sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: As a result, we can reduce the problem of synthesizing programs in the Strategy language to the problem of synthesizing branches with predicates over a set of candidate strategies.
Sentence 2: stratEVO has several limitations -it performs search using an evolutionary algorithm which does not incorporate any form of learning, the search does not depend on the actual formula, and local mutations tend to get stuck in local minima (as we show in our experiments in Section 5).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Additionally, and although it achieves 0% error, it is learning very complicated and unintuitive coordinate transformations to represent the data in a linearly separable form.
Sentence 2: they often overlook the fundamental process taking place, where successive layers successively warp the coordinate representation of the data manifold with nonlinear transformations into a form where the classes in the data manifold are linearly separable by hyperplanes.
Relationship: neutral
Domain: NeurIPS
Sentence 1: If Q and P factorize, then each of these two terms further breaks into a sum of individual terms, each quantifying the aspects of the fit of the data point with respect to different parts of the model.
Sentence 2: the free energy for a single sample t can be expressed as the sum where all the free energy pieces f t i, θ derive from the finest decomposition ( 5) or ( 4).
Relationship: neutral
Domain: NeurIPS
Sentence 1: However, a caution should be exercised when using the measure of performance at the bag level as a surrogate for the measure of performance at the individual level: in order to perform well at the bag level, one can simply utilise spatial coordinates and ignore other covariates, as malaria intensity appears to smoothly vary between the bags (Left of Figure 2).
Sentence 2: y a ∼ P oisson( i p a i λ a i ), where λ a i is the underlying pixel incidence rate of malaria per 1000 people that we are interested in predicting.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For this paper, we restrict our case to A, B ∈ R n×n being positive definite matrices so that the objective function is strongly convex.
Sentence 2: one such point set in [0, 1] n is called the (t, m, n)-net in base η, which is defined as a set of N = η m points in [0, 1] n such that any axis parallel η-adic box with volume η t−m would contain exactly η t points.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Due to linearity the PRG Pareto front is convex and easily constructed by visual inspection.
Sentence 2: score 0.76 corresponds to β 2 = (1 − 0.76)/0.76 = 0.32 and score 0.49 corresponds to β 2 = 1.04, so the point closest to the Precision-Recall breakeven line optimises F β for 0.32 < β 2 < 1.04 and hence also F 1 (but note that the next point to the right on the convex hull is nearly as good for F 1 , on account of the connecting line segment having a calibrated score close to 1/2).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Our method supports inequality, equality, and mixed constraints, and to our knowledge this updated ALBO procedure is unique in the BO literature in its applicability to the most general mixed constraints problem (1).
Sentence 2: in the latter iterations stark distinctions emerge, with Slack-AL+optim and PESC, both leveraging L-BFGS-B subroutines, outperforming.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Whereas previous methods single out a single "best" correspondence across images, in a Bayesian framework we are interested in characterizing our knowledge about the unknowns conditioned on the data only, averaging over all possible correspondences.
Sentence 2: the structure from motion problem is this: given a set of images of a scene, taken from different viewpoints, recover the 3D structure of the scene along with the camera parameters.
Relationship: neutral
Domain: NeurIPS
Sentence 1: To demonstrate this, we adapt the proposed DirBN structure to the following models: PFA+DirBN Poisson Factor Analysis (PFA) is a popular framework for topic analysis (DPFA , DPFM , GBN (Zhou et al., 2016) can be viewed as a deep extension to PFA).
Sentence 2: here DirBN is a new hierarchical approach of constructing WDs, detailed as follows.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The last term in Eq.4 is nonnegative and integrates to one over R d , and is thus a proper probability distribution in Ω d . Furthermore, p is the fixed point of operator Φ g p,A , as Φ g p,A {p} = p. The following result shows that both Φ m A and Φ mg p,A are KL contraction operators, and that they are in a sense complementary to each other.
Sentence 2: the KL contraction in Eq.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Step 4 is carried out in Corollary 2, which follows.
Sentence 2: a careful reader may notice that the procedure for FREEREXMOMENTUM involves computing argmin W √ 5ψ(at(w−wt) atηt + g 1:t • w , which may not be easy if the solution w t+1 is on the boundary of W . When the w t+1 is not on the boundary of W , then we have a closed-form update: when w t+1 lies on the boundary of W , it is not clear how to compute it for general W . In this section we offer a simple strategy for the case that W is an In this setting, we can use the standard trick (e.g.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We expect pooling methods one and two to be more powerful than the third pooling method used in the current study, since the Bonferroni and Hommel tests are conservative compared to using  We showed that multivariate K-sample and independence tests can be performed by comparing the univariate distributions of the distances from center points, and that favourable properties of the univariate tests can carry over to the multivariate test.
Sentence 2: as an interesting example we discuss the energy test of ( 25) and ( 1) for the two-sample problem.
Relationship: neutral
Domain: NeurIPS
Sentence 1: . . , π K } that is shared for all {x i } i=1,N B , it is expected that there may be a mixture of π vectors, corresponding to different segments in the image.
Sentence 2: note that the model in (2) may be employed for unlabeled data, and the extension above may be employed for the available labeled data; consequently, all data (labeled and unlabeled) may be processed jointly to infer D. In the above discussion, we implicitly assumed all data D = {x i , y i } i=1,N are used together to infer the dictionary D. in some applications N may be large, and therefore such a "batch" approach is undesirable.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In SVMs, approaches based on optimizing generalization bounds [4], such as the radius/margin-bound [15] or maximal discrepancy criterion [2] have been proposed.
Sentence 2: the partial derivative of 1 2 w T Cw + ℓ T (w) with respect to w i vanishes when w = w ⋆ , so where C T i denotes the ith row of the C matrix.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Under our learning problem setup, after nearly log(km) initial time steps, the proposed algorithm achieves near-optimal performance relative to an oracle algorithm that recommends all likable items first.
Sentence 2: instead of maximizing the expected reward r (T ) , we aim to maximize the expected number of likable items recommended up to time T : where X ut is the indicator random variable for whether the item recommended to user u at time t is likable, i.e., X ut = +1 if p uπut > 1/2 and X ut = 0 otherwise.
Relationship: neutral
Domain: NeurIPS
Sentence 1: To deepen the LSTM, the popular Stacked LSTM (sLSTM) stacks multiple LSTM layers [20]; however, runtime is proportional to the number of layers and information from the input is potentially lost (due to gradient vanishing/explosion) as it propagates vertically through the layers.
Sentence 2: during the online inference when the input presents sequentially, temporal computations can no longer be parallelized and will be blocked by deep computations of each timestep, making these methods potentially unsuitable for real-time applications that demand a high sampling/output frequency.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This is in analog to structured output learning (e.g.
Sentence 2: we can train the LSVM by an iterative algorithm that alternates between inferring h on positive examples and optimizing the model parameter w. Dual form with fixed h on positive examples : Due to its nature of non-convexity, it is not straightforward to derive the dual form for the general LSVM.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Our contributions are the following: (i) we design a randomized hierarchical data structure with which we can do NN search using the comparison oracle (ii) we develop the first lower bound for the search complexity in the combinatorial framework of [1,12], and thereby demonstrate the importance of combinatorial disorder.
Sentence 2: the main question we ask in this paper is to design a (approximate) nearest neighbor retrieval algorithm while minimizing the number of questions to such an oracle.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The first is a function which is a measure of the distance of the current state ω from the solution state ω * (in the next section we will require ω * = arg min ω D(P 0 (x)||P (x|ω))).
Sentence 2: younes [7] gives convergence results when the gradient of the energy ∂E(x; ω)/∂ω is bounded by a term that is linear in ω (and hence unbounded).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The authors thank Yevgeny Seldin for finding a bug in an earlier version of the paper.
Sentence 2: y, s(f ) 2 {0, 1} d is an indicator vector whose i th component indicates whether f is sensitive to its i th input (in particular, s i (f ) = 0 by definition when f (x 1 , .
Relationship: neutral
Domain: NeurIPS
Sentence 1: First, recall that Lemma 3 argues that the regret of just the A(Σ) ∩ B(Σ) game is infinite.
Sentence 2: this paper is concerned with the minimax strategy s * , which is the strategy that minimizes the worst case regret over all possible covariate and outcome sequences in some constraint set, i.e.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Comparing the IQRP to the current state-of-the-art randomized algorithms that were reviewed in Section 2.2 we observe that the IQRP is competitive in terms of the number of passes and appears to outperform these algorithms in terms of the number of IO-passes.
Sentence 2: it is based on the following two ideas: 1.
Relationship: neutral
Domain: NeurIPS
Sentence 1: . . , µ K , and second, construct a ray emanating from a (weighted) center of the polytope and extending through each of the centroids µ k until it intersects with a sphere of radius R k or with the simplex ∆ V −1 (whichever comes first).
Sentence 2: the connection between extension parameters and the thresholding is the following: if the cluster centroid assigns probability to a word smaller than the whole data does on average, this word will be excluded from topic k with large enough m k . the extension parameters can as well be used to control for the sparsity of the inferred topics.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In this graph, one host is "connected" to another host if there is at least one link from a web-page in the first host to a web-page in the other host.
Sentence 2: if, while ascending from v i we enter a vertex c which has previously been visited during the ascent from another vertex w (line 6) then we compute r T (v i , w) as r T (v i , c)+r T (c, w).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Here, we take a pragmatic approach, and constrain ourselves to solutions of the form μ(y , where b µ : R !
Sentence 2: thus our model is misspecified in the sense that there is no function f such that a current prediction y(t) i increased the log-odds of the next prediction by f (y (t) i ), and so this example can be taken as a stretch case for our method.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Building on the Sparse Subspace Clustering (SSC) algorithm [26], we cast the problem as finding a completion of the data so that each complete point can be efficiently reconstructed using a few complete points from the same subspace.
Sentence 2: in the low-rank regime, while LRMC and SSC-lifting have almost zero error for ρ ≤ 0.35, the performance of LRMC quickly degrades for larger ρ's, while SSC-lifting performs well for ρ ≤ 0.6.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Nevertheless, it has to be said that in some of our experiments the model selected by the bounds resulted as accurate as the ones selected by ten-fold cross-validation in terms of test error rate on a separate test.
Sentence 2: 0.025 ± 0.004 0.030 ± 0.007 0.038 ± 0.005 0.036 ± 0.007 0.038 ± 0.005 10 0.020 ± 0.007 0.021 ± 0.007 0.021 ± 0.007 0.025 ± 0.008 0.026 ± 0.008 100 0.020 ± 0.007 0.021 ± 0.007 0.021 ± 0.007 0.025 ± 0.008 0.026 ± 0.008 500 0.020 ± 0.007 0.021 ± 0.007 0.021 ± 0.007 0.025 ± 0.008 0.025 ± 0.005 the comparison with Table 5 points out that the PAC-Bayes bound is not as accurate as Ten Fold cross-validation when it comes to selecting a model that yields a low test error rate.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The posterior distribution of x for given observed syndrome z is derived from the prior p(x) and the conditional p(z | x) by applying the Bayes formula, and the result is where we let The objective of decoder of Gallager codes is to obtain the marginal-posterior-mode (MPM) estimate based on the posterior p(x|z): The MPM estimation provides the Bayes-optimum decoder minimizing expected bit-error probability of the decoding results.
Sentence 2: aG T = O mod 2 holds by definition.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For this model, the theoretical receptive field of the top convolutional layer units is quite big at 505 × 505.
Sentence 2: if we make an extra simplifying assumption that the gradients σ are independent from the weights and g in the upper layers, we can simplify the variance as Var 2 ] = Var[σ p i,j ] = 1/4 is a constant factor.
Relationship: neutral
Domain: NeurIPS
Sentence 1: If not specified otherwise, x next = argmax x a(x) is computed using gradientbased search with multiple restarts (see supplementary material for details).
Sentence 2: for a fully-Bayesian treatment of hyperparameters (summarized here by θ alone), it is desirable to marginalize over hyperparameters and compute the integrated acquisition function: where a(x) depends on θ and all of the observations.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We are given a fixed set of sequentially labeled data pairs (so D 0 = ∅), where each x (n) ∈ R V0 is a data point, and each y (n) is a label taken from a binary set Y ⊂ {−1, +1} V L . For brevity, we will sometimes suppress the sample index n, where it is clear from the context.
Sentence 2: in order to reduce the computational complexity in cases that Σ m is not diagonal, we will approximate the distribution of u m with its factorized ('mean-field') version.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Denote by z t the point played by Algorithm 3 in round t. Suppose we set ⌘ = ↵ 4/3 LRT 2/3 , ✏ = ↵R T and = 2/3 d T 1/3 in Algorithm 3 (assuming T > 2 d 3 so < 1).
Sentence 2: recent advancements on black-box online-to-offline reductions were made in (Kakade et al., 2009;Dudík et al., 2016;Garber, 2017).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The allowable drift is restricted by ensuring that consecutive probability distributions are close in total variation distance.
Sentence 2: [Bar92,BL97] study the problem of learning a subset of a domain from randomly chosen examples when the probability distribution of the examples changes slowly but continually throughout the learning process; they give upper and lower bounds on the best achievable probability of misclassification after a given number of examples.
Relationship: neutral
Domain: NeurIPS
Sentence 1: ( 1) can be rewritten as the Bellman equation The advantage of using the Bellman equation is that it describes the relationship between the value function at one state s and its immediate follow-up states s ∼ p(s |s, a).
Sentence 2: hence, DSDP can be regarded as a special case of NPDP for discrete state systems.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We currently lack an accurate meta-inference algorithm for MCMC samplers that do not employ annealing, and therefore AIDE is not yet suitable for use as a general MCMC convergence diagnostic.
Sentence 2: it is typically intractable to compute output probabilities for sampling-based inference algorithms like MCMC and SMC, because that would require marginalizing over all possible values that the random variables drawn during the algorithm could possibly take.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Concerning ⇤ (c) , rather than explicitly imposing a sparse diagonal via b, we may also draw , with shrinkage priors employed on the ck (i.e., with the ck drawn from a gamma prior that favors large ck ; which encourages many of the diagonal elements of ⇤ (c) to be small, but typically not exactly zero).
Sentence 2: recall that G (c) In the original construction, (4) and ( 5), in which refractory-time violations are not account for, the Gibbs update equations for {U (c) ji } are analytic, due to model conjugacy.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Without losing any generality, we assume the instances are arranged in the order of [U, L], such that The instance selection problem formulated in (6) selects a subset of b instances indexed by Q from U and moves them into the labeled set L. This problem is actually equivalent to partitioning matrix Σ into submatrices Σ L L , Σ U U , Σ L U and Σ U L by reordering the instances in U .
Sentence 2: a kernel function is first learned from a mixture of labeled and unlabeled examples, and then is used to effectively identify the informative and diverse instances via a min-max framework.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In theorem 2 we show that for a fixed t, as the number of points |X| = N tends to infinity, with probability 1, β(t, X) tends to α(t).
Sentence 2: even more importantly, this paper establishes an explicit connection between the point-cloud Laplacian applied to such characteristic functions (weighted graph cuts) and a geometric quantity, which is the weighted volume of the cut boundary.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Since the mix of content in the available pool already incorporates constraints like voice, topicality, etc., we focus on choosing articles to maximize overall click-through rate (CTR), which is the total number of clicks divided by total number of views for a time interval.
Sentence 2: whenever polar articles were in the pool of candidate articles, there was usually a non-polar one in the pool that was more popular than the polar ones across all segments.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Then, there is a single solution to q l log q l PS l + (1 − q l ) log 1−q l PS h = Λ, where q l and 1 − q l are the probabilities that Q assigns to the events from S l and S h , respectively.
Sentence 2: let P (x) be the underlying distribution of the target class, defined over some space Ω.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The hybrid method consisting of SVRG interleaved with Oja's algorithm for Hessian descent also improves upon SGD but does not match the performance of our method or AdaGrad.
Sentence 2: in the limit where ✏ is small the result simplifies: o , then under the settings of Theorem 1 we can conclude that Algorithm 1 will output an ✏-second-order stationary point of f with probability at least 1 within total stochastic gradient and Hessian-vector product evaluations.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Thus, we see that our optimization approach generally achieves competitive results with random features at lower computational costs, and it offers the most improvements when either the base kernel is not well-suited to the data or requires a large number of random features (large D) for good performance.
Sentence 2: in under 0.2 seconds, we whittle down the original feature space to 379 important features.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The number of function evaluations required by the algorithm to achieve this ε-approximation of the maximum defines the sample-complexity.
Sentence 2: a necessary condition for an interval [a, b] to be split is that max(W (a), W (b)) ≥ M − η which means that either a or b or both are η-near-optimal which is the key point of Lemma 2.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Suppose that we are given a hierarchical decomposition of a control task in the form of two MDP\Rs -a low-level and a high-level MDP\R, denoted M = (S , A , T , H , D ) and M h = (S h , A h , T h , H h , D h ) respectively -and a partitioning function ψ : S → S h that maps low level states to high-level states (the assumption here is that |S h | |S | so that this hierarchical decomposition actually provides a computational gain).
Sentence 2: often the reward function R can be represented more compactly as a function of the state.
Relationship: neutral
Domain: NeurIPS
Sentence 1: To derive the submodularity cut, we use the following definition: Definition 3 (γ-extension) Let g : R n → R be a convex function, x ∈ R n , γ be a real number satisfying γ ≥ g(x) and t > 0.
Sentence 2: as stated below, this approach is inefficient even for moderate problem sizes.
Relationship: neutral
Domain: NeurIPS
Sentence 1: First, the fixed objective f gives us a more direct way of tracking the change in comparator through f (u t ) − f (u t+1 ), which might often be milder than u t − u t+1 .
Sentence 2: statistically, using stages guarantees that the monomials added in the previous stage have an opportunity to have their corresponding parameters converge.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the presence of gross errors, the classical Principal Component Analysis (PCA) method, which is probably the most widely used tool for data analysis and dimensionality reduction, becomes brittle -A single gross error could render the estimate produced by PCA arbitrarily far from the desired estimate.
Sentence 2: capital letters such as M are used to represent matrices, and accordingly, [M ] ij denotes its (i, j)th entry.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We examine two special cases: (1) MNNs with real weights (2) MNNs with binary weights (and real bias).
Sentence 2: the output of the network is therefore We assume that the weights are constrained to some set S, with the specific restrictions on each weight denoted by S ij,l , so W ij,l ∈ S ij,l and W ∈ S. If S ij,l = {0}, then we say that W ij,l is "disconnected".
Relationship: neutral
Domain: NeurIPS
Sentence 1: Such models take the form where, for FilterBoost, f t (x) = α t h t (x).
Sentence 2: on each round, a boosting-by-filtering algorithm draws a set of examples from D t via the filter, trains the weak learner on this set, and receives a weak hypothesis h t . Though a batch booster would estimate t using the fixed set S, filtering boosters may use new examples from the filter.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In general, these matrices describe how the state of a dynamical system autonomously evolves with time, A; how inputs affect system states, B; how states are mapped into outputs, C; and how inputs instantaneously affect the system's output D (Astrom and Murray, 2012).
Sentence 2: in order to generate optimal eligibility traces, synapses must include a forward model of the controlled subsystem.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Freund and Schapire [10] showed that MW updates converge to NE empirically in any two-person zero-sum game using the notion of regret; their analysis also yields a simple and beautiful proof of John von Neumann's Minimax Theorem.
Sentence 2: such bad accumulation exists even when ε is arbitrarily tiny, which might be surprising to people not familiar with numerical methods, since they might have the misconception that once the step-size ε is sufficiently small, the discretization would always yield a good approximation of its continuous counterpart.
Relationship: neutral