sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Upon receiving x n , it learns two hypotheses, h ŷ = LEARN H (S ∪ {(x n , ŷ)}, T ) for ŷ ∈ {±1}, and then compares their empirical errors on S ∪ T . If the difference is large enough 3 , it is possible to infer how h * labels x n (as we show in Lemma 3).
Sentence 2: in this case, the algorithm adds x n , with this inferred label, to S. Otherwise, the algorithm requests the label y n and adds (x n , y n ) to T . S contains examples with inferred labels consistent with h * , and T contains examples with their requested labels.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 5 and 6, are Markovian and thus we can update individual intensities and opinions in O(1) -let t i and t i+1 be two consecutive events, then, we can compute λ * (t i+1 ) as (λ * (t i ) − µ) exp(−ν(t i+1 − t i )) + µ and x * (t i+1 ) as (x * (t i ) − α) exp(−ω(t i+1 − t i )) + α, respectively; and, (ii) social networks are typically sparse and thus both A and B are also sparse, then, whenever a node expresses its opinion, only a small number of opinions and intensity functions in its local neighborhood will change.
Sentence 2: we can reuse the majority of samples from the intensity functions and sentiment distributions for the next new sample.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, the assumption on the approximation error function is satisfied for β := s/m, whenever f * L,P ∈ W s (X) and s ∈ (d/2, m].
Sentence 2: the resulting learning rate is n − 2sα 2s+d+2ds/m , which in the i.i.d.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: According to ( 16) and ( 17), we have ỹjii = where |R l i | denotes the number of elements in R l i , the subtree of T l rooted at v i .
Sentence 2: from the nature of the maximal root-tree T l , l ∈ I 1 , we have we can construct from T l a new root-tree Tl by removing the subtree of T l rooted at v i , so that Tl achieves a larger value than T l . This contradicts with the argument that T l , l ∈ I 1 is the maximal root-tree of the working tree T . it follows from ( 18) that ( 14) holds.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let us also denote by φ φ φ T (x, y), the projection of φ φ φ(x, y) on the edges of T . Namely, (φ φ φ T (x, y)) i,j = φ φ φ i,j (x, y i , y j ) if (i, j) ∈ T , and (φ φ φ T (x, y)) i,j = 0 otherwise.
Sentence 2: recall that φ φ φ i,j (x, for all (x, y) ∈ X × Y and for all T ∈ S(G), we have We now establish how F (w, x, y) can be written as an expectation over all the spanning trees of G. Let U(G) denote the uniform distribution on S(G).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is usually infeasible to calculate (7) exactly due to expectation operator.
Sentence 2: the empirical risk for ( 7) is used as a cost function as follows: where {(x (1) , y (1) ), • • • , (x (N ) , y (N ) )} are the N pairs of a training dataset, sampled from the joint distribution of x (j) ∼ p(x) and n (j) ∼ N 0,σ 2 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Our analysis differs from all of these, by focusing on the maximum likelihood situations noted above as convex optimization problems, and using SGD convergence techniques [29] for the sequence of iterates M Recall that the (negative) precision is a natural parameter of the normal distribution written in exponential family notation, and that the log-likelihood is a concave function of the natural parameters for this family; this makes max-likelihood a convex optimization problem over the precision, even in the presence of linear constraints [30,31].
Sentence 2: this implies that the problems (3), ( 4) have a unique maximum, denoted by M * I above.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From this, we can see that it is not possible to achieve a O(1/n) rate unless c ≥ 1/(2(λ 1 − λ 2 )).
Sentence 2: we will assume this when stating our final results, although most of our analysis is in terms of general γ n . An interesting practical question, to which we do not have an answer, is how one would empirically set c without prior knowledge of the eigenvalue gap.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, p(x ′ |x, u) is generally unknown, and it is a non-trivial problem to identify this distribution in large-scale applications.
Sentence 2: it is desirable to find parameter θ in model-free ways, i.e., parameter is estimated from the sample sequences alone, instead of using p(x ′ |x, u).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Each product z e(i) j z j k (1−z k ) appearing in (3) results in a sum of O(2 D ) terms, each of which has (up to a sign) the form q M ≡ m∈M z m for some set M . It is still the case that q M ∈ {0, 1}.
Sentence 2: objective function (3) can be interpreted as being linear on a set of binary variables {{z}, {q}}.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, we also show that the gap between the elastic net and the k-support norm is at most a factor of √ 2, corresponding to a factor of two difference in the sample complexity.
Sentence 2: our work can also be interpreted as justifying the use of the elastic net, viewing it as a fairly good approximation to the tightest possible convex relaxation of sparsity intersected with an 2 constraint.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: which can be equivalently stated as The numerator of the second term above can be rewritten as where X i is the i th column vector of X.
Sentence 2: the second term of (21) can be written as In ( 23), Since each matrix Z i is a symmetric projection matrix and X i 1 and X i 2 are orthogonal to each other when i 1 = i 2 , Z is a projection matrix of the form X(X T X) −1 X.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The monotonicity of the objective function with respect to ε i implies that the constraints ) are saturated at the solution of (3) for t i = 0 or t i = 1 respectively.
Sentence 2: the loss in (3) is the neg-log-likelihood of the lower or the upper bound on p(1|x i ; θ) respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that, in this new graphical model, the global parameters θ are independent of x tr and y tr and are independent of x test when y test is not observed.
Sentence 2: we can now write down a variational lower bound for the logarithm of the approximate likelihood, which is given by In this bound, we essentially perform approximate inference via MAP on φ i to obtain p(φ i |x tr i , y tr i , θ), and use the variational distribution for θ only.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As mentioned above, the affect of altering the position of a control point is to change the shape of the contour locally since the basis functions have limited support.
Sentence 2: when evaluating p(x~lxt-t}p(ZtlxD for a proposed particle, we only need to reexamine measurement lines and evaluate p(zm,t Ix~,t) for lines in the effected interval and similarly for p(x~,t IXn,t-l).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Evidence from the cognitive science literature suggests humans find it difficult to predict the motion of objects when multiple information dimensions are involved [40].
Sentence 2: we arrive at an alternative explanation for why humans may behave near-optimally with respect to a dynamics model that differs from the real dynamics: even if users have perfect knowledge of the real dynamics, they may not have the computational resources to plan under the real dynamics, and instead choose to plan using a simplified model.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, F ( x T ) = f ( x T ) + γ ln 1 + exp λ 0 g( x T ) γ ≥ f ( x T ) + max (0, λ 0 g( x T )) .
Sentence 2: with the value of γ = ln T /T , we have f ( x T ) + λ 0 g( x T ) ≤ f (x * ) + O ln T T . Applying the inequalities ( 13) and ( 14) to (16), and noting that γ = ln T /T , we have For λ 0 > G 1 /ρ, we have x T − x T 2 ≤ (1/(λ 0 ρ − G 1 ))O(ln T /T ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The function C λ turns out to have intrinsic connections with statistical leverage scores, as the quantity 1/C λ corresponds precisely to a notion of leverage used in [6,2,28,7].
Sentence 2: we uncover a variational formulation for leverage scores which helps elucidate connections with the RKHS and the density p on R d . Our main contribution is a precise asymptotic expansion of C λ as λ → 0 under restrictions on the RKHS.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Specifically, we suggest using a SGD-like low-regret update for the variable w, while updating the variables ξ and b via a PST-like step; the dual update of our method is similar to that of CHW.
Sentence 2: our algorithm enjoys the benefits of both methods, each in its respective domain, and avoids the problem originating from the "size" of K. We defer the detailed description of the method to the following section.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This set is convex and is a superset of the original non-convex set.
Sentence 2: the above SDP is a relaxation of the QCQP in 3 with f SDP ≥ f QCQP , providing an upper bound on y (x,ȳ) that could serve as a certificate of robustness.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Formally, let T R→Φ be a |Φ (i−1) | by |S| matrix for which T R→Φ (φ, s) = |φ| −1 if s ∈ φ, and 0 otherwise, and let T Φ→R be a |S| by |Φ (i−1) | matrix for which T Φ→R (s, φ) = 1 if s ∈ φ, and 0 otherwise.
Sentence 2: t Φ→R T R→Φ R is a vector where the reward in each state is the average over all rewards in the feature that state belongs to.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Now, consider the case that W > W is non-singular (note that N VOX N CELLS ).
Sentence 2: this implies that there is a one-to-one mapping between the RNN state and predicted neural activity: we can rewrite equation 8 in terms of the effect changing r t1 has on the predicted neural activity u v t in each voxel at time t times what a change in u v t implies about a change in ⇡ t2 , operating implicitly via what the change in u v t tells us about a change in h t . We can write this as, Note that this is a correlational relationship -the direction of causality is from h t to u t . Nevertheless the individual terms in this sum: combine the influence that voxel u v t at time t receives from the reward at time t 1 (which is @u v t /@r t1 ), with the covariation between the voxel activity and the action at time t 2 (which is @⇡ t2 /@u v t ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This spiking condition can be written as E(no spike) > E(with spike).
Sentence 2: using equation 6, this can be rewritten as x, x x x cancelling terms, and using equation 2, we can write our spiking condition as the spiking threshold for each neuron must be set to T k ≡ −Ω kk /2, though this condition can be relaxed considerably if our loss function has an additional linear cost term 1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The weighted estimates P s w (σ 0:t ) for each context s depend on the ratio of the probability of the observed data at s itself, P s e (σ 0:t ), and that of the data observed at its children, P s w (σ 0:t ) at s = σs ∀σ ∈ Σ.
Sentence 2: these estimates depend on the number of times each symbol followed a context, implying that c s,t = σ∈Σ c s ,t . the weighting in (3) assumes that each symbol that was observed to follow the non-leaf context s was also observed to follow exactly one of its children s . If this was not so and, e.g., s was created at time 0 but its children only at τ > 0, then, since P s w (σ τ :t ) ≥ P s w (σ 0:t ), the weighting would be biased towards the children, which would have been exposed to less data.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Hence, the expected number of iterations E[T ] for finding the desired solution is at most 2ek 2 n. Comparing with the approximation guarantee of FR, (1 − e −γ S F R ,k ) • OP T [3], it is easy to see that γ ∅,k ≥ γ S F R ,k from Definition 3.
Sentence 2: pOSS with the simplest configuration of the isolation function can do at least as well as FR on any sparse regression problem, and achieves the best previous approximation guarantee.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, we can see that X T g Θ ⊆ Ξ g , g = 1, . . . , G. To develop the TLFre rule by (R1 * ) and (R2 * ), we need to solve the following optimization problems: t * gi (λ, λ; α) = sup θ {|x T gi θ| : θ ∈ Θ}, i = 1, . . . , n g , g = 1, . . . , G. (18) Solving problem (17) We consider the following equivalent problem of ( 17): We can see that the objective function of problem ( 19) is continuously differentiable and the feasible set is a ball.
Sentence 2: ( 19) is a nonconvex problem because we need to maximize a convex function subject to a convex set.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that this is not equivalent to adding λ k to the Fisher matrix.
Sentence 2: nevertheless, it does become equivalent to a small SGD step in the limit of large damping factor λ k . at iteration k we update the synaptic weights in layer i according to We update λ k in each iteration to reflect the ratio ρ k between i) the actual decrease in the loss resulting from the latest damped NG parameter update, and ii) the decrease predicted by a quadratic approximation to the loss 1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The decision function is of the form where x E ~d is the d-dimensional input vector of a test example, y E {-I, I} is a class label, Xi  O:S ai :S C Vi.
Sentence 2: (4) The kernel K(X,Xi) can have different forms, such as the Radial Basis Function (RBF): K(Xi, Xj) = exp (-llxi(T~ Xj112) (5) with parameter (T. to train an SVM, we need to solve a quadratic optimization problem, where the number of parameters is N. This makes the use of SVMs for large datasets difficult: computing K(Xi' Xj) for every training pair would require O(N2) computation, and solving may take up to O(N3).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given v = [1, 1, −ǫ, −ǫ] T ∈ Null(Φ), this can be accomplished by computing ∂g (I) (x (2) + βv)/∂β = h ′ (|α 1 − α 2 + β|) + h ′ (|β|) + 2ǫh ′ (|ǫα 2 − ǫβ|).
Sentence 2: in the limit as β → 0 (from the right or left), this expression will always be positive for ǫ < 0.5 based on the concavity of h. x (2) must be a minimum.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By direct analogy to FA, we defined the prior distribution of the neural state x :,t at each timepoint t to be N (0, I) by setting Furthermore, because we seek to extract smooth neural trajectories, we set σ 2 n,i to a small value (10 −3 ).
Sentence 2: the timescale τ i is the only (hyper)parameter of the SE covariance that is learned.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The meta-algorithm tracks the state reached in the WFST and maintains a set of external regret minimizing algorithms that help the learner perform well at every state.
Sentence 2: here, we need one external regret minimization algorithm A u,i , for each state u reached at time t after reading sequence x 1:t 1 and each i 2 ⌃ labeling an outgoing transition at u.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For the replication-based scheme, since the number of extra workers n−k < k, there is a non-zero probability (which does not change with T dl ) that the n − k extra workers do not replicate the computation in the slowest one among the first worker.
Sentence 2: replication when n − k < k does not improve the error exponent, because the error is dominated by this slowest worker.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This result is based on recent approximation bounds dealing with the eigenvectors of the kernel matrix which show that if a function can be reconstructed using only a few kernel PCA components asymptotically, then the same already holds in a finite sample setting, even for small sample sizes.
Sentence 2: the use of a kernel function not only greatly increases the expressive power of linear methods by non-linearly transforming the data, but it does so ensuring that the high dimensionality of the feature space will not become overwhelming: the relevant information for classification will stay confined within a comparably low dimensional subspace.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , Z m be a sequence of random variables sampled uniformly without replacement from a fixed set of m + u elements Z, and let φ : Then, for all ǫ > 0, the following inequality holds: where α(m, u) We define the selection matrix corresponding to a sample of m columns as the matrix S ∈ R n×m defined by S ii = 1 if the ith column of K is among those sampled, S ij = 0 otherwise.
Sentence 2: c = KS is the matrix formed by the columns sampled.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The first component was grown around the point P in Figure 4 a.
Sentence 2: the second component was grown around point Q in Figure 4 e. The symbolic signatures inside the region around P encode the geometric configuration of three of the four components learned by the first module (healthy heads), while the symbolic signatures around Q encode the geometric configuration of three of the remaining four components (malformed heads), Figure 4 b and f . the points of the output mesh of the second module will be set to "+1" if they belong to learned symbolic signatures associated with the healthy heads (Figure 4 c) , and "-1" otherwise (Figure 4 g).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When the And-Or fragment is added into the grammar, the size of the grammar is changed in two aspects: first, the size of the grammar is increased by the size of the And-Or fragment; second, the size of the grammar is decreased because of the reductions from configurations of multiple nodes to the new And-node.
Sentence 2: the prior gain of learning from the And-Or fragment is:   where s a and s o are the number of bits needed to encode each node on the right-hand side of an And-rule and Or-rule respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , x N ) be any multivariate distribution over real-valued random variables, then there exists a copula function such that The constructive converse which is of central interest from a modeling perspective is also true: since for any random variable the cumulative distribution F(x i ) is uniformly distributed on [0, 1], any copula function taking the marginal distributions {F(x i )} as its arguments, defines a valid joint distribution with marginals F(x i ).
Sentence 2: copulas are "distribution-generating" functions that allow us to separate the choice of the univariate marginals and that of the dependence structure expressed in the copula function C, often resulting in an effective real-valued construction.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let the optimal transport cost in (18) be the W 1 distance, and suppose F is a convex subset of 1-Lipschitz functions.
Sentence 2: then, the minimax problem (18) will reduce to the moment-matching interpretation also holds for WGAN: for a convex set F of 1-Lipschitz functions WGAN finds the generative model with minimum W 1 distance to the distributions penalized to share the same moments over F with the data distribution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Other parameters describing the appearance of an object in an image can be added.
Sentence 2: the image features most commonly used for describing local structures are the energy outputs of oriented band-pass filters, as they have been shown to be relevant for the task of object detection [9,10] and scene recognition [2,4,8,12]~the local image representation at the spatial location (x) is given by the vector VL(X) == {v(X,k)}k==l,N with: where i(x) is the input image and gk(X) are oriented band-pass filters defined by gk(i) == e-llxI12/u~e27fj<f~,x>.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The bigger the value D ii (corresponding to y i ) is, the more "important" is y i .
Sentence 2: we impose a constraint as follows: y T Dy = 1 ⇒ a T XDX T a = 1 Finally, the minimization problem reduces to finding: The transformation vector a that minimizes the objective function is given by the minimum eigenvalue solution to the generalized eigenvalue problem: It is easy to show that the matrices XLX T and XDX T are symmetric and positive semidefinite.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Further, one can use the conjugate gradient method to solve for (W t K −1 u,u W t +σ 2 I) −1 v which only involves computing the matrix-vector product In practice, the conjugate gradient method converges within only a few iterations.
Sentence 2: approximating the posterior mean µ using SKI takes only O(n + d + m log m) time to compute.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When used as a tool to assess an ANN index before actually putting it to work, it is also important that the tester actually reduces the total computation time compared to observing poor results at the end of the data processing pipeline (e.g., bad classification results) and starting over.
Sentence 2: we seek to answer the following questions: Setup We implemented our property tester in C++ and integrated it into the Python framework ANN-Benchmarks [2,4].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: £ ge £ ¦ 4 ne £ n 0 § fY "£ ny @ 2 # Now for each & (' v v Q£ ) , the term £ ge £ ¦ 2 4 u ge f£ n t 0 § fY "£ ¦y @ 2 is non-negative, and equal to zero only when y @ belongs to ¢ 0 ¤£ B£ ge f£ n t 2 .
Sentence 2: the bound is met with equality if and only if y @ achieves the maximum defining £ ge £ ¦ 2 for all trees & (' v v Q£ ) .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: sparse or Gaussian) can determine the type of structure captured in B, we discovered that it does not affect the nature of the lower-level representation.
Sentence 2: for the results reported here, we assumed a Gaussian prior on v. as in other multi-variate Gaussian models, the precise directions of B are not important; the learned vectors only serve to collectively describe the volume of the space.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The key to the utility of this technique is that the number of compound parameters can be polynomial in L even when the number of possible topologies is exponential in L. Previous analytic techniques [13] based on Kruskal's theorem [10] cannot be applied here because the possible topologies are too many and too varied.
Sentence 2: note that the mixing equation µ = M Ψ holds for each sentence length L, but many compound parameters p appear in the equations of multiple L. we can combine the equations across all observed sentence lengths, yielding a more constrained system than if we considered the equations of each L separately.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Beginning with only the root ∅, AOBFS expands the search tree in a best-first manner.
Sentence 2: more precisely, it maintains explicit AND/OR search tree of visited nodes, denoted S. For each node n in the AND/OR search tree, AOBFS maintains u n , an upper bound on v n , initialized via a pre-compiled heuristic v n ≤ h + n , and subsequently updated during search using information propagated from the frontier: the upper bound at the root, u ∅ , is an anytime deterministic upper bound of Z.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Usually the optimal policy is defined as the policy that maximizes the value conditioned on the initial state, i.e., for any realization s of S 0 [10].
Sentence 2: the policy that maximizes the conditional value given each realization of the initial state also maximizes the value averaged over the initial state, and it is the unique maximizer if the distribution of the initial state D is positive over S. when optimizing over all possible policies, the maximization of ( 1) and (2) are equivalent.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The minimax problem ( 15) is the f-GAN problem introduced and discussed in [3].
Sentence 2: theorem 2 reveals that f-GAN searches for the generative model minimizing the f-divergence to the discriminator moment matching models specified by discriminator set F. The following example shows the application of this result to the vanilla GAN introduced in the original GAN work [1].
Relationship: reasoning