sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: The parameter updates ( 4)-( 7) imply µ This follows from the strong law of large numbers, as the updates are recursive implementations of the sample mean and sample covariance matrix.
Sentence 2: the large-sample approximation to the conditional likelihood becomes: where we used lim u→∞ (1+ c u ) u = e c . The conditional likelihood (15) corresponds to the multivariate Gaussian distribution with mean µ h and covariance matrix Σ h . A similar asymptotic normality result was recently obtained in Tsiligkaridis & Forsythe (2015) for Gaussian observations with a von Mises prior.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, as V → ±∞, the function h(V ) goes to positive infinity and it thus has a global minimum at some value Ṽ ∈ R. At the global minimum, its derivative vanishes, .
Sentence 2: at the global minimum of the polynomial h, all terms except the highest order term cancel, and we find  Avg h( Ṽ ) is strictly positive, and since Ṽ is a global minimum of h, we have h(V ) ≥ h( Ṽ ) > 0 for all V ∈ R. Inserting into the expression for L(λ * , V * 0 ) concludes the proof that the lower bound at the optimum is positive.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Here, the apparently small effect on the number of LPs for low ratio schedules actually arises because of the large amount of time spent eating.
Sentence 2: according to the model DA is not really allowing animals to cope with higher work requirements, but rather is important for optimal choice of vigor at any work requirement, with the slowing effect of DA depletion more prominent (in the crude measure of LPs per session) when more time is spent leverpressing.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: • We show in Section 3.3 that our choice to study the mean estimator here is not somehow "unlucky" (it is not a particularly bad linear smoother, nor is the upper bound on its MSE loose): the minimax linear risk over T d (C n ) is on the order C 2 n /n, for all d ≥ 2.
Sentence 2: even the best linear smoothers have the same poor performance as the mean over T d (C n ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To establish convergence of the off-policy update in the dual representation, first define the maxnorm for state-action visit distribution as Then one can simply reduce the dual to the primal case by appealing to the relationship MHr where the second equality holds since we assumed (1 − γ)q (sa) = [Hr] (sa) for all (sa).
Sentence 2: given convergence of Mq to a fixed point Mq * = q * , the same must also hold for MH.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Furthermore, one sees that lim µ→0 w µi = 1 if and only if Λ i > 1, and lim µ→0 w µi = 0 if and only if Λ i < 1.
Sentence 2: lim µ→0 w µ = w * holds if and only if Λ i > 1 for all i with w * i = 1 and Λ i < 1 for all i with w * i = 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: defined on the same probability space) in such a way that G n → G and Q n → Q almost surely as n → ∞ for fixed T . Under assumption (5), it is easy to show that ( 11) holds for all n > n 0 with n 0 a sufficiently large constant (for a proof see the provided supplementary material).
Sentence 2: by the proof of Theorem 3.1, the conditions in Proposition 4.1 hold for gradient G n and hessian Q n for any n ≥ n 0 , with probability larger than 1 − δ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Applying stable random projections on By the properties of Fourier transforms, the projected differences, v 1,j −v 2,j , j = 1, 2, ..., k, are i.i.d.
Sentence 2: samples of the stable distribution S(α, d (α) ), i.e., the task is to estimate the scale parameter from k i.i.d.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It returns the state x(n) with highest evaluation.
Sentence 2: note that Assumption 2 implies that the b-value of any cell containing x * upper bounds f * , i.e., for any cell a node (h, i) such that f (x h,i ) + δ(h) < f * will never be expanded (since at any time t, the b-value of such a node will be dominated by the b-value of the leaf containing x * ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that a Gaussian with covariance matrix @7 has zero variance along directions that are orthogonal to the images of the data.
Sentence 2: in order to compute the mutual information, we only need to compute the covariance matrix of the projection of ¥ onto the linear span of the data, that is, for all CB ED ¦ 0F : where © G denotes the § F ¥ vectors with only zeros except at position D , and I ¡ is the Gram matrix of the centered points, the so-called centered Gram matrix of the -th component, defined from the Gram matrix P ¡ of the original (non-centered) points as , where Q is a § F ( § matrix composed of ones [8].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: p k is a continuous quantity whereas F is defined on at discrete 2D locations.
Sentence 2: we use the differentiable bilinear sampling operation to sample from the discrete grid [25] to obtain the feature at X k w . Such an operation has the highly desirable property that features from pixels in multiple images that may correspond to the same 3D world point unproject to the same location in the 3D gridtrivially enforcing epipolar constraints.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The following corollary is straightforward.
Sentence 2: corollary 1 For any s ∈ S * , we have: the homogeneity testing problem may be seen as closely related to the problem of estimating the optimal AUC * , since it may be re-formulated as follows: "H 0 : AUC * = 1/2" versus "H 1 : AUC * > 1/2".
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since the connection between rank and exact inference is obvious from Theorem 2, the more interesting question in Q2 is whether Boolean rank is indicative of the complexity of approximate lifted inference as well.
Sentence 2: we investigate its influence on the Lifted MCMC algorithm (LMCMC) [29] with Rao-Blackwellized probability estimation [30].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, when the network is sparse, whenever we sample a new node (or link) event from the model, only a small number of intensity functions, in the local neighborhood of the node (or the link), will change.
Sentence 2: we can reuse most of the samples from the intensity functions for the next new sample and find which intensity functions we need to change in O(log m) operations, using a heap.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This stretching does multiplicatively attenuate any curvature in layer l − 1 by a factor 1/χ 1 (see the update equation for κl in ( 8)), but new curvature is added in due to a nonzero χ 2 , which originates from the curvature of the single neuron nonlinearity in (9).
Sentence 2: unlike in linear expansion, extrinsic curvature is not lost, but maintained, and ultimately approaches a fixed point κ * .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , π t−N ∈ σ(u t−N ) and µ N t can be computed using where P a is the transition probability matrix when in every state action a is used and e x is the unit row vector corresponding to x (and we assumed that X = {1, . . . , |X |}).
Sentence 2: moreover, a simple but tedious calculation shows that ( 3) and ( 4) ensure the conditional unbiasedness of our estimates, that is, It then follows that E[ ρt |u t−N ] = ρ t , and, hence, by the uniqueness of the solutions of the Bellman equations, we have, for all (x, a) ∈ X × A, we also have, for all (x, a) ∈ X × A, t ≥ N + 1, and The bandit algorithm that we propose is shown as Algorithm 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The task of learning is to find the model parameters Θ and the assignments Y which maximize the likelihood of the complete dataset {I, Y , A}.
Sentence 2: the joint probability of {I, Y , A} given Θ from equation (1) can be written as  to minimize J , each latent variable Y i must belong to the set of possible assignments A i .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Even in the corrupted setting, the difference in leverage scores between corrupted and noncorrupted points is small (see §6).
Sentence 2: the main contribution to the influence for each point will originate from the residual error, e 2 i . Consequently, we propose sampling with probability inversely proportional to the approximate residual, 1 ẽ2 i .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: log where ∑ x q(x) = 1, and so x) .
Sentence 2: the lower bound for the log-likelihood is given by The maximum of F[q(z)] with respect to q(z j,i = t) = ϕ j,i,t and β is given by The updates of α are the same as Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Notice that Fourier coefficients of p are the expectations p(z) = EX∼p [ψ z (X)].
Sentence 2: p(z) := 1 n n j=1 ψ z (X j ) and q(z) := 1 n n j=1 ψ z (Y j ) are independent unbiased estimates of p and q, respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The wrapper method, on the other hand, is defined as a search through the space of feature subsets using the estimated accuracy from an induction algorithm as a measure of goodness of a particular feature subset.
Sentence 2: one approximates T(O', a) by minimizing Twrap(O', a) = min Talg(O') IT (2) subject to 0' E {a, l}n where Talg is a learning algorithm trained on data preprocessed with fixed 0'.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: permutation matrices since they corresponds to ordering K elements.
Sentence 2: even though {q i,l } can be obviously decomposed into a linear combination of permutation matrices, it is not clear how to compute them without computing the set of all permutation matrices that are exponentially large in K. Algorithm 2 solves this problem: Let Ni,l be a temporal variable that is initialized by Ñi,l at the beginning.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Yue and Joachims (2009) showed that a stochastic gradient descent algorithm can be applied to dueling bandit.
Sentence 2: it is naturally expected that a stochastic mirror descent algorithm, which achieves the (near) optimal order in convex optimization with one/two-point bandit feedback, can be applied to dueling bandit setting and achieves good performance.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In a POMDP, the agent reasons about the current state based on an observation.
Sentence 2: pOMDP can be regarded as an MDP with observations, Z and an observation function O : Z × A × S → [0, 1], which determines P (z|a, s), the probability of observing z after performing action a in state s. In a POMDP, instead of knowing the current state, s t , the agent computes the belief state, b t , which is the posterior probability over states given all past observations and actions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ¡ 1 2 d ¢ 2C 8 3 ¨is sampled by executing action 3 in state ¨and thereafter following the policy.
Sentence 2: in the episodic formulation, where the agent executes a maximum of 9 steps during each episode, at the end of each episode, ¡ 1 2 d ¢ 2¨6 8 6 ¨for step D can be evaluated as follows: ¡ 1 2 d ¢ ©¨6 8 6 ¨A £ $ ¥ @9 § 9 b 6 87 i ¨6 A ¡C 26 A § 8 given that the agent executes a complete episode ¢ ¢ 229 I9 ¨ ! "
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . ×N U N , the root mean square error (RMSE) is a frequently used measure of the difference between the recovered tensor and the true one: RMSE := 1 √ I N ∥D − A∥ F . [25] analyzes the statistical performance of the convex tensor Schatten 1-norm minimization problem with the general linear operator X : R I1×...×IN → R m . However, our model ( 6) is non-convex for the LRTC problem with the operator P Ω .
Sentence 2: we follow the sketch of the proof in [26] to analyze the statistical performance of our model ( 6).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: At iteration t, it requires two passes over the network: in the forward pass, the activations of all layers are computed from the bottom layer l = 1 to the top layer l = L following: ; in the backward pass, it applies the chain rule and propagates error gradients through the network from the top layer l = L to the bottom layer l = 1 following: According to (3), computing gradients for the weights w l of the layer l is dependent on the error gradient ∂f x t (w t ) ∂h t l from the layer l + 1, which is known as backward locking.
Sentence 2: the backward locking prevents all layers from updating before receiving error gradients from dependent layers.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By Lemma 4, if a round t ends after I(⌧, t , N t ) queries are answered, then the total revenue collected from queries in that round is at least 6N t , so the "high price" at the end of the round is 0.
Sentence 2: a query q uj from the non-adaptive user costs the low price 96/(⌧ 2 u j ) unless it causes an instantiation of VALIDATIONROUND to halt prematurely.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Regression with Brownian Sheets: When one uses Brownian sheets for regression with a target function f * = i α * i ϕ i that lies in the Cameron-Martin space K defined previously (i.e.
Sentence 2: such that ||α * || < ∞), then the term ||α * || 2 sup x∈X ||ϕ(x)|| 2 that appears in Theorems 1 and 2 is bounded as: from Theorem 2, ordinary least-squares performed on random subspaces spanned by P Brownian sheets has an expected excess risk (and a similar bound holds in high probability).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For applications involving high-dimensional data, such as face recognition, σ is usually large (typically ranging from thousands to hundreds of thousands [13]), and the condition in Lemma 4.1 holds if we restrict our search space to the convex hull of each class in the original space.
Sentence 2: the global minimum of the optimization problem in (7) can be found very efficiently using Newton's or gradient decent methods.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For any matrix norm II• II it holds that e(A) :S IIAII [5].
Sentence 2: we know that ehIID~P) :S IbIID~PIID~' where the vector norm   Besides the iterative solution of (6) often a direct solution by matrix inversion is computed using equation ( 9) in the fixed point form AiFwiF + bIt = O.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Each edge (i, t) is associated with exactly one ground set element i ∈ V, and selecting i (i ∈ T s ) is equivalent to cutting the edge (i, t).
Sentence 2: the cost of edge (i, t) will model the cost g(i) of its element i ∈ V. Let E t be the set of such edges (i, t), and denote, for any subset C ⊆ E t the set of ground set elements adjacent to C by We define the cost of C to be the cost of its adjacent ground set elements, h g (C) g(V (C)); this implies h g (δ s (S ∩ E t )) = g(S).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: So far the segmentation S was assumed to be given, and we estimated GES(Q, H ref , S).
Sentence 2: in order to obtain the global evidence score of Q, we marginalize over all possible segmentations S of Q: Namely, the likelihood P (S|H ref ) of a segmentation S can be interpreted as a weight for the likelihood ratio score of Q induced by S. we would like P (S|H ref ) to reflect the complexity of the segmentation S (e.g., its description length).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As we employ normalized kernels (where the diagonal is 1), it is sufficient to measure the pairwise kernel similarity.
Sentence 2: for a test point x, the nearest neighbor classifier reduces to: The USPS hand written digits dataset Hull (1994) consists of n = 7291 training (and 2007 test) greyscale images of 10 handwritten digits from 0 to 9.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, both convolutional and recurrent operations process a local neighborhood (several nearest neighboring neurons) in either space or time.
Sentence 2: the long-range dependencies can only be captured when these operations are applied recursively, while those long-range dependencies are sometimes significant in practical learning problems, such as image or video classification, text summarization, and financial market analysis [1,6,22,27].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is easy to see that X S X S is always invertible.
Sentence 2: furthermore, one can show that if , we have |µ j | < 1, ∀j ∈ S c and sgn(β * i )γ i > 0, ∀i ∈ S. by our assumptions, the preconditions of Lemma 1 are satisfied for the original Lasso problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A classifier f is a function f : X → Y, and the true risk of f w.r. where : Y × Y → R + is a given loss function, for example the 0/1 error.
Sentence 2: given a labeled set [5] is a pair (f, g), where f is a classifier, and g : X → {0, 1} is a selection function, which serves as a binary qualifier for f as follows, the selective classifier abstains from prediction at a point x iff g(x) = 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus, we obtain that as {p k i } is bounded (Proposition 8 in Supplement F).
Sentence 2: we conclude that, for all k ∈ K, k ≥ k = max( ǩ, k) and for all i ∈ {1, • • • , n}, at least one of the following three cases must happen: It follows that there exists a constant ᾱ > 0 such that Thus, considering According to Proposition 8 (in Supplement F), we know that the largest eigenvalue of H k is uniformly bounded from above by some positive constant M . Thus, we have which together with Eqs.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, the basis of such eigenfunctions can be learned using only the unlabeled data by constructing the Nystrom extension of the eigenvectors of an appropriate kernel matrix.
Sentence 2: given unlabeled data we construct the basis of eigenfunctions and then apply L 1 penalized optimization procedure Lasso [Tib96] to fit a sparse linear combination of the basis elements to the labeled data.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A key example is when h : R d → R describes the least-squares Using that this h is twice differentiable with ∇ 2 h(Ax) = I n , it is easy to see that we can track the evolution of all gradient entries, when performing CD steps, as follows: for i k being the coordinate changed in step k (here we also used the separability of the regularizer).
Sentence 2: all gradient changes can be tracked exactly if the inner products of all datapoints are available, or approximately if those inner products can be upper and lower bounded.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By applying the discussion following Theorem 1 when G ?
Sentence 2: is rank d, we can state an upperbound of  This shows that D is uniquely determined as a function of C. since (D) = (C) and because C is orthogonal to the kernel of , the distance matrix D can be recovered from (D), even though the linear operator is non-invertible.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Theorem 1 implies that the stronger column-wise independence constraint (10) does not degrade approximation accuracy, and the VB solution under matrix-wise independence (5) essentially agrees with the simpleVB solution.
Sentence 2: we can obtain a global analytic solution for VB, by combining Theorem 1 above with Theorem 1 in [15]: Let γ h be the second largest real solution of the following quartic equation with respect to t: where the coefficients are defined by Let Then, the global VB solution under matrix-wise independence (5) can be expressed as Theorem 1 holds also in the empirical Bayesian scenario, where the hyperparameters (C A , C B ) are also estimated from observation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The algorithms in [4,13,5] all fall into this category.
Sentence 2: if we choose I to be the set {a, a + 1, . . . , b} for some integers 0 ≤ a < b < m, and A updates its hypothesis k times during rounds a + 1 through b, then h A I is defined by at most B + k support patterns.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In addition, suppose there exists c ∈ K such that A t F t ≤ K c for all t, then where κ is given by κ c,ψ,ψ S = inf{r | c, ∇ψ S (0) − ∇ψ S (u) ≤ rψ(u), u ∈ K} Proof: Since ψ S satisfies Assumption 1, we have ŷt+1 ≤ K * ŷt .
Sentence 2: we can write: Now by combining the duality gap given by Lemma 1 with 11, we get The conclusion follows from the definition of α ψ,ψ S , κ c,ψ,ψ S and the fact that D seq ≥ D . Based on the result of the previous theorem we can modify the optimization problem set up in Section 3.2 for separable functions on R n + to maximize the lower bound on the competitive ratio of the sequential algorithm.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since no prior knowledge of α is known, the prior of each α i , i ∈ {w, u, θ}, P (α i ) is chosen to be a constant on log-scale or flat.
Sentence 2: the value of α maximizing the posterior of α P (α|Y, X) can be found by maximizing By Laplace's approximation [14], this integral is approximated around the MAP estimation of weights.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The number of calls k to a blackbox vector-valued function f needed to compute an approximate Jacobian via finite difference technique in the noiseless setting satisfies k ≤ χ(G int ), where G int is the corresponding intersection graph.
Sentence 2: blackboxes whose unknown Jacobian happens to be associated with intersection graphs of low chromatic number admit accurate Jacobian estimation with few function calls.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We remark that the detailed problem instance and the greedy regret can be found in Theorem E.2 of the supplementary material.
Sentence 2: furthermore, we may also restrict the maximum gap ∆ max to Θ(1), and the lower bound R G (T ) = Ω( mW ∆max ln T ∆ 2 ), for any sufficiently large T . For the upper bound, OG-UCB (Theorem 3.1) gives that R G (T ) = O( mW ∆max ∆ 2 log T ), our upper bound of OG-UCB matches the lower bound within a constant factor.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From Proposition 1, any continuous PWL function can be expressed by the virtual unit, a difference of two convex PWL functions.
Sentence 2: then, according to Proposition 2, any continuous function can be approximated arbitrarily well by this network to achieve the desired degree of with sufficiently large n 1 and d 2 on the compact domain C. a two-layer neural network satisfies Theorem 3 and then the Theorem holds.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As the variables φ n,t are expected to be large when xn,t cannot explain the observation x n,t -typically due to a corruption or a high noise -the weights will accordingly suppress the importance of the particular point x n,t .
Sentence 2: the overall approach will be more robust to corrupted data than the Gaussian models where all weights would be deterministic and equal to 0.5.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Figure 3 shows the results obtained with binary descriptors of varying dimensionality.
Sentence 2: while our approach can accommodate easily the case were the number of feature dimensions (C) is different from the number of feature-training categories (K), the classeme learning method can only produce descriptors of size K. the descriptor size is typically reduced through a subsequent feature selection stage [27,19].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The first term minimizes the distance between positive pairs and the second term pushes negative pairs to be at least margin m away from each other.
Sentence 2: the second term is only active when the distance between the features F I (x i ) and F I (x i ) are smaller than the margin m. Such boundary defines the metric space, so it is crucial to find the negatives that violate the constraint and train the network to push the negatives away.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Σ = Σ + αϕ i ϕ T i , where α = 1 2 ( si − s i )σ 2 i .
Sentence 2: to compute ∆E x we can take advantage of the Sherman-Morrison formula and of a similar formula for the log det term Using ( 1) and ( 2) ∆E x can be written as The transition probabilities can thus be computed efficiently, and if a new state is accepted we update Σ and Σ −1 using (1).
Relationship: reasoning