sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: A distinctive feature in convolutional phase retrieval is that L is not a polynomial.
Sentence 2: hence, it may be challenging to posit a Q L dec which "recouples" back to L. in the existing form, we need to tolerate an approximation error as L ̸ = L. By the triangle inequality, ( As discussed above, the sup w∈S L(a, w) can be sharply controlled via its moments bound in (19).
Relationship: entailment
Domain: NeurIPS
Sentence 1: For a new problem, the input may be encoded differently (e.g., different image size, different pixel ordering, different color representation), but there is a transformation T that maps this problem into the same latent space as the previous problem so that the prior solution can be re-used.
Sentence 2: for each problem category i = 1, 2, . . . , N , there is a latent problem space Λ i and a solver S i : Λ i → Y i .
Relationship: entailment
Domain: NeurIPS
Sentence 1: To construct directionally selective place field estimates in 2D we took inspiration from recent analyses of 2D phase procession [22] conditioning the responses on the main direction of motion within the place field.
Sentence 2: we used our estimation of a traditional 2D place field to define a region of interest (ROI) that covers 90% of the field for each cell (Fig.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We propose a balance-based approach that, like the existing weighting-based methods, also reweights the historical data to make it look as though they were generated by the policy being evaluated or learned and potentially denoises outcomes in a doubly robust fashion, but rather than doing so circuitously via a plug-in approach, we do it directly by finding weights that optimize for balance between the weighted data and the target policy in the given, finite sample.
Sentence 2: we formalize balance as a discrepancy between the reweighted historical covariate distribution and that induced by the target policy and prove that it is directly related to the worst-case conditional mean square error (CMSE) of any weighting-based estimator.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, it is straight-forward to sample from this distribution without recourse to f (x) or p(dx).
Sentence 2: the second term can be accessed through the law of total probabilities: where the first term [dp | φ 1:n ] is the stick-breaking construction and the term [φ 1:n | X, θ] can be targeted with a Gibbs sampler.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Two inconvenient facts make it difficult to realize this goal in a computationally efficient manner: 1) model complexity -we typically need a large number of parameters to accurately model a system's response p(r t |{ x t }, {r t−1 }, θ); and 2) stimulus complexity -we are typically interested in neural responses to stimuli x t which are themselves very high-dimensional (e.g., spatiotemporal movies if we are dealing with visual neurons).
Sentence 2: it is computationally challenging to 1) update our a posteriori beliefs about the model parameters p( θ|{r t }, { x t }) given new stimulus-response data, and 2) find the optimal stimulus quickly enough to be useful in an online experimental context.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To gain some intuition on the behavior of these interpolationnorms, first note that for any p ≥ 1 and any v ∈ R m it holds that max In other words, the interesting range of t for the Next, we prove a theorem which states that interpolating a p-norm with the ∞-norm is approximately equivalent to restricting that p-norm to the absolutely-largest components of the vector.
Sentence 2: the 1 − ∞ interpolation norm with parameter t (with t chosen to be an integer in [1, m]) is precisely equivalent to taking the sum of the absolute values of the t absolutely-greatest elements of the vector.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Next, we provide a mechanism for bounding the Gaussian width w(Ω R ) of the unit norm ball in terms of the Gaussian width of a suitable cone, obtained by shifting or translating the norm ball.
Sentence 2: the result involves taking any point on the boundary of the unit norm ball, considering that as the origin, and constructing a cone using the norm ball.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we design, to the best of our knowledge, the first online learning algorithm for the triggering functions in the nonparametric regime.
Sentence 2: we tackle the challenges mentioned above, and the only assumption we make is that the triggering functions f i,j (t)s are positive, have a decreasing tail, and that they belong to an RKHS.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let us imagine that the gambler, with the benefit of hindsight, considers what could have happened had he always bet a constant fraction of his wealth on heads.
Sentence 2: a simple exercise shows that his wealth would become (1 ) This is optimized at = G T , which gives a simple expression in terms of KL-divergence for the maximum wealth in hindsight, exp , and the former is well-approximated by exp(O(G 2 /T )) when G is not too large relative to T . with knowledge of the final G, a naïve betting strategy could have earned the gambler exponentially large winnings starting with constant capital.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Linear regression (shown as the dotted lines in Figure 1(a)) gives the function y = 0.0094x + 0.3656 for Diverse (R 2 = 0.9206, p = 0.0024) and y = 0.0050x + 0.3542 for Uniform (R 2 = 0.8712, p = 0.0065).
Sentence 2: the linear approximation for the winning rate of Diverse increases roughly twice as fast as the one for Uniform as the number of agents increases.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Theorem 4 implies the following "worst-case" upper bounds on influence, which only depends on |A|: Theorem 6.
Sentence 2: the influence of A ⊆ V in an independent cascade model satisfies if λ ∞ < 1, we have Note that when λ ∞ > 1, the bound (8) exceeds n for all large enough n, so the bound is trivial.
Relationship: entailment
Domain: NeurIPS
Sentence 1: PLANET for high-dimensional datasets, we measured the training time on a series of synthetic datasets parameterized by p. For each dataset, approximately p 2 features were categorical, while the remaining features were continuous.
Sentence 2: from Figure 4, we see that, YGGDRASIL scales much more effectively as p increases, especially for larger D. for D = 15, YGGDRASIL is initially 3× faster than PLANET for p = 500, but is more than 8× faster for p = 4000.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The assumption r k=1 n −α k = o(1) is tantamount to saying that the number of tiny communities cannot be too large (e.g., the number of polylogarithmic-size communities cannot be a power of n).
Sentence 2: one needs to have mostly large communities (growing like n ǫ , for some ǫ > 0) for this assumption to be satisfied.
Relationship: entailment
Domain: NeurIPS
Sentence 1: With these goals in mind, we devise an empirical approach to generate the pairs (x ni , y ni ) ∈ N . The main idea is to sample a series of candidate neighborhoods for each instance x ni , evaluate their relative success at predicting the training instance's label, and record the best candidate.
Sentence 2: for instance x ni , we first compute its proximity to the M − 1 other training images in the feature space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the sample compression setting, each classifier is described by a subset S i of the training data, called the compression set, and a message string σ that represents the additional information needed to obtain a classifier.
Sentence 2: in this setting, there exists a reconstruction function R that outputs a classifier R(σ, S i ) when given an arbitrary compression set S i and a message string σ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This risk-sensitive MDP becomes equivalent to the robust MDP having the objective (2) when the objective of the risk-sensitive MDP is to find the optimal π with respect to an iterated conditional tail expectation (ICTE) [13]: where ICTE (N ) α denotes the ICTE defined for N decision epochs with parameter α.
Sentence 2: iCTE (N ) α is defined recursively with conditional tail expectation (CTE) as follows [13]: where we define The right-hand side of (4) evaluates the CTE of this random CTE is also known as conditional value at risk or average value at risk and is formally defined as follows for a random variable Y : where V α ≡ min{y | F Y (y) ≥ α}, and F Y is the cumulative distribution function of Y .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The proof uses the PAC-Bayes bound, which states that for all prior distributions P, Pr (VQ: KL(eQlleQ) ~KL(QIIP) A choice of the "posterior" Q completes the proof.
Sentence 2: the Q we choose depends upon the direction w, the margin 'Y, and the stochastic error E. Q equals P in every direction perpendicular to w, and a rectified Gaussian tail in the w direction 2 • The distribution of a rectified Gaussian tail is given by R(p,) = 0 for x < p, and R(p,) = F(p,~.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Due to this equivalence between saliency and discrimination, the principle of discriminant saliency can be easily translated into an optimality criteria for the design of saliency algorithms.
Sentence 2: it is naturally formulated as an optimal feature selection problem: optimal features for saliency are the most discriminant features for the one-vs-all classification problem that opposes the class of interest to all remaining classes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The rich body of work on welfare economics offers several axioms to characterize the set of all welfare functions that pertain to collective rationality or fairness.
Sentence 2: any such function, W, must satisfy the following axioms [Sen, 1977;Roberts, 1980]: if everyone is better off under b 0 , then W should strictly prefer it to b.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Natural candidates for parametrizing the agent's dynamics model are Bayesian neural networks (BNNs) [19], as they maintain a distribution over their weights.
Sentence 2: this allows us to view the BNN as an infinite neural network ensemble by integrating out its parameters: we utilize a BNN parametrized by a fully factorized Gaussian distribution [20].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In general each time that d i=1 u 2 i t∈M∪U x 2 t,i ≤ s u 2 with s small enough, it is possible to show that, with an optimal tuning of r, this bound is better of the Perceptron's one.
Sentence 2: using a proof similar to the one in [1], in the conservative version of this algorithm, it is enough to have s < M R 2 2d , and to set r = sM R 2 M R 2 −2sd .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the second, simpler model, that we are also interested in here, there is no side-information, but the agent is given a model that describes the possible relations between the arms' payoffs.
Sentence 2: in "linear bandits" [10,8,11,12], each arm a ∈ A is associated with some d-dimensional vector m a ∈ R d known to the agent.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 1. and Ω = ℜ r+ , leads to MIL versions of the Quadratic-Programming-SVM [3].
Sentence 2: the optimization problem reduces to: min where T is the within class scatter matrix, µ j = 1 rj X j e is the mean for class j. X j ∈ ℜ rj×n is a matrix containing the r j representative points on ndimensional space such that the row of X j denoted by b i j = B i j λ i j is the representative point of bag i in class j where i = {1, . . . , r j } and j ∈ {±}.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The collection of all permutations of n objects forms a non-abelian group under composition, called the symmetric group of order n, denoted as S n . Let S n−k denote the subgroup of S n consisting of all permutations whose first k positions are fixed: ( The right coset S n−k π = {σπ|σ ∈ S n−k } is a subset of permutations whose top-k objects are exactly the same as in π.
Sentence 2: • All the probabilistic model based methods are better than BordaCount, the heuristic method.
Relationship: neutral
Domain: NeurIPS
Sentence 1: To circumvent these difficulty, we devised a modified algorithm called PMED-Hinge (Algorithm 1) that involves extra exploration.
Sentence 2: assuming that each convex optimization yields an optimal value, such an alternate optimization Algorithm 3 Cutting-plane method for obtaining {q i,l } on Line 12 of Algorithm 1 1: Input: the number of iterations S, nominal constraint {θ i }, .
Relationship: neutral
Domain: NeurIPS
Sentence 1: In many real world applications, this property may be crucial.
Sentence 2: ( 14) can be re-written as follows The key point of this paper, is to note that the matrix K = K ij = K(x i , x j ), i, j = 1, . . . , N , is the data affinity matrix, and that K(x i , x j ) is a Gaussian RBF kernel function.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The distinction here is significant, as junction points where two decision boundaries meet might actually have a very large (or infinite) curvature (even in linear classification settings), and the curvature condition in Corollary 1 typically does not hold for this global curvature definition.
Sentence 2: hence, for semi-random noise that is mostly random and only mildly adversarial (i.e., the subspace dimension is small), affine classifiers remain vulnerable to such noise.
Relationship: neutral
Domain: NeurIPS
Sentence 1: So we can compute local minimums in successive regions [s l , s l+1 ] (l = 1, . . . , d s − 1) and get the global minimum over region [c ik , e ik ] by comparing all local optima.
Sentence 2: for those two problems, we also compare our proposed coordinate descent method described in Table 1 with some off-the-shelf solvers such as the CVX solver [11] with respect to the running time.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This material is based upon work supported in part by the U.S. National Science Foundation Grant 1330008, by the Department of Homeland Security, Science and Technology Directorate, Office of University Programs, under Grant Award 2013-ST-061-ED0001, by ONR Grant 50202168 and US AF contract FA8650-14-C-1728.
Sentence 2: nearly all previous approaches require solving a non-convex problem and resort to alternating optimization [25,19] or greedy approximation [11].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Specifically, the natural modification of CAL produces a method that (in the realizable case) makes the same number of label requests as before, except that now it makes zero mistakes, since CAL will request a label if there is any uncertainty about its label.
Sentence 2: the following algorithm is essentially taken from [DHM07,Han11], adapted here for this streambased setting.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We showed in Section 3 that SE is more expressive than our proposal.
Sentence 2: another, secondary, motivation comes from the recent work of [8], in which the authors learn word embeddings from free text, and some 1-to-1 relationships between entities of different types, such "capital of" between countries and cities, are (coincidentally rather than willingly) represented by the model as translations in the embedding space.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the following theorem statement, I(•) denotes the indicator function, π(a|x) the probability that the logging policy chooses action a on input x, and V h π our estimator as defined by Equation 2 based on parameter τ . Lemma 3.1.
Sentence 2: the techniques reported here lift these restrictions, allowing the learning of a policy for choosing actions given features from historical data where no randomization occurred or was logged.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Major differences in task design complicate direct comparisons among the studies.
Sentence 2: while it seems sub-optimal, and thus mysterious, that humans have an underestimated prior reward expectation, our simulations show that an underestimated prior mean helps to maximize long-term gain, if the observer assumes volatility when reward rates are stable, and utilizes a softmax decision policy instead of the optimal one (obtainable by dynamic programming).
Relationship: neutral
Domain: NeurIPS
Sentence 1: To do so, for each time t, we build a weighted bipartite graph where each human decision maker v j is connected to each individual (x i (t), z i (t)) with weigh: otherwise, Note that the maximum weighted bipartite matching is the optimal assignment as defined by Eq.
Sentence 2: moreover, they also reveal that our algorithms benefit from higher diversity across the pool experts, being able to ensure fairness even if a significant percentage of judges are biased against a group of individuals sharing a sensitive attribute value (e.g., race).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The best k-group approximation problem is the discrete optimization problem arg min where the goal is to compute the projection of a vector y onto a union of subspaces each defined by a subcollection of k groups.
Sentence 2: for the unit ball of Ω LGS-OWL , we can exploit the fact that the atoms of Ω LGS-OWL are obtained from Ω LG(k) across different k at different scales.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We then perform binary classification for each noun-phrase , where w k is the weight vector corresponding to the class N P k and σ denotes the sigmoid function.
Sentence 2: competitive results are obtained for CompCap trained using in-domain and out-of-domain data, which suggests the benefit of disentangling semantics and syntax, as the distribution of semantics often varies from dataset to dataset, but the distribution of syntax is relatively stable across datasets.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Nonnegative matrix factorization (NMF) is a popular data analysis method, the objective of which is to approximate a matrix with all nonnegative components into the product of two nonnegative matrices.
Sentence 2: as the overall mfNMF can be solved by optimizing each individual factor in an SMS problem, we focus here on the case where the sparsity regularizer of each factor is introduced in (4), to solve max As we have mentioned, the typical choice of (•) as the matrix 1 norm is problematic in (10), as if we treat each column of X as a point on a probability simplex, as their elements are nonnegative and sum to one, then we can induce a sparse regularizer from the Dirichlet distribution.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Although the act of recognising the tennis ball is seemingly independent of our intuition of Newtonian dynamics [31], very little of this assumption has yet been captured in the end-to-end models that presently mark the path towards artificial general intelligence.
Sentence 2: deep neural networks can parameterize the transition and emission distributions of different variants of deep state-space models [8,16,19].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Given a particular state, an action modifies the setting to a subset of the hidden variables; therefore, an action can also be defined as a setting to all the hidden variables Y =y .
Sentence 2: even though reinforcement learning's policy requires it to be greedy with respect to the q-function, we observe that it is able to better escape the random initial configuration than the Metropolis-Hastings method.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In our experiments, iVSGPR shows better performance than the direct adoption of stochastic variational inference to solve variational sparse GPs.
Sentence 2: a Gaussian process GP(m, k) has a dual representation in an RKHS H [12]: there exists µ ∈ H and a positive semi-definite linear operator Σ : H → H such that for any x, x ∈ X , ∃φ x , φ x ∈ H, the mean function has a realization µ in H, which is defined by the reproducing kernel k(x, x ) = ρ 2 φ T x φ x ; the covariance function can be equivalently represented by a linear operator Σ.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We are interested in the multi-agent MDP: , where S is the state space and A i is the action space for agent i.
Sentence 2: π(a|s) is the conditional probability of taking joint action a given the current state s. We define the reward function of joint policy π as an average of the local rewards: That is, R π c (s) is the expected value of the average of the rewards when the agents follow policy π at state s. Besides, any fixed policy π induces a Markov chain over S, whose transition matrix is denoted by P π .
Relationship: neutral
Domain: NeurIPS
Sentence 1: a(λ) ≤ Cλ β for some C > 0 and all λ > 0.
Sentence 2: if H approximates P with exponent β then the rate in Theorem 3.1 becomes n − 4β (2+p)(1+β) which is the rate we established in Theorem 2.3 for the original SVM.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Ic): Aj(Xj) = L:L: ... L: L: ... L:(IT Ak(y))P(ylxl, ... ,XJ)(IT 7I"i(Xi)).
Sentence 2: exact local computation for the sigmoid function, P(y lXI, ... ,XJ) = 1/(1 + exp[-Bo -L: j BjXj]) , requires the full exponential sum.
Relationship: neutral
Domain: NeurIPS
Sentence 1: To show the scalability, we report the running time using 1, 2, 4, 8, 16 machines, respectively.
Sentence 2: in terms of the amount of data (in bits) sent over the network, the communication cost of the cleaning stage grows as O(dnmT ), where d is the number of features selected by the screening stage and n is the sketch size.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For the NIR dataset, we follow the steup for the original dataset: 40 instances for training and 32 instances for test.
Sentence 2: recently, some nonconvex alternatives, such as the bridge penalty, the nonconvex exponential penalty (EXP) [3,8], the logarithmic penalty (LOG) [19,13], the smoothly clipped absolute deviation (SCAD) penalty [6] and the minimax concave plus (MCP) penalty [20], have been demonstrated to have attractive properties theoretically and practically.
Relationship: neutral
Domain: NeurIPS
Sentence 1: ( 7), starting at the stage when the pair entered the race, and ending with the stage when the pair either dropped out or wins the stage.
Sentence 2: this model assumes that each point on the tree is associated with a D dimensional multinomial vector, and each entry of this vector on each branch of the tree evolves independently (thus independent-sites), forward in time, and with mutations occurring at rate λ d on entry d. When a mutation occurs, a new value for the entry is drawn from a distribution φ d , independently of the previous value at that entry (thus parentindependent).
Relationship: neutral
Domain: NeurIPS
Sentence 1: For any δ such that 0 < δ < 1, with probability at least 1 − δ, for any k ∈ {1, . . . , K} e(f, ĥk ) ≤ 2 While the use of data-dependent complexity measures can be expected to yield more accurate bounds and thus better decisions about the number k * of piles to use, it is not without its costs in comparison to the more standard data-independent approaches.
Sentence 2: here our target model f is any function from an input class X into some bounded subset of R. (Frequently we will have X ⊆ R d , but this is not required.)
Relationship: neutral
Domain: NeurIPS
Sentence 1: When the minority classes form compact clusters and the majority class distribution is locally smooth, the method will select examples both on the boundary and in the interior of the minority classes, and is proved to be effective theoretically.
Sentence 2: if we select a point at random from B 2 , the probability that this point is from the minority class is at least In subsection 2.2, we have discussed rare category detection for the binary case.
Relationship: neutral