).

novel tools, that can handle generic posteriors, are needed for efficient analysis.


We would like the label complexity of MARMANN to depend only on N ( t) (where t is the selected scale), which is of the order m Ĝ.

during model selection we can only invest a bounded number of labels in each tested scale.


The following lemma gives some useful properties about W n and W n . Lemma 1 Both random variables W n and W n are bounded by Haus( M n , M ).

namely, The cumulative distribution function (CDF) for W n and W n are P(W n ≤ r| R n ) is the ratio of R being covered by padding the regions around R n at distance r. This lemma follows trivially by definition so that we omit its proof.


Integer programming is an NP-hard problem.

the first step toward solving this problem in practice is to relax it into a continuous optimization by replacing the integer constraints (6) with continuous constraints 0 ≤ µ ≤ 1, yielding the relaxed formulation max µ (1 − µ j e)H(y|x j , w t+1 ) (10) µ j e ≤ 1, ∀j If we can solve this continuous optimization problem, a greedy strategy can then be used to recover the integer solution by iteratively setting the largest non-integer µ value to 1 with respect to the constraints.


2) System Reward: When soliciting reviews on businesses, the decision maker expects more businesses covered under the budget constraint.

individual user's marginal quality is maximized if each of them reviews a distinct business.


Thus, the Dirichlet process has probability mass only on the K spike words, and is described by a (finite dimensional) Dirichlet distribution, In the absence of data, the parametric base measure controls the mean of this nonparametric model, regardless of ↵.

we loosely say that ⇡ is "centered" around G ✓ .


First, note that the value of the objective function f ( S) is independent of how long each metaaction ãt takes.

the only potential danger is that in making this assumption we have overlooked a constraint violation of the form E S The argument bounding E [R cost ] is similar, although somewhat more involved (for details, see [18]).


These knowledge sets, which can be as simple as cubes, are supposed to belong to one of two categories into which all the data is divided.

a single knowledge set can be interpreted as a generalization of a training example, which typically consists of a single point in input space.


The size of each cropped image is 32 × 32 pixels, with 256 gray levels per pixel.

each image can be represented by a 1024-dimensional vector.


Since I/O → TRACE independently predicts traces for each I/O pair, we hypothesize that they lack consistency with each other compared to the gold traces.

for future work we suggest to investigate inferring the execution trace for a given I/O pair, conditioned on already generated execution traces for the same underlying program but on different I/O pairs.


Notice that the (unnormalized) posterior is maximized at its mean m c since this corresponds to the mode for a multivariate Gaussian distribution.

the solution of the problem of reconstructing stimuli from responses reduces to the following simple expression: 3 Results We used the following datasets in our experiments: fMRI dataset.


However, the GPUs used are capable of about half the FLOPs compared to our P100.

we corrected this value to 3840 hours*GPUs.


It is unnatural to do this with Q-learning, as the Q function generally cannot contain different information at training and test time.

we propose a simple extension of actor-critic policy gradient methods where the critic is augmented with extra information about the policies of other agents.


In contrast, when we set the block size to over 32, the speedup is approaching the ideal case.


Our RTDP based planner was not able to perform any significant optimisation in 20m due to memory problems.

to demonstrate FPG-Planning is having some effect, we compared the optimised policies to two simple policies.


Without the loss of generality, we assume that the (M l + 1)-th neuron in the l-th layer is the augmented exponential neuron.

the output of the network f is expressed by where σl : R M l−1 +1 → R M l +1 is a vector-valued activation function with the first M l components being the activation functions σ in the network f and with the last component being the exponential function, i.e., σl (z) = (σ(z), ..., σ(z), exp(z)).


While these approaches require labels for the in-distribution data to train the classifier network, our method does not use label information.

it can be applied for the case when in-distribution data is represented by one class or label information is not available.


random variables 6, ... ,em is FO m , where F(•) is the cdf of ei.

in order to find a column to add to P that is with probability 0.95 among the best 0.05 of all such columns, a random subsample of size ilogO.05/log0.951 = 59 will suffice.


To enhance the clustering structure of the documents, we update θ by minimizing the Wasserstein distance between the documents and their closest topics.

the documents belonging to different clusters would be far away from each other.


Similarly, the ending position d j is determined using another independent Ptr-Net module.

we have two Ptr-Net modules for determining the locations of the beginning and ending positions.


α f (y A f ) when one of its incoming message m jf or m if is 0 (see details in Appendix C-2).

we can find the greedy factor (18) by maintaining an ordered multimap for the scores of outcome y A f in each factor {v(y A f , x f )} f ∈F .


To apply such a framework as a model for cortical processing, then, we take the sensory cortical activity to represent the inferred values of the latent variables.

perceptual inference in this framework involves estimating the values of the causal variables that gave rise to the sensory input, while developmental (unsupervised) learning involves discovering the correct causal structure from sensory experience.


Thus the Equal Spacing CTC deformation (EsCTC) guides the end-to-end model training by optimizing the loss function: We consider the EsCTC as complementary to EnCTC that explicitly rules out all unreasonable alignments.

we also combined the Equal Spacing CTC deformation with maximum conditional entropy regularization defined in section 3.3.


The high level features here (in R 7 ) have been shown to have good discriminative properties in [1].

we expect them to have different distributions across two processes.


Therefore x k is likely to be discrete at every step.

k Mx k ; and D < 0, when the quadratic function f q (t) is convex with the maximum in the domain of Problem 2 attained at point the algorithm is guaranteed to increase the score at every step.


For a composite entity that consists of two entities of the same type, we'd like to capture its correlation with its twinthe composite entity made of the same basic entities but in reversed order.

we add the T win relation between all pairs of twin entities: e.g., from ⟨p1, p2⟩ to ⟨p2, p1⟩, and vice versa.


Each agent has a local field-of-view (making the world partially observable), sees 17 spaces in front, 10 to either side, and 2 spaces behind.

observations are encoded as 21x20x3 RGB tensors with values 0 -255.


( 7) and 1 = (1, 1, . . . ) ∈ R N . Note that the Wiener series is represented only implicitly since we are using the RKHS representation as a sum of scalar products with the training points.

we can avoid the "curse of dimensionality", i.e., there is no need to compute the possibly large number of coefficients explicitly.


The policy π instead uses updated belief-mean m t|t (smoother and better-informed than z t ) to decide the action: u t = π(m t|t , ψ).

the joint distribution over the updated (random) belief and the (non-random) action is Next, the filtering 'prediction step' computes the predictive-distribution of b t+1|t = p(x t+1 |z 1:t , u 1:t ) from the output of dynamics model f given random input bt|t .


Choice of Lag Index Set L. Unlike most approaches mentioned in Section 2.2, the choice of L in TRMF is more flexible.

tRMF can provide important advantages: First, because there is no need to specify the weight parameters W, L can be chosen to be larger to account for long range dependencies, which also yields more accurate and robust forecasts.


According to this method, at each step, the weak threshold ranker is selected so as to maximize the AUC of the weighted distribution.

with this method, the global objective of obtaining the best AUC is obtained by selecting the weak ranking with the best AUC at each step.


For computational convenience, we enforce this rotational invariance by randomly rotating our images.


Since, we are already tracking where along the desired trajectory the helicopter is, we chose a potential function that increases along the trajectory.

whenever the helicopter's $ ¡ ¤£ ¡ §¦ ¡ ( makes forward progress along this trajectory, it receives positive reward.


Consider first f . Due to memoization, the algorithm executes at most two basic operations (one comparison and one multiplication) per triplet (S, R, C), with S and R disjoint, S ⊂ C ⊆ S ∪ R, and |C| ≤ w. Subject to these constraints, a set C of size i can be chosen in n i ways, the set S ⊂ C in at most 2 i ways, and the set R \ C in 2 n−i ways.

the number of basic operations needed is at most Similarly, for h the algorithm executes at most two basic operations per triplet (C, R, S), with now C and R disjoint, |C| ≤ w, and S ⊂ C. A calculation gives the same bound as for f . Finally consider g. Now the algorithm executes at most two basic operations per triplet (C, U, R), with C and U disjoint, |C| ≤ w, and ∅ = R ⊆ U .


We note that during the construction of the column pcp, the j th column, if sampled, is rescaled by 1 √ s1qj .

the resulting matrix, AS may no longer be a distance matrix.


The parameter λ minimizing the RHS of the previous equation depends on ρ.

we need to get a version of this inequality which holds uniformly in λ.


knl means expectation wrt q conditioned on the variables in k n l) we can precompute (!l</>k)knl for all pairs of overlapping cliques k, l, for all states in knl.

the worse case complexity of the second order term is less than ninoverlap exp(c).


Multi-label data refers to data sets in which an image can be associated with more than one semantic concept, which is more challenging but closer to real world applications than single-label data [21].

we evaluate the proposed method in multi-label image categorization tasks.


. . , t N } such that |t i − t j | = |i − j|τ for all i and j, we consider a draw from the multivariate normal distribution: where each observation is generated from the cluster indicated by z (ar) w (described in Section 2.2), and each cluster is represented uniquely by a covariance matrix, Σ , whose elements are defined through the covariance kernel k(τ ; θ , γ).

the parameters θ z (ar) w describe the autocorrelation content associated with each y (ar) w . We address two concerns with this formulation.


We can easily extend our methods to handle this case for appropriate h. Recall that the auxiliary function that we used, depended upon the convexity of ϕ.

if (ϕ •h) is a convex function, whose derivative ∇(ϕ •h) is "factorizable," then we can easily derive algorithms for this problem with link functions.


Note that the eigenvalues correspond by λ , which means the order of eigenvalues is reversed (λ SF A i > 0).

the subspace spanned by the slowest features is the same that optimizes separability in terms of Fisher's Discriminant, and the slowest feature is the weight vector which achieves maximal separation.


On the other hand, variable A does little in the PGM in (b) since W and Y are nearly independent.

a similar argument can be made about variable B. variable C in the PGM in (a) can be used to explain the weak interactions between variables, whereas in the PGM in (b), variable C essentially has the task of explaining the interaction between all the variables.


The proof can be found in the extended technical report [11].

incremental and best-k problems for DAG(∨) are PP-Hard as well.