. . , (i t−1 , C it−1 , a t−1 ) , we take the general approach of [1], and assume that for any fixed i ∈ V and x ∈ R d , the variable i (x) is conditionally sub-Gaussian with variance parameter σ 2 > 0, namely, E t exp(γ i (x)) ≤ exp σ 2 γ 2 /2 for all γ ∈ R and all x, i.

u i x is the expected reward observed at node i for context vector x.


However, motivated from the study of homometric structure and the fact that each element of FGSD encodes some local or global sub-structure information of the graph, inspired us to define the graph spectrum as R = {S f (x, y)|∀(x, y) ∈ V }.

comparing two R's implicitly evaluates the sub-structural similarity between two graphs.


We formally express this behavior in the following Corollary.

for policy π ∈ Π, the posterior belief process µ t is a supermartingale with respect to Ft , where unlike classical Bayesian learning models with a belief martingale [18,[21][22][23], the belief process in our model is a supermartingale that leans toward decreasing over time.


Specifically, l ia (1) t (t) > 1/2 indicates that arm i beats a (1) t with high probability.

comparing a (1) t and arm i brings little information gain and thus should be eliminated to minimize the regret.


Given a fixed adaptive control algorithm f and a fixed sequence of disturbance terms w 0 , w 1 , . .

., the sequence of states x t visited is exactly determined by the equations ) for fixed f , we can think of the (controlled) dynamical system as a mapping from the sequence of disturbance terms w t to the sequence of states x t . We now give the most commonly-used definition of stability, called BIBO stability (see, e.g., [6]).


The idea is that as ν increases, it will be harder to distinguish between P and Q for a fixed sample size.

using this setup we can verify whether the adaptive bandwidth selection achieved by γ (as the test statistic) helps to distinguish between P and Q at higher ν compared to γ k with a heuristic σ.


After n steps, PCA will then be dominated by dimensions for which α i is small enough that n ≪ n c,i .

even if relatively few 10 1 10 2 10 3 PCA component 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 ResNet-50-v2 dimensions have small curvature they will come to dominate the PCA projected trajectory after enough steps.


We adopt a collapsed, blocked Gibbs sampling approach, where θ and B have been integrated out.

only the community indices s need to be sampled.


The proof, which is provided in the appendix, shows that in the noiseless case, for each principal submatrix, A l , of A, step 2a of Algorithm 1 will learn an exact Ĉl such that A l = Ĉl ĈT l . Further, when assumptions A3 and A4 are met, step 2b will correctly learn the orthonormal transformation to align each Ĉl to the previously added rows of Ĉ.

progressive iterations of step 2 correctly learn more and more rows of a unified Ĉ.


In this formulation, the mixing function, α(β, τ ), is by definition a member of L 2 .

if f t (β, τ ) ∈ L 2 , then P (t) is finite by the Cauchy-Schwartz inequality since α(β, τ ), For ease of readability we shall henceforth suppress the domain variables in f t (β, τ ) and α(β, τ ) and refer to them as f t and α. for some > 0.


On the other hand, the evaluation of a stochastic gradient is usually significantly more efficient than that of a full gradient.

replacing full gradients with stochastic gradients essentially trades the number of iterations with a low computational cost per iteration.


we obtain the following pairs: Obviously, (RKM) is equivalent to (GKM) when Φ(x) = e F λ (x).


This number indicates the fold-change increase in drug concentration that is necessary to inhibit viral replication by 50%, as compared to a fully susceptible reference virus.

higher fold-changes correspond to increasing levels of resistance.


Proposition 3.2 implies that at any fixed time, the likelihood of observing ⌧ given a candidate source is proportional to the number of feasible orderings originating from that candidate source.

an ML estimator (timestamp rumor centrality) counts the number of feasible orderings at estimation time t. Timestamp rumor centrality is a message-passing algorithm that proceeds as follows: for each candidate source, recursively determine the set of feasible times when each node could have been infected, given the observed timestamps.


In the regions where ∇Q < η(n, f ) • √ d • σ, the Byzantine workers may take advantage of the noise (measured by σ) in the gradient estimator G to bias the choice of the parameter server.

proposition 2 is to be interpreted as follows: in the presence of Byzantine workers, the parameter vector x t almost surely reaches a basin around points where the gradient is small ( ∇Q ≤ η(n, f ) • √ d • σ), i.e., points where the cost landscape is "almost flat".


Moreover, these applications use confidential dataset and use of ad hoc mechanism can lead to serious privacy leaks [47].

32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montréal, Canada.


Fourth, we show that these learning rules can also be applied to a network that feeds a time-delayed version of the network output back into itself.

the network starts to efficiently represent both its signal inputs and their history.


First, the total reward on the agent's single trajectory is a random variable, and may deviate from the expected value of its policy.

it is generally impossible to decide if the agent's policy is near-optimal, and instead we assume that the human can check if each action that the agent takes in the trajectory is near-optimal: when the agent takes a at state s, an error is counted if and only if Q (s, a) < V (s) − .


This model assumes that the action of the player in each round is a function of their action in the previous round (the "Previous Action" model).

to fit the data we need to estimate p(a i t |a i t−1 ).


( 6) with current estimate X and θ, the maximization over A becomes a quadratic programming problem as follows: which is identical to solving the dual problem of the standard SVMs.

we can obtain optimal A by employing an existing SVM package.


A policy learned on one setting of the acrobot will generally perform poorly on other settings of the system, as noted in [3].

subtle changes in the physical parameters require separate policies to adequately control the varied dynamical behavior introduced.


In this work, we show that choosing a subset of the elements in the context can improve performance when the objects in the subset are truly related to the object to be modeled.

the embedding vectors can reflect co-occurrence relations with higher fidelity compared with the base embedding model.


The location of the non-zero entries of M r depends on (X 2 ) and is independent of b :,r . Proof The product of the (k + (i − 1)K)-th row of (X 2 ) and b :,r is the (k + (i − 1)K)-th element of v :,r . And, this element is the (i, k)-th entry of M r by definition of unvec (K,I) (•) .

if all the entries in the (k + (i − 1)K)-th row of (X 2 ) are zero, then the (i, k)-th entry of M r is zero regardless of b :,r . Consequently, the location of the non-zero entries of M r is independent of b :,r , and is only determined by (X 2 ) .


Blei and Jordan [7] define an explicit truncation level L ≡ T for the variational mixture in (1) by setting q vT (v T = 1) = 1 and assuming that data-cases assign zero responsibility to components with index higher than the truncation level T , i.e., q zn (z n > T ) = 0.

in their model only components of the mixture up to level T need to be considered.


In our model, each leaf-node is mapped to several feature dimensions of the vector ϕ(X, y, H * ).

the process of reconfiguration is equivalent to reorganizing the feature vector ϕ(X, y, H * ).


Additionally, in [2], no prior is put on φ (t) , unlike in GC-LDA.

when using GC-LDA with a single multivariate Gaussian and γ = 0, it is equivalent to a smoothed version of Correspondence-LDA.


This additional error (due to the numerical approximation) can be made arbitrarily small, e.g.

o(N −1/2 ), whenever H ≥ log N d . using P = O( √ N ) random features, we deduce that the complexity of building the matrix Then in order to solve the least squares system, one has to compute Ψ T Ψ, that has numerical cost O(P 2 N ), and then solve the system by inversion, which has numerical cost O(P 2.376 ) by [5].


If this vector is not in T , then the steering policy that satisfies (2) with x = mn is selected for the next cycle.

the average reward is "steered" towards T , and eventually converges to it.


Notice also that a i p,l = k(x p , m i l ) measures a proximity between point x p and the mode obtained at the previous iteration.

given the current assignment z i p , the modes can be obtained as a proximal optimization, which seeks a high-density data point that does not deviate significantly from the mode obtained at the previous iteration: Now observe that the z-updates we obtained in Eq.


Indeed, recent empirical studies studies [17] show that belief propagation methods perform considerably worse than min-cut-based methods when applied to a variety of (purely) regular MRFs.

falling back on belief propagation methods for these MRFs may result in poor performance.


Unfortunately, the limiting distribution of such V -statistics is quite complicated -it is an infinite sum of dependent χ 2 -distributed random variables, with a dependence determined by the temporal dependence structure within the process {Z t } and by the eigenfunctions of a certain integral operator associated with h 2 [5,8].

we propose a bootstrapped version of the V -statistics which will allow a consistent approximation of this difficult limiting distribution.


This is expected, as a large degree of model uncertainty will yield a large variation in p(µ|x ⇤ , ✓), and large uncertainty in µ will lead to a large uncertainty in estimates of data uncertainty.

model uncertainty affects estimates of data and distributional uncertainties, and distributional uncertainty affects estimates of data uncertainty.


Our novel distribution is based on the parametric form of the Poisson MRF model [1] but is fundamentally different because of the domain restriction to a fixed-length vector like in a Multinomial where the number of trials is fixed or known.

we propose the Fixed-Length Poisson MRF (LPMRF) distribution.


Since B_j and Xj are independent B-"J Xj has the same distribution as the projection of a test example x*, B_; x*.

b_jB_; Xj and B_jB_; x* have the same distribution as well.


The Hawkes process is equivalent to the union of these Poisson processes -each event is sampled from one of these processes.

it is natural to introduce a latent cause variable z for each event: z = 0 if the event is caused by the base rate; z = n if the event is caused by a preceding event s n . For clearer description, let us consider the flattened single event sequence S = [(s 1 , i 1 ), .


The gradient estimator is biased because each min-/max-problem is in fact nonconvex and cannot be solved in one-shot.

our framework is more stable and efficient than the mini-max formulation for learning point processes.


Lacking prior knowledge we assume that x takes any value in the interval uniformly at random.

the  probability of x 1s occurring inside the knot, given the value of y (i.e., 1s prior to the knot) is: Based on this conditional probability we can write the probability of each value of x as in which P k (x|y) is computed by Equation ( 4) and Q k (y) refers to the probability of having y 1s prior to knot k. In order to evaluate Equation ( 5), we need to compute the values of Q k (y).


However, this choice of λ depends on β which is not known in practice (although η 0 is known as it is determined by the choice of g λ ).

λ is usually learnt from data through cross-validation or through Lepski's method [24] for which guarantees similar to the one presented in Theorem 5 can be provided.


Given a non-planar graph G over binary variables with a vector of interaction potentials θ, we wish to use the exact planar computation to obtain a bound on the partition function of the MRF on G. We assume for simplicity that the potentials on the MRF for G are given in the form of Equation 2.

g violates the assumptions of the previous section only in its non-planarity.


The rule r 2 should pay more for the erroneous prediction than the rule r 1 .

we generally want each row of C to be V-shaped.


However, majority voting can not be applied when there are clustering models because the correspondence between clusters and classes is unknown.

we compare BGCM with two clustering ensemble approaches (MCLA [26] and HBGF [12]), which ignore the label information from supervised models, regard all the base models as unsupervised clustering, and integrate the output of the base models.


If we want a single non-zero entry per column of H i , to achieve success probability 1 − O(1/s) so that we can union bound over all s servers succeeding, we naively would need to increase the number of rows of H i by a factor linear in s. We give a general technique, which takes a subspace embedding that succeeds with constant probability as a black box, and show how to perform a procedure which applies it O(log 1/δ) times independently and from these applications finds one which is guaranteed to succeed with probability 1 − δ.

in this setting the players can compute a subspace embedding of their data in nnz(P i ) time, for which the number of non-zero entries of H i P i is no larger than that of P i , and without incurring this additional factor of s. This may be of independent interest.


The second step of the iHMM construction involves defining the transition distributions T (•|s) ∼ DP(α, T ) for each state s, where α, the concentration parameter for the DP, determines how closely the sampled distribution T (•|s) matches the mean transition distribution T . Because T puts higher probabilities on states with smaller indices, T (s ′ |s) will also generally put more mass on earlier s ′ (see lower rows of figure 2).

the generating process encodes a notion that the agent will spend most of its time in some local region.


Initial theoretical work (see [4]) shows that increasing the number of bootstraps in the selection module (B 1 ) increases the amount of feature compression (primary controller of false positives), while increasing the number of bootstraps in the estimation module (B 2 ) increases feature expansion (primary controller of false negatives), and we observe this empirically.

neither should be too large, and their relative values express the balance between feature compression and expansion.


Let a convolution be defined as Considering a naive convolution algorithm, the number of MACs for a convolutional layer is equal to PCKd h d W where P is the number of pixels in the output feature map.

it is important to reduce CK whenever P is large.


The third key obstacle is that each of the latent variables, Φ and Θ, simultaneously serve as the inputs and output targets for successive layers.

it is necessary to reformulate the connecting problem (2) so that it is jointly convex in all three components, U , Φ and Θ; and unfortunately (3) is not convex in Θ.


Typically, weight decay is to regularize the kernel by minimizing its 2 norm, while there is no regularization on the direction of the kernel.

mHE completes this missing piece by promoting angular diversity.


Although we now have an appealing distribution, the integral at the denominator of ( 1) is intractable to compute even for simple Q-function models.

we propose a variational approximation q ξ by considering a simpler family of distributions parameterized by ξ ∈ Ξ.


Using standard concentration inequalities one can show that the training error of the true risk minimizer, h * , cannot be "too far" from the training error of the empirical risk minimizer, ĥ.

we can guarantee, with high probability, that the class of all hypothesis with "sufficiently low" empirical error includes the true risk minimizer h * .


Then, for the pair (G, θ), the distribution on X p is given as: θi,j x i x j where x ∈ X p and Z is the normalization factor, also known as the partition function.

we obtain a family of distributions by considering a set of edge-weighted graphs G θ , where each element of G θ is a pair (G, θ).


the model is trained so that given a sample set or task, produces a classifier for that specific task.

the model is exposed to different tasks (or episodes) during the training phase, and it is evaluated on a non-overlapping set of new tasks [33].


The advantage of the proposed approach is that we perform GP modeling only either in the spatial or temporal domain at a time.

the dimensionality can be remarkably reduced and modeling large datasets becomes feasible.


Here, we depart from the approach of [1] by sampling a local split of topic a into topics b and c. Temporary parameters, {π b , πc , θb , θc }, and topic assignments, ẑ, are sampled according to We note that a sample from q(ẑ|v, v) is already drawn from the restricted Gibbs sampler described in Equation ( 9).

no additional computation is needed to propose the split.


However, new experience may quickly render all of the current model samples unlikely.

we must periodically resample a new set of models directly from the current belief.


( 9)) of different methods, since their values are calculated by different metrics.

we make use of the normalized reconstruction error defined as N R = Rt−Rinit Rinit , where R init and R t are corresponded to the reconstruction errors at the initial step and the t-th epoch, respectively.


Then the objective function in Definition 4 becomes which is exactly the objective function in Definition 6.


As an alternative, in Sections 3.1 and 3.2 we propose soft penalty functions that encode the belief that points near the origin are more likely to be optimal (we can always re-center the problem to match our beliefs in this regard), but do not rule out any x 2 R n a priori.

one of our contributions is showing that interesting results can be obtained by choosing L differently than in Eq.


For any staircase mechanism Q, there exists a possibly different staircase mechanism Q ∈ [Q] such that Q = S (k) Θ for some diagonal matrix Θ with nonnegative entries.

to solve optimization problems of the form (5), we can restrict our attention to such representatives of equivalent classes.


At the traversed node G i j , we update u G i j according to the operation in (5), which reduces the Euclidean norm of u G i j by at most λ i j .

the time complexity of MY tgLasso is O( the time complexity of MY tgLasso is O(pd).


RADAGRAD therefore makes use of the corrected update γ t is the projection of the current gradient onto the space orthogonal to the one captured by the random projection of G t . This ensures that important variation in the gradient which is poorly approximated by the random projection is not completely lost.

if the data has rank less than τ , γ ≈ 0.


The algorithm, MaxFlow, returns a maximal set, Z E , for which the equations, A • Λ T a(E),y = b, are linearly independent, regardless of whether |Z E | = |E| and E is identified or not.

if we are able to find an allowed node w that satisfies the conditions below, then the equation a w • Λ T a(E),y = b w will be a linear combination of the equations, A • Λ T a(E),y = b. Theorem 2.


In multi-view learning [4,13], a joint model is learnt from both labeled and unlabeled data from multiple sources.

it can be regarded as a semi-supervised ensemble requiring access to the raw data.


However in our setting, computing them naively is prohibitively expensive as the number of agents is large (in thousands).

we use similar technique as for WLQ by normalizing counts, and computing differentials lim ∆=1/M→0 Qw(ñ sa , d)−Qw(ñ sa +∆•(e ij −e ij ), d) , final estimate is (proof in appendix): Crucially, the above computation is independent of agent population M , and is thus highly scalable.


After 1000 iterations, the perplexity of Async-LDA has converged to the standard LDA perplexity.

in this experiment, learning in an online fashion does not adversely affect the final model.


We further assume that the kernel is bounded, that is, |k ψ (x, x')| ≤ M for some M > 0 for all x, x' ∈ X .

the problem relaxes to the following, Using the Representer theorem, we have that the optimum solution for the above is of the form v * = m i=1 α i ψ(x i ) for some α ∈ R n . Denoting the sample kernel matrix be K such that K i,j = k ψ (x i , x j ), the above optimization problem is equivalent to the following optimization problem, where Y is the vector corresponding to all y i and ||Y || ∞ ≤ 1 since ∀i ∈ [m], y i ∈ [0, 1].


These models are known to be over-parametrized, and large quantities of data are needed for them to reach their full potential [12].

the training time can be very long (up to weeks) even when vast computational resources are available [31].


Formally, on each iteration T + 1, player i selects actions a ∈ A(I) according to probabilities If a player plays according to CFR in every iteration, then on iteration T , R T (I) So, as T → ∞, In two-player zero-sum games, if both players' average regret [15].

cFR constitutes an anytime algorithm for finding an -Nash equilibrium in zero-sum games.


A downside of this parametric approach, however, is that the feature maps are chosen manually by the designer.

any such model is biased to certain image structure, and therefore discriminates features that might not seem plausible at first sight, but may well play a significant role.


The fact that F µ ( ) is indeed smooth with modulus µ follows from Lemma 6.1 in the Supplement.

one is able to apply an accelerated gradient descent algorithm on this modified Lagrange dual function, which is detailed in Algorithm 1 below, starting from an initial primal-dual pair (e x, e ) 2 R d ⇥ R N . Algorithm 1 Primal-Dual Smoothing: PDS ⇣ e , e x, µ, T Our proposed algorithm runs Algorithm 1 in multiple stages, which is detailed in Algorithm 2 below.


The relevant differences between similar actions in terms of the actor body configuration can be at a fine level of detail.

one cannot rely on a fixed pre-determined number of configuration types [7,9,11]; rather, one needs to be able to make as fine discriminations as required by the task.


Note that z is continuous and, intuitively, indicates the distance of a given point x from the unknown safety boundary in the input space.

given the function g -or an estimate of it -we can evaluate the level of safety for a trajectory τ .


Clearly, In a similar way we can write, L p K φ L,2 0 (x) = π 2 λ 2 0 φ L,2 0 + T 2 (x) where, T 2 (x) = π1 π2λ 2 0 k(x, z)φ L,2 0 (z)p 1 (z)dz.

when T 1 (x) and T 2 (x) are small enough then φ L,1 0 and φ L,2 0 are eigenfunctions of L p K with corresponding eigenvalues π 1 λ 1 0 and π 2 λ 2 0 respectively.


The sample complexity of an algorithm A is the function m A defined as follows: For , δ > 0, m A ( , δ) is the smallest integer such that for every m ≥ m A ( , δ) and every distribution D on X × Y, with probability of > 1 − δ over the choice of an i.i.d.

sample S of size m, The first term on the right-hand side is the approximation error of H. the sample complexity is the number of examples required to ensure that the estimation error of A is at most (with high probability).


Following (21), the gradient of the KL divergence functional F (q) := KL(q || p) is grad H KL(q || p) = ∇ • (φ * q,p q).

the SVGD-Valsov equation ( 13) is a gradient flow of KL divergence under metric W H (•, •): ∂q t ∂t = −grad H KL(q t || p).


Given a MDP M and a homomorphism, an aggregated MDP M ′ is given by (S ′ , A, {P(C, a, D) : a ∈ A;C, D ∈ S ′ }, {R(C, a) : a ∈ A,C ∈ S ′ }, ρ, g s : s ∈ S) where S ′ is a partition of S, ρ : S → S ′ maps states to their aggregates, each g s : A → A relabels the action set and we have that ∀C, D ∈ S ′ and a ∈ A, Note that all the states in a partition have actions that are relabelled specifically so they can exactly match each other's behaviour.

a policy in the aggregate MDP can be lifted to the original MDP by using this relabeling.


These methods make detection by finding the common characteristics of the attack profiles that differ from the normal profiles.

they have a common assumption that the attack profiles are produced by the same attack strategy.


However, the output uncertainties are highly correlated since different tuples share their lineage.

inspecting a tuple also gives us information about the correctness of other tuples.


We provided a key result stating that a set can be approached to a given precision using only a finite number of steering policies, which may be learned on-line.

an interesting observation regarding the proposed learning algorithm is that the learned optimal polices in each direction are essentially independent of the target set T . the target set need not be fixed in advance and may be modified on-line without requiring a new learning process.


For instance, it is difficult to maximize or integrate the above log-sum quantities.


SURE-T methods yielded a slightly better denoising performance than SURE-based methods (approximately 0.02 -0.06 dB and 0.04 -0.12 dB for BSD68 and Set12 datasets, respectively) with considerably increased the overall inference time.

at this moment, SURE-T methods do not seem to have considerable benefits compared to SURE-based methods.


This constraint on the entropy is crucial since otherwise the solution of the LMO would be a degenerate distribution as the authors of [5] have also argued.

we now replace this problem with its regularized form using Lagrange multipliers and solve for s given a fixed value of : the regularized LMO problem is equivalent to the following minimization problem: where Z is the normalization constant of q p q t . From this optimization problem, we can write what we call the Residual Evidence Lower Bound (RELBO) as: Discussion.


During training, the predicate representations of parameterized rules are optimized jointly with all other subsymbolic representations.

the model can adapt parameterized rules such that proofs for known facts succeed while proofs for sampled unknown ground atoms fail, thereby inducing rules of predefined structures like the one above.


However, there is no way to invert the modeling process to determine parameter ranges given a fixed sum of squared error.

we use the algorithm detailed in §2 to map out the confidence surface as a function of the input parameters; that is, we use the algorithm to pick a location in the seven dimensional parameter space to perform an experiment, and then run CMBFast [12] to create simulated power spectrum given this set of input parameters.


[In digital systems] Round-off error does not accumulate significantly for many computations.

complex systems with many stages are easy to build" [27].


When the largest element of this matrix is found and the template subtracted, no values of dC need to change except those very close in time to the fitted template (t s samples away).

instead of finding the global maximum of dC, we can find local maxima above the threshold criterion, and impose a minimal distance (t s ) between such local maxima.


However, some regions outside the grid cells are also raised, but were supposed to be zero.

we address this issue via another, separate level set property: there is a threshold T such that a) the function value inside each I δ k is larger than T ; b) the function values outside the grid cells are smaller than T . the desired grid indicator function can be obtained by performing a max operator with the threshold T , i.e., cutting off the smaller values and setting them to zero (see Appendix C).


Since ρ is continuous, V is the image of a compact set under a continuous map, and thus V is compact and the inverse of the bijective map I • ρ : X → W is continuous.

there is a one-to-one relationship between the continuous functions f X on X and the continuous functions f W on W , namely C(X) = C(W ) • I • ρ, see also the discussion following (7).


. . , N K . Although there are several approximation algorithms [8,21,17], their computational costs are still intensive.


This observation parallels one of the key insights of [10], here applied to M instead of K. Unfortunately, even though we can pose a convex objective, it does not allow us to immediately solve our problem because we still have to relate M to y, and M = yy is not a convex constraint.

the main challenge is to find a way to constrain M to ensure M = yy while respecting the class balance constraints − ≤ e y ≤ .


This bound will depend on the empirical risk R S (G Q ) which relates to the the empirical loss ζ Q (measured on the training sequence S of m examples) through the equation where Note that Equations 7 and 8 imply that Hence, any looseness in the bound for R(G Q ) will be amplified by the scaling factor c on the bound for ζ Q .

within this approach, the bound for ζ Q can be tight only for small values of c. Note however that loss functions having a small value of c are commonly used in practice.


This allows to reparameterize w qm according to w qm = s qm w qm and assign the above prior distributions on s qm and w qm .

the reparameterized spike and slab prior takes the form: Notice that the presence of w qm in the likelihood function in (1a) is now replaced by the product s qm w qm .


In terms of a flat MDP, OPI can be interpreted as sometimes picking a greedy off-policy action while evaluating a fixed policy, when the value function of the greedy policy is at least as good and more compact than that of the given policy.

oPI may be viewed as asynchronous policy iteration ( [9]).


our goal -to synchronize topics across different languages -and there's no reason to believe that likelihood measures would assess that: a model where topics are synchronized across languages is not necessarily more likely than a model that is not synchronized.

to evaluate our model we compare the topic distributions of each English document with its corresponding French pair (and analogously for the other combinations: English/Portuguese and French/Portuguese), with these metrics: where L 1 denotes the set of documents in the first language, F a mapping from a document in the first language to its corresponding translation in the second language and θ d the topic distribution of document d. Mean Hellinger distance: Agreements on first topic: where I is the indicator function -that is, the proportion of document pairs where the most likely topic is the same for both languages.


1 1/ poly(n), the algorithm can compute marginal contribution estimates ṽ that are all a 1/ poly(n)-additive approximation to the true marginal contributions v (See Appendix B for formal analysis of estimates).

we give the analysis for the true marginal contributions, which, with probability 1 1/ poly(n) over the samples, easily extends for arbitrarily good estimates.


Furthermore, this problem does not afflict smooth nonconvex problems (h ⌘ 0), where convergence with constant minibatches is known [6,21,22].

there is a fundamental gap in our understanding of stochastic methods for nonsmooth nonconvex problems.


In fact, any minimum of (1) will be such that rank[Z] + max j e •j 0 ≤ n and rank[Z] + max i e i• 0 ≤ m, regardless of Y. Secondly, unlike any separable penalty function (4) that retains the correct global optimal as (1), Proposition 2 implies that (14) need not be locally minimized by every possible support pattern for outlier locations.

premature convergence to suboptimal supports need not disrupt trajectories towards the global solution to the extent that (4) may be obstructed.


As this formulation only considers real-valued cross spectral densities, the authors propose a multivariate covariance function by including a complex component to the cross spectral densities to cater for phase differences across channels, which they call the Cross Spectral Mixture kernel (denoted CSM).

this multivariate covariance function can be seen as the proposed MOSM model with µ i = µ j , Σ i = Σ j , θ i = θ j ∀i, j ∈ {1, . . . , m} and φ i = µ i ψ i for ψ i ∈ R n . the SM-LMC is a particular case of the proposed MOSM model, where the parameters µ i , Σ i , θ i are restricted to be same for all channels and therefore no phase shifts and no delays are allowed-unlike the MOSM example in Fig.


In fact, if f ✓ is a bijection between samples in high probability regions, then f 1 ✓ is automatically also such a bijection.

we ignore f 1 ✓ during training and only train f ✓ (x, v) to reach the target distribution p(x, v) = p d (x)p(v).


Therefore, if we consider our algorithm to take samples in blocks of size Poi(m), the probability that we have not found an r-wise collision after t blocks is at most (1 − c r ) t . Thus, the expected number of blocks until we have an r-wise collision is O r (1).

the expected number of samples is O r (m) = O r (1/ p r ) completing the proof.


However, with the submodular function for g, the average ratio of HM's k-means-score to SM's is 1.635 ± 0.038.

sM does a significantly better job optimizing the submodular loss.


The time of computing and transmission can thus overlap.

in our implementation we divide the gradients for each layer of the model into k * N pieces in a BCube(n,k) network (with N = n k ), instead of simply dividing the gradients for the entire model into k * N pieces.


Conversely, if the current entropy is below h, then we pick an assignment which is guaranteed to increase the entropy.

when h = 0, there is no adjustment made.


(Sketch) For each ℓ = 1, . . . , n, observe that T ℓ is x ℓ -tight.

we can show by induction that any cluster in P ℓ is x ℓ -tight for each ℓ = 1, . . . , n. f λ The procedure FINDPARTITION(λ) returns h λ ∈ R and P λ .


A key property of differential privacy in this context is that it composes adaptively: namely if each of the algorithms used by the analyst is differentially private, then the whole procedure will be differentially private (albeit with worse privacy parameters).

one way to avoid overfitting in the adaptive setting is to use algorithms that satisfy (sufficiently strong) guarantees of differential-privacy.


Our initialization further adds another 0.87% (42.60%).

both the diversity loss and the clustering help to improve the performance.


The components of , defined in this way, are Gaussian basis vectors slightly narrower than those in £ (this is easily seen by noting that the eigenvalue spectrum for the Toeplitz matrix £ is given by the Fourier transform, and that the square-root of the Gaussian function in the Fourier space is a Gaussian of larger width, corresponding to a smaller width in the original space).

weight vectors obtained through ARD The results of this procedure (labelled ASD/RD) on our example recording are shown in the rightmost panel of figure 1.


This negative result holds even if the distributions are independent, binary-valued, the difference is restricted to P (X 0 = 0), and, finally, for a smaller family of processes (B-processes).

for now what we can say is that there is no test for independence available that would be consistent under ergodic sampling.


In this case, note that x = X T X(X T X) −1 x = X T q.

x can be written as a linear combination of the rows of X, whose coefficients make up q, of previously experienced input vectors.


Once i ng is fixed, however, the relationships between all the other variables are conjugate in the exponential family.

we alternate our inference between the extraction-related variables {i ng , f g , l d } and the selection-related variables {y g , γ g , t gk , β gk , z n }.


2 in Appendix D. Note that unlike IPS, which divides by probabilities of whole slates, the PI estimator only divides by probabilities of actions appearing in individual slots.

the magnitude of each term of the outer summation is only O( m), whereas the IPS terms are m Ω( ) .


These functions can be shown to be δ-related as well as λ-strongly convex.

moreover, the optimum of F (w) = 1 2 (F 1 (w) + F 2 (w)) equals we see that the optimal point w * depends on the j-th column of M . Intuitively, the machines need to approximate this column, and this is the source of hardness in this setting: Machine 1 knows M but not j, yet needs to communicate to machine 2 enough information to construct its j-th column.


In the previous section, we showed the equivalence between the MHMM and an HMM with a blockdiagonal transition matrix.

it should be possible to use an HMM learning algorithm such as spectral learning for HMMs [1,2] to find the parameters of an MHMM.


The difference between the free energy and the true likelihood is given by the KL divergence between the approximate and true posterior.

the freeenergy bound is tightest in regions where q(V, Θ) is a good match to the true posterior, and loosest in regions where it is a poor match.


In particular, if the distance of q from monotone distributions is more than "/2, we declare that p is not monotone.

property 1 in Theorem 1 is satisfied, and the lemma states that Property 2 holds with probability at least 5/6.


Otherwise, the buyer is largely indifferent to being offered prices p or p , but distinguishing between the two prices is essential for the seller during exploit rounds.

we assume that the value distribution D is K-Lipschitz, which eliminates this problem: Defining F (p) ≡ Pr v∼D [v ≥ p], we assume there exists K > 0 such that This assumption is quite mild, as our Phased algorithm does not need to know K, and the dependence of the regret rate on K will be logarithmic.


Given N observations, inference of GPR involves inverting an N × N covariance matrix which requires O(N 3 ) operations and O(N 2 ) storage.

gPR for large N is infeasible in practice.


Sensors such as the spectrometer are particularly expensive to use because the robot must be maneuvered to bring the rock sample into the sensor manipulator workspace.

it is desirable that initial classifications be made using data from cheap long range sensors, such as a color camera, before final verification using expensive sensors on promising rock samples.


For example, during the ascent from vertex v k ∈ V m to the root we will compute {r The computational complexity is obtained by noting that every ascent to the root requires O(S T ) steps and along each ascent we must compute up to max(m, S T ) resistances.

the total complexity is O(m 2 + mS T ), assuming that each step of the algorithm is efficiently implemented.


We found that the Optimal policy, which corresponds to normalized reward of 1.0, uses approximately 648M simulations on Playground domain.

mCTS+NSGS performs almost as well as the Optimal policy with only 0.005% simulations compared to the Optimal policy.


If we apply a policy trained on one robot to a different robot it will fail to generalize.

in this paradigm, one would need to collect millions of examples for each task and each robot.


Θ 0 places Θ 0 in a region where the chaser Θ n τ can efficiently chase the leader Θ n+s τ in n SVGD-update steps starting from Θ 0 .


If the r(•) is non-negative, it is easy to see that this diversity function is monotone and submodular -adding a new window never hurts, and since the marginal gain is the score contribution of the new regions not covered by previous window, it is naturally diminishing.

even though this heuristic not was presented as such, the authors of [26] did in fact formulate a near-optimal greedy algorithm for maximizing a monotone submodular function.


If R/Q is large, the variance in the noisy updates Var[ λt ] is assumed large.

the filter uses a smaller step size, yielding more bias (Eq 5), but with lower overall error.


Let's look at the gradients of L q loss to see this: where f yi (x i ; ✓) 2 [0, 1] 8 i and q 2 (0, 1).

relative to CCE, L q loss weighs each sample by an additional f yi (x i ; ✓) q so that less emphasis is put on samples with weak agreement between softmax outputs and the labels, which should improve robustness against noise.


By definition of I , we know r ∈ I =⇒ V (r, y 1 ) < 0 < V (r, y 0 ) and the orientation of V then implies Γ(δ y1 ) > r > Γ(δ y0 ).

γ(δ y1 ) = γ(1) ≥ b > a ≥ Γ(δ y0 ) = γ(0), with the strict inequality since I is nonempty.


Then we propose an alternative active learning method which can also be theoretically justified for approximately correct models, but the condition on the approximate correctness of the models is weaker than that for the existing method.

the proposed method has a wider range of applications.


When T = ( Ṽ , Ẽ) is a part of a "larger" tree T = (V, E), i.e., Ṽ ⊆ V and Ẽ ⊆ E, we can treat T as a "super-node" of the tree T with value m( T ).

we have the following definition of a super-tree (note that a super-tree provides a disjoint partition of the given tree): Definition 5.


Let ǫ > 0 be given, and let (f, g) be a selective regressor chosen by Strategy 1 after observing the training sample S m . Let f be an ERM.

then, We thus have, by the definition of fǫ,x , and using the convexity of ℓ, together with Jensen's inequality, As for the other direction, if So far we have discussed the case where ǫ is given, and our objective is to find an ǫ-pointwise optimal regressor.


We would like to optimize the margin by solving the optimization problem max Introducing one additional variable ζ k per classifier, problem (1) is equivalent to: Considering the unnormalized rows W k /ζ k , we obtain the following equivalent formulation: When y = −1 and z(x i ) satisfy the margin constraint in (3), we have that the constraint holds for every sub-classifier k since y i W k x i is minimal at k = z(x i ).

when y = −1, we can enforce the constraint for all k. We can also smooth the objective into a convex, defined everywhere one by minimizing the sum of the inverse squares of the terms instead of maximizing the sum of the terms.


The Low Approximate Regret property differs from previous properties such as RVU in that it only requires that the learner's cost be close to a multiplicative approximation to the cost of the best action in hindsight.


On the other hand, since ⌘ 1  1, it follows that 1 0.

there indeed exists a non-negative integer-valued vector s 1 , such that x 1 = 2 1 s 1 .


All of these methods produce sentence representations that are passed to a supervised task and depend on a class label in order to backpropagate through the composition weights.

these methods learn highquality sentence representations but are tuned only for their respective task.


Figure 4 shows that for this range of , and ( ) 2 the deviations of the inverse square error from linearity are small.

in the regime 1 is given by the asymptotic behavior, Eq.


latent vector z) and parameter spaces are also initially aligned, since the current generator is initialized with the same parameters of the replay generator.

we can synchronize both the replay generator and current one to generate the same image by the same category c and latent vector z as inputs (see Fig.


Whereas the original ME formulation used expectations of assignment variables called responsibilities, this is inadequate for inference in the mixture of GP experts.

we directly represent the indicators, ' , and Gibbs sample for them to capture their dependencies.


Now, it can be checked that for a, b ∈ A, we have L(V, a)/ L(V, b) = i∈{2,...,m} (p i /p 1 ) SC APP (b)−SC APP (a) .

sC APP (a) ≥ SC APP (b) if and only if L(V, a) ≥ L(V, b), as required.


Their algorithm can also be formulated into the statistical query model; we use the simpler version in (3.16) for ease of presentation.

with more sophicated proof techinique, it can be shown that � s 2 /n ∧ (1/α 2 • s log d/n) is the critical threshold for asymptotically powerful detection with computational efficiency.


This means that the model focuses on learning to output the next token given the current state of the model AND the previous token.

the model represents the probability distribution of sequences in the most general form -unlike Conditional Random Fields [8] and other models that assume independence between between outputs at different time steps, given latent variable states.


Once an embedding is found based on a subset of judgments, the relative locations of the objects, at least at a coarse level, are constrained.

many other judgments (not yet collected) can be predicted from the coarse embedding, while others are still highly uncertain.


In this case, each read produced by the sequencer may correspond to a different type of bacteria.

each DNA sample (column) may contain genes (rows) from different types of bacteria, which is precisely the model that MMC describes.


. . , b n are different, the polynomial c ⊤ ν n (x) has no repeated factor.

there is a δ > 0 and a T > 0 such that for all t > T we have Dν n (x(t)) ⊤ c ≥ δ and Dν n (x(t)) ⊤ ĉ(t) ≥ δ (see proof of Theorem 3 in [9] for the latter claim).


By adding the regularization , and substituting G = X T B, we get the regularized where XDX T , and η is the positive weighting factor.

different from the existing regularizations [8,9,14] for SIR, which are constructed as deterministic terms, the manifold regularization in ( 6) is a random term that involves two data dependent variables (matrices) Q and S. it is necessary to discuss the convergence property of the proposed manifold regularization.


For η = 1 (homogeneous models) and regular graphs ∆ min = ∆ max = ∆, the sample complexity simplifies to n = Ω ∆ 2 ρ −2 (log p) 3 .

we can characterize an explicit dependence on the fraction of observed nodes ρ.


To arrive at a sparse approximation of the stimuli either one of the dictionary elements could potentially be used, but due to the high cost of non-sparseness both of them together are not likely to be chosen a given representation.

small changes in the image, as might arise due to various noise sources, might cause one of the coefficients to be preferred over the other in an essentially random fashion, potentially resulting in very different coefficient values for highly similar images.


We can also use mini-batches within the SAG iterations, and for problems with dense gradients this decreases the storage requirements of the algorithm since we only need a y k i for each mini-batch.

for example, using mini-batches of size 100 leads to a 100-fold reduction in the storage cost.


We next consider the posterior on the track meta-states: i,k (s where (18) follows from (2).

if P (s i,• ) follows a Markov chain then the form for q(s i,• ) is the same as a hidden Markov model (HMM) with emission log likelihoods i,k ∈ [R − ] N S . the meta-state posterior q(s i,• ) update is implemented using the forward-backward algorithm [21].


However, the expression of Var[ Dj ] in Lemma 3.5 also indicates that the degree of improvement diminishes quickly as N j grows.

collecting more observations is not always an efficient way to reduce the skewness.


We consider that x i is far from hyperplane k if ln(1+e β k xi ) is close to zero.

as x i moves away from hyperplane k, that proximity measure monotonically increases on one side of the hyperplane while decreasing on the other.


The green lines are the standard deviation around the ML values and the red represent three times that.

the effect of the BHS on control is captured by γ that of BDIa on reward sensitivity is captured by β as predicted.


As a consequence, the subspace spanned by the orientation lld, locally at q, is likely to contain points having the same class label as q .

when applying a nearest neighbor rule at q, we desire to stay close to q along the lld direction, because that is where it is likely to find points similar to q in terms of class posterior probabilities.


Another challenge with modeling the strategic settings is that they are invariably unsupervised.

standard learning techniques such as structured prediction that enforce global consistency constraints fall short in such settings (cf.


However, different values of k can be tested and an Op1q-estimate on k is enough for the proofs to hold.

it is possible to run Algorithm 1 Oplog nq times with different "guesses" for k and take the best of these runs.


A vector field V on the manifold is called a gradient field if there exists a function f on the manifold such that ∇f = V where ∇ is the covariant derivative on the manifold.

gradient fields are one kind of vector fields.


In case of information propagation, as bloggers discover new information, they write about it without explicitly citing the source [15].

we only observe the time when a blog gets "infected" but not where it got infected from.


Interestingly, we can show that this distribution is itself a conditional DPP, and hence can be compactly described by an N × N kernel matrix.

to complete the E-step, we simply need to construct this kernel.


Since both the KLdivergence and the U function in P1 are convex, and the constraints in F 1 are linear, P1 is a convex program.

one can apply the calculus of variations to the Lagrangian to obtain a variational extremum, followed by a dual transformation of P1.


To show this, we express the Fisher discriminant ratio f as the composition where g(u, t) = u 2 /t and H is the function The function H is linear (as a mapping from µ x , µ y , Σ x , and Σ y into R 2 ), and the function g is convex (provided t > 0, which holds here).

the composition f is a convex function of µ x , µ y , Σ x , and Σ y .


In this work, we aim towards a more modest goal and search for a locally optimal θ.

the main problem that we are trying to solve in this paper is how to calculate the gradients of the SRP's and DRP's objective functions We are interested in non-trivial cases in which the gradients cannot be calculated analytically.


In the view of the long-term controller, Algorithm 1 is the same as the UCB policy in [2], and its regret has been proven to be O .

we focus on the compensation upper bound, which is shown in Theorem 2.


We employ RJ-MCMC, which extends the capability of conventional MCMC to allow sequential samples of different dimensions to be drawn (Green, 2003).

a posterior distribution is not limited to coefficients and other parameters of a fixed basis expansion, but can represent a changing number of basis functions, as required by the description of Lévy processes described in the previous section.


The idea is to design hash functions that deliberately collide symmetric entries, i.e., (i, j, k), (j, i, k), etc.

we only need to consider entries T ijk with i ≤ j ≤ k when building tensor sketches.


The only difference is the factor of n in the denominator in the case of ARD, but this can be offset by an appropriate rescaling of the FOCUSS λ trade-off parameter (analogous to Σ ).

aRD can be viewed in some sense as taking the recursive FOCUSS update rules and including the sLORETA normalization that, among other things, allows for depth bias compensation.


the output consists of states in the order of their first-visit; the underlying entire walk trajectory is hidden.


Geodesics on D are realized as exponential maps from tangent spaces to D. The intricate geometry of D disallows explicit analytic expressions.

we adopt an iterative strategy, where in each step, we first flow infinitesimally in the prescribed tangent direction in the space L 2 , and then project the end point of the path to D. Next, we parallel transport the velocity vector to the new point by projecting the previous velocity orthogonally onto the tangent space of D at the new point.


Throughout the paper, we use capital letters to refer to random quantities and lower case letters to refer to deterministic quantities.

we use Z to refer to the decision randomly assigned by the (unknown) historical policy and z to refer to a specific action.


[21] showed that for bounded losses the Bernstein condition implies stochastic mixability.

when starting from a Bernstein condition, Theorem 5 offers a different path to fast rates.


At the same time, the distributed representations help generalization, because they can represent graded similarities between clusters and fine differences between objects in the same cluster.

given a set of relations, we expect the model to find both meaningful clusters of objects and relations, as well as predictive distributed representations.


In this step of the proof we generalize the technique of [19] to multi-kernel classes.

first we note that, since the (centered) covariance operator E φ m (x) ⊗ φ m (x) is also a self-adjoint Hilbert-Schmidt operator on H m , there exists an eigendecomposition is an orthogonal basis of H m . Furthermore, the no-correlation assumption (U) entails E φ l (x) ⊗ φ m (x) = 0 for all l = m. for all j and m, Let now h 1 , .


First, we do not know how the energy changes with the varying locations of ROIs.

techniques like Newton's method cannot be used.


More generally, experience replay mechanisms that compose well with safe interruptibility could allow to compensate for the extra amount of exploration needed by safely interruptible learning by being more efficient with data.

they are critical to make these techniques practical.


However, with the advent of deep scoring functions s(x, y; w), learning is no longer convex.

it is worthwhile to rethink the architecture of structured prediction models, and consider models that map inputs x to outputs y directly without explicitly maximizing a score function.


The marginal of λ over the initial state is the initial state distribution, so the second term of left hand side, s v(s)η(s), converges to E π i ,π −i [r i ], which is the second term of right hand side.

the left hand side and right hand side of Equation ( 8) are the same as t → ∞.


Unfortunately, thus far there is no evidence or mathematical argument that DNN training is one of these special cases.

even though empirically it has been successfully applied to DNN training [34; 41], it still lacks of convergence guarantee.


Even though this pairwise correlation clustering takes a rich pairwise feature vector and a trained parameter vector (which will be presented later), it often produces incorrectly predicted segments due to the segment boundary ambiguities caused by limited pairwise relations of neighboring superpixels (see Figure 2).

to incorporate higher-order relations, we develop higher-order correlation clustering by generalizing the correlation clustering over a hypergraph.


Since P contains at most B solutions for each possible size {0, . . . , 2k − 1}, |P | ≤ 2Bk.

j max can increase by at least 1 in one iteration with probability at least , where 1 |P | is the probability of selecting x in line 3 of Algorithm 3 due to uniform selection and 1 n (1 − 1 n ) n−1 is the probability of flipping only a specific bit of x in line 4.


In this semi-autonomous local government, county managers act as executives, and the individual departments are synonymous with the individual departments and agencies in, for instance, the U.S. federal government.

not only does this email data set offer a view into the communication patterns of the managers of New Hanover County, but analyses of it also serve as case studies in modeling inter-agency communications in the U.S. federal government administration.


We refer to the spectrum in such neighbourhood as the local spectrum and define it through the Fourier transform as where since the max(|f |) is finite a.s. due to the GP prior.

the local spectrum F c {f (t)} exists and it is finite.


L(x, p, q) lower-bounds the marginal log-likelihood for any choice of q, and the bound is tight when q is the true posterior p(z|x).

the joint optimum of L(x, p, q) in p and q is the MLE.


This is true especially for components with small variance.

after removing this noise, a finite number of components suffice to represent the optimal decision boundary.


For the 148/176 images with faces, SM+VJ was better than SM alone for 144/148 images (p < 10 −29 ), whereas VJ alone (equal to the face conspicuity map) was better than SM alone for 83/148 images, a fraction that fails to reach significance.

although the face conspicuity map was surprisingly predictive on its own, fixation predictions were much better when it was combined with the full saliency model.


The underlying hypothesis is that the true/latent score matrix is lowrank and we observe its partial, noisy version.

matrix completion algorithms are used for learning, cf.


It thus avoids repeated applications of Bayes rule, which is expensive for all but the simplest priors over the MDP.

our algorithm is particularly well suited to support planning in domains with richly structured prior knowledge -a critical requirement for applications of Bayesian reinforcement learning to large problems.


If sample size n > c(max i w 2 (C i ∩ S p−1 ) + log k)/ρ 2 , then we have with probability at least for constants c, C > 0 that depend on sub-Gaussian norms |||Z||| φ2 and |||ω||| φ2 .

assuming the SC condition in ( 9) is satisfied, the sample complexity and error bound of the estimator depends on the largest Gaussian width, rather than the sum of Gaussian widths.


However, an optimization problem with a BMI constraint is NP-hard [34].

we relax the optimization problem to that of a singular value constraint as in eq.


At the same time, the intra-personal variations decrease in both the diversity and magnitude, as shown by the green curve in the right.

both the inter-and intra-personal variations changes in a direction that makes face verification easier.


While [25] considers generic methods for handling this nonstationarity, we focus on the special case in which all A i are policy gradient methods.

theorem 3 of [25] states that the policy gradient of M can be decomposed into the policy gradients for all of the CoMDPs, M i : if each coagent computes and follows the policy gradient based on the local environment that it sees, the coagent network will follow its policy gradient on M . Thomas and Barto [25] also show that the value functions for M and all the CoMDPs are the same for all s t , if the additional state components of M i are drawn according to the modular policy: The state-value based TD error is therefore the same as well: This means that, if the coagents require δ t , we can maintain a global critic, C, that keeps an estimate of V θ M , which can be used to replace every C i by computing δ t and broadcasting it to each A i .


For a Lipschitz continuous function φ : R → R, φ ∞ ≤ 1, and constants r, s, t > 0 let F Θ ⊂ C (X), Θ = (φ, r, s, t), be a set of ridge-wave functions defined on the set X, From this definition, it follows that for all g ∈ F Θ it holds g ∞ ≤ r. As a ridge-wave function g ∈ F Θ is bounded and Lipschitz continuous, it is also square integrable in the measure ρ and g ∈ L 2 ρ (X).

f Θ is a subset of the Hilbert space of square integrable functions defined on X with respect to the probability measure ρ, i.e., F Θ ⊂ L 2 ρ (X).


We introduce a new notion of data-driven distributional robustness: the uncertain problem parameter is a probability distribution from which samples can be observed.

it is natural to take as the uncertainty set a set of functions, such as a norm ball around an estimated probability distribution.


We will see examples of problems requiring other likelihoods in Section 4.

to some extent, BWGP can be regarded as likelihood learning tool.


Even given a fixed uncertainty about where in space an object will end up, the odds of another object appearing therein is greater, again limiting our ability to infer γ.

both increasing velocity variance and decreasing spatial variance will make tracking harder, and to achieve a particular level of performance the two must trade off.


Besides encoding dependencies, in a variety of real-world applications we often want consistent solutions that are physically plausible, e.g., when jointly reasoning about multiple tasks or when enforcing geometric constraints in 3D indoor scene understanding applications [18].

various methods [22,13,16,12,1] enforce consistency structure during inference by imposing constraints on the feasible instances.


Given that all the weights w(u), u 2 U in (3) are non-negative, we can construct a concave upper bound for the multilinear extension F (x) using the following Lemma.

proofs can be found in the Appendix A. given a weighted coverage function f with F (x) represented as in (3), we can define and conclude using Lemma 1 that (1 1/e) F (x)  F (x)  F (x), as desired.


all the backward states may be concentrated around the zero vector.

we additionally train the backward network to predict the output variables in reverse (see Figure 1): −D KL q φ (z t |x 1:T ) p θ (z t |x 1:t , z 1:t−1 ) .


(i) Since D is ultra-metric, √ D is 2 embeddable by lemma 1, and D c is negative (semi)definite by lemma (2).

s c = − 1 2 D c is positive (semi)definite.


However, the features' spatial distribution is not taken into account.

the optimal margin hyperplane is often scattered and lacks spatial coherence, making its anatomical interpretation difficult.


In addition, EM started with random initialisation still yielded worse performance than with PLDSID initialisation even after 50 iterations (see Figure 2).

even with a conservative estimate, PLDSID initialisation reduces computational cost by at least a factor of 50 compared to random initialisation.


Instead of requiring relevance labels, we use the pseudoinverse estimator to convert page-level reward into per-slot reward components-the estimates of φ x (j, a)-and these become targets for regression.

the pseudoinverse estimator enables pointwise L2R to optimize whole-page metrics even without relevance labels.


At the other end of the spectrum, if we are dealing with a deterministic bug 3 , false positives should have a probability of zero: if the buggy feature is observed to be true, then the program has to crash; if the program did not crash, then the bug must not have occurred.

for a deterministic bug, any false positives during the training process should incur a much larger penalty compared to any false negatives.


6 can be started at the leaf nodes and carried out in a bottom up manner.

the split node parameters can be learned by standard back-propagation.


However, Mira employs a different approach in which there is a single input instance (instead of the set X t ) and constructs multiple predictors (instead of a single vector ω).

mira employs a larger hypothesis space which is more difficult to learn in online settings.


if r R = 0.01, nearly a million samples are required in just a three dimensional space.

a sample efficient alternative is needed for the algorithmic assurance.


Conversely, the sparse long-range contacts are the most informative and also the most difficult to predict.

as in the CASP experiments, we focus primarily on long-range contact prediction for performance assessment.


It is proven in [6] that lim K→∞ V (D (K) ; Θ) is the expected sum of discounted rewards by following the RPR policy parameterized by Θ for an infinite number of steps.

the RPR resulting from maximization of V (D (K) ; Θ) approaches the optimal as K is large (assuming |Z| is appropriate).


Algorithm 2: Reduction from from pricing with switching cost to strategic buyers Input:T, A % A is an algorithm with bounded regret for strategic buyers; Output:p The above section concluded that switching cost for pricing may be reduced to pricing with strategic buyers.

our next step would be to show that we can produce a sequence of non-strategic buyers with high switching cost regret.


3) as a hyperparameter, since it plays a similar role to C in determining the posterior and evidence.

for the (D = 2) examples considered here, there are 12 hyperparameters, including scalars σ 2 and ρ, two hyperparameters each for ν s and ν f , and three each for symmetric matrices M s and M f . Note that although the conditional ALD prior over k|θ assigns high prior probability to smooth and sparse RFs for some settings of θ, for other settings (i.e., where M s and M f describe elliptical regions large enough to cover the entire RF) the conditional prior corresponds to a simple ridge prior and imposes no such structure.


Specifically, once EP has converged, the gradient of the natural parameters of the gj 's with respect to these hyper-parameters is zero [21].

the gradient of log Z with respect to σ 2 , ρ 2 , γ 2 and P can be computed in terms of the gradient of the exact factors.


That is, there is an extra additive error term that depends on the ratio of lies to number of learning rounds.

if no lies are told, then there is no additive error.


During the pass over the corpus the list of equivalence sets is updated continuously; the identification of new significant patterns is done using thecurrent equivalence sets (Figure 3(d)).

as the algorithm processes more and more text, it "bootstraps" itself and enriches the RDS graph structure with new SPs and their accompanying equivalence sets.


For a review of properties of Bregman divergences, see Chapter 11.2 in [11], or Appendix A in [2].

by definition of the Bregman divergence, we have if the dual variable Z obeys the dynamics Ż = −∇f (X), then and by the same argument as in the gradient descent ODE, V is a Lyapunov function and The mirror descent ODE system can be summarized by Note that since ∇ψ * maps into X , X(t) = ∇ψ * (Z(t)) remains in X .


First, it is worth emphasizing once more that the equality constrained program ( 8) is a nonconvex program.

although we overcome the difficulties of combinatorial optimization, our formulation still inherits the difficulties associated with nonconvex optimization.


We hypothesize that this is why this approach is not studied in the literature (as far as we know), a notable exception being the work of Baird [5].

our main contribution, presented in Sec.


We used the ratings of all the rest of the people who viewed enough movies as features.

the goal is to learn to predict the "taste" of a random user using the user's past ratings as a feedback and the ratings of fellow viewers as features.


Otherwise the intermediate result needs to be stored in high precision, before reading the new exponent and quantizing the result, which would negate much of the potential savings in hardware.

intelligent management of the exponents is required.


For a fixed v, problem (4) has the same form as problem (3).

the optimization for self-paced learning is as easy (or as difficult) as the original parameter learning algorithm.


. . , (i * , d)} for some i * , there is a consistent function in F with error strictly greater than on D i * , namely, the function that maps (i, j) to 1 if and only if i = i * and (i * , j) is not in the sample set.

to achieve uniform convergence, at least d/2 elements from X \ {⊥} must be drawn from each distribution.


While execution traces can provide highly detailed guidance for a program synthesis method, they are more difficult to obtain than more basic forms of specification such as input/output pairs.

we use the insight that we can split the process into two parts: infer traces from input/output examples, then infer programs from traces.


Inference in the IWSGD is same as the standard dropout; input activations to each dropout layer are scaled based on dropout probability, rather than taking a subset of activations stochastically.

compared to the standard dropout, neither additional sampling nor computation is required during inference.


Based on the open-world assumption, if a web page does not link to another, it does not mean that they are unrelated.

we employ the likelihood p(r i,n |t i,n ) = Φ(r i,n t i,n − b) for those observed links r i,n = 1, where b is an offset.


( 1) as where we have conditioned ρ(y, x) = ρ(y|x)ρ X (x) and used the linearity of the integral and the inner product.

any function f * : X → Y minimizing the above functional must satisfy the following condition where we have introduced the function g * : X → H that maps each point x ∈ X to the conditional expectation of ψ(y) given x.


We recall that when P = I ∈ R p×p , our norm is equal to the 1 -norm, and we want to understand its behavior when P departs from the identity.

we compute a second order approximation of our norm around the Lasso: we add a small perturbation ∆ ∈ S p to the identity matrix, and using Prop.


The marginalization condition is important because for some stochastic differential equations there exists a ρ that makes (9) hold even though the marginalized distribution is not the target distribution.

care must be taken when designing the dynamics.


Generalizing Theorem 1, following a coordinate descent principle, now we present an alternating minimization (EM-style) procedure for P2: Step 1: keep p(z) fixed, infer p(w) by solving the following problem: where F 1 = {p(w) : p(w)E p(z) [∆F i (y, z; w) − ∆ i (y)] dw ≥ −ξ i , ∀i, ∀y}, which is a generalized version of F 1 with hidden variables.

we can apply the same convex optimization techniques as being used for solving the problem P1.


The trust region radius k is positive almost surely by construction.

k is bounded above almost surely by the constant sup L. Let the constant c , sup L 0 .


is a diagonal matrix with at most K nonzeros elements.

w ⇤ will have rank at most K, which means that we do not need the rank constraint on W. This leads to the following lemma.


Initialization Without proper initialization, variational methods can be easily trapped into local optima.

using pure randomization methods, one cannot rely on a single result, but has to run the algorithm multiple times and then either pick the best result [12] or average the results [11].


Note that if any eigenvalue has multiplicity greater than one then the corresponding eigenspace is well defined but not the individual eigenfunctions.

assumption 2 enables us to compare how close each ψ i is to some other function in L 2 (X , P X ) in L 2 (X , P X ) norm sense.


Problem (1) thus imposes the constraint that the solution lies at one coordinate of the lift curve, just as problems (2) and (3) place the constraint that the solution lies at one coordinate of the ROC curve.

a solution to problems (2) or (3) will also serve as a solution to (1).


They conclude from a study of a number of trials, including IST, that heparin provides little therapeutic benefit, with the caveat that the trial evidence base is lacking for the highest-risk patients where heparin may be of benefit.

our robust method appropriately treats those, and only those, who stand to benefit from the more aggressive treatment regime.


However, the enormous amount of content in high-resolution images makes generative learning difficult [6,7].

generative models have found most success in learning to model small patches of natural images and objects: Zoran and Weiss [4] learned a mixture of Gaussians model over 8×8 image patches; Salakhutdinov and Hinton [2] used 64×64 centered and uncluttered stereo images of toy objects on a clear background; Tang et al.


Interactions are then induced only by the interactions of highaffinity active motifs in the two proteins.

in our example, the motif d in p 2 is inactive, and hence does not lead to an interaction between p 2 and p 4 , despite the affinity between the motif pair c, d. We note that Deng et al.


Note that not only is the covariance matrix K x an explicit function of X, but also fi and W i are also implicitly functions of X -when X changes, the optimum of the posterior fi changes as well.

log q(y i |X) contains an implicit function of X which does not allow for a straightforward closed-form gradient expression.


Note that the probabilities in the optimal offline distribution may be complex functions of all the entries in the matrix.

regardless of computational complexity, the optimal distribution might be impossible to compute in the streaming model.


However, the receptive fields size of a nine-layer dilated CNN is 512, and is insufficient to cover the sequence length of 784.

we added one more layer to the dilated CNN, which enlarges its receptive field size to 1,024.


Suppose we choose γ = 2 and η = c/σ * 1 for any c ≤ 1/36.

there exist constants c 1 , c 2 such that when α ≤ c 1 /(κ 2 µr), given any using proper initialization and step size, the gradient iteration converges at a linear rate with a constant contraction factor 1 − O(1/κ).


For b k it is enough for our needs to consider an upper bound; since we have b k+1 = b k + k/2 + 3/2, one can easily verify that k 2 /2 is an upper bound for k ≥ 3.

in order to provide an upper bound on the distance form the center grid u we would like to find an initial d such that and d is minimal.


If the budget allocated to context j is sufficient, then the agent may take action k to maximize the expected reward.

to approximate the oracle, the ALP algorithm in this case needs to solve an LP problem accounting for all context-action pairs with an additional constraint that only one action can be taken under each context.


Since x q is the primary eigenvector for Z q , we have [10]: .

to a first degree of approximation, and thus x Q is approximately an eigenvector for Z Q .


Structurally, their EM algorithm is similar to ours, but they do not account for the normalization constants for different mixture components.

their approach is inconsistent with the probabilistic formulation.


1 correspond to the directions exploited by various first-order methods.

we hypothesise that to identify such a basis, one need actually only perform SVD on a matrix of stacked class-score gradients 5 .


On the other hand, when dealing with real data where observed entries are corrupted by noise, exact reconstruction is infeasible or comes at the price of losing the sparsity of the solution, which is undesired.

to deal with noisy incomplete data, we consider approximate reconstruction by selecting ρ(•) = ρ a (•).


When the upper and lower Minkowski dimensions coincide-which is the case in many natural spaces, such as normed vector spaces-the latter bound matches a lower bound of [10] that holds even when there are no switching costs.

a surprising implication of our result is that in infinite actions spaces (of bounded Minkowski dimension), adding movement costs do not add to the complexity of the MAB problem!


Given observations O i and O j of differing length (and, thus, different underlying graphs in the case of dynamic graphical models), a kernel-based classifier over these instances is trained using U Oi and U Oj in the feature-space.

a similarity measure is learned in the gradient space, under the intuition that objects which induce similar likelihoods will induce similar gradients.


Then, using Jensen's inequality for convex functions, we obtain a lower bound on the objective function: It can be shown that, provided both functions are continuous and differentiable, local maxima of the "free-energy" F with respect to all of its arguments correspond, in their optimal values of θ, to local maxima of L [5].

any hill-climbing technique applied to the freeenergy functional can be used to find parameters that maximise the objective.


Introducing more structure into a distribution typically reduces its entropy.

we expect that the entropy can vary considerably for different dependence strengths, even though the correlation is always zero.


Suppose two data points u 1 and u 2 are independent (or less strictly, uncorrelated to the second order), it is easy to show that the variance of CRS is always smaller: even if we ignore the data sparsity.

cRS will be much better for estimating inner products in nearly independent data.


The asymptotics mn−1(h) as n → ∞ imply that the mixture distribution Ln,K+ in (10) converges to the true Gaussian mixture distribution p T of (9).

for any small δ, we expect D(p T Ln,K+ ) ≤ δ for all n ≥ N , validating the assumption of Theorem 2.


In ZO-SVRG (Algorithm 2), the total number of gradient evaluations is given by nS + bT , where T = mS.

by fixing the number of iterations T , the function query complexity of ZO-SVRG using the studied estimators is then given by O(nS + bT ), O(q(nS + bT )) and O(d(nS + bT )), respectively.


We observe that sometimes these definitions lead to significantly different bounds.

this study advocates for the use of the error-region definition, even though other definitions, in other contexts with context-dependent assumptions, may coincide with the error-region definition.


1(a), our first problem is to select a set of image patches that depict the same visual part of these objects.

an object part is regarded as a collection of image patches e.g., Fig.


The general goal of DPS is to find an optimal policy parameter θ * ∈ Θ that maximizes the expected reward defined by where In general, the direct maximization of objective function (2) forces us to solve a non-convex optimization problem with a high non-linearity.


2 we cannot expect to improve the dependency on Δ min .

on the one hand, we need to use the tighter bounds in Eq.


If we solve for N in 7N < N 2 , we get N > 7.

the parameterization (3) cannot represent all unitary matrices for dimension N > 7.


The success of such algorithms depends heavily on the choice of the metric, but this choice is generally not treated as part of the learning problem.

time-consuming manual feature selection and weighting is often a necessary precursor to the use of spectral methods.


It turns out that the problem we consider is equivalent to that of recovery of a function over a symmetric group using the first order Fourier coefficients.

our problem is thematically related to the recovery of functions over non-commutative groups using a limited set of Fourier coefficients.


Thus for d ≥ 15 log(4n/δ), we have ≤ 3/4 and as a result 2 /4 − 3 /6 ≥ 2 /8 and d ≥ log(4n/δ) 2 /4− 3 /6 .

from Proposition 1 in [14], for all From this result, we deduce that with probability Proof of Theorem 1.


Yet, we can say that if, among the tested elements, there is one that gives a non-zero change in s I, then one of such elements will be the one that gives the maximal change in s I (provided, of course, that we have enough data for the estimates s I to be close enough to the theoretical values s I).

we keep each split that arises from such a maximal-change element, resulting in O(N 2 ) candidate splits for the case of 2 clusters.


Although the field of robust regression is well established, it has not considered computational complexity analysis to be one of its central concerns.

none of the standard regression estimators in the literature are both robust and tractable, even in a weak sense: it has been shown that standard robust regression formulations with non-zero breakdown are NP-hard [1,2], while any estimator based on minimizing a convex loss cannot guarantee bounded response to even a single leverage point [3] (definitions given below).


However, depending on the network's traffic, each node may use different routes to communicate at different times.

the same column in X may contain measurements from different low-rank matrices.


The features at "pixel" (i, j) correspond to the weights of a (complete) bipartite undirected edgeweighted graph K 21,21 with 21 + 21 vertices, with the first disjoint set of 21 vertices representing the 21 amino acid types at position i, the second set representing the 21 amino acid types at position j, and the edge weights representing co-evolution of the respective variants.

b = (J i,j,k,l ) k,l∈{1,...,21} is the biadjacency matrix of this graph, i.e.


However, for all these versions of the PAC-Bayes theorem, the prior P must be defined without reference to the training data.

these theorems cannot be applied to the sample-compression setting where classifiers are partly described by a subset of the training data (as for the case of the SCM).


It is easy to see that if 7 or A changes sign in (9), also changes sign.

the spectrum of the problem (9) has paired positive and negative values between W V and V . Cross-linguistic retrieval with KCCA.


Due to the invertibility of <I> we have [<I>] = ~N.

the optimal solution V* is exactly representable because V* E [<I>].


We proved R T (I, a) ≤ 0 for T 0 ≤ T ≤ T 0 + m, so R T + (I, a) = 0.

for all We can improve this approach significantly by not requiring knowledge beforehand of exactly how many iterations can be skipped.


Note that the steps performed by Algorithm 1 on the rounds indicated in S are precisely the steps that the standard EXP3 algorithm would perform if it were presented with the loss sequence ( t ) t∈S .

theorem 1 guarantees that Taking expectations on both sides of the above and noting that E[|S|] ≤ T /4 proves the lemma.


First, we will upper-bound (in the Loewner ordering sense) the matrices Σ j in Algorithm 2.

since ∥f xj (y)∥ 2 ≤ r for all y ∈ Ω j and since µ j in Algorithm 1 is a convex combination of f xj (y), the outer-product terms in the update for Σ j satisfy Σ j ≼ F(α 1 , .


If one interpolates the look-up table with the standard multilinear interpolation (by choosing the multilinear interpolation kernel φ), then the fitted surface forms a multilinear polynomial over [0, 1] S , and is thus ceterus paribus linear in each feature (but nonlinear overall), and thus the lattice layer does not affect convexity/concavity (note this only holds because W acts as a feature selector, (see Appendix B) for more on that).

what is needed to guarantee that f (x; β, θ) is convex/concave with respect to the dth input is that each of the calibrators {c β d (x[d])} is convex/concave, and each of the T look-up tables is monotonically increasing with respect to x [d].


10) include all possible distributions, the maximized entropy (variance for quadratic loss) and thus the value of F θ would be infinity.

given a parameter ρ, we restrict the second moment of distributions in and then apply (10).


We cannot expect the error reduction values to be very high, partly because our images are highly complex natural scenes, and also because some noise was present in both the range and intensity images.

it is difficult to assess how much of the remaining error could be recovered by a superior algorithm, and how much is simply due to sensor noise.


is convex for all y ∈ P, then the inner risk L(p, .)

is also convex for every distribution p ∈ P. This implies that M (p, δ) is a convex subset of R n . if is r-calibrated, then Theorem 2 implies that A r (p) = arg sort(M (p, δ)) is a set of rankings induced by a convex set of R n . The following theorem presents a condition that the set A r (p) must satisfy if it is generated by a convex set of scores: if there exists at least one pair of items (i, j) which are inverted in two rankings of A r (p), then i and j are "indifferent" in A r (p): Theorem 3.


In the monochromatic case, | | = |ℛ| = , = ℛ = and the degree of bichromaticity = 1 since the query and the reference tree are the same.


In addition, p 2 = 1 − p and µ 2 = β 4 w 2 = ( 1−pα 2 1−p ) 2 p 2 α 4 η 2 ≤ p 2 α 2 σ 2 4(1−p) 2 .

the oracle risk is for the active learner, with probability 1 − δ, In contrast, consider any passive learner that receives m labeled examples and outputs a predictor ŵ.


In lifted decomposition, the MLN is partitioned into a set of MLNs that are not only identical (up to a renaming) but also disjoint in the sense that they do not share any ground atoms.

unlike the propositional procedure which creates n disjoint MLNs and searches over each, the lifted procedure searches over just one of the n MLNs (since they are identical).


In our application, any two citations may or may not refer to the same paper.

for citations C 1 and C 2 , there is uncertainty as to whether the corresponding papers P 1 and P 2 are in fact the same object.


Smola and Schölkopf [8] state in section 4.2 of their paper that their algorithm 2 finds a low rank approximation of the kernel matrix such that it minimises the Frobenius norm X− X 2 Frob = tr{K− K} where X is the low rank approximation of X.

we need to prove that Algorithm 1 also minimises this norm.


For example, replacing partial stochastic transitivity by interval stochastic transitivity (now a condition on quadruples instead of triplets) leads to a characterization of interval orders, a subclass of strict partial orders; a partial order Q on [M ] 2 is called an interval order if each i ∈ [M ] can be associated with an interval Our main theoretical results below state that thresholding (4) yields a strict partial order relation Q, both for the PL and the Mallows model.

we can guarantee that a strict partial order relation can be predicted by simple thresholding, and without the need for any further reparation.


We simulated (i) by setting ∆t stim = 5ms, thus producing larger and more infrequent jumps in discrete OVs than with ∆t stim = 1ms (default).

ϑ(t) grows more slowly (delayed build up of inhibition), and the peak occurs later (t max ≡ t c − t = 10ms with everything else identical with figure 1b).


Noting that the statistics vectors appear only as an inner product in (11), we apply the "kernel trick" to obtain where the kernel function K(u, v) = Φ(u), Φ(v) defines the inner product of some high-dimensional mapping Φ(•) of its inputs.

4 In particular, choosing a Gaussian (RBF) kernel, K(u, v) := exp(−γ||u − v|| 2 ), gives a non-parametric representation for g: g(u ki ) is a weighted combination of the values {α jk x (j) i }, where the weights depend exponentially on the squared 2 -distance of u ki to each of the statistics vectors {u  To validate our method, we evaluated its ability to learn parameter functions on a variety of email and webpage classification tasks in which the number of classes, K, was large (K = 10), and the number of number of training examples per class, m/K, was small (m/K = 2).


If we select t satisfying −(v + 1)/2 = 1/(1 − t) and denote, then by some simple but tedious calculation (included in the supplementary material) where λ = 2Ψ (t − 1)vσ and gt = Ψ − 1 t − 1 .

we work with the Student's-t prior in our setting: Here, the degree of freedom for Student's-t distribution is chosen such that it also belongs to the exp t family, which in turn yields v = (3 − t)/(t − 1).


Each row i of the parameter matrix B represents the degrees to which the entropy estimates on each projection contribute to the entropy estimator of point x i .

the sum over each row of B is 1, and the regularization penalty applies to the number of non-zero columns in B. subject to The problem with this optimization is that it is not convex.


However, we weren't able to finish in one week.

we considered a smaller dataset, by training and testing with only 40 images from each category.


Note that this measure cannot be used to compare variabilities across feature-sets with different number of features.

we cannot directly compare contribution of variabilities in time and frequency domains.


Given K multi , this is simply a standard GP.

the complexity still grows cubically in the total number of observations.


NPB hierarchical clustering models are an important regime of such models, and have been shown to have superior performance to alternative models in many domains [8].

further advances in the applicability of these models is important.


Hence, any predictor w can be written as w = (w i,j ) (i,j)∈G where w i,j is w's weight on φ φ φ i,j (x, y i , y j ).

for any w and any (x, y), we have where we denote by F i,j (w i,j , x, y i , y j ) = w i,j , φ φ φ i,j (x, y i , y j ) the score of labeling the edge (i, j) by (y i , y j ) given input x.


3) from the mean parameters {r (i) } is not straightforward in the general case since Z(η) is intractable.

it is not immediately clear how a DDC representation can be used for learning.


Choose the step size γ k ≡ γ = 2c 2τ +1 for arbitrary fixed 0 < c < 1.

result: we can choose ε > 0 to obtain So we have that the smallest gradient obtained by step k decays faster than O(1/ √ k).


Thus, if H 0 is the set of all stationary ergodic distributions on pairs, we have found a distribution ρ ∈ cl H 0 such that W ρ (H 0 ) = 0.

there is no consistent test that could provide a given level of confidence under H 0 , even if only asymptotic consistency is required under H 1 .


We model X via a structural equation model (SEM) defined by a weighted adjacency matrix W 2 R d⇥d .

instead of operating on the discrete space D, we will operate on R d⇥d , the continuous space of d ⇥ d real matrices.


These events were selected on the basis of representing transitions between different task phases.

we hypothesized that such events should correlate with changes in the relative strengths of different influences on visual attention, and that we should be able to detect such changes using the previously described models as diagnostic after annotating the video clips with the times of each event of interest, we subsequently aligned further analyses on a temporal window of -5s/+5s around each event (shaded background regions in Figure 2, rows b-e).


Furthermore, model uncertainty is easily quantified through the BNN's output variance.

we can scale to larger domains than previously possible.


First, if D max (S, k min ) < D(S * ), we know that no subregion of S can be optimal; we can prune the region completely, and not search its (gridded or non-gridded) children.

second, we can show that (for 0 < k < n) any region of size 2 k + 1 or less is contained entirely in an odd gridded region of size 3 2 × 2 k . if D max (G, 2 n−1 + 2) < D(S * ) for the entire grid G, any optimal non-gridded region must be contained in an odd gridded region.


Notice that the objective E X [1/(1 − P (x))] is already a barrier at P (x) = 1.

to enforce the lower bound (2), we modify the objective to where µ is a parameter chosen momentarily to ensure P (x) ≥ P min,m for all x ∈ D m . the modified goal is to minimize (7) over non-negative P subject only to (1).


Independent (additive) rewards: R( s Interactions between agents are given in terms of a set of dependencies between certain state-action transitions for one agent, and events featuring transitions involving the other agent.

if a history contains one of the primitive events from the latter set, this can have some direct effect upon the transition-model for the first agent, introducing probabilistic transition-dependencies.


The associated residual Jacobian matrix J can be given in the block-angular form below left, and thus the (full) Hessian matrix H has the n × n sparse block arrow form below right (× denotes some non-zero block) as well as the GN-Hessian J T J: Here in  k have no effect on r k ], and F vertical B k blocks corresponding to the n B hidden parameters " B that contribute to minimizing all the residuals r k (k=1, • • • , F ) evaluated at all F terminal nodes.

the posed problem is overdetermined when "m > n" (namely, "d > CA + 1 F nB") holds.


Unlike the diffusion kernels of [7], the kernels we investigate here are over continuous parameter spaces even in the case where the underlying data is discrete.

some of the machinery that has been developed for analyzing the generalization performance of kernel machines can be applied in our setting.


advocate instead using a trust region approach, whereby the update is scaled down to modify the policy distribution (in terms of KL divergence) by at most a specified amount.

we adopt the trust region formulation of K-FAC introduced by [2], choosing the effective step size η to be min(η max , 2δ ∆θ F ∆θ ), where the learning rate η max and trust region radius δ are hyperparameters.


. . , Z t+m+k ) have the same distribution.

the index t or time, does not affect the distribution of a variable Z t in a stationary sequence (note that this does not imply independence).


Secondly, it is infinitely exchangeable: given a genealogy drawn from an n-coalescent, the genealogy of any m contemporary individuals alive at time t ≤ 0 embedded within the genealogy is a draw from the m-coalescent.

taking n → ∞, there is a distribution over genealogies of a countably infinite population for which the marginal distribution of the genealogy of any n individuals gives the n-coalescent.


We have For the proof of the above result and for the general algorithm (when d ≥ r), see [9].

if we have a general product distribution mixture over nodes in V , we can learn the parameters by performing the above spectral decomposition over different triplets {u, v, w}.


The complexity of solving a discounted MDP with K states, each one connected to M next states, is O(KM), as is the complexity of computing the influences.

the overall complexity of this algorithm is O(NKM).


While stochastic SPML does not scale with the size of the input graph, evaluating distances using a full M matrix requires O(d 2 ) work.

for high-dimensional data, one approach is to use principal component analysis or random projections to first reduce dimensionality.


Within each game, subjects learned to play three game levels, and during eye tracking, each subject played each game level twice.

in total, our recorded data set consists of video frames and eye tracking data from 60 clips (5 subjects × 2 games per subject × 3 levels per game × 2 clips per level) covering 4.7 hours.


One of the key properties of circulant matrices is that they all share the same eigenvectors.

we can multiply them in any arbitrary order and obtain the same result.


In other words, the number of machines M has dual effect on the lower bound: with more machines, local data size becomes smaller which reduces the accuracy of local voting, however, it also leads to more copies of local votes and thus increase the reliability of global voting.

in terms of accuracy, there should be an optimal number of machines given a fixed-size training data.


From the lemma we see that the quantity mi / √ ṽi = m i / √ v i is invariant to a.

the behavior of the algorithm in general, and its updates and mistakes in particular, are independent to the choice of a.


Because the pool of available modules is shared between processing stages (or time steps), modules can be used at multiple locations in the network.

the algorithm can learn to share parameters dynamically depending on the context.


This is because MMV prox is a first-order method, which has a slower convergence rate than the second-order method SOCP.

there is a tradeoff between scalability and convergence rate.


The spike counts were uncorrelated but had a strong dependence.

relying solely on the correlation coefficient would lead to an oversight of such dependencies.


the entries of M , hence they are also polynomial w.r.

the coefficients of the characteristic polynomial are continuously dependent on (c 1 , • • • , c n ).


Hence, we have where a = P n i=1 a i /n, s = p P n i=1 a 2 i /n a 2 .

for SGD to be stable around x = 0, we not only need |1 ⌘a|  1 (stability condition of GD), but also (1 ⌘a) 2 + ⌘ 2 s 2  1.


That is, the outputs are mixture density networks [6], with a shared hidden layer, using the same parameter tying as NADE.

our Real-valued Neural Autoregressive Density-Estimator or RNADE model represents the probability density of a vector as: where p M is a mixture of Gaussians with parameters θ d . The mixture model parameters are calculated using a neural network with all of the preceding dimensions, x <d , as inputs.


With the DP table, it takes only low polynomial time to construct the optimal BN structure over the set of all vertices [22], i.e., a BN extension of ∅.

we can obtain an initial lower bound solution G * for the branch and bound as follows.


The mosaic is very difficult to interpret, due to abrupt changes at the boundaries between station coverage areas.

although we are very accustomed to seeing nationwide composites of radar reflectivity, radial velocity data is not presented or analyzed in this way.


Additionally, TPGM shows some advantages in high-dimensional settings.

this likely results from a facet of its node-conditional distribution which places larger mass on strongly dependent count values that are close to R. the TPGM method may be better able to infer edges from highly connected networks, such as those considered.


Currently, it is only known how to solve conjunctive queries in ACs.

we will answer complex queries by asking multiple conjunctive queries.


The higher resolution features operated by the encoder contain both shared and exclusive information, but the decoder must receive only the shared part of the representation from the encoder.

instead of using skip connections, we reduce the architectural bottleneck by increasing the size of the latent representation.


Note that the accuracy of Mallows(d t ) is the same as that of CPS-G(d t ) because of the mathematical equivalence of the two models.


Further, as the number of variables in the models increase, the query aware samplers not only outperform the baseline for longer periods of time, but also exhibit more dramatic convergence rate improvements.

query specific sampling is a promising approach for approximately answering queries on realworld probabilistic databases (and relational models) that contain billions of variables.


The proof is along the lines of [Hazan, 2015, Theorem 5.2], except that the loss function does not take a raw state as input, and our domain for optimization is complex.

the mean value theorem does not hold, which means we need to approximate the Bregman divergence instead of replacing it by a norm as in the original proof.


Thus, precipitation nowcasting can be viewed as a type of video prediction problem with a fixed "camera", which is the weather radar.

methods proposed for predicting future frames in natural videos are also applicable to precipitation nowcasting and are related to our paper.


The elements of x c t can be history-based estimations or truly observed values, depending on the presence of the observations.

we learn the weight β t by considering both the temporal decay γ t and the masking vector m t as shown in Eq.


We find in our experiments that some tasks require more, but not necessarily larger, memories, and others such as language modeling require fewer, larger memories.

we have a number of tune-able parameters: the number of memories, the size of each memory, the number of attention heads, the number of steps of attention, the gating method, and the postattention processor g ψ .


The critic corresponds to a conventional value function, mapping states to expected cumulative future reward.

the critic addresses a problem of prediction, whereas the actor is concerned with control.


2 In particular, if a learner employs a ( , L)-private strategy, then no adversary estimator can be close to the target within an absolute error of /2 with a probability great than 1/L.

for any fixed , the parameter L can be interpreted as the level of desired privacy.


Let z k be the sequence generated by Algorithm 1.

then, for any k 1, we have where the sequence of the last iterates z k globally converges to a solution z ?