sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: These computations, appearing as T a→b c (α b,i,k , α c,k+1,j ), T b→c a (1,2) (β b,k,j , α c,k,i−1 ) and T b→a c (1,3) (β b,i,k , α c,j+1,k ) output a vector of length m, where computation of each element in the vector is O(m 2 ).
Sentence 2: this multiplicative factor can be prohibitive for a large number of hidden states; various heuristics are used in practice to avoid this problem [16].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Because according to Algorithm 1 in [11], the output of input Θ weighted by the scalars s = Φ g can be approximated to first glance, the scalars s is totally erased by BN in this mathmatical process.
Sentence 2: we have introduced a simple approximated formulation of compact generalized non-local operation, and have validated it on the task of fine-grained classification and action recognition from RGB images.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For the third case, let λ 1 = T α and λ t = 0 for 1 < t ≤ T . Note that The original online gradient descent (OGD) algorithm as analyzed by Zinkevich [2] used the Euclidean distance of the current point from the optimum as a potential function.
Sentence 2: it is natural to ask if this is essential: is there an algorithm that can adapt to the convexity of the loss functions and achieve the same regret rates in both cases-O(log T ) for uniformly convex functions and O( √ T ) for arbitrary convex functions?
Relationship: neutral
Domain: NeurIPS
Sentence 1: Written under partial support by the Austrian Science Fund FWF, project # S9102-N04, project # IST2002-506778 (PASCAL) and project # FP6-015879 (FACETS) of the European Union.
Sentence 2: whereas previous models for discrete or continuous attractors in recurrent neural circuits required that the whole dynamics of such circuit was entrained by the attractor, our new model predicts that persistent firing states can co-exist with other high-dimensional attractors and with responses to time-varying afferent inputs within the same circuit.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We found empirically that the Monte-Carlo procedure is more accurate as the simulated PIP losses concentrate tightly around their means across different runs.
Sentence 2: given an embedding matrix E ∈ R n×d , define its associated Pairwise Inner Product (PIP) matrix to be PIP It can be seen that the (i, j)-th entry of the PIP matrix corresponds to the inner product between the embeddings for word i and word j, i.e.
Relationship: neutral
Domain: NeurIPS
Sentence 1: ( 9)] associated with the four weights [p j , p k , v 0,j , v 0,k ] T ; they may be expressed in a compact form as where the entries are given below with B ≡ Notice here that the subscript d implies datum d (d = 1, ..., m); hence, h d is the hidden-node output on datum d (but not the dth hidden-node output) common to both nodes j and k due to Theorem 1: When e = 0, the n-by-n Hessian ∇ 2 E and its block H in Eq.
Sentence 2: singularities have been well discussed in the nonlinear least squares literature also: For instance, Jennrich & Sampson (pp.65-66 [5]) described an overlap-singularity situation involving a redundant model; specifically, a classical (linear-output) model of exponentials with h i ≡ φ(v i x) and no thresholds in Eq.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The results are shown in Figure 3 and Table 1.
Sentence 2: assuming perfect transitions, if θ(β) and the Fisher information matrix G θ (β) = cov x∼p θ (∇ θ log p θ (x)) are continuous and piecewise smooth, then as K → ∞ the bias δ behaves as follows: where θ(β) represents the derivative of θ with respect to β.
Relationship: neutral
Domain: NeurIPS
Sentence 1: [32], because the ones they used were much smaller, resulting in an expensive cross-validation procedure involving a separate hyperparameter search for each fold.
Sentence 2: straightforward recurrent autoregressive models would update a hidden state sequentially for every variable, requiring D sequential computations to compute the probability p(x) of a D-dimensional vector, which is not well-suited for computation on parallel architectures such as GPUs.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This bound can be expressed on a scale that is independent of the temperature parameter γ; intuitively, ∆ max corresponds to a utility difference so slight that the user identifies the best item only with probability 0.56 under R L with temperature γ.
Sentence 2: a number of these focus directly on (myopically or heuristically) reducing uncertainty regarding utility parameters as quickly as possible, including max-margin [10], volumetric [12], polyhedral [22] and entropy-based [1] methods.
Relationship: neutral
Domain: NeurIPS
Sentence 1: A number of algorithms have been proposed to solve such systems of linear inequalities for feasible points d. In our sampling algorithm we implement the relaxation method introduced by Agmon [11] and Motzkin and Schoenberg [12] due to its simplicity.
Sentence 2: the Z-test scores on the 4 differently sized splits are reported in Table 2 b) and allow us to reject the null hypothesis with very high confidence.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The tradeoff is controlled by varying the level of discretization.
Sentence 2: the coin-flipping process captures the live-edge characterization of the Independent Cascade model [7], in which the presence of edge (u, v) in G allows influence (e.g., behavior, disease, or some other spreading phenomenon) to propagate from u to v. Influence maximization seeks a seed set S of at most k nodes to maximize the expected number of nodes reachable from S, which is easily modeled within our model by assigning appropriate rewards and actions.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We assume that all edges have a minimum influence strength, and that even when all neighbors of a node u are influenced in a time step, there is a small probability of u not being influenced in the next step; we consider a specific seed distribution, where each node has a non-zero probability of (not) being a seed node.
Sentence 2: we stress that a node is influenced at time t if it changes its opinion from 0 to 1 exactly at t. Also, in both the LT and IC models, no node gets influenced more than once and hence an influence cascade can last for at most n time steps.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As observed in Figure 4, we see that the PG sampler applied to the iHMM and sticky iHMM, converges far more quickly from both "small" and "large" initialization of K = 3 and K = 30 "active" states to the true value of K = 10 hidden states, as well as converging in JLL more quickly.
Sentence 2: we can compute p(y t |s n t−1 ) = K+1 k=1 π(k | s n t−1 )f (y t | φ k ) for each particle s n t where n ∈ 1, ..., N − 1.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Since there are exponentially many constraints, we follow [10] in adopting a constraint generation strategy, which starts by solving the problem with no constraints and iteratively adding the most violated constraint for the current solution of the optimisation problem.
Sentence 2: we can partition the optimisation problem into k optimisation problems conditioned on the sets Y k := {y : |y| = k}: where A [k],n denotes the particular matrix A n that we obtain when |y| = k. However the inner maximization above, i.e., the problem of maximising a supermodular function (or minimising a submodular function) subject to a cardinality constraint, is itself NP-hard [19].
Relationship: neutral
Domain: NeurIPS
Sentence 1: The learned policy is the result of minimum-cost planning using these cost functions.
Sentence 2: the algorithm in its most basic form is given by the update rule w t+1 ← w t − γ t g t , where {γ t } ∞ t=1 is a prespecified stepsize sequence and g t is a subgradient at the current timestep t. Note that computing the subgradient requires solving the problem µ * = arg min µ∈Gi (w T F i − l T i )µ for each MDP.
Relationship: neutral
Domain: NeurIPS
Sentence 1: . . , d M ) where each d m ∈ n . We wish to impose that any x ∈ n may be represented approximately as x = Dα, where α ∈ M is sparse, and our objective is to also minimize the 2 error x − x 2 .
Sentence 2: the DP still assumes that the members of {x i } i=1,N B are exchangeable.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In this section we formally introduce the Random Recursive SVM model, and discuss the motivation and justification behind it.
Sentence 2: linear SVMs often fail to solve complex problems however, and with non-linear kernels, SVMs usually suffer from speed and memory issues when faced with very large-scale data, although techniques such as non-convex optimization [6] or spline approximations [19] exist for speed-ups.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This work was supported in part by the European Commission under Grant MIRG-268398, ERC Future Proof, SNF 200021-132548, ARO MURI W911NF0910383, and DARPA KeCoM program #11-DARPA-1055.
Sentence 2: further assumptions on the multivariate functions beyond smoothness are needed for the tractability of successful learning [13,14,16,19].
Relationship: neutral
Domain: NeurIPS
Sentence 1: We assume that the observed tensor T ∈ R I1×I2•••×IN can be decomposed as a true tensor D with rank-(r 1 , r 2 , .
Sentence 2: aDMM decomposes a large problem into a series of smaller subproblems, and coordinates the solutions of subproblems to compute the optimal solution.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Kearns [11] extended Valiant's model and introduced the statistical query oracle STAT(c, D).
Sentence 2: the intuition that given I V x = h, the strings with x h+1 ∈ U +1 have higher probability of positivity than that of the strings with x h+1 ∈ U +1 is still true under product distributions.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We also note that under the relatively minor assumption of the existence of an oracle for offline optimization, the algorithm of Kalai and Vempala [2005] is an efficient algorithm for this setting.
Sentence 2: there is only one (non-trivial) special case [Gyorgy et al., 2007] in the literature where an O * (poly(n) √ T ) regret has been established, and this case assumes significantly more feedback than in the partial information case -their result is for Path Planning (where D is the set of paths on a graph and n is the number of edges) and the feedback model assumes that learner receives the weight along each edge that is traversed (significantly more information than the just the scalar loss).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We will take advantage of this fact to develop an efficient stochastic inference algorithm within the framework of G-flow.
Sentence 2: based on its hypothesis about pose history, each expert has an opinion about the current pose of the object, U t , and the texture maps of the object and background, V t and B t . Each expert also has a credibility, a scalar that measures how well the expert's opinion matches the observed image y t . (5) can be interpreted as follows: The filtering distribution at time t is obtained by integrating over the entire ensemble of experts the opinion of each expert weighted by that expert's credibility.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The algorithm starts with some U 0 and V 0 , and given a new observation (M) ij , SGD updates the i th -row and the j th -row of the current iterates U t and V t respectively by t , and, where η is an appropriately chosen stepsize, and U (i) denote the i th row of matrix U.
Sentence 2: σ min (X U) not too small requires both σ min (U U) and σ min (X W) are not too small, where W is top-k eigenspace of UU .
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the case of a sparse use of trajectories to cover a space, the cost of the approach is dominated by the E © costs of updating second derivative matrices, and thus the cost of the trajectory-based approach increases quadratically as the dimensionality increases.
Sentence 2: this is a model of a robot that walks along the trusses of a large structure such as a bridge, much as a monkey brachiates with its arms.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Note that this joint distribution is for a general Lévy measure ρ, density f ρ and it is conditioned on the valued of the random variable T . We can recover the well known Stick breaking representations for the Dirichlet and Pitman-Yor processes, for a specific choice of ρ and if we integrate out T , see the supplementary material for further details about the latter.
Sentence 2: draw with a simple rejection sampler [2] where the rejection constant is bv and the proposal is U p0, vq.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We compared the proposed model with probabilistic canonical correlation analysis (PCCA), horizontal anomaly detection (HOAD) [13], consensus clustering based anomaly detection (CC) [16], and one-class support vector machine (OCSVM) [18].
Sentence 2: because the proposed model simultaneously estimates the parameters and infers anomalies, the estimated parameters are not contaminated by the anomalies.
Relationship: neutral
Domain: NeurIPS
Sentence 1: It is computed by taking the expectation over output values H(X|R) = E r − p(x|r) ln p(x|r)dx .
Sentence 2: although this is large relative to the variance of a pixel, as a result of strong spatial correlations in the input, some projections of the data (low frequency components) had SNR over 40dB.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Finally, the IL penalties are τ j (z j + |s j /σ| 2 ) 1/2 , enforcing sparsity more strongly for smaller z j .
Sentence 2: while this suboptimal behaviour of our optimization will be analyzed more closely in future work, it is our experience so far that the gain in using greedy sequential Bayesian design optimization over simpler choices is generally largest below 1/2 Nyquist.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For each node i, we denote P i , S i respectively as the set of direct predecessors and successors of node i.
Sentence 2: m i = |P i |, n i = |S i | imply the in-degree and out-degree of the node i. Matrix A denotes the corresponding adjacency matrix where each entry a ij ∈ [0, ∞) is the weight of link (i, j).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If we approximate the true posterior p(Θ|y, O, m) by the estimated variational posteriors q(Θ|y, O, m), p(x t |y, O, m) can be calculated by p(x t |y, O, m) ≈ p(x t |y, Θ, m)q(Θ|y, O, m)dΘ.
Sentence 2: the optimal class y can be obtained by In the calculation of ( 8), the integral over Gaussian means and covariances for a frame can be solved analytically to be Student distributions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Starting at t = 1, since the initial distribution is π 1 ∈ CRP(S, µ, 0), the conditional cluster assignment of the sequence i in π 1 is given by the CRP probabilities from (1): To find the conditional distribution of b t given a t , we use the definition of the fragmentation operation as independent CRP partitions of each cluster in π t . If a t = ∅, then the sequence i is in a cluster by itself in π t and so it will remain in a cluster by itself after fragmenting.
Sentence 2: b t = ∅ with probability 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Is is not hard to see that for every φ : [d + 1] → {±1}, the approximation error of the class of halfspaces with respect to D φ is zero.
Sentence 2: in order to ensure a large approximation error for every distribution, the number of classes must be at least linear in the dimension, so in this sense, the lemma is tight.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given a feature φ, our set of classifiers H φ consists of all one-vs-one classifiers, all one-vs-all classifiers, and all node classifiers from a tree-based method [13], where a node classifier can be trained to distinguish two arbitrary clusters of classes.
Sentence 2: for a K-class problem, the number of classifiers given a single feature is |H φ | = (K−1)K 2 + K + N φ,tree , where N φ,tree is the number of nodes in the tree model.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Using the expression (9) of V , we have V, V = η 2 (t)∆ T (t)Σ(X, t)∆(t), where ∆(t) = ∇ψ * (Z(t)/s(t)) − ∇ψ * (z ).
Sentence 2: since the mirror map has values in X and X is assumed compact, the diameter D = sup x,x ∈X x − x is finite, and ∆(t) ≤ D for all t. dβ(t) ≤ D 2 η(t) 2 σ 2 * (t)dt, and integrating, Since β(t) is a non-decreasing process, two cases are possible: if lim t→∞ β(t) is finite, then lim sup t→∞ |V(t)| is a.s. finite and the result follows immediately.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For any set cover instance, we will put all elements to be the input of our multi-input-probabilistic-AND gate, and connect the output with a large number of additional nodes.
Sentence 2: if k sets can cover all elements, all of those addition nodes will be activated, on contrast, if at least one of the elements cannot be covered, almost all of the additional nodes will remain inactive.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since the latent variables are not directly connected, the complexity of inferring the posterior distribution of H is the same as in EFH when all the input data are observed, as reflected in the factorized form of p(h|x, z).
Sentence 2: multi-view latent space MNs do not increase the complexity on testing if our task depends solely on the latent representation (i.e., expectation of H), such as information retrieval [26], classification, clustering etc.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Although this worst case formulation (15) might appear to be conservative compared to working with a known distribution on R and using (2), when additional information about the distribution is available, such as symmetry or unimodality, (15) can be brought very close to using a Gaussian distribution, as shown in Figure 1.
Sentence 2: given reasonable constraints, the minimax approach does not have to be overly conservative, while providing robustness and tractability.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A key factor in understanding this algorithm is the observance that a strategy a receives higher reward than a strategy b if and only if the gradient at a is in the direction of b (i.e., r t • (b − a) > 0).
Sentence 2: the step (3) adjustment is in the direction of the gradient if and only if z t received higher reward than x t . Notice also that, as long as x t is not near the boundary, the change due to step (3) is of lower magnitude than the change due to step (1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Our approach extends Tenenbaum and Freeman's work on bilinear models for learning content and style [12] by casting the problem within probabilistic sparse coding framework.
Sentence 2: whereas prior work on bilinear models used global decomposition methods such as SVD, the approach presented here emphasizes the extraction of local features by removing higher-order redundancies through sparseness constraints.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since the rewards are independent of whether we choose to explore, one sees that E[μ i (t " i (k))] = µ i .
Sentence 2: we can decompose E[max j μj (t " i (k)) μi (t " i (k))] as follows: The second term in (2) is bounded by i = µ 1 µ i . Summing over all these steps and all arms, we obtain the first term in our bound (1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It can be shown that with high probability, we always have |✓ i (t) μi (t)|  r i (t), where r i (t) = q 2 log t Ni(t) is defined in Algorithm 1.
Sentence 2: we can focus on the case that |✓ i (t) μi (t)|  r i (t) for any i and t. Note that we do not need to pay compensation in empirical steps.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Furthermore, we have K g (x, z) = k(x, •)g(x, x) dx, k(z, •)g(z, z) dz H . Thus, it is clear that the feature map of x depends not only on the kernel k, but also on the density g(x, x).
Sentence 2: by virtue of Lemma 4, the kernel K g allows the SVM to place different kernels at each data point.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, if A i,j = d(p i , q j ) is large, then since d(p i , q j ) ≤ d(p i , q 1 ) + d(q 1 , p 1 ) + d(p 1 , q j ), at least one of d(p i , q 1 ), d(q 1 , p 1 ), d(p 1 , q j ) is large, and further, all these distances can be found by reading the first row and column of A.
Sentence 2: large entries cannot hide in the matrix.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Now, suppose w * t is the optimum of Eq.
Sentence 2: ( 9) at iteration t, the item r t (w) we added can be shown to be the delta over w * t , that is, r t (w) = I(w = w * t ); in addition, we have q t (w) = 1 t t s=1 I(w = w * s ) when the step size is taken to be β t = 1 t . this Frank-Wolfe update can be naturally interpreted as greedily inserting a hidden unit into the current model p(v | q t , ϑ t ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Finally, our algorithm was successful in identifying the following types of "adversaries": (1) uniform workers who always label 1 or 1 (in temp, task2, stage2), (2) malicious workers who mostly label incorrectly (in bluebird, stage2) and (3) random workers who label each task independent of its true label (such workers were identified as "spammers" in [13]).
Sentence 2: our algorithm is able to identify a broad set of adversary strategies in addition to those detected by existing techniques.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Formally, let G = (V, E) be the random graph over n nodes obtained from an independent cascade process over the graph generated by the stochastic block model.
Sentence 2: similarly as for the stochastic block model, edge probabilities for the independent cascade model may vary between different communities and are identical within a single community C, where all edges have weights q ic C . an edge e between two nodes in a community C is in E with probability p C := q ic C • q sb C , independently for every edge, where q ic C and q sb C are the edge probabilities in the independent cascade model and the stochastic block model respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On optimizing A: By fixing β, the optimization problem ( 17) is reduced to where The problem ( 18) is a trace ratio problem, i.e., min A trace(A ⊤ S β W A)/trace(A ⊤ S β W ′ A).
Sentence 2: a closedform solution can be obtained by transforming (18) into the corresponding ratio trace problem, i.e., min the columns of the optimal A * = [α 1 α 2 • • • α P ] are the eigenvectors corresponding to the first P smallest eigenvalues in On optimizing β: By fixing A, the optimization problem (17) becomes min where The additional constraints β ≥ 0 cause that the optimization to (22) can no longer be formulated as a generalized eigenvalue problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For N large we can replace L with L = {x p(x) > λ} with small loss in accuracy, where p is an estimate of p using the unlabeled data; see Rigollet (2006) for details.
Sentence 2: let k j = n i=1 I (X i ∈ C j ) and for x ∈ C j define m(x) simply averages the labels of the data that fall in the set to which x belongs.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Namely, a (k+1) = w (k) and Recall that the LOG and EXP penalties are differentiable and strictly concave in |b| on [0, ∞).
Sentence 2: the above algorithm enjoys the same convergence property of the LLA was studied by Zou and Li [23] (see Theorem 1 and Proposition 1 therein).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This algorithm, straightforward as it is, provides (asymptotically) optimal regret guarantees (see Section B in the appendix for a brief review) and is arguably one of the most well-studied and widely-used algorithms in online learning theory and applications [24,39,57].
Sentence 2: several natural questions arise in multi-agent online learning : if each agent adopts OGD to minimize regret, what is the resulting evolution of the joint action?
Relationship: reasoning