sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Suppose rank(X * ) ≤ k and ε > 0.
Sentence 2: if each V t computed in blockFW is a ( 1 2 , ε 8 )approximate solution to (3.1), then for every t, the error h it takes O(κ log h1 ε ) iterations to achieve the target error h t ≤ ε.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, implicit regularization has been shown to play a crucial role in training deep models [14,13,18,11]: deep models often generalize well even when trained purely by minimizing the training error without any explicit regularization, and when there are more parameters than samples and the optimization problem is underdetermined.
Sentence 2: there are many zero training error solutions, all global minima of the training objective, many of which generalize badly.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, observing target i at track j will zero out p everywhere outside the coset µ j S n−k µ −1 i , which is difficult for the Fourier approach to handle.
Sentence 2: in fact, by analogy with (7), denoting the operator that projects to the space of functions supported on this coset by P i j , the new distribution will just be P i j p. if we set ν 0 = ξ, after any single observation, our distribution will lie in V 1 of the corresponding Bi-CMRA.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, by making novel use of the Sherman-Morrison-Woodbury formula , a much smaller matrix of the order of the original input space is inverted at each step.
Sentence 2: a problem with a 32-dimensional input space and 7 million points required inverting positive definite symmetric matrices of size 33 x 33 with a total running time of 96 minutes on a 400 MHz Pentium II.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Reliabl v inference from the observed number of spikes about the underlying firing rate of a neuronal response, however, requires a sufficiently long time interval, while integration times of neurons in vivo [3] as well as reaction times of humans or animals when performing classification tasks [4,5] are known to be rather short.
Sentence 2: it is important to understand, how neural rate coding is affected by a limited time window available for decoding.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By equation ( 16), the number of mistakes will be upper bounded by 10p + 5 because there is a cut of size 1 and the diameter is 2p.
Sentence 2: for balanced octopi O m,p with p ≥ 3, as m grows the number of mistakes of the kernel perceptron will be bounded by a fixed constant.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let X § mf % g0 d k (1 be the number of times that a reduced joint action Young showed that AP in a weakly acyclic game converges to a strict Nash equilibrium w.p.1.
Sentence 2: aP on the VG for the game in Table 1 leads to an equilibrium with payoff 1 which is actually an optimal Nash equilibrium for the original game.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider a setting with binary relevance judgement labels, so that Y = {0, 1} r with n = 2 r . The prediction space is T = S r (group of permutations on r objects) with k = r!.
Sentence 2: for y ∈ {0, 1} r and σ ∈ S r , where σ(i) denotes the position of document i under σ, the Precision@q loss for any integer q ∈ [r] can be written as follows: by Theorem 3, for the r-dimensional surrogate ψ * P@q : {0, 1} r × R r → R+ and pred * P@q : R r →S r defined as we have that (ψ * P@q , pred * P@q ) is � P@q -calibrated.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If (a) the weight functions w n and the sampling densities q n are pointwise strictly positive and such that w n (x)q n (x) converges in the L ∞ -norm, and (b) E qn(x) w 2 n (x) is bounded , then θn − θ n converges to zero in probability and we have where From Proposition 1, it is worth noting that in general θ n will not converge to the population maximum likelihood estimate θ 0 , i.e., using a different sampling distribution than p 0 (x) may introduce a non asymptotically vanishing bias in estimating θ 0 .
Sentence 2: active learning requires to ensure that (a) our estimators have a low bias and variance in estimating θ n , and (b) that θ n does actually converge to θ 0 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: During training, the network function f θ evolves along the (negative) kernel gradient with respect to the neural tangent kernel (NTK) However, in contrast to F lin , the realization function F (L) of ANNs is not linear.
Sentence 2: the derivatives ∂ θp F (L) (θ) and the neural tangent kernel depend on the parameters θ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: These computations, appearing as T a→b c (α b,i,k , α c,k+1,j ), T b→c a (1,2) (β b,k,j , α c,k,i−1 ) and T b→a c (1,3) (β b,i,k , α c,j+1,k ) output a vector of length m, where computation of each element in the vector is O(m 2 ).
Sentence 2: the inside-outside has a multiplicative O(m 3 ) factor in its computational complexity, which we aim to reduce.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Finally, a Dirichlet prior with parameter ) is placed on the topics & 0' (¨¦ .
Sentence 2: the generative process is given by ¥ ©¦ ¨ 21 43 65 07 & 8' (¨¦ 91 43 65 ) @7 A 1 B¥ ¦ ©¨ " 8 1 C& #' (¨D FE HG Given the observed words I CP " Q ¡ , the task of Bayesian inference is to compute the posterior distribution over the latent topic indices R BP ¡ , the mixing proportions ¥ !¦ ©¨ , and the topics & #' (¨¦ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, to ensure fairness of comparison, we also allow each baseline method to access and include all training datasets of all training multitask problems, besides the training datasets in a test multitask problem at hand.
Sentence 2: we report the better performance of each baseline method when it learns on the test multitask problem only and on all the training and test multitask problems, respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, the on-policy operator is a contraction with respect to • z,r .
Sentence 2: proof: once again by the contraction map fixed point theorem there exists a fixed point of O among row normalized matrices H, and repeated applications of O will converge to a matrix H Π such that OH Π = H Π ; that is, H Π satisfies the policy based Bellman equation for dual representations.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that Sobolev spaces are large: In fact, Sobolev spaces are more flexible than Hölder spaces (a generalization of Lipschitz spaces to higher order smoothness) in that in these spaces the norm measures the average smoothness of the functions as opposed to measuring their worst-case smoothness.
Sentence 2: functions that are smooth most over the place except for some parts that have a small measure will have small Sobolev-space norms, i.e., they will be looked as "simple", while they would be viewed as "complex" functions in Hölder spaces.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For a good generative model, we expect that x 0 = g ✓ (z 0 , y source ) is diverse for randomly sampled z 0 and o(x 0 ) = y source holds with high probability.
Sentence 2: by reducing the distance between z and z 0 , L 1 has the effect of generating more diverse adversarial examples from class y source .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In addition, for every γ ∈ (0, 1) define the auxiliary measure where Q S γ is constructed according to (15) with Applying Lemma 2 with φ(u) = u log u, we can write Next we apply Lemma 3.
Sentence 2: defining η = 1+h 2 and using the easily proved fact that combining Eqs.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: That is, we would like to learn a mapping f : X → {±1} k over an infinite domain X using only a finite training set of objects, and then apply the mapping to obtain binary codes f (x) for future objects to be encountered, such that S(x, x ) ≈ sign( f (x), f (x ) − θ).
Sentence 2: the mapping f : X → {±1} k is usually limited to some constrained parametric class, both so we could represent and evaluate it efficiently on new objects, and to ensure good generalization.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: is the mean of n independent Bernoulli(0.5) random variables (since the rewards are chosen randomly), and has expected value 0.5.
Sentence 2: by the Hoeffding inequality, taking a union bound over all policies π ∈ Π M , we have (3) Here, the probability is over the random MDP M . But since L outputs a policy in Π M , the chance of L outputting a policy π with U M (π) ≥ 1 − 2ǫ is bounded by the chance that there exists such a policy in Π M . ) By setting the right hand side to 1/4 and solving for m, we see that so long as we have that P (U M (π) ≥ 1 − 2ǫ) < 1/4.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We also know from the proof of Theorem 2 that lim k→∞ lim h→∞ lim i→∞ λ i k,h = λ * ∞ .
Sentence 2: there exist k 0 , h 0 , i 0 such that |λ i k0,h0 − λ * ∞ | < δ for any i > i 0 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The model is illustrated in Figure 1 jk ] of matrix Θ k . This way linking tendency from node i to node j is reflected based on their membership to group k. We then determine the overall link probability p (t) ij by combining the link affinities via a logistic function g(•) 1 .
Sentence 2: where ǫ t is a density parameter that reflects the varying link density of network over time.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Eliminating x i in (2.3) by the representation above yields j .
Sentence 2: x ij can be selected as a neighbor of x i if the orthogonal projection θ (i) j is small and (2.4) Assume all the x ij satisfy the above inequality, then we should approximately have We will use (2.5) as a criterion for adaptive neighbor selection, starting with a K-NN at each sample point x i with a large enough initial K and deleting points one by one until (2.5) holds.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: of the target signal, it can be seen that the solution ŵ(t) of deterministic equation corresponding to (14) (which is of the same form as (10) with the obvious sign changes) converges to an eigenvector of the covariance matrix C 0 .
Sentence 2: for β = 0 we expect the learning rule (14) to perform PCA for small learning rates α.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The primary motivation in considering probably conditionally concave energies is that they enjoy (with high probability) the same properties as the conditionally concave energies, i.e., (i)-(iii).
Sentence 2: locally minimizing probably conditionally concave energies over F can be done also with the Frank-Wolfe algorithm, with guarantees (in probability) on the feasibility of both the optimization result and the solution set of this energy.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [11, Lemma 1] The shrinkage operator S ν (•) is non-expansive, i.e., for any M and Proof.
Sentence 2: for any given entry index (i, j), there are four cases: • Case 3: The sequence {M k } generated by the projected gradient descent iterations in Algorithm 1 with 0 < τ < min(2, 2 ρ ) converges to M * , which is an optimal solution of (2).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Under the following conditions, the rounded (re-optimized) estimator maintains bounded response: (i) If either y 1 remains bounded, or κ(x Note that the latter condition causes any convex loss to demonstrate unbounded response (see proof of Theorem 5 in Appendix B).
Sentence 2: the approximate estimator is strictly more robust (in terms of bounded response) than regularized empirical risk minimization with a convex loss .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: They are in different scales: λ i (K) ≤ 1 (resulting from the fact that κ(x, y) ≤ 1 for all x and y), and λ i (A/ρ(n)) ≈ λ i (K/n) ≈ λ i (K) if n and ρ(n) are sufficiently large.
Sentence 2: λ d (K) are independent of n for a fixed d and should be treated as Θ (1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given K i g , K max g and K i H , K max H , and 0 < δ, δ g , δ H < 1, we define At any x k ∈ M, suppose that the sampling is done uniformly at random to generate S g and S H . Then, we have From Theorem 4.1, it can be easily seen that Assumption 4 follows with the same probability with It should be emphasized that if we use the typical condition (5) instead of Assumption 4, we obtain, e.g., for the sub-sampled Hessian H k . Considering that ∥η k ∥ goes to nearly zero as the iterations proceed, this obtained bound indicates that |S H | increases accordingly.
Sentence 2: the size of the sub-sampled Hessian needs to be increased towards the convergence.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Knowing that and by the assumption of ||w|| 1 = 1, after simple calculations, we find that ρ ≤ h = ρ/||w|| 2 ≤ √ M ρ.
Sentence 2: by minimizing d ∞ (0, P ) = ρ, (and ||w|| 1 = 1), h will be bounded and the volume of the simplex considered, as well.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: One can show that the only function (up to a multiplicative constant) that can serve as penalty without breaking the property of scale equivariance is the logarithmic function.
Sentence 2: we introduce an additional tuning parameter µ > 0 and look for minimizing the criterion If we make the change of variables φ 1 = Mβ/σ, φ 2 = Nβ/σ and ρ = 1/σ, we get a convex function for which the first-order conditions [20] take the form Thus, any minimizer of (2) should satisfy these conditions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First, notice that (by negation) the feasibility constraints in (1) can also be defined as: In other words, a solution w is feasible iff there exist no pair of samples from the two classes such that f w ( ) orders them incorrectly.
Sentence 2: second, we will make the constraints in (1) more stringent: instead of requiring that equation (1) be satisfied for each possible pair (x + ∈ A + , x − ∈ A − ) in the training set, we will  Next, notice that all the linear inequality and equality constraints on (λ + , λ − ) may be conveniently grouped together as Bλ ≤ b, where, ) our constraints on w can be written as: Where the second equivalent form of the constraints was obtained by negation (as before), and the third equivalent form results from our third key insight: the application of Farka's theorem of alternatives [9].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Both the encoder and decoder cells also use the information that is propagated horizontally.
Sentence 2: at every iteration t, an encoder cell in the l-th layer receives three inputs: 1) the output of the encoder cell from the level below e l 1 (t), 2) the output d l (t 1) of the decoder cell from the same level from the previous iteration, 3) the encoder state s l (t 1) from the same level from the previous iteration.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 20) tells us that when β is sufficiently small, The term in the denominator, b (0), is relatively easy to compute, and we show in Appendix C (in the supplementary material) that it is given by The key result of the perturbative analysis is that when the true distribution is out of the model class, the normalized bias can be increased by a term proportional to b (0) 1/2 .
Sentence 2: the size of b (0) is crucial for telling us how big the bias really is.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Simplified equation When the tuning curves of the sensory cells are uniformly distributed Gaussians (e.g., spatial receptive fields), namely λ k (x) = λ max exp(−(x−k∆x) 2 /2σ 2 ), it can be shown [13] that for small enough ∆x, and a large number of sensory cells, Therefore the matrix Λ has no effect on the solution of (4), except for an exponential attenuation that is applied to all the cells simultaneously.
Sentence 2: in cases where the number of sensory cells is large, Λ can be omitted from (4).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that whereas (J2 was previously described as a human-specified parameter, the analysis also suggests a particularly simple way of choosing it automatically: For the right (J2, Theorem 2 predicts that the rows of Y will form k "tight" clusters on the surface of the k-sphere.
Sentence 2: we simply search over (J2 , and pick the value that, after clustering Y 's rows, gives the tightest (smallest distortion) clusters.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It only requires O(log(1/δ)) power iterations to compute a top SVD solution of accuracy δ.
Sentence 2: for a sufficiently small δ > 0, the overall per-iteration complexity of the distributed method at the tth iteration is reduced to O(|Ξ| + max{n, p} log(1/δ)) at the central server, and O(|Ω k |(max{n, p} log(1/δ) + q)) at the kth worker.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: One can show that the Haldane prior is equivalent to putting a flat prior on the log-odds log[q j /(1 − q j )]; also, it has the largest variance (and hence is "most uninformative") among all the possible distributions of q j .
Sentence 2: although appearing to be extremely dichotomous, it is well known in Bayesian statistics as an uninformative prior of binomial distributions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The optimization problem defined in Equation 5 is very likely to overfit if we cannot precisely estimate the hyperparameters, which automatically control the generalization capacity of the MRMA model.
Sentence 2: for instance, σ U and σ V will control the regularization of U and V . it is more desirable to estimate the parameters and hyperparameters simultaneously during model training.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Evidently θ * = 0, and by an asymptotic expansion we have that E[θ 1 ] = Ω(n − 1 2 ) (see the long version of our paper [22,Appendix D] for this asymptotic result).
Sentence 2: the bias of θ 1 is Ω(n − 1 2 ), and the AVGM algorithm using N = mn observations must suffer mean squared error E[(θ 1 − θ * ) 2 ] = Ω(n −1 ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Our goal is to construct a mapping f : V w2v → V rnn parameterized by a matrix W such that v = Wv for v ∈ V w2v and v ∈ V rnn .
Sentence 2: inspired by [15], which learned linear mappings between translation word spaces, we solve an un-regularized L2 linear regression loss for the matrix W. any word from V w2v can now be mapped into V rnn for encoding sentences.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then, the column space of P contains an approximately optimal solution for A, since AS − P N T 2 F and AS is a column pcp for A.
Sentence 2: it follows from Lemma A.1 that with probability at least 98/100, there exists a good solution for A in the column space of P . Since we cannot compute this explicitly, we set up the following regression problem: min X A − P X 2 F . Again, we sketch the regression problem above by sampling columns of A according to the leverage scores of P . We can then compute X A = argmin X E A − E P X 2 F with probability at least 99/100, where E is a leverage score sketching matrix with poly k rows.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A second key limitation is that the LCMS method does not allow for dependencies between multiple biomarkers, measurement noise, or latent variables (such as unobserved confounders of the biomarker-outcome associations).
Sentence 2: for instance, without allowance for noise in the biomarker measurements, non-zero conditional mutual information I(g i , y|x j ) will be interpreted as evidence of pleiotropy or reverse causation even when the relation between the underlying biomarker and outcome is causal.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Additionally, this means that p(S) ≥ 2/3, so again with high probability, at least half of our m samples will lie in S. These first m /2 samples from S will be independent samples from (p|S), which is uniform, and therefore with 99% probability will pass the uniformity tester.
Sentence 2: in this case, our algorithm will return "YES" with probability at least 9/10.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then T j,j,j = Θ(v 3 1,j ) + E j,j,j , and one can show T j, * , * ) if and only if T j,j,j = ω(1/n), and therefore one would notice this by reading T j,j,j .
Sentence 2: there can only be o(n 2/3 ) slices j for which T j, * , * for each of them we can afford to take O(n 2 ) 2 -samples and still have an O(n 2+2/3 ) = o(n 3 ) sublinear running time.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given a set A, we define SampleSubset(A,p) as the process of applying a Bernoulli trial with success probability p to each element of A and returning the subset of elements for which the trial was successful.
Sentence 2: sampling ϕ global and ϕ eventual is accomplished by performing SampleSubset(τ , p G ) and SampleSubset(Ω, p E ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given the rapid advances in data collection technologies for biological systems, we expect that complex, high-dimensional, and feature rich data from complex dynamic biological processes, such as cancer progression, immune responses, and developmental processes, will continue to grow.
Sentence 2: we believe our new method is a timely contribution that can narrow the gap between imminent methodological needs and the available data and offer deeper understanding of the mechanisms and processes underlying biological networks.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this paper, following [5,30], we use the hinge loss: to measure the misclassification cost.The expected and empirical risks associated with the hinge loss are defined respectively as: and In theory, the excess misclassification error R(sgn(f )) − R(f c ) can be bounded by the excess convex risk E(f ) − E(f c ) [29,1,7].
Sentence 2: the classification algorithm usually is constructed under structural risk minimization [22] associated with E z (f ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We can then consider a quadratic regularizer of the form cV n /2kwk 2 to define the regularized empirical risk R n (w) := L n (w) + (cV n /2)kwk 2 and the corresponding optimal argument Since the regularization in ( 4) is of order V n and (3) holds, the difference between R n (w ⇤ n ) and L(w ⇤ ) is also of order V n -this may be not as immediate as it seems; see [22].
Sentence 2: we can say that a variable w n satisfying R n (w n ) R n (w ⇤ n )  V n solves the ERM problem to within its statistical accuracy.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The seller can compute an offline estimate of both this projection and by collecting a dataset of context vectors before starting to offer prices to the buyer.
Sentence 2: in view of Proposition 1 and the strong convexity assumption, we see the learning phase of the LEAP algorithm is conducting a stochastic gradient descent to minimize the -strongly convex function F , where at each time step we update w t+1 = ⇧ W (w t 1 t gt ) and gt = g t is an unbiased estimate of the gradient.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In practice, there may be fewer nodes since one need only to allocate space for contexts found in the data at least once, but their total number may still grow linearly with the length of the input string.
Sentence 2: for problems such as partially observable RL, in which the amount of input data is unbounded, or for large |Σ| and K, the memory used by CTW can quickly become unreasonable.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let N (F, r) be the size of the smallest -net over F with respect to the infinity norm at scale r = random landmark pairs then we have the following with probability greater than (1 − δ) This result tells us that in a large enough landmarked space, we shall, for each function f ∈ F, recover close to the best classifier possible for that transfer function.
Sentence 2: if we iterate over the set of transfer functions (or use some gradient-descent based optimization routine), we are bound to select a transfer function that is capable of giving a classifier that is close to the best.
Relationship: reasoning