sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: The exact solution to this problem would require, in the worst case, the solution of |Γ a | × |Γ a ′ | LPs of the form (11) for every pair of actions with conflicting value bounds.
Sentence 2: the resulting equation can be rewritten as: with β ∈ R n , γ ∈ R m and δ ∈ R. By maximizing (or minimizing) the terms associated with the potentially free variables, we can use this form to establish the maximum (and minimum) value that can be attained at b L . Theorem 1. . . , m and β ∈ R m : β i = min j∈Ii β j , i = 1, . . . , m. The maximum achievable value for a local belief point, b L , according to α, is: Analogously, the minimum achievable value is Proof.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Benefits of this untying are twofold: (1) Now the weight matrices W xh • , and W hh • are of size |h| × |x| and |h| × |h| which means that we can use large pretrained word vectors and a small number of hidden units without a quadratic dependence on the word vector dimensionality |x|.
Sentence 2: number of hidden units in a single hidden layer).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Formally, a set function , where the marginal gain F (i | X) is defined as F ({i} ∪ X) − F (X).
Sentence 2: as a concrete example, the graphic matroid of a graph G = (V, E) is generated by the (arbitrarily oriented) edge-vertex incidence matrix U ∈ {0, ±1} (|V|−1)×|E| of G after removing an arbitrary vertex.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Once our model is trained, we can obtain a compact network for faster and more memory-efficient inference by making use of a post-processing step.
Sentence 2: in essence, and by contrast with methods that aim to learn uncorrelated units to prevent overfitting [5,54,40], we seek to learn correlated ones, which can then easily be pruned in a second phase.
Relationship: neutral
Domain: NeurIPS
Sentence 1: More importantly, for the robust matrix decomposition problem considered [1], only the sum of error bound of two matrices (low-rank matrix and the sparse corruption matrix) can be obtained under the assumption of restricted strongly convexity.
Sentence 2: in this paper, we focus on the following questions: under what conditions for the size of the tensor, the rank of the tensor, and the fraction (sparsity level) of the corruption so that: (i) ( 2) is able to recover W * and V * with small estimator error?
Relationship: neutral
Domain: NeurIPS
Sentence 1: Also, as the results show, learning a purely discriminative model at the top level of a DBN gives much worse performance.
Sentence 2: the two types of update for the c th slice of the tensor W •,•,c are then combined by a weighted sum: where λ is a parameter that sets the relative weighting of the generative and discriminative updates, and η is the learning rate.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Problems may arise if the search engine deploys the classifier, and starts featuring search results that are predicted to be newsy for some queries: promoting the search result may lead to a higher CTR, which in turn leads to higher newsiness predictions, which makes the result be featured even more.
Sentence 2: much of the work on model feedback in fields like finance, education, or macro-economic theory has focused on negative results: there is an emphasis on understanding when feedback can happen and promoting awareness about how feedback can interact with policy decisions, but there does not appear to be much focus on actually fitting feedback.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Our result supports the use of a local approximation based on representative states spread over the state space S. This is in line with the quantization strategies used in batch-mode kernel-based reinforcement learning to define the states s j [4,5].
Sentence 2: we compare iKBSF with fitted Q-iteration using an ensemble of 30 trees generated by Ernst et al.
Relationship: neutral
Domain: NeurIPS
Sentence 1: RSALP outperforms established policies: Approaches such as the Max-Weight or 'parametric' ADP with basis spanning polynomials have been previously shown to work well for the problem of interest.
Sentence 2: a non-parametric approach would, in principle, permit the user to select a rich, potentially full-dimensional architecture (e.g., the Haar basis).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We propose to overcome this limitation by using likelihood-free inference approaches (also known as Approximate Bayesian Computation, ABC) to perform full Bayesian inference on single-neuron models.
Sentence 2: many models of interest -such as multi-compartment biophysical models [2], networks of spiking neurons [3] or detailed simulations of brain activity [4] -have intractable or computationally expensive likelihoods, and statistical inference has only been possible in selected cases and using model-specific algorithms [5,6,7].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Finally, alterations in resting state activity patterns have recently been shown to be diagnostic for clinical conditions such as neuropsychiatric disorders [18], and have been shown to alter with increasing age [34].
Sentence 2: experimental results show that one of the primary assumptions of semi-supervised learning, the manifold assumption, holds well for this data, and that the marginal distribution of unlabeled resting state data is observed to augment that of labeled data to consistently improve regression performance.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Special cases When r = 2, the SD is an instance of a kernel Stein discrepancy (KSD) with base reproducing kernel k(x, y) = R (x, z) (y, z) dz.
Sentence 2: moreover, the underlying principle used to generalize the KSD could also be used to develop fast alternatives to maximum mean discrepancies in twosample testing applications [6,13].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Constant memory backprop through reversibility Recent work developed reversible versions of residual networks (Gomez et al., 2017;Chang et al., 2017), which gives the same constant memory advantage as our approach.
Sentence 2: for reference, a neural net with a single hidden layer of 300 units has around the same number of parameters as the ODE-Net and RK-Net architecture that we tested.
Relationship: neutral
Domain: NeurIPS
Sentence 1: With the added layers for the classifier, the conjugate-exponential character of the model is retained, sustaining the ability to perform VB or MCMC inference with analytic update equations.
Sentence 2: for Gibbs rounds 16, 32 and 64 the corresponding PSNR values were 26.78 dB, 28.46 dB and 29.31 dB.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the code-based approach, we simply ignore the fact that the quantized data actually intervals and them precisely in same way as full-precision Recovery of cosine similarity is performed by means of the estimator in §2 with z, z replaced by q, q .
Sentence 2: the plots highlight the reduction in bias compared to ρ lin and the fact that the variance is decreasing in ρ as for b = ∞.
Relationship: neutral
Domain: NeurIPS
Sentence 1: A portfolio is characterized by a vector of investment allocations, w = (w 1 , .
Sentence 2: under this model, the covariance matrix of the stock returns is given by ) where B = [b jk ] is a d × 3 matrix consisting of the factor loadings, Σ f is the covariance matrix of the three factors, and σ 2 j is the variance of the noise ε i .
Relationship: neutral
Domain: NeurIPS
Sentence 1: To simplify expressions, we will often omit the arguments from φ(x, z) and ψ(z, y), and use S and ¬S for the events [S(z, y) = 1] and [S(z, y) = 0].
Sentence 2: the relaxation parameter β provides a trade-off between faithfulness to the original objective (large β) and tractability (small β).
Relationship: neutral
Domain: NeurIPS
Sentence 1: For a semi-deterministic approach, when the dynamics matrix A is block-diagonal, a simple heuristic is given in Remark ??
Sentence 2: there are two key challenges with these approaches: the first is ensuring the scalability of the model to large scale phenomena, which manifests due to the fact that the hyperparameter optimization problem is not convex in general, leading to methods that are difficult to implement, susceptible to local minima, and that can become computationally intractable for large datasets.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We are interested in deriving the FIM, J(s), for our correlated neural population encoding the stimuli s. The significance of the FIM is that the inverse of the FIM provides a lower bound on the covariance matrix of any unbiased estimator of s and also expresses the asymptotic covariance matrix of the maximum-likelihood estimate of s in the limit of large K 1 .
Sentence 2: [3] found that subjects' performances were significantly better in the minimal color change condition than in the single probe condition.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We will seek an interval regression tree model T : R p → R that minimizes the total hinge loss on data set S: where ∈ R + 0 is a hyperparameter introduced to improve regularity (see supplementary material for details).
Sentence 2: in the typical supervised regression setting, we are given set of learning examples, each associated with a real-valued output.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As for parameter τ , it can be seen as an upper bound for P x∼P [x ∈ supp R] since in this case, we have supp R ⊂ {R(x) = 1}.
Sentence 2: we further focus solely on the last two terms and, similar to the previous case, provide a result based on both the L 1 and χ 2 distances.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We introduced the teaching risk, a quantity that depends on the worldview of the learner and the true reward function and which (1) measures the degree to which policies which are optimal from the point of view of the learner can be suboptimal from the point of view of the teacher, and ( 2) is an obstruction for truly optimal policies to look optimal to the learner.
Sentence 2: the discussion in the last section shows that, under our assumptions on how L learns, a teaching scheme in which T solely provides demonstrations to L can generally, i.e., without any assumption on the teaching risk, not lead to reasonable guarantees on the learner's performance with respect to the true reward.
Relationship: neutral
Domain: NeurIPS
Sentence 1: • We numerically demonstrate the effectiveness and robustness of our algorithms in recovering the minimizer of ridge logistic regression on several real datasets (Figures 1 and 2).
Sentence 2: the following theorem indicates that if the blocks of the augmented matrix of {A i (w)} (see Assumption 3) are sampled based on block norm squares or block partial leverage scores with large enough sampling size, (C1) or (C2) holds, respectively, with high probability.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Sections 4 and 5 give precise statements and proofs of the results in Section 3.
Sentence 2: our estimator undersmooths the density, and converges at a slower rate of O n −β/(β+d) when β < d (and the parametric rate O n −1/2 when β ≥ d), but obeys an exponential concentration inequality, which is not known for the estimators of [8].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Given a proposal q(θ |θ), the MH algorithm starts its chain at a user-defined θ 0 , then at iteration k + 1 it proposes a candidate state θ ∼ q(•|θ k ) and sets θ k+1 to θ with probability while θ k+1 is otherwise set to θ k . The core of the algorithm is thus to draw a Bernoulli variable with parameter α = α(θ, θ ) for θ, θ ∈ R d . This is typically implemented by drawing a uniform u ∼ U [0,1] and checking whether u < α.
Sentence 2: such a process is called the determinantal point process associated to kernel L and base measure µ.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We let kwk denote the Euclidean norm of w, and supp(w) denotes the support of w. For any vector w 2 R p , [8] defined the overlapping group norm as We also introduce the notion of "group-support" of a vector and its group-`0 pseudo-norm: where a Gi satisfies the constraints of (2).
Sentence 2: our result holds for arbitrarily large , and arbitrary group structures.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 7  Spectral algorithms for Kalman filters typically use the left singular vectors of the covariance between history and future features as a basis for the state space.
Sentence 2: to address this problem, we present a new view of dynamical system learning: we show how to learn dynamical systems by solving a sequence of ordinary supervised learning problems, thereby allowing users to incorporate prior knowledge via standard techniques such as L 1 regularization.
Relationship: neutral
Domain: NeurIPS
Sentence 1: . . , cm, ym if there exists a threshold Θ such that yi = sign(ci • w − Θ) for i = 1, . . . , m. We define sign(z) = 1 if z ≥ 0 and sign(z) = 0 otherwise.
Sentence 2: we say that a vector w ∈ R n linearly separates the list c1, y1 , .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Following the usual set-up of composite optimization [10], no smoothness assumption is made on M and hence on f or g. Many learning problems are formulated as convex optimization problems, and hence by duality as saddle-point problems.
Sentence 2: . . , i m ∈ I from the probability vector (π i ) i with replacement end for end for Output: Approximate solution (x, y) (x old(i) , y old(i) ), and we consider the estimate of B(x, y): πi B i (x, y) − 1 πi g i , which has the correct expectation when i is sampled from I with probability π.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Consider the following class of exchangeable ranking functions f (x) where the score for the ith document is given by: where b(x i ) is the score provided by the base ranker for the i-th document, and S k are pairwise functions ("kernels") applied to x i and x j .
Sentence 2: in the following, we extend the theory above to the partially symmetric case (proof in Appendix B).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Similarly, the presented bound for r ∞ also implies sparsity, though it is the sparsity of w k − w. In fact, this is directly related with the sparsity of w k , considering that the classifiers can be postprocessed to subtract their average simultaneously whilst the classification decision won't change for any possible input.
Sentence 2: within each iteration, a portion (ρ) of nonzero entries of w, whose magnitudes are relatively small in comparison with the others, will be directly set to zero and shall never be activated again.
Relationship: neutral
Domain: NeurIPS
Sentence 1: , the shortest-distance from the initial states of to its final states using the shortest-distance algorithm described in the previous section. .
Sentence 2: the -sum and ¡ -multiplications of two transducers c g and c w are defined for each pair E 4 ¤ ~7 R by: This section introduces rational kernels, presents a general algorithm for computing them efficiently and describes several examples of rational kernels.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We can also show that combining graphs improves labelled sample complexity (Claim 7, Suppl.).
Sentence 2: we investigate LS labelling [16]; given a graph G, LS labelling K LS ∈ K(G) is defined as LS labellinghas high Rademacher complexity on a large class of graphs, in particular Corollary 4.3.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 3(b), we added a curve labeled squared MLP with Ensemble, which appears to be better than the MLP ensemble though it is the same curve with softmax scores scaled by simply squaring them.
Sentence 2: second, we expand this loss into the Objectosphere loss, which requires the samples of D c to have a magnitude above a specified minimum while driving the magnitude of the features of samples from D b to zero, providing a margin in both magnitude and entropy between known and unknown samples.
Relationship: neutral
Domain: NeurIPS
Sentence 1: If we believe that P "closely approximates" P f , and know the probability p xi i ≡ P(M xi a→i = 1 for all a ∈ C(i)) that x i is in the consistent domain for variable i of f , for every variable i, clause a and possible assignment x i , we can use them to dynamically order both the variables and the values they can take during search.
Sentence 2: hsu and McIlraith [11] give an intuitive explanation of the behavior of SP and BP from the perspective of traditional local search methods.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Color constancy methods perform worst on images that are dominated by a small number of materials with ambiguous chromaticity, and our results indicate that end-to-end training increases the reliability of our estimation method in these cases.
Sentence 2: while pschophysical experiments have demonstrated that the human visual system is remarkably successful at achieving color constancy [1], it remains a challenging task computationally.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The straddle score for a point may be negative, which indicates that the model currently estimates the probability that the point is on a boundary is less than five percent.
Sentence 2: the ideas of root bracketing and function transversal are not well defined [2]; given a particular bracket of a continuous surface, there will be an infinite number of solutions to the equation f ( x) − t = 0, since the solution in multiple dimensions is a set of surfaces, rather than a set of points.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In this section, we study the infinite horizon LQR problem specified by where toeplitz(r, c) denotes the Toeplitz matrix with first row r and first colum c. This is the same problem studied in [17, §6] (for n x = 3), where it is noted that such dynamics naturally arise in consensus and distributed averaging problems.
Sentence 2: though better suited to optimization than (2), which is almost surely infinite, this integral cannot be evaluated in closed form, due to the complexity of J(•|✓) w.r.t.
Relationship: neutral
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: [2] and applied in [11,4,5,7] to various mean-reward distributions are variants of the 1-failure strategy where each arm is played until the first failure, called a run.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In practice, sEM can take a minibatch of data instead of a single datum per iteration, but we stick to a single datum for cleaner presentation.
Sentence 2: var t [μ t ] is always quadratic with respect to the step size ρ t , so we divide it by ρ 2 t to cancel the effect of the step size, and just study the intrinsic variance.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For all x ∈ U ∩ {x : Finally, we note that the required number of iterations per-stage t for finding an accurate solution depends on an unknown constant c and sometimes θ.
Sentence 2: by adding a strongly convex prox function in terms of u with a smoothing parameter µ into the definition of f (x), one can obtain a smoothed approximation of the original objective function.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We apply this logic to all comparisons, where the goal is to determine the heaviest path edge e and to compare it to the candidate e. If we are lucky enough that all of these comparisons are conclusive, the amount of work we save is related to how much data was used in computing the confidence intervals -the rest of the data for the attribute-pair that is represented by the eliminated edge can be ignored.
Sentence 2: we then proceed to describe its use in the case where some edge weights are known not exactly, but rather only to lie within a given interval.
Relationship: neutral
Domain: NeurIPS
Sentence 1: given Ξ i , and the null matrix of size n × k. Moreover, by singular value decomposition, we , for i = 1, . . . , n. Using the fact that V (p) T V (p) = I, and combining the above equations, we get ) T V (p) Λ (p) −1 .
Sentence 2: we start by factorizing Ξ using SVD, obtaining Ξ = VΛU T where V ∈ R n×n is an unitary matrix, Λ ∈ R n×s is a rectangular diagonal matrix with nonnegative real numbers on the diagonal with λ 1,1 ≥ λ 2,2 ≥ • • • ≥ λ n,n (the singular values), and U T ∈ R s×n is a unitary matrix.
Relationship: neutral
Domain: NeurIPS
Sentence 1: By zooming in and drawing many more samples we would see the same behavior in both plots.
Sentence 2: at any location x, the latent function f (x) conditioned on past observations D n is then Gaussian with marginal mean µ n (x) and variance v n (x) given by where k n (x) is a vector of cross-covariance terms between x and {x 1 , .
Relationship: neutral
Domain: NeurIPS
Sentence 1: p k can be identified with a probability distribution on Ω, because l p l k = lm γ lm k = 1, and γ k can be viewed as a joint probability distribution on Ω 2 with marginal distributions Q k and p k . We can eliminate variables p 1 , p 2 by substituting p k with γ k using the last constraint, so that γ 1 , γ 2 are the only decision variables.
Sentence 2: note that ω l are parameters, but not decision variables, thus ω l − ω m can be computed before solving the program.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Interestingly, all methods quickly found good configurations of the hyperparameters as shown in Figure 2(left), with BOHAMIANN reaching the validation performance of the manually-tuned baseline ResNet after 104 function evaluations (or approximately 27 hours of total training time).
Sentence 2: obs ) and n k = |D k | points have already been evaluated for function f k . BO then requires a probabilistic model p(f | D) over the K functions, which can be used to transfer knowledge from related tasks to the target task t (and thus reduce the required number of function evaluations on t).
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the proposed framework, modal decomposition is first obtained in terms of learned observables g; thus, the values of g must be back-projected to the space of the original measurements y to obtain a physically meaningful representation of the dynamic modes.
Sentence 2: we focus on a (possibly nonlinear) discrete-time autonomous dynamical system where M denotes the state space and (M, Σ, µ) represents the associated probability space.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Though an improvement upon the naive message passing scheme, backpropagation remains biologically implausible because it would require rapid transmission of information backwards along the axon, which has not been observed [8,28].
Sentence 2: we assume that a critic exists that broadcasts the TD error, so a module's local information would consist of its input x i , which is not necessarily a Markov state representation, its output a i , and the TD error.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Two variables are called equivalent if there is a bijection from Unify(X) to Unify(Y ) such that for any Z 1 ∈ Unify(X) and its image Z 2 ∈ Unify(Y ), Z 1 and Z 2 always occur together.
Sentence 2: we define a set of rules that look only at the logical representation to identify models for which exact efficient inference is possible.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This parametrization results in the new objective function: In the objective function, although we minimize the discrepancy between the transformed labels and the predicted labels for only the labeled points in the test domain, we put a regularization term on the transformation for all X te to ensure overall smoothness in the test domain.
Sentence 2: sMS with active selection is better than SMS with random selection, especially in the Active Learning result.
Relationship: neutral