sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Nonetheless, as we will show in this section, NW and PL are deeply connected to the screening rule, and when given a large enough regularization parameter, the application of the screening rule to NW and PL can be lossless in practice (see Section 5).
Sentence 2: when applied to NW and PL, the proposed screening rule can be considered as a strong rule (i.e., a rule that is not foolproof but barely makes mistakes) and an optimal solution can be safeguarded by adjusting the screened solution to optimality based on the KKT conditions of the inexact problem [Tibshirani et al., 2012].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For the base case, suppose V is a leaf with child W; we want to describe the Nash breakpoint policy for V. If for all w E [0,1], the function ~v(w) is non-negative (non-positive, respectively), V can choose 1 (0, respectively) as a best response (which in this base case is an upstream Nash) to all values W = w. Otherwise, ~ v (w) crosses the w-axis, separating the values of w for which V should choose 1, 0, or be indifferent (at the crossing point).
Sentence 2: this crossing point becomes the single breakpoint in Fv(w).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Finally, if the gradient in this context is not stochastic (i.e., the parameter Q = 0), then linear convergence to the optimal value is achieved, which matches known rates of convergence with (centralized) gradient descent under strong convexity and smoothness assumptions.
Sentence 2: , the sequence of objective function values are themselves upper bounded as follows: using Theorem 1 we can establish analogous convergence rates in terms of the true objective function values { N n 1 T F(x k )} as well.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, as mentioned earlier for large dimensions, this is infeasible.
Sentence 2: we need to have an alternative search approach similar to the version for non-parametric version, except that in each round, for each variable X 2 ⇡ t , quantity T IB X, 1 (D), given in line 6 of Algorithm 1, is calculated.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since v σ i (I) ≥ L(I) and v σ i (I, a) ≤ U (I, a), so from (4) we get r t (I, a) ≤ U (I, a) − L(I).
Sentence 2: for iteration T 0 ≤ T ≤ T 0 + m, R T (I, a) ≤ 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To avoid confusion on this point, we also introduce the notion of the observed denotation set of a query, denoted q train , which corresponds to the set of nodes that exactly satisfy q according to our observed, training edges.
Sentence 2: our goal is to train using example queryanswer pairs that are known in the training data, i.e., (q, v * ), v * ∈ q train , so that we can generalize to parts of the graph that involve missing edges, i.e., so that we can make predictions for query-answer pairs that rely on edges which are unobserved in the training data (q, v * ), v * ∈ q \ q train .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, the other papers which have up to now discussed global optimality for neural networks [11,8] have not included any results on real datasets.
Sentence 2: up to our Nonlinear Spectral Method for 1 hidden layer Input: Model n 1 ∈ N, p w , p u ∈ (1, ∞), ρ w , ρ u > 0, α 1 , .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is because the hovering controller is always trying to "catch up" to the moving $ S£ ¡ £ $£ ¡ §¦ p£ ¡ £ ( .
Sentence 2: ` e£ may remain large, and the helicopter will continuously incur a ` S£ cost, even if it is in fact flying a very accurate trajectory in the increasing -direction exactly as desired.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: During inference, 2 the neuron will fire strongly for its preferred stimulus (or stimulus interpretation), but then its firing probability will decrease as its excitability and synaptic efficacy drop, allowing the network to discover potential alternative interpretations of the stimulus.
Sentence 2: in the case of sparse activity, equations 6-8 implement a form of neuronal fatigue and synaptic depression.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We add initial structure to q φ by noticing that the prior p θ d (d 1:T |u 1:T , d 0 ) in the generative model is a delta function over d 1:T , and so is the posterior p θ (d 1:T |x 1:T , u 1:T , d 0 ).
Sentence 2: we let the inference network use exactly the same deterministic state setting d 1:T as that of the generative model, and we decompose it as .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Hence, using induction, we can argue that after the first mistake, the learning can be done with k − 1 mistakes and k(|H| = |H| 1/k+1 number of ⊥'s before the first mistake.
Sentence 2: the total number of ⊥'s will not exceed Before moving to the next section, we should mention that Algorithm 2 is not computationally efficient.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that the larger the region R ⊆ Q is, the higher its evidence score LES(R|H ref ) (and therefore it will also provide higher statistical evidence to the hypothesis that Q was composed from ref ).
Sentence 2: for example, assume for simplicity that R has a single identical copy in ref , and that T is restricted to shifts with uniform probability (i.e., P (T |H ref ) = const), then P (R|H ref ) is constant, regardless of the size of R. On the other hand, P (R|H 0 ) decreases exponentially with the size of R. the likelihood ratio of R increases, and so does its evidence score LES.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 4 can only vanish in the limit 0: --+ 0 if N(~-~ = 0 while N(~-l) > 0; in other Xi , 7r i 7ri words, the parent-configuration 7rf must already have occurred previously along the sequence (7rf is "old"), while the child-state xf occurs simultaneously with this parent-state for the first time (xf, 7rf is "new").
Sentence 2: in this case, the leading polynomial order of the ratio (for given k and i) is linear in 0:, assuming P(Xi' IIi) > 0; otherwise the ratio (for given k and i) converges to a finite positive value in the limit 0: --+ O. the dependence of the marginal likelihood in leading polynomial order on 0: is completely determined by the number of different configurations in the data.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Bowling and Veloso showed that in selfplay, a restricted version of WoLF-PHC always reaches a stationary Nash equilibrium in two-player two-action games, and that the general WoLF-PHC seems to do the same in experimental trials.
Sentence 2: in the long run, a WoLF-PHC player achieves its stationary Nash equilibrium payoff against any other PHC player.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The goal is to optimize these operators-along with embeddings for all graph nodes z v ∈ R d , ∀v ∈ V-so that the embedding q for any query q can be generated and used to predict the likelihood that a node v satisfies the query q.
Sentence 2: in particular, we want to generate query embeddings q and node embeddings z v , so that the likelihood or "score" that v ∈ q is given by the distance between their respective embeddings: our goal is to generate an embedding q of a query that implicitly represents its denotation q ; i.e., we want to generate query embeddings so that score(q, z v ) = 1, ∀v ∈ q and score(q, z v ) = 0, ∀v / ∈ q .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Fix π ∈ Π HR , µ ∈ C S and a history h t where t < T , denote r = E µ (r), p = E µ (p), then we have: From Lemma 1, by backward induction, one can show the following lemma holds, which essentially means that for any policy, the expected performance under an admissible distribution µ only depends on the expected value of the parameters under µ.
Sentence 2: the distributionally robust MDP reduces to a robust MDP.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By the strict convexity of f , we then have f α(q y , w) + (1 − α)(q y , w ) < αf (q y , w)+(1−α)f (q y , w ) = f (q y , w), for 0 < α < 1, contradicting the global optimality of f (q y , w).
Sentence 2: any convex relaxation of (1) that uses a distribution q y over missing values and does not make arbitrary distinctions can never do anything but produce a uniform distribution over the hidden variable values.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Graphically, we can see that both the exponential family Harmonium (EFH) [26] and its extension of dual-wing Harmonium (DWH) [28] are special cases of multi-view latent space MNs.
Sentence 2: it is not surprising to see that multi-view MNs inherit the widely advocated property of EFH that the model distribution can be constructively defined based on local conditionals on each view.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let N (i) = {j|{i, j} ∈ E} denote the neighbors of i in the graph.
Sentence 2: the mean µ ≡ E{x} and covariance P ≡ E{(x − µ)(x − µ) } are given by: The partial correlation coefficients are given by: J ij = 0 if and only if x i and x j are independent given the other variables x V \{i,j} .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This expression can be simplified because e T i is a row of W * = C −1 , so that e T i C is the unit vector of the form (0,0,...,1,0,...) where the position of the "1" indicates the solution number 0.
Sentence 2: we have e T i CΛ(τ )C T e k = Λ(τ ) ik .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , w K ) ∈ R K . Note minimizing the cross-entropy loss is equivalent to maximizing the likelihood of , which is biased towards fitting neither the data with y i = 1 nor these with y i = 0, since NLL(x i ) = ln(e −yiw ln(1+e Bx i ) + e (1−yi)w ln(1+e Bx i ) ).
Sentence 2: while iSHM is structurally similar to an MLP, it is distinct in its unbounded layer width, its positive constraint on the weights r k connecting the hidden and output layers, its ability to rigorously define whether a hyperplane is active or inactive, and its inductive bias towards fitting the data labeled as "1."
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the general case, it is not obvious what aspects of the quality of a population code are captured by the above measure.
Sentence 2: both Fisher Information and the class-conditional second-order approximation used by Averbeck and Lee have shortcomings: The latter does not account for information originating from changes in the covariance matrix as is quantified by J cov (cf.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By applying the Taylor expansion to KL divergence between p(x; η) and p(x; η + dη), ds 2 can be represented as follows: , where G * is the FIM.
Sentence 2: by comparing these equations, it is clear that the metric matrix consisting of the inner products of the basis vectors corresponds to the FIM: the FIM is a proper metric for the feature vectors (the expectation parameters) obtained from the GMRF.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Observing Eq.3, L i can be reformulated as: Xi + e Wy i Xi ), then performing the first order Taylor Expansion for the second term, a term of W yi X i shows up.
Sentence 2: minimizing Eq.3 is to minimize W yi X i to some extend, and it can be viewed as a coupling decay term, i.e.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given the asymptotic normality of G m , the same arguments used to derive the multivariate delta method can be used, with some modification, to show that ).
Sentence 2: asymptotically, f i ( G m ) is normally distributed around its true value with a variance that depends on both the gradient of f and the covariance matrix of the approximated arguments in G m . This being the case, uniform sampling of the recursively estimated f i is equivalent to sampling from a distribution Pf that gives weight 1 n to a normal distribution centered on each f i .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By definition of GREEDY for any pair of points v, v ∈ ∪ m i=1 C i it must be that d(v, v ) ≥ d X (u) > 2OPT.
Sentence 2: in the set {u} ∪ X there are k+1 points all of distance greater than 2OPT from each other.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: While strong faithfulness requires that for a node pair (i, j) that are adjacent in the DAG, the partial correlation corr(X i , X j |X S ) is bounded away from zero for all sets S 2 {S ✓ [p] \ {i, j}}, RSAF only requires non-zero partial correlations with respect to a subset of sets in {S ✓ [p] \ {i, j}}.
Sentence 2: rSAF is strictly weaker than strong faithfulness.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 5) is a simple fixed-point type update; in each iteration, only one component (i.e., p t s ) is modified while the others remain the same.
Sentence 2: it is motivated from using a descent direction to strictly decrease l(p): where p t,n s − p t s is a descent direction in optimization since a sufficiently small step along this direction guarantees the strict decrease of the function value.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We observe that the third case is symmetric to the second case, and thus we can carry out very similar calculations in the third case to show that the right hand side is at most 6 hñ .
Sentence 2: we have that for any T and T , provided the event E holds, |q(F (T, h, α, R), V ) − q(F (T , h, α, R), V )| ≤ 6 hñ The theorem now follows by combining Equation 21with Equation 19.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For ML estimation, the lower bound on the number of measurements required by ML decreases to k − 1 as SNR → ∞.
Sentence 2: 1 In contrast, with thresholding, increasing the SNR has diminishing returns: as SNR → ∞, the bound on the number of measurements in (12) approaches even with SNR → ∞, the minimum number of measurements is not improved from m = Ω(k log(n − k)).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Indeed, in (3) all summands corresponding to m > max i=1,2 n i equal 0; moreover, all summands corresponding to l > s min are equal, where s min corresponds to the partition in which each cell has at most one point in it s min := min i,j∈1..n, Xi =Xj |X i − X j |.
Sentence 2: even with a most naive implementation the computational complexity of the algorithm is at most polynomial in all arguments.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For ¡ ¢ , we can see intuitively that a boosting algorithm will not let any observation have exceptionally low probability over time since that would cause this observation to have overwhelming weight in the next boosting iteration and hence the next selected ¢ is certain to give it high probability.
Sentence 2: after some iterations we can assume that we would actually have a threshold ¨independent of the iteration number and hence the loss would decrease at least as the sum of squares of the "weak learnability" quantities # ¢ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is important to note that we do not make any assumptions on the "wrong" models (those that do not have Markovian dynamics).
Sentence 2: we are not able to test which model is Markovian in the classical statistical sense, since in order to do that we would need a viable alternative hypothesis (such as, the model is not Markov but is K-order Markov).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: f does not depend on i), and g i is a convex function.
Sentence 2: after t online rounds, the amount of intrinsic strong convexity we have in the primal objective t i=1 t (w) is at least σ t. In particular, this explains the learning rate of 1 σ t proposed in the gradient descent algorithm of Hazan et al.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When the difference between two stimuli is small, the conditional probability p(r|s + ∆s) can be expanded with respect to ∆s as p(r|s+∆s) = p(r|s)+p ′ (r|s)∆s+ 1 2 p ′′ (r|s)(∆s) 2 +..., where ′ represents differentiation with respect to s. Using the expansion, to leading order of ∆s, we can write mutual information I as where dr p ′ (r|s) 2 p(r|s) is the Fisher information.
Sentence 2: we can see that the mutual information is proportional to the Fisher information when ∆s is small.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider also that there exists a reward function, g, which informs the agent of the scalar goodness of taking an action with respect to the goal of some multi-step decision task.
Sentence 2: for each time, t, s(t + 1) = f (s(t), a(t)), and (2) r(t + 1) = g(s(t), a(t)).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This approximation mitigates multiple issues that would arise with sampling approaches: By forcing π βp m (θ m |y 1:t ) into the conjugate closed form, steps (II) and (III) are solved analytically.
Sentence 2: inference is orders of magnitude faster, while the resulting approximation error remains negligible (see Figs 2B, 3).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that the stability measures the disagreement between labels on training data and test data, both assigned by C . This asymmetry arises naturally and directly measures the generalization performance of C . Furthermore, the stability can be interpreted as the expected empirical risk of C with respects to the labels computed by itself (compare (1) and ( 2)).
Sentence 2: stability measures the self-consistency of C . This interpretation is also valid in the semi-supervised and unsupervised settings.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Suppose H k is a reproducing kernel Hilbert space (RKHS) with reproducing kernel k : X × X → R [4,6].
Sentence 2: if k is translation invariant (i.e., there exists κ ∈ L 2 µ such that, for all x, y ∈ X , k(x, y) = κ(x − y)), then Bochner's theorem (see, e.g., Theorem 6.6 of [51]) implies that, up to constant factors, in the setting of Theorem 1, we have It is well-known known that MMD can always be estimated at the parametric rate n −1/2 [18]; however, to the best of our knowledge, only recently has it been shown that any probability distribution can be estimated at the rate n −1/2 under MMD loss [41], emphasizing the fact that MMD is a very weak metric.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We can compute its mode, its marginal distributions, and draw samples from it, all using similar message passing algorithms.
Sentence 2: by optimizing the variational bound, we obtain the desired approximate posterior over discrete and continuous variables q(z s , x s , h s ) = p θ (z s | x s , h s , i s , y s ) q φ (x s , h s ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Prediction reliability is orthogonal to prediction accuracy,  and [22] showed that state-of-the-art deep networks are not reliable as they are not well-calibrated to correlate model confidence with model strength.
Sentence 2: to demonstrate the reliability of our uncertainty-aware attention, we evaluate it for the uncertainty calibration performance against baseline attention models in Table 3, using Expected Calibration Errors (ECE) [22] (Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We will conduct our primary evaluations on multiclass classification problems, where output vectors y encode target classes by indicator vectors y 2 {0, 1} m⇥1 such that y 0 1 = 1.
Sentence 2: although it is common to adopt a softmax transfer for f 2 in such a case, it is also useful to consider a perceptron model defined by f 2 (ẑ) = indmax(ẑ) such that indmax(ẑ) = 1 i (vector of all 0s except a 1 in the ith position) where ẑi ẑl for all l. for multi-class classification, we will simply adopt the standard large-margin multi-class loss [29]: (3) Intuitively, if y c = 1 is the correct label, this loss encourages the response ẑc = y 0 ẑ on the correct label to be a margin greater than the response ẑi on any other label i 6 = c. Second, consider the first layer model.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This scheme works as follows: Let k be a parameter with k being a divisor of m, and pick a random hash function h : To estimate the similarity of two sets A and B we simply take the fraction of indices, i, where This is, however, not an unbiased estimator, as there may be empty bins.
Sentence 2: [31,32] worked on handling empty bins.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When µ B is small (note that µ B 1), this model for the low-rank matrix L ensures that its singular vectors are not sparse.
Sentence 2: this can be seen in the case of the a singular vector u by noting that 1 = Rearranging terms enables us to write kuk 0 µ B controls the sparsity of the matrix L by bounding the sparsity of its singular vectors.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is due to the fact that given functions f 1 , f 2 , .
Sentence 2: . . , f k for the personalized setting, where f i is the function assigned to player i, we can combine these functions into a single function f ∈ F k for the centralized setting by defining f (⊥) = 0 and f without loss of generality we focus below on the centralized setting.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For non-uniform absolutely-continuous P , P(r 1 (x) > ε) → 0 as n → ∞ for any x in the support of p(x) and any ε > 0.
Sentence 2: as n tends to infinity p(arg 2 )) → p(x) and the limiting distribution of p(x)/ p 1 (x) is a unit-mean exponential distribution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, by Carathéodory's theorem [32, §17], any matrix K ∈ S can be written as the conic combination of finitely many rank one matrices of the form {cc : c ∈ C(Γ)}.
Sentence 2: conceptually, the recovery problem has been reduced to finding a sparse set of non-negative weights, µ, over the set of feasible basis vectors, c ∈ C(Γ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Large deviation principle [16] states that, the probability that an opponent with strategy q behaves like p * is roughly exp (− .
Sentence 2: we need to continue exploration of the actions until ∑ i N i (t)D(p * i ∥S i q) ∼ log t holds for any q ∈ C c 1 to reduce the risk to exp (− log t) = 1/t.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: (1) We assume that f (x i ) < f (x j ), and define f ′ as the function such that f According to the conditions of RDPS, the requirements of the weight function D in Section 2 and the assumption that ϕ is a non-increasing function such that ϕ(z) < ϕ(−z), ∀z > 0, we can obtain This is a contradiction with R Φ (f )=inf h∈F R Φ (h|x).
Sentence 2: we have proven that f (x i )≥f (x j ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Synchronization occurs for normalized weights, unnormalized ones do not synchronize [6].
Sentence 2: for discrete weights, we introduce a restriction in the space of possible vectors and limit the components w In order to obtain synchronization to a parallel -instead of an antiparallel -state w A = w B , we modify the learning rule (3) to: Now the components of the random input vector x are binary x i ∈ {+1, −1}.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 1)) is a regularly-defined smooth manifold works in pair with the ambition that the result should hold for (almost) all cost matrices C. The starting point is that, for a given non-convex smooth optimization problem-even a quadratically constrained quadratic program-computing local optima is hard in general [Vavasis, 1991].
Sentence 2: we wish to restrict our attention to efficiently computable points, such as points which satisfy first-and second-order KKT conditions for (P)-see [Burer and Monteiro, 2003, §2.2] and [Ruszczyński, 2006, §3].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, in the situation of domain adaptation, we only know the information of source domain, while the expectation EN (F, ξ, 1 (Z N 1 )) is dependent on both distributions of source and target domains because z = (x, y).
Sentence 2: the covering number is no longer applicable to our framework to obtain the generalization bounds for domain adaptation.
Relationship: reasoning