sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: To enable a faster local update, we replace q ijk (x, y, z | φ ijk ) in (1) with a simpler "mixture-of-deltas" variational distribution, where A is a randomly chosen set of triples (a, b, c) with size O(K), and a δ aaa ijk + (a,b,c)∈A δ abc ijk = 1.
Sentence 2: we assume the probability mass of the variational posterior q(s i,jk , s j,ik , s k,ij ) falls entirely on the K "diagonal" role combinations (a, a, a) as well as O(K) randomly chosen "off-diagonals" (a, b, c).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Partitioning the workers into three disjoint and non-empty groups G 1 , G 2 and G 3 , the outline of this stage is the following: we use the spectral method to estimate the averaged confusion matrices for the three groups, then utilize this intermediate estimate to obtain the confusion matrix of each individual worker.
Sentence 2: for g ∈ {1, 2, 3} and j ∈ [n], we calculate the averaged labeling within each group by Denoting the aggregated confusion matrix columns by µ gl := E(Z gj |y j = l) = 1 |Gg| i∈Gg π i µ il , our first step is to estimate C g := [µ g1 , µ g2 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, the noisy ICA model X = A(S + ξ) + η in which ξ is considered part of the latent source signal S = S + ξ, and the model X = AS + (Aξ + η) in which ξ is part of the noise are indistinguishable.
Sentence 2: the latent source S and its covariance are ill-defined.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We begin by specifying a model of decision-making that makes the standard assumptions that decision makers tend to choose things with greater utility and that utilities are additive.
Sentence 2: if f j is a binary vector indicating the effects produced by option o j and u is a vector of utilities assigned to each of the m effects, then the total utility associated with option o j can be expressed as U j = f j T u.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the second step we "pull" μ back into the feasible regime by taking a convex combination with the uniform distribution u (see [3] for a related approach).
Sentence 2: this step solves the simple problem of finding the smallest λ ∈ [0, 1] such that 0 ≤ μ ≤ 1 (where μ = (1 − λ)μ + λu).
Relationship: entailment
Domain: NeurIPS
Sentence 1: If for a given neuron x t−1 k is always positive, then the gradient ( 4) is only strictly positive when h t−1 k = 1 and h t j = 1 and strictly negative when h t−1 k = 1 and h t j = 0.
Sentence 2: the connection R jk is strengthened when neuron k appears to cause neuron j to activate and inhibited if neuron k fails to activate neuron j.
Relationship: entailment
Domain: NeurIPS
Sentence 1: First, instead of feeding a single sampled word as the input to the generator RNN, we use the softmax distribution over words instead.
Sentence 2: during the generating process of transferred x 2 from G(y 1 , z 2 ), suppose at time step t the output logit vector is v t . We feed its peaked distribution softmax(v t /γ) as the next input, where γ ∈ (0, 1) is a temperature parameter.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Suppose `t is convex and L-Lipschitz; in particular, for every x 2 R, there is a sub-derivative `0 t (x) such that |`0 t (x)|  L. Then there is an explicit learning strategy that guarantees regret R T = O(L p T n ) for all T . This is so even assuming the measurement E t and loss function `t are chosen adaptively, in response to the learner's previous behavior.
Sentence 2: the algorithm applies to L 1 loss and L 2 loss, and achieves regret O( p T n ) for both.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Because the size of the smallest interval known to contain X ⇤ is halved with each additional query, this yields the query complexity Unfortunately, once the level of privacy L increases above 1, the bisection strategy is almost never private: it is easy to verify that if the adversary sets b X a to be the learner's last query, Q n , then the target is sure to be within a distance of at most ✏.
Sentence 2: the bisection strategy is not ( , L)-private for any L > 1, whenever ✏ < /2.
Relationship: entailment
Domain: NeurIPS
Sentence 1: From (2) we see the high level and low level rewards are a linear combination of the same set of reward weights w. This allows us to combine both types of expert advice presented above to obtain the following unified optimization problem min w,η,ξ This optimization problem is convex, and can be solved efficiently.
Sentence 2: even though the optimization problem has an exponentially large number of constraints (one constraint per policy), the optimum can be found efficiently (i.e., in polynomial time) using, for example, the ellipsoid method, since we can efficiently identify a constraint that is violated.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Standard LSH approaches to the nearest neighbor problem work by scoring points in the set That is, LSH considers only those points in W that are covered by at least one of the same α sets as x. Predictive indexing, in contrast, maps each cover set Q i,J to an ordered list of points sorted by their probability of being a top-10 nearest point to points in Q i,J .
Sentence 2: the lists are sorted by h Q i,J (p) = Pr q∼D|Q i,J (p is one of the nearest 10 points to q).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The inverse of the ancestor function is the descendent function τ , which maps u ∈ U to the subset {u ′ ∈ U : u ∈ π ⋆ (u ′ )}.
Sentence 2: u is a descendent of u ′ if and only if u ′ is an ancestor of u. Graphically, we can depict T as a rooted tree: U defines the tree nodes, ALL is the root, and { u, π(u) : u ∈ U \ ALL} is the set of edges.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2 Where we define , and D KL [q||p] for q, p ∈ [0, 1] is defined as the divergence of Bernoulli distributions with parameters q and p. From now on, following [2], we will define the clustering distance function d D of Eq.
Sentence 2: ( 1) as: the clustering distance is the probability that two independently drawn instances from D will be in the same cluster under one clustering, and in different clusters under another clustering.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our general result is stated in terms of a maximization over a vector parameter `= (`1, . . . , `d) with `t 2 {0, . . . , ⇡ t p} for all t. We will see in the proof that `t represents the number of items of type t that are unknown to the decoder after p⇡ t `t are revealed by a genie.
Sentence 2: we define the following: • Given `and , we let S `be a random set of indices in [p] such that for each t 2 [d], the set contains `t indices corresponding to entries where equals t. we define S `to be uniformly distributed over all such sets.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We present an alternative probabilistic formulation, called semi-parametric PCA (SP-PCA), where no assumptions are made about the distribution of the latent random variable θ. Non-parametric latent distribution estimation allows us to approximate data density better than previous schemes and hence gives better low dimensional representations.
Sentence 2: multi-modality of the high dimensional density is better preserved in the projected space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Proposition 4.1 (The SVAE objective lower-bounds the mean field objective) The SVAE objective function L SVAE lower-bounds the mean field objective L in the sense that for any parameterized function class {r(y; φ)} φ∈R m . Furthermore, if there is some φ * ∈ R m such that ψ(x; y, φ * ) = E q(γ) log p(y | x, γ), then the bound can be made tight in the sense that Thus by using gradient-based optimization to maximize L SVAE (η θ , η γ , φ) we are maximizing a lower bound on the model log evidence log p(y).
Sentence 2: by optimizing over φ we are effectively learning how to condition on observations so as to best approximate the posterior while maintaining conjugacy structure.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Those assumptions, including low-rankness, incoherence and uniform sampling, are now standard and widely used in the literatures, e.g., [14,17,22,24,28,33,34,36].
Sentence 2: the analyses in [17,33,36] show that, in terms of theoretical completeness, many nonconvex optimization based methods are as powerful as the convex program (2).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition to knowing the description of the algorithm, an adaptive adversary is stronger and can record all of the past actions of the algorithm, and use these arbitrarily to decide the future payoff vectors and site allocations.
Sentence 2: communication: We do not explicitly account for message sizes, since we are primarily concerned with scaling in terms of T and k. We require that message size not depend k or T , but only on the number of experts n. we assume that n is substantially smaller than T and k. All the messages used in our algorithms contain at most n real numbers.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to encode the image label x cls , the encoder r cls = φ cls enc (x cls ) takes the vector of C cls binary probabilities x cls ∈ R C cls , one for each of the C cls possible object classes, and broadcasts the corresponding values to all H × W spatial locations (u, v) in h. Formally r cls ∈ R H×W ×C cls and ∀u, v, c : r cls uvc = x cls c . Encoding the object detection label x det is similar, but reflects the geometric information captured by such labels.
Sentence 2: each bounding box b m of the M extracted by SSW is associated to a vector of C cls + 1 probabilities (one for each object class plus one more for background) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, we observe that such an operation can be further combined using a regular 3-D convolution [10].
Sentence 2: the m d f ⇥ d f input feature maps can be viewed as a single 3-D feature map with a size of d f ⇥ d f ⇥ m. The special depth-wise convolution, or equivalently the global average pooling layer, is essentially a 3-D convolution with a kernel size of d f ⇥ d f ⇥ 1, where the weights in the kernel is fixed to 1/d 2 f . Moreover, in this view, the channel-wise convolution is a 3-D convolution with a kernel size of 1 ⇥ 1 ⇥ d c . These two consecutive 3-D convolutions follow a factorized pattern.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Consider the following problem: given two matrices A 2 R d⇥n1 and B 2 R d⇥n2 that are stored in disk, find a rank-r approximation of their product A T B.
Sentence 2: we are interested in the setting where both A, B and A T B are too large to fit into memory.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We show how this scheme can be used to perform Bayesian inference, and how simple extensions of this basic idea make it possible to combine sources of information and to propagate uncertainty through multiple layers of random variables.
Sentence 2: we focus on one Monte Carlo method, namely importance sampling with the prior as a surrogate, and show how recursive importance sampling approximates hierarchical Bayesian inference.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Unlike in the synthetic experiments training data was either used for building models, or associating models to tasks.
Sentence 2: we set |S 1 t | = |S 2 t | = 0.5|S t | for η = 0.5, and used one half of the examples to build models (set the weights of prediction functions), and the remaining half to evaluate each of the k models on the T tasks, and associating models to tasks.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , T k } ∈ O k (S) is an optimal k-clustering, let r be the number of clusters of T that "do not respect" the partition {S 1 , S 2 }.
Sentence 2: r is the number of clusters of T that intersect both S 1 and to have the smallest r. If r = 0, then T is a refinement of S and there is nothing to show.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In fact, many stochastic algorithms assume that S t is a uniform subset of [n], because in this case the sub-sampled Hessian is an unbiased estimator of the full Hessian.
Sentence 2: 8✓ 2 C, E [H St (✓)] = H [n] (✓), where the expectation is over the randomness in S t . We next show that for any scaling matrix Q t that is formed by the sub-samples S t , iterations of the form Eq.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Contrastive loss [3,7] takes pairs of examples as input and trains a network to predict whether two inputs are from the same class or not.
Sentence 2: the loss is written as follows: where m is a margin parameter imposing the distance between examples from different classes to be larger than m. Triplet loss [27,2,19] shares a similar spirit to contrastive loss, but is composed of triplets, each consisting of a query, a positive example (to the query), and a negative example: Compared to contrastive loss, triplet loss only requires the difference of (dis-)similarities between positive and negative examples to the query point to be larger than a margin m. Despite their wide use, both loss functions are known to suffer from slow convergence and they often require expensive data sampling method to provide nontrivial pairs or triplets to accelerate the training [2,19,17,4].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given a Gaussian policy π(a|s, θ) ∼ N θ T φ(s), σ 2 , under the assumption of uniformly bounded rewards and basis functions (Assumption 4.1), we have the following upper bound to the variance of the i-th component of the PGT gradient estimate ∇ θi J P GT µ (θ): As expected, since the variance of the gradient estimate obtained with PGT is smaller than the one with REINFORCE, also the upper bound of the PGT variance is smaller than REINFORCE one.
Sentence 2: while the variance with REINFORCE grows linearly with the time horizon, using PGT the dependence on the time horizon is significantly smaller.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Three experiments simulating various levels of uncorrelated input/output activity were conducted.
Sentence 2: excitatory Poisson inputs at 2, 20 and 40 Hz were balanced by inhibitory Poisson inputs at 6.3, 63 and 124 Hz to generate output rates of approximately 2, 20 and 40 Hz, respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As shown in Figure 2, training with = 2 • 10 4 converged to a fixed solution, which provided an estimate f close the original function f . In contrast, training with = 3 • 10 4 converged to an oscillation and not to a fixed point.
Sentence 2: after sufficient training, the estimate kept switching between fodd and feven at each iteration of the gradient descent algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Consider a discrete-time stochastic process X = {X t : t ≥ 0} on the state space N = {1, 2, . . . , n}, where the initial state X 0 is given, say X 0 = i, the next state X 1 is determined by the transition probability P(X 1 = j|X 0 = i) = p ij , and the subsequent states are determined by the transition probabilities where t ≥ 0.
Sentence 2: note that the process X is time homogeneous, i.e., the transition probabilities in (1) are independent of t. if the previous and current states are the same, the process will remain in the current state forever.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For any k = 1, . . . , K, the statistical error of the sample-based minimization function defined in (3.3) satisfies that, for any fixed where 2) and (3.3), and m = Theorem 3.4 establishes the statistical error associated with c M k (⌦ [K] k ) for arbitrary ⌦ j 2 B(⌦ ⇤ j ) with j 6 = k. In comparison, previous work on the existence of a local solution with desired statistical property only establishes theorems similar to Theorem 3.4 for ⌦ j = ⌦ ⇤ j with j 6 = k. The extension to an arbitrary ⌦ j 2 B(⌦ ⇤ j ) involves non-trivial technical barriers.
Sentence 2: we first establish the rate of convergence of the difference between a sample-based quadratic form with its expectation (Lemma B.1) via concentration of Lipschitz functions of Gaussian random variables [24].
Relationship: entailment
Domain: NeurIPS
Sentence 1: We will show how to translate each instance of X3C, (U, S), to an instance of k-means clustering in the Euclidean plane, X.
Sentence 2: x has a grid-like structure consisting of l rows (one for each S i ) and roughly 6m columns (corresponding to U ) which are embedded in the Euclidean plane.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To make this precise, we introduce a time after which for each arm, the empirical proportion allocated to it is accurate.
Sentence 2: given β ∈ (0, 1) and > 0, we define It is clear that P(M β < ∞) = 1 for all > 0 if and only if T n,i /n → w β i with probability 1 for each arm i ∈ A.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the presence of outlier data points, there is an inherent unidentifiability between AB and E which can prevent the detection of outliers and hence reduce the performance of subspace clustering.
Sentence 2: an outlier y i can be included in the sparse component as e i = y i or included in the dictionary D with its own subspace, which leads to (AB) ii ≈ 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: y is the sum of relatively few shifted copies of waveforms, so the coefficients of most dictionary elements will be zero), CBP draws on the basis pursuit denoising, which has been effective in the compressive sensing setting and elsewhere [10], [1].
Sentence 2: cBP (with a Taylor basis) recovers coefficients using: Here we denote by F the matrix with columns {f n,j (t)} and F the matrix with columns {f n,j (t)}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The bad news is that, in general, condition Q s,q (κ), as well as RE and Compatibility conditions, cannot be verified.
Sentence 2: given a sensing matrix A and a representation structure B, it seems to be difficult even to verify that a pair (H, • ) satisfies condition Q s,q (κ) associated with A, B, let alone to synthesize such H which satisfies this condition and results in the best possible error bounds (4), ( 5) for the regular and the penalized 1 -recoveries.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As a consequence of the gradient uniform convergence bounds, we show how to use any algorithm that finds approximate stationary points for smooth functions in a black-box fashion to obtain optimal sample complexity for these models-both in high-and low-dimensional regimes.
Sentence 2: standard algorithms including gradient descent [33], SGD [10], Non-convex SVRG [38,3], and SCSG [26] enjoy optimal sample complexity, even when allowed to take multiple passes over the dataset.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Most importantly, this function U (S, c) is identical to V(S, 1 2 (T c)), the Black-Scholes value of the option in (4) when the GBM volatility parameter is in the Black-Scholes analysis.
Sentence 2: analogous to to (3), U (S, c) satisfies a differential equation: The following is our main result of this paper.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This paper demonstrates a more principled approach to address this problem, by constructing new bases that emerge from investigating the links between basis construction methods and different power series expansions of value functions.
Sentence 2: instead of using the eigenvectors of the transition matrix, the proposed approach uses the average-reward or gain as the first basis vector, and dilates the reward function by powers of the average-adjusted transition matrix.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let Y s,a,π be a random variable defined as Y s,a,π = (X s,a,π − R(s, a))/γ.
Sentence 2: we can see that the pdf of Y s,a,π is a convex combination of the pdfs of X s ,π for each s ∈ S. we have f Ys,a,π (y) = s ∈S T (s |s, a)f X s ,π (y).
Relationship: entailment
Domain: NeurIPS
Sentence 1: A key property of the ANOVA kernel is multi-linearity [4]: where p ¬j denotes the (d − 1)-dimensional vector with p j removed and similarly for x ¬j .
Sentence 2: everything else kept fixed, A m (p, x) is an affine function of p j ∀j ∈ [d].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this work, we compare several different techniques for neural program induction, with a particular focus on how the relative accuracy of these techniques differs as a function of the available training data.
Sentence 2: if technique A is better than technique B when only five I/O examples are available, does this mean A will also be better than B when 50 I/O examples are available?
Relationship: entailment
Domain: NeurIPS
Sentence 1: In contrast, the cost (4) will heavily penalize the classifier if it does not predict the output for which p LM (•) is positive, and will penalize less if it predicts outputs for which p LM (•) is zero.
Sentence 2: this cost will encourage p ✓ (y|x) to cover as much of p LM (•) as possible, a behavior called "coverage-seeking" in [25].
Relationship: entailment
Domain: NeurIPS
Sentence 1: This restriction rules out a variety of renewal densities that are commonly employed to model neural data [13,14,15].
Sentence 2: the log-normal and inverse-Gaussian densities both have increasing regimes on a subset of [0, ∞), as does the gamma density q(u) ∝ u κ−1 e −uκ when κ > 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As a result, we can perform stochastic approximation of (7) while monitoring the KL divergence between the posterior approximates due to the change of m, S, and X across iterations.
Sentence 2: we use stochastic mirror ascent [17] in the space of probability densities in RKHS, which was recently proven to be as efficient as stochastic natural gradient ascent [19].
Relationship: entailment
Domain: NeurIPS
Sentence 1: their gradients are L i -Lipschitz), and x is a minimizer of (almost) all f i (x) (i.e.
Sentence 2: p i (∇f i (x ) = 0) = 1), then E x k − x goes to zero exponentially, rather then polynomially, in k. reaching a desired accuracy of E x k − x 2 ≤ ε requires a number of steps that scales only logarithmically in 1/ε.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Thus if 1 − µ < 2 n−t , it is preferable not to explore new arms and to play the best known arm, with mean reward µ, until time n. A fortiori, the best known arm is played until time n whenever its parameter is larger than 1 − 2 n . We denote by A n the first arm whose parameter is larger than 1 − 2 n . We have K n ≤ A n (the optimal policy cannot explore more than A n arms) and The parameter θ An of arm For all k = 1, 2, . .
Sentence 2: ., let L 1 (k) be the length of the first run of arm k. We have: using the fact that: and To conclude, we write: Observe that, on the event {L 1 (1) + .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In contrast to previous uses of graph-based priors, we define f as the real function f (defined over the entire observation space) evaluated at the graph nodes.
Sentence 2: f is defined as a linear function of x, and at the graph node i, Xw, and p(f ) induces a Gaussian prior on w, with precision matrix A = X T ∆X, ) Notice that since ∆ is singular, A may also be singular, and the corresponding prior may therefore be improper.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Once we could ensure that a pair of corresponding images are mapped to a same latent code and a same latent code is decoded to a pair of corresponding images, (x 1 , G 2 (z 1 ⇠ q 1 (z 1 |x 1 ))) would form a pair of corresponding images.
Sentence 2: the composition of E 1 and G 2 functions approximates F ⇤ 1!2 for unsupervised image-to-image translation discussed in Section 2, and the composition of E 2 and G 1 function approximates F ⇤ 2!1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We will also consider a vector restricted on a support set.
Sentence 2: for a d-dimensional vector x and a support set T ⊂ {1, 2, . . . , d}, depending on the context, x T can either be a |T |-dimensional vector by extracting the elements belonging to T or a d-dimensional vector by setting the elements outside T to zero.
Relationship: entailment