sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Here, we show that the Z statistic, Z = i , which is the core of our testing algorithm, can accurately distinguish whether two words are very similar based on surprisingly small samples of the contexts in which they occur.
Sentence 2: for each pair of words, a, b that we consider, we select m 1 random occurrences of a and m 2 random occurrences of word b from the Google books corpus, using the Google Books Ngram Dataset.
Relationship: entailment
Domain: NeurIPS
Sentence 1: By sampling a recommendation from a joint probability distribution over Σ 1 , Σ 2 , the coordination device introduces a correlation between the strategies of the team members that is otherwise impossible to capture using behavioral strategies.
Sentence 2: in general there exists no behavioral strategy for the team player that is realization-equivalent to the normal-form strategy induced by the coordination device, as the following example further illustrates.
Relationship: entailment
Domain: NeurIPS
Sentence 1: [24] bypassed their lower bounds under a stronger assumption known as singular value separation: the difference between k-th singular value and all k -th singular values for k > k is at least ω( √ n).
Sentence 2: our result shows that we do not need singular value separation while using significantly less space-O(nk/α 2 ) as compared to O(n 2 )-if we are ready to pay for a small multiplicative error.
Relationship: entailment
Domain: NeurIPS
Sentence 1: After the n-th sample is received, the posterior is updated according to Bayes rule: for n = 1, . . . , N . Note that the MNN is deterministic, so the likelihood (per data point) has the following simple form 3 Therefore, the Bayes update in Eq.
Sentence 2: 3.3 simply makes sure that P (W|D n ) = 0 in any "illegal" configuration (i.e., any W 0 such that g x (k) , W 0 = y (k) ) for some 1 ≤ k ≤ n. the posterior is equal to the prior, restricted to the "legal" weight domain, and re-normalized appropriately.
Relationship: entailment
Domain: NeurIPS
Sentence 1: PPM-α Noise Model For α > 0, a neutral noise model P is called positional pairwise majority preserving (or PPM-α) if for every ground truth θ (and the corresponding true ranking σ * ) and every i < j, the quantity is non-negative for every i < j , and at least α for some i < j .
Sentence 2: for i < j , the probability that σ * (i) and σ * (j) go to positions i and j respectively in a vote should be at least as high as the probability of them going to positions j and i respectively (and at least α greater for some i and j ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, by relating risk to robustness, we derive a novel result that further motivates the usage of a CVaR objective in a decision-making context.
Sentence 2: we show that the CVaR of a discounted cost in an MDP is equivalent to the expected value of the same discounted cost in presence of worst-case perturbations of the MDP parameters (specifically, transition probabilities), provided that such perturbations are within a certain error budget.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The rise of this error as k increases captures the decay of memory traces in (1).
Sentence 2: the central goal of this paper is to obtain a deeper understanding of the memory properties of neural networks for sparse signals by studying the memory curve E(k) and especially its dependence on W. we are interested in classes of network connectivities W and input statistics for which E(k) can remain small even for k N . Such networks can essentially perform compressed sensing of their past inputs.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that unlike RKHS, this space of functions, K p X H, is density dependent.
Sentence 2: this now allows us to formulate the following optimization problem for semi-supervised classification/regression in a way similar to many supervised learning algorithms: The Fredholm learning framework solves the following optimization problem 1 : The final classifier is c(x) = (K pX f * ) (x), where K pX is the operator defined above.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The resulting coreset consists of O(dk/ε 2 ) points that approximates the sum of square distances for any k-piecewise linear function (k segments over time).
Sentence 2: we can use this coreset to compute the k-piecewise linear function that minimize the sum of squared distances to the input points, given arbitrary constraints or weights (priors) on the desired segmentation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we extend these ideas, and we show that-from the statistical perspective-we are able to obtain a low-rank approximation that comes with improved statistical guarantees by using a variant of this more traditional notion of statistical leverage.
Sentence 2: we improve the recent bounds of Bach [3], which provides the first known statistical convergence result when substituting the kernel matrix by its low-rank approximation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We have recently been exploring the use of PG algorithms in an incredibly difficult and exciting control domain -fluid dynamics -and as such applied the shell distribution to a fluid dynamical system.
Sentence 2: we applied learning to a system used to sudy the dynamics of flapping flight via a wing submerged in water (see Figure 4 for a description of the system (Vandenberghe et al., 2004)).
Relationship: entailment
Domain: NeurIPS
Sentence 1: One of the challenges of using latent variable models for exploratory data analysis, however, is developing models and learning techniques that accurately reflect the intuitions of the modeler.
Sentence 2: when analyzing multiple specialized data sets, it is often the case that the most salient statistical structure-that most easily found by fitting latent variable models-is shared across all the data and does not reflect interesting specific local structure.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Before proceeding, we recall a few definitions from convex analysis [17].
Sentence 2: a convex function f (x) with a bounded (sub • if there exists a constant β > 0 (often called the modulus of strong convexity) such that, for any α ∈ [0, 1], it holds: In the sequel, we use the standard Euclidean norm to define Lipschitz and strongly convex functions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: ( 11) plays a fundamental role.
Sentence 2: as pointed out in [8], the gradient of the regularized Sinkhorn approximation can be obtained directly from the dual solution as for any a ∈ R n and b ∈ R m . This characterization is possible because of well-known properties of primal and dual optimization problems [17].
Relationship: entailment
Domain: NeurIPS
Sentence 1: This distribution now has two modes, one representing the posterior given that a jump occurred, the other representing the posterior given no jump.
Sentence 2: importantly, we are more likely to infer a jump, and resample from scratch, if the observation r t+1 is far from that expected under the hypothesis of no jump, x t+1 • w L t . the probability that no jump occurred (and that we therefore resample according to the posterior distribution given drift -effectively, the chance that the sample "survives" as it would have in the no-jump Kalman filter) -is proportional to P (r t+1 | x t+1 , w t = w L t , no jump).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Further, joint parameter fits to 2AFC and GNG choice data indicated that the principal difference in the two tasks was in the nondecision time and decision threshold; the rate parameter (representing the evidence accumulation process) was similar in both tasks.
Sentence 2: they suggested that the nondecision time was shorter, and the decision threshold higher than in the 2AFC task (Figure 4B).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here we make use of the discriminative pose and shape estimate from Section 4 to give us the initial distribution for the posterior.
Sentence 2: given the discriminative model for the shape, p(ν|X), position, p(τ |X), and articulated pose, p(θ|X), of the body, we can let (with slight abuse of notation) y The outlined stochastic optimization framework also requires an image likelihood function, p(I|Y), that measures how well our model under a given state Y matches the image evidence, I, obtained from one or multiple synchronized cameras.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While these methods can obtain good class posteriors, we note that in order to obtain better accuracy/coverage, it is not sufficient to take naive Bayes' output p(y|x) and find a monotone mapping from that to a set of hopefully better class posteriors (e.g., [16]).
Sentence 2: in order to obtain better accuracy/coverage, it is also important to rearrange the confidence orderings that naive Bayes gives to documents (which our method does because of the normalization).
Relationship: entailment
Domain: NeurIPS
Sentence 1: • Under the canonical Erdös-Rényi sampling, we show that the Schatten k-norm of an incoherent rank-r matrix can be approximated within any constant multiplicative error, with number of samples scaling as O(dr 1−2/k ) (see Theorem 1).
Sentence 2: this is strictly smaller than the number of samples necessary to complete the matrix, which scales as O(dr log d).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since ( unh , F H,λ ) is SLN-robust, it follows that for hinge : (y, v) → max(0, 1−yv), ( hinge , F H,λ ) is similarly SLN-robust provided λ is sufficiently large.
Sentence 2: strong 2 regularisation (and a bounded feature map) endows the hinge loss with SLN-robustness 4 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: While previous studies (by Mirrokni and Gomes, and by Niazadeh et al) focused on revenue-sharing schemes in static double auctions, in this paper, we take advantage of the repeated nature of the auctions.
Sentence 2: we introduce dynamic revenue sharing schemes where we balance the two constraints over different auctions to achieve higher profit and seller revenue.
Relationship: entailment
Domain: NeurIPS
Sentence 1: That might allow us to reformulate the construction algorithm for G x in a way that allows for a more straightforward approach as compared to the parametric ansatz that we have proposed.
Sentence 2: our method relies upon the requirement that the target function can be expressed in terms of an underlying potential function g. This leads to the intriguing and nontrivial question: Is it possible to mathematically guarantee the existence of such a potential?
Relationship: entailment
Domain: NeurIPS
Sentence 1: Second, in many recent models, loss is measured not with L p distances (as is conventional in nonparametric statistics [49,47]), but rather with weaker losses, such as where F D is a discriminator class of bounded, Borel-measurable functions, and P and Q lie in a generator class F G of Borel probability measures on a sample space X .
Sentence 2: gANs often use losses of this form because (1) can be approximated by a discriminator neural network.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This makes 5 H the natural scale for measuring $ ¢ 5( and suggests that using far fewer than H kernel evaluations we can get good approximations of $ &¢ ( .
Sentence 2: for a chosen (fixed) value of " let us say that 2 $ ¢ 5( is trivial if 2 $ &¢ ( #" ¡ %$ 7 29 '@ H § Having picked some threshold " (for SVM expansions " is related to the classification offset) we want to determine whether $ &¢ ( is non-trivial and, if so, we want to get a good relative error estimate for it.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our approach relies on a blocked Gibbs sampling algorithm that alternates between sampling the number of considered chains and the global variables conditioned on the current value of matrices S and X, and sampling matrices S and X conditioned on the current value of the remaining variables.
Sentence 2: the algorithm proceeds iteratively as follows: • Step 1: Add M new new inactive chains using an auxiliary slice variable and a slice sampling method.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We followed [3,4], where the motivation was to aid the prediction of a variable that is expensive to measure by using abundant measurements of correlated variables which are less expensive to acquire.
Sentence 2: we estimated Cadmium and Copper at the validation locations using measurements of related variables at the training and test locations: Nickel and Zinc for Cadmium; and Lead, Nickel and Zinc for Copper.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Recent successes of deep neural networks have spanned many domains, from computer vision [1] to speech recognition [2] and many other tasks.
Sentence 2: sequence-to-sequence recurrent neural networks (RNNs) with long short-term memory (LSTM) cells [3] have proven especially successful at natural language processing (NLP) tasks, including machine translation [4,5,6].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, after drawing a new c t , we increment the relevant counts accordingly.
Sentence 2: if c t = K + 1 (i.e., a new ancestor is to be drawn), we update n = n + 1, set n K+1 = 1, m ct = m ct + 1, m ct,K+1 = 1, and set up a new (empty) HMM urn with color K + 1 (i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To mitigate this problem one can leverage several tactics commonly used in the GP literature, including the use of pseudo-inputs (Csató and Opper, 2002;Gal et al., 2014;Hensman et al., 2013;Quiñonero-Candela and Rasmussen, 2005;Titsias, 2009), Kronecker-factorized covariances (Casale et al., 2017;Stegle et al., 2011;Rakitsch et al., 2013), and low rank structures (Casale et al., 2015;Lawrence, 2005).
Sentence 2: in the instantiations of GPPVAE considered in this paper, we focus on low-rank factorizations of the covariance matrix.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Suppose that γ(r) ≥ C r 1+κ , ∀ r ≥ 0 for some C > 0 and κ > 0.
Sentence 2: then if u t * ≤ M for all t and η t = η t −β , then Assuming h is bounded, optimizing over β yields a rate of R t (x) = O(t κ 1+κ ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The position of c i is equal to a convex combination of the 5 pentagon corners' (x, y) coordinates, where the coordinates are weighted by the elements of θ i .
Sentence 2: circles c i at the pentagon's corners represent single-membership θ i 's, while circles on the lines connecting the corners represent θ i 's with mixedmembership in 2 communities.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In maximum coverage problems, for instance, it is quite reasonable to assume that the covered area is only a function of the chosen items and their states.
Sentence 2: the gain can be computed as ḡi (y) = f (dom (y) ∪ {i} , φ) − f (dom (y) , φ), where φ is any state such that φ ∼ y and φ[i] = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We set L = 3 for all the experiments and use the same hyperparameters in both algorithms.
Sentence 2: the stick-breaking prior parameter γ is set to 1.0; the symmetric Dirichlet prior parameter for the topics is set to 1.0; the prior for level proportions is skewed to favor high levels (50, 20, 10).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our results showed that for the first time an ADP algorithm (CBMPI) performed extremely well in both small 10×10 and large 10×20 boards and achieved performance either better (in the small board) or equal with considerably fewer samples (in the large board) than the state-ofthe-art CE methods.
Sentence 2: the best policy learned by CBMPI obtained the performance of 51, 000, 000 lines on average, a new record in the large board of Tetris.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If additionally is strongly convex over the path of the iterates ↵ t then there exists ⌧ > 0 and ⌫ > 0 such that for all t > ⌧, Illustration of the abstention stumps on a variable X.
Sentence 2: this theorem holds for the exponential loss (u) = exp(u) and the logistic loss ) since they are strongly convex over the compact set containing the ↵ t s. We first define a family of base hypotheses, abstention stumps, that can be viewed as extensions of the standard boosting stumps to the setting of classification with abstention.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Moreover, even though the infimum above is not attained in general, the problem in W resulting after partial minimization over D admits a minimizer.
Sentence 2: since the first term in (2.1) is independent of D, we can first optimize the second term with respect to D. we can compute the infimum In this way we could end up with an optimization problem in W only.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In our experiments, we are also able to predict surprisingly well the relative performances of trained residual networks based only on their initialization hyperparameters, in a variety of settings.
Sentence 2: we find that the quality of initialization for tanh resnets is determined by trainability (how much gradient explosion on average) while that for (↵-)ReLU resnets is determined by expressivity (how far can two different input vectors be pulled apart) (see Section 6).
Relationship: entailment
Domain: NeurIPS
Sentence 1: One-point Gradient Estimates [FKM05]: Suppose one has to estimate the gradient of a function f : R p → R at a point w ∈ R p via a single query access to f . [FKM05] showed that one can approximate f (w) by p β f (w + βu)u, where β > 0 is a small real parameter and u is a uniformly random vector from the p-dimensional unit sphere S p−1 = {a ∈ R p : a 2 = 1}.
Sentence 2: more precisely, For finite, nonzero values of β, one can view this technique as estimating the gradient of a smoothed version of f . Given β > 0, define f (w) = E v∼B p [f (w + βv)] where B p is the unit ball in R p . f = f * U βB p is the convolution of f with the uniform distribution on the ball βB p of radius β.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since the vectors u, v used by hash function h u,v are sampled independently, then for a query hyperplane vector w and a database point vector x, Next, we use the following fact proven in [25], where u is sampled as defined above, and θ a,c denotes the angle between the two vectors a and c. Using (4) and ( 5), we get: Hence, when θ x,w − π We note that unlike traditional LSH functions, ours are asymmetric.
Sentence 2: to hash a database point x we use h u,v (x, x), whereas to hash a query hyperplane w, we use h u,v (w, −w).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the Atari benchmark, the system is able to achieve high reward on the task with extremely low sample complexity.
Sentence 2: the model is able to learn the paddle parameters quickly from random trajectories, improving the control precision, and leading to high reward, as shown in Figure 6 Figure 5: Even though the baseline is able to achieve lower MSE over one-step predictions of the dynamics of the Cartpole environment (left), the physics engine-based controller is able to achieve a higher reward very quickly (right).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Specifically, MAML performs the following optimization: where φ i is used to denote the parameters updated by gradient descent and where the loss corresponds to negative log likelihood of the data.
Sentence 2: in the case of supervised classification with inputs {x j }, their corresponding labels {y j }, and a classifier f θ , we will denote the negative log likelihood of the data under the classifier as L(θ, D) = − (xj ,yj )∈D log p(y j |x j , θ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let B be a d q × k matrix whose columns form an orthonormal basis of V . Thus, we have The condition that xS = (1 ± ) x simultaneously for all x ∈ V is equivalent to the condition that the singular values of B T S are bounded by 1 ± .
Sentence 2: by Lemma 2, for m ≥ (2 + 3 q )/(( /k) 2 δ), with probability at least 1 − δ, we have the squared singular values of B T S are bounded by 1 ± , implying that the singular values of B T S are also bounded by 1 ± .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, it is important to stress that the -LDP guarantee established in Theorem 4 is not a panacea, and in particular it is a weaker guarantee provided in a much more challenging setting than just the -LDP guarantee across all users that we provide for a single round of data collection (an easier setting).
Sentence 2: while LDP across all population of users is resilient to any attack based on auxiliary information, LDP across a sub population may be vulnerable to such attacks and additional levels of protection may need to be applied.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Theorem 2 Let x q q−1 ≤ a for q ∈ (1, 2] for any x ∈ X and f * ∈ F. Under Assumptions 1-2, for q ∈ (1, 2], by taking λ = σ 2 = O(n − q 4q+3 ), we have: with confidence at least 1 − δ.
Sentence 2: for q = 1 and x ∞ ≤ a, choosing λ = σ 2 = ( ln p n ) 1 7 , we have: with confidence at least 1 − δ, Here C 1 , C 2 is a constant independent of n, δ. Theorem 2 shows that the excess risk of R(f * ) − R(f z ) → 0 with the polynomial decay and the estimation consistency is guaranteed as n → ∞.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We use the COMPAS recidivism prediction dataset compiled by ProPublica [8], which comprises of information about all criminal offenders screened through the COMPAS (Correctional Offender Management Profiling for Alternative Sanctions) tool in Broward County, Florida during 2013-2014.
Sentence 2: for each offender, it contains a set of demographic features (gender, race, age), the offender's criminal history (e.g., the reason why the person was arrested, number of prior offenses), and the risk score assigned to the offender by COMPAS.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A principled way to tackle this problem is to use Bayesian inference.
Sentence 2: we first compute the posterior distribution over the model parameters, p(θ|D N ) ∝ p(θ) N i=1 p(y i |x i , θ), where D N = {(x i , y i )} N i=1 , x i ∈ X D is the i'th input (where D is the number of features), and y i ∈ Y is the i'th output.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The noise variance has some low intrinsic level , which under ¦ ¡ -but not ¦ ¨is increased by some fraction © of the vibrational energy ¤ .
Sentence 2: ¥ ¦ £ ¦ ¡ ¨¤ E¦ ¢ ¦ T£ ! )
Relationship: entailment
Domain: NeurIPS
Sentence 1: In our sampling scheme, we sample a fixed number of example queries for each possible query DAG structure.
Sentence 2: given a DAG structure with E edges-specified by a vector d = [d 1 , d 2 , ..., d E ] of node out degrees, which are sorted in topological order [42] -we sample edges using the following procedure: First we sample the query target node (i.e., the root of the DAG); next, we sample d 1 out-edges from this node and we add each of these sampled nodes to a queue; we then iteratively pop nodes from the queue, sampling d i+1 neighbors from the ith node popped from the queue, and so on.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We follow the same procedure as RootSIFT-HQE [43] method, by replacing RootSIFT with our learned HardNet++ descriptor.
Sentence 2: we use: (i) weighting of the votes as a decreasing function of the Hamming distance [44]; (ii) burstiness suppression [44]; (iii) multiple assignments of features to visual words [34,45]; and (iv) QE with feature aggregation [43].
Relationship: entailment
Domain: NeurIPS
Sentence 1: We consider the simplified case of a state space comprised of all the parameters relevant to the task.
Sentence 2: the state space includes the identity of the previous action, an indicator as to whether a reward is available in the food magazine, and, as necessary, the number of LPs since the previous reinforcement (for FR) or the elapsed time since the previous LP (for RI).
Relationship: entailment