sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Return-based methods (where return refers to the sum of discounted rewards � t γ t r t ) offer some advantages over value bootstrap methods: they are better behaved when combined with function approximation, and quickly propagate the fruits of exploration (Sutton, 1996).
Sentence 2: 3 Analysis of Retrace(λ) We will in turn analyze both off-policy policy evaluation and control settings.
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Domain: NeurIPS
Sentence 1: Proof Technique As a fundamental step in the proof of Theorem 2 we prove tight asymptotic bounds for high-order norms of the approximation factor.
Sentence 2: for a synthetic set of generated bit-vectors, we show that whenever ε 2 δ the constant hidden by the Θ-notation is at least 0.75 (except for very sparse vectors, i.e.
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Domain: NeurIPS
Sentence 1: high-dimensional), ǫ ∼ N (0, σ 2 I n ), and the columns of X are normalized to have ℓ 2 norm √ n. Given such a linear model, two key tasks are to identify the relevant set of variables and to estimate β with bounded ℓ 2 loss.
Sentence 2: a special case of design matrices that satisfy the Restricted Eigenvalue assumptions are the random design matrices.
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Domain: NeurIPS
Sentence 1: In the full version of the paper, for the specific problem of portfolio selection, where f t (x) = − log(r t • x), we give a faster implementation whose per iteration running time is independent of the number of iterations, using the more sophisticated "online Newton method" of [HKKA06].
Sentence 2: another approach to connecting worst-case to average-case investing was taken by Jamshidian [Jam92] and Cross and Barron [CB03].
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Domain: NeurIPS
Sentence 1: The reward received at transition s a − → s is given by the random variable R(s, a, s ); usually one is interested in the expected value of this variable, which we will denote by r(s, a, s ) or by r(s, a) = E S ∼p(•|s,a) [r(s, a, S )].
Sentence 2: by appropriately defining the rewards one can induce different task decompositions.
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Domain: NeurIPS
Sentence 1: If the discount factor ϑ satisfies Assumption 4.1 and α 2 + ϑ 2 β 2 < 1 2 , then the operator T π is a contraction with respect to • ξ .
Sentence 2: it is known that if ξ is the steady state distribution of the policy π under evaluation, then Π is non-expansive in • ξ [4,2].
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Domain: NeurIPS
Sentence 1: An agent using normal Q-learning in a finite MDP repeatedly observes a state s, chooses a legal action a, and then observes an immediate reward r and a transition to a new state s . The Q-learning equation is given by: ∆Q where γ is a discount parameter, and α(t) is an appropriate learning rate schedule.
Sentence 2: nevertheless Hyper-Q does exhibit substantial reduction in Bellman error, and also significantly exploits PHC in terms of average reward, as shown in figure 3.
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Domain: NeurIPS
Sentence 1: This section shows that the choice of a good parameterisation is not obvious, and that the use of the Riemannian geometry of the simplex [Ama95, GC11] is important in designing Langevin MCMC algorithms.
Sentence 2: note that the β−1 term in Equation 9 has been replaced with β in Equation 10 as the −1 cancels with the curvature terms as detailed in Table 1.
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Domain: NeurIPS
Sentence 1: Existing VMMs address these problems by allowing models to differentiate between contexts at nonconsecutive past timesteps, ignoring intermediate observations [13,22,10,24,4].
Sentence 2: conversely, the frequency of each particular context in the data decreases as its length increases, so it may be difficult to accurately model long-term dependencies without requiring prohibitive amounts of data.
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Domain: NeurIPS
Sentence 1: In the next section we will derive the general (algebraic) nonlinear family which is based on "lifting" the input matrices A, B onto higher dimensional spaces and feeding the result onto the < •, • > machinery developed in this section, i.e., sim(A, B) =< ψ(A), ψ(B) >.
Sentence 2: this dichotomy is probably most emphasized in the area of computer vision, where image understanding from observations involve data instances of images or image sequences containing huge amounts of data.
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Domain: NeurIPS
Sentence 1: For the probability simplex with d vertices, the pyramidal width is actually the same as its width, which is 2/ √ d when d is even, and 2/ d−1/d when d is odd [2] (see Appendix B.1).
Sentence 2: the Frank-Wolfe (FW) optimization algorithm [9], also known as conditional gradient [23], is particularly suited for the setup (1) where M is only accessed through the linear minimization oracle.
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Domain: NeurIPS
Sentence 1: Since the non-zero elements of the optimal solution correspond to the k ⇤ smallest values of , then Equation ( 3) is equivalent to the following quadratic equation in : Solving for and neglecting the solution that does not agree with The above implies that given k ⇤ , the optimal solution (satisfying KKT) can be directly derived by a calculation of according to Equation ( 5) and computing the ↵ i 's according to Equation (1).
Sentence 2: for any nonzero weight ↵ ⇤ i > 0 the following holds: Squaring and summing Equation ( 2) over all the nonzero entries of ↵, we arrive at the following equation for : Algorithm 1 k ⇤ -NN Input: vector of ordered distances 2 R n , noisy labels y 1 , .
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Domain: NeurIPS
Sentence 1: Using the notation above, assume F and κ are well-conditioned, and C(n) and m/n are Ω(log c n) for some suitably large c. The algorithm for the simplified model shown in Figure 1 and that for the bipartite model (appears in [19]) give us an (1/ log 2 n, , O(1/ log n))approximation algorithm w.h.p.
Sentence 2: our goal is to estimate {x i } i∈[n] based on B for suitably large n. Simplified Model.
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Domain: NeurIPS
Sentence 1: . . , ŵ CT , obtained e.g., by ordinary grid-search, our first problem is to provide a theoretical approximation level ε in the sense of (3) 2 .
Sentence 2: given a lower and an upper bound of the score of each validation instance, a lower bound of the validation error can be computed by simply using the following facts: ) Furthermore, since the bounds in Lemma 1 linearly change with the regularization parameter C, we can identify the interval of C within which the validation instance is guaranteed to be mis-classified.
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Domain: NeurIPS
Sentence 1: Multi-task learning (MTL), transfer learning (TL), and domain adaptation (DA) techniques have recently been utilized to leverage the relevant data and knowledge of different domains to improve the predictive power in all domains or one target domain [Pan andYang, 2010, Patel et al., 2015].
Sentence 2: the key question is whether we may integrate available data from all different sources or domains to achieve reproducible disease prognosis based on NGS count data.
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Domain: NeurIPS
Sentence 1: When ∇ 2 f LI n (i.e., LI n − ∇ 2 f is positive semidefinite), if ∆t/t small and ∆t < 2/ √ L, we see that all the roots of (3.2) lie in the unit circle.
Sentence 2: although it has been empirically observed that this method significantly boosts convergence, there is no general theory characterizing the convergence rate.
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Domain: NeurIPS
Sentence 1: Can we recover the column-space of the low-rank matrix L 0 , and the identities of the non-zero columns of C 0 , exactly and efficiently?
Sentence 2: finally, for any matrix X let P I ′ (X) denote the matrix with all columns in I ′c set to 0, and the columns in I ′ left as-is.
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Domain: NeurIPS
Sentence 1: A finite convex combination of copulas is a copula, so r (a) is a copula density.
Sentence 2: 2 shows an average log-likelihood of models trained on training sets with N = 50, 100, 200, 500, 1000, 2000, 5000, 10000 and evaluated on 2000-example test sets (averaged over 10 training and test sets).
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Domain: NeurIPS
Sentence 1: Based on the peaked prior assumption, p(µ) ≈ δ(µ− μ), we have p(x(t), φ) = p(x(t)|µ, φ)p(µ)p(φ) ≈ p(x(t)|φ, μ).
Sentence 2: the intermediate layers demonstrate a multiplicative gain modulation by attention, as observed in primate V4 neurons [8]; and the temporal behavior of the final layer, representing the marginal posterior, qualitative replicates the experimental observation that LIP neurons show noise-dependent firing rate increase when aligned to stimulus onset, and noise-independent rise when aligned to the decision [13].
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Domain: NeurIPS
Sentence 1: The only lower bound of which we are aware is due to [20], which shows a bound of Ω(kd/ε 2 ) for learning mixtures of spherical Gaussians (and hence for general Gaussians as well).
Sentence 2: density estimation has a long history in the statistics literature, where the focus is on the sample complexity question; see [6,7,19] for general background.
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Domain: NeurIPS
Sentence 1: Several recent papers have considered ways to alleviate this burden by incorporating prior knowledge into the metric, either in the setting of K-means clustering [1,2] or spectral clustering [3,4].
Sentence 2: we have observed that for small q, the function to optimize is smoother and thus easier to optimize-in particular, the long plateaus of constant values are less pronounced.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Hence, by grouping kernels generated from the same feature descriptor together and employing the new MKL formulation, classifiers which are potentially well-suited for object categorization tasks can be built.
Sentence 2: the mirror-descent based algorithm presented in the paper not only solves the proposed formulation efficiently but also outperforms simpleMKL in solving the traditional l 1 regularization based MKL.
Relationship: neutral
Domain: NeurIPS
Sentence 1: , 0: = L Xi ,11"i O:Xi ,11"i ' The average parameter value OXi l11" i ' which typically serves as the regularized parameter estimate given a network structure m , is given by where N Xi ,11"i are the cell-counts from data D; E[•] is the expectation.
Sentence 2: given a uniform prior over the network structures, p(m) =const, the posterior distribution p(mID) over the graphs thus becomes increasingly spread out as 0: grows, permitting more variable network structures.
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Domain: NeurIPS
Sentence 1: Suppose player i is to play at s: under perfect recall, player i remembers the sequence of their own states reached, which is the same for all h ∈ s, since they differ only in private information seen by opponent(s); as a result For some history h and action a, we call h a prefix history h ha, where ha is the history h followed by action a; they may also be smaller, so h ha hab ⇒ h hab.
Sentence 2: (i) both required Poker-specific domain knowledge, and (ii) neither were model-free, and hence are unable to learn directly from experience, without look-ahead search using a perfect model of the environment.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Essentially, after spending a certain cost we report a point (BO) or a classification (LSE), but there is no preference on the values of f (x t ) for the points x t chosen before coming to such a decision (in contrast with other notions such as cumulative regret).
Sentence 2: in this paper, we provide a unified treatment of the two via a new algorithm, Truncated Variance Reduction (TRUVAR), which enjoys theoretical guarantees, good computational complexity, and the versatility to handle important settings such as pointwise costs, non-constant noise, and multi-task scenarios.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We consider learning a weight matrix consisting of the weight vectors for m linear predictive models: In this paper, we propose a non-convex multi-task feature learning formulation to learn these m models simultaneously, based on the capped-ℓ 1 ,ℓ 1 regularization.
Sentence 2: moreover, the parameter estimation error decreases quickly and converges in a few stages.
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Domain: NeurIPS
Sentence 1: Observe that when the matching graph M is such that deg M (u) > T n for some user u ∈ B ∪ G, no algorithm will be able to uncover all M matches in expectation, since Steps (1 B ) and (1 G ) of our learning protocol entail that the expected number of times each user u logs into the system is equal to T n . In fact, this holds even for the Omniscient Matchmaker A * , despite the prior knowledge of σ.
Sentence 2: the following theorem guarantees that EM T (OOMM) = Θ T n 2 M , which is as if we were able to directly sample in most of the T rounds pairs of reciprocal edges.
Relationship: neutral
Domain: NeurIPS
Sentence 1: More precisely, assuming the system is initialized to a safe state (i.e., y 2 Y 0 = [y max /2, y max ]), then it should avoid an unsafe region (i.e., y = 0 ^(x  x p L _ x x p + L), where L is half the paddle length).
Sentence 2: according to the 0-1 loss g(⇡), the worse policy ⇡ left (J(⇡ left ) = 0) is better, whereas according to our loss `(⇡), the better policy ⇡ right (J(⇡ right ) = (T ↵)) is better.
Relationship: neutral
Domain: NeurIPS
Sentence 1: By arguments similar to those in Theorem 4, it can be shown that any stable non-positive steady-state is unique, attains the minimum total opinion among steady-states, and is smooth and convex for decreasing h. The above result paints a significantly simplified picture of the MF-IIM problem when J is stronglyconnected and there exists a stable non-negative steady-state m(b).
Sentence 2: theorem 3 shows that the objective M M F (b + h) is well-defined.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The width η of the negative Gaussian is set to be twice as large as the width σ of the positive Gaussian.
Sentence 2: after spikesorting, spike trains from 32 neurons were binned at 2 0ms resolution, and the response of a neuron to a stimulus at time t was defined to consist of the the spike-counts in the 1 0 bins between 40ms and 240ms after t. each population response Y t is a 3 20 dimensional vector.
Relationship: neutral
Domain: NeurIPS
Sentence 1: . . , ŵTn } of unit L 2 norm weight vectors where each ŵTi operates on a unit L 2 norm feature vector φ φ φ Ti (x, y).
Sentence 2: but the support vector machine (SVM) substitutes the hinge for the discrete loss and, modulo a margin assumption, can nonetheless efficiently find a hyperplane with a guarantee of good generalization.
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Domain: NeurIPS
Sentence 1: The update of θ is based on the samples generated from q φ (H|V), and is identical to (10).
Sentence 2: in the work presented here, we introduce the Temporal Sigmoid Belief Network (TSBN), which can be viewed as a temporal stack of SBNs, where each SBN has a contextual hidden state that is inherited from the previous SBNs and is used to adjust its hidden-units bias.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 1 2 mln-w w 2 (1) For linearly non-separable problems, a generalization of the optimal hyperplane method has appeared in [2], where a slack variable f.i is introduced for each data point (xi, yi) for i = 1, ... ,n. We compute a hyperplane Wn that solves min~wTw+CLf.i s.t.
Sentence 2: it is suitable for non-separable problems.
Relationship: neutral
Domain: NeurIPS
Sentence 1: First, the group additive structure may not be unique, which will lead to the nonidentifiability problem for the group additive model.
Sentence 2: thanks to the additive structure, the nonparametric estimation of f or equivalently the individual f i 's for 1 ≤ i ≤ p becomes efficient and does not suffer from the curse of dimensionality.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The asymptotics mn−1(h) as n → ∞ imply that the mixture distribution Ln,K+ in (10) converges to the true Gaussian mixture distribution p T of (9).
Sentence 2: we developed a fast online clustering and parameter estimation algorithm for Dirichlet process mixtures of Gaussians, capable of learning in a single data pass.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As a result, if max θ∈A λ | θ, xj | < mλ, we can also conclude that [β * λ ] j = 0 by (R1).
Sentence 2: for j ∈ {1, . . . , p} and Px j = 0, let x = −ξx j . Denote , the dual problem of (UBP ) is equivalent to: We can now solve problem (UBP ) in the following theorem.
Relationship: neutral
Domain: NeurIPS
Sentence 1: (2) The second category [18,23,27,51,58,65] involves using the predicted labels of the unseen class data in an iterative model update/adaptation process as in self-training [62,63].
Sentence 2: one approach to overcoming the above challenge is zero-shot learning (ZSL) [48,25,46,50,8,69,10,1,61].
Relationship: neutral
Domain: NeurIPS
Sentence 1: This can be done via randomized rounding [Park and Boyd, 2015, Goemans andWilliamson, 1995]: Instead of letting x 2 [0, 1] n , the integrality constraint x 2 {0, 1} n in (3) can be replaced by the inequalities x i (x i 1) 0 for all i 2 [n].
Sentence 2: we replace the quadratic programming relaxation of Kolter and Jaakkola, 2012 with a relaxation to an semi-definite program (SDP), which, based on the literature of relaxations is expected to be tighter and thus better.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 2b: Bounding p t ∞,u from a warm start Unfortunately, we cannot apply our results of step 2a directly since we are only assuming that X 0 has a warm start, not a stationary start.
Sentence 2: very recently, [33] made some progress towards this conjecture by proving that O * (d 1 4 ) gradient evaluations are required in the special case where U is separable into orthogonal O(1)-dimensional strongly convex components satisfying Lipschitz gradient, Lipschitz Hessian and fourth-order regularity conditions.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Remark 2 (ALTERNATIVE LOSS FUNCTIONS) For simplicity, all our results are stated for the case of the 0-1 loss I{g(X i , X j ) = e i,j }, but they straightforwardly extend to more practical alternatives such as the convex surrogate and cost-sensitive variants used in our numerical experiments.
Sentence 2: we also showed how sampling schemes applied to the population of edges (not nodes) can be used to scale-up such ERM-based predictive methods to very large graphs by means of a detailed rate bound analysis, further supported by empirical results.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Evidently, for this problem the likelihood-based functions are poor surrogate measures for per-label accuracy: Figures 2c and 2d show declines in training and testing set accuracy, despite increases in the objective function.
Sentence 2: to obtain a smooth approximation to this objective function, we can express the condition that the algorithm predicts the correct label for y (t) j in terms of the posterior probabilities of correct and incorrect labels as Substituting equation ( 8) back into equation ( 7) and replacing the indicator function with a generic function Q(•), we obtain When Q(•) is chosen to be the indicator function, Q(x) = 1{x > 0}, we recover the original objective.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We can compute wR t naively in O(N ) time by calculating w R t and keeping the K largest values.
Sentence 2: though we can only guess at what class of memory models will become staple in machine learning systems of the future, we argue in Supplementary A that they will be no more efficient than SAM in space and time complexity if they address memories based on content.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The RBM also plays a critical role in deep belief networks [4], [5], but we do not use this connection in this paper.
Sentence 2: the algorithm maintains a term ∂O/∂r t which is computed from ∂O/∂r t+1 and ∂ log Q(v t+1 |r t )/∂r t using the chain rule, by the equation where a.b denotes component-wise multiplication, the term r t .(1 − r t ) arises from the derivative of the logistic function s ′ (x) = s(x).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The areas where these relations are particularly strong are Shannon's coding theory [2] and classical spin systems with quenched disorder, which is the replica theory of disordered statistical systems [3].
Sentence 2: the paramagnetic solution, which is also valid for general β, K, and C, is in the form of π(x) = δ(x) and π = δ(x); it has the lowest possible free energy per bit f PARA = −1, although its entropy s PARA = (R−1) ln 2 is positive only for R ≥ 1.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Let q ∈ {1, . . . , t} and P q be the matrix comprised by the top q eigenvectors of P . Then max argmax Proof See [15] for a proof of a slightly more general result (Theorem 3.4).
Sentence 2: to find such a Λ and therefore recover the correct M , we need to enforce that we are at the optimal c (c * ), i.e.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Let argmin i 1 d i and For q ∈ {2, ∞}, when λ ≥ λ max,q , the solution of the optimization program (4) is equal to Z = e 1 , where e denotes the vector whose elements are all zero except its -th element, which is equal to 1.
Sentence 2: to do so, we propose a row-sparsity regularized trace minimization program on Z that consists of two terms.
Relationship: neutral
Domain: NeurIPS
Sentence 1: At each iteration, SG-HT only evaluates gradient of one (or a mini-batch) randomly selected sample for variable update and hard thresholding.
Sentence 2: as a direct consequence, the statistical error bound O( ∇ I f (x * ) ) of AHSG-HT can be improved in the sense that the cardinality | I| = 3k + k * has better dependency on the restricted condition number κ s . To better illustrate the boosted efficiency, we establish the computational complexity of AHSG-HT in IFO and hard thresholding in Corollary 2, whose proof is given in Appendix C.2.
Relationship: neutral
Domain: NeurIPS
Sentence 1: . . , α d ), j ∈ [n], Following Scheme 2.1, we first describe the relevant parametrized subset of RLM problems.
Sentence 2: due to difficulties which arise when studying the complexity of general optimization problems under discrete computational models, it is common to analyze the computational hardness of optimization algorithms by modeling the way a given algorithm interacts with the problem instances (without limiting its computational resources).
Relationship: neutral
Domain: NeurIPS
Sentence 1: After x k visits the terminal state, it enters a recurring sink state x S at the next time step, incurring zero cost, i.e., C(x S , a) = 0, ∀a ∈ A.
Sentence 2: we may rewrite (13) as From ( 14), it is easy to see that λ − λ1{s T ≤ 0}/(1 − α) is an unbiased estimate of the sub-gradient of L(θ, ν, λ) w.r.t.
Relationship: neutral
Domain: NeurIPS
Sentence 1: When Hybrid-MST is in MST mode, the runtime is approximately n times more efficient than Crowd-BT and GM method.
Sentence 2: recently, with the access of big data, preference aggregation from pairwise labeling has been widely applied in recommendation systems such as on movie, music, news, books, research articles, restaurant, products according to user's preference selection; or in social networks for aggregating social opinions; or in sports race, chess and online games to infer the global ranking of the players, etc.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As the covariance matrix K n is fully occupied, the input points x are fully correlated within a piecewise trajectory, as well as across different trajectories; this enables the exploitation of high capacity correlations.
Sentence 2: ( [23], Theorem 5) the maximum information gain satisfies γ n = O log(n) d+1 , i.e.
Relationship: neutral
Domain: NeurIPS
Sentence 1: SDP-cert provides non-vacuous certificates for all networks considered.
Sentence 2: robust training on CNNs leads to better error rates: for example, adversarial training against the PGD adversary on a four-layer feedforward network has error 9% against the PGD attack, while a four-layer CNN trained using a similar procedure has error less than 3% [21].
Relationship: neutral