sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: We propose a generalization of a well known extremality result for orthogonal univariate polynomials on the real line (or the complex plane) [16,Theorem 3.1.2].
Sentence 2: the distinguished SOS polynomial of interest in this paper is understood as the unique optimal solution of a convex optimization problem: minimizing an average value over a structured set of positive polynomials.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Applying Chernoff bounds, Ç ÐÓ ´½ AEµ ¯ examples are sufficient that, with probability at least ½ AE ¾, this classifier is incorrect on at most a fraction ¯ ¾ of the training data.
Sentence 2: the training error of the hypothesis output by is at most ¯ ¾ with probability at least ½ AE ¾.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, as shown in [13], the Euclidean distance between points mapped to these eigenvectors is equal to a so called 'diffusion distance' between points on the graph, where p(z, t | x) is the probability distribution of a random walk at time t given that it started at x, π s is the equilibrium distribution, and • L2(w) is the weighted L 2 norm with weight w(z).
Sentence 2: the eigenvalues and eigenvectors {λ j , v j } for j > 1, capture the characteristic relaxation times and processes of the random walk on the graph towards equilibrium.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 3 Because the Hessian of L(W ) is Lipschitz continuous, if the Hessian at a saddle point has a negative eigenvalue, it starts appearing as we approach the saddle point.
Sentence 2: theorem 2.3 and Corollary 2.4 suggest that for 1-hidden layer networks, training can be done in polynomial time with a second order method or even with a modified stochastic gradient decent method, as discussed in (Ge et al., 2015).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In [PBM16], the authors assume that .
Sentence 2: for variable E and any set S, if , then S is not the parent set of E. Similarly, the authors of [WSBU18] consider the case that a (i) 6 = a (j) and either E . under this assumption, in this case for variable X 2 {C, E} and set , then S is the parent set of X.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is our second important system of spaces.
Sentence 2: to understand the interplay between modules and window spaces, observe that each coset µ i1...i k S n−k has an internal notion of left-translation which fixes S i1...i k . S i1...i k must be decomposable into a sum of irreducible Furthermore, the modules of different window spaces can be defined in such a way that is an S n−k -module in the sense of being invariant to the internal translation action (4), and this action depends on i 1 . . . i k .) Now, for any fixed t, the space U = i1,...,i k M i1...i k t , is fully S n -invariant, and therefore we must also have U = α∈A M α , where the M α are now irreducible S n -modules.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Recall that uPU overfitted seriously on all the benchmark datasets, and note that the larger π p is, the more different nnPU is from uPU.
Sentence 2: the replacement of π p with some π p > π p introduces additional bias of R pu (g) in estimating R(g), but it also pushes R pu (g) away from R pu (g) and then pushes nnPU away from uPU.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As mentioned before, let o l i (•) denote the output function of this locally learned KM, then the good performance of local learning methods mentioned above implies that o l i (x i ) is probably a good guess of f l i , or the proper f l i should be similar as o l i (x i ).
Sentence 2: a good SPM F should have the following property: For any x i (1 ≤ i ≤ n) and any cluster C l (1 ≤ l ≤ c), the value of f l i can be well estimated based on the neighbors of x i .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: end for end while data that survived upto stage k. On the other hand, the optimization problem for g k is exactly in the form where Lemma 2.1 applies.
Sentence 2: we can also reduce this problem to a supervised learning problem: where The composite classifier F (x) based on the reject and region classifiers can be written compactly as follows: Observe that if the kth region classifier correctly classifies the example x i , i.e., f k (x i ) = y i then this would encourage g k (x i ) = 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then, by the iterative applications of Eqs.
Sentence 2: ( 2) and (3), we obtain where f l is the l-time compositions of f . λ j characterizes the temporal behavior of the corresponding Koopman mode u j , i.e., the phase of λ j determines its frequency, and the magnitude determines the growth rate of the dynamics.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If µ is a compactly supported distribution satisfying the two-subspaces criterion for γ > 0, then there exists a constant α 0 ≡ α 0 (µ, D, ) > 0 such that for any > 0 and N > 2(D −1) the following estimate holds: where .
Sentence 2: (1.9) Intuitively, α 0 represents a lower bound on the directional second derivatives of F . α 0 should affect sample complexity because the number of random samples taken to approximate a minimum of F should be affected by how sharply F increases about its minimum.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is of course possible to select more than one K, but only if the data indeed supports both clusterings.
Sentence 2: indirectly, the BTR can provide a theoretically sound method of selecting K. The author gratefully acknowledges Maryam Fazel for her early interest in this work, for reading a previous version of this paper and for many discussions; the Simons Institute for the Theory of Computing where part of this research was performed, and partial support from the NSF DMS PD 08-1269 and NSF IIS-0313339 awards.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: At the level of generality at which (VC) has been stated, it is unlikely that any convergence rate can be obtained, because unlike in the convex case, one has no handle on measuring the progress of mirror descent updates (recall that in (VC), only non-negativity is guaranteed for the inner product).
Sentence 2: we focus here on the class of strongly coherent problems (a generalization of strongly convex problems) and derive a O(1/ √ T ) convergence rate in terms of the squared distance to a solution of (Opt).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The second KL term vanishes as the two distributions are equivalent.
Sentence 2: our final maximization objective is: where we first sample random weights with dropout masks ω ∼ q M (ω|X, Y) and sample z such that z = g(x, ε, ω), ε ∼ N (0, I), with a pathwise derivative function g for reparameterization trick.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Applying the feature-based model analysis from Experiment 1 to the current results revealed that the weighted feature model required an average of 6.0 features to match the performance of the absolute utility model for participants in Cluster 1, and an average of 3.9 features to match the performance of the relative utility model for participants in Cluster 2.
Sentence 2: although a single model did not fit all participants well in the current experiment, many participants were fit well by one of the two inverse decision-making models, suggesting that this general approach is useful for explaining how people reason about negative effects as well as positive effects.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, using a standard covering argument and a union bound [23] shows that if (9) holds for a sufficiently large constant c 1 > 0 then we have (4) for a sufficiently small constant δ 2k with probability exceeding 1 − 2e −cm for some constant c > 0 that depends only on δ 2k .
Sentence 2: theorem 1 yields the desired result which holds with probability exceeding 1 − 2e −cm − e −C3n ≥ 1 − 3e −c2m for some constant c 2 > 0 depending only on δ 2k .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let -AI > -A2 > -A3 > ... be the poles of J (z) and mk be the order of -Ak.
Sentence 2: then, by using the inverse Mellin tansform, it follows that v(t) has an asymptotic expansion with coefficients {Ckm}, k=lm=1 also Fo (n) has an asymptotic expansion, by putting A = Al and m = ml, which ensures the asymptotic expansion of F(n) by eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, raw features are typically not realizable, i,.e., the assumption that e t (s t , a) = θ * a T x(s t ) + η t does not hold.
Sentence 2: we propose learning a set of features φ(s t ) that are realizable, i.e., e(s t , a) = θ * a T φ(s t ), using neural networks (using the activations of the last layer as features).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As a first step, we translate the Boolean alphabets Z = {0, 1} to the "Ising" ones, S = {+1, −1}.
Sentence 2: we need to translate the additive operations, such as, Similarly, we translate the Boolean y i (t)s into the Ising J i (t)s. For simplicity, we omit the subscript i, which labels the L agents, in the rest of this section.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , q ri ) is an orthonormal basis to the first r i columns in S. Thus, v i (r i ) is the (squared) error of approximating x i with the first r i columns in S. In each pass the algorithm identifies the l candidate columns x i corresponding to the l largest values of v i (|S|).
Sentence 2: that is, the v i values are computed as the error of predicting each candidate by all columns currently in S. The identified l columns with the largest v i (|S|) are stored in the list L. In addition, the value of the l+1'th largest v i (|S|) is kept as the constant B F . after a pass is terminated the following condition holds: The list L and the value B F can be calculated in one pass using a binary heap data structure, with the cost of at most n log(l + 1) comparisons.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Here, our approach is based on a simple theoretical connection between GDA and the softmax classifier: the posterior distribution defined by the generative classifier under GDA with tied covariance assumption is equivalent to the softmax classifier (see the supplementary material for more details).
Sentence 2: the pre-trained features of the softmax neural classifier f (x) might also follow the class-conditional Gaussian distribution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The fixed point equation for the Using the definition of U and S, we have is the ridge regression solution of where Tr is the trace.
Sentence 2: after multiplying out all i T i in 1/n n i=1 i T i , we obtain: For the fixed point of Ψ, the update rule gives: diag (Ψ) = diag 1 n n i=1 i T i + W ΣW T . Thus, W minimizes Tr (Ψ) given µ i and Σ. Multiplying the Woodbury identity for W W T + Ψ −1 from left and right by Ψ gives Inserting this into the expression for diag (Ψ) and taking the trace gives for Ψ W W T the error is quadratic in Ψ. W U T = W SW T = U W T follows from fixed point equation U = W S. Using this and Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, we will see shortly that the PAC-Bayes theorem offers a better guarantee for another type of deterministic classifier.
Sentence 2: with these choices for Q i (ρ ρ ρ), P I (i), and P i (ρ ρ ρ), the KL divergence between Q i and P is given by Notice that the KL divergence is small for small values of |i| (whenever p(|i|) is not too small) and for large margin values (b i − a i ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: While we prove that the learning bounded vertex cover Bayesian network structures can be done in polynomial time, the unavoidable dependence on k in the degree the polynomial makes the algorithm of our main theorem infeasible for practical usage when the vertex cover number k increases.
Sentence 2: we investigate using an integer linear programming (ILP) formulation as an alternative way to find optimal bounded vertex cover Bayesian networks in practice (Section 4).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Recall that our generalization bounds use the posterior's divergence from a fixed prior to penalize the posterior for overfitting the training data.
Sentence 2: to connect Algorithm 1 to our bounds, we analyze the adaptive posterior's divergence from a uniform prior on {1, . . . , n} T . This quantity reflects the potential cost, in generalization performance, of adaptive sampling.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, since the outgoing edges of v are divided by c, we get γ p ( w(t) , e) = γp(w (t) ,e) c 2 and ∂L ∂we ( w(t) ) = ∂L c∂we (w (t) ).
Sentence 2: a similar argument proves the invariance of Path-SGDupdate rule for outgoing edges of v. Path-SGDis rescaling invariant.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Each observation n belongs to exactly one latent cluster k, indicated by z n . The binary variable i ng indicates whether group g is present in observation n. Finally, the probability of some observation w nd = 1 depends on whether its associated group g (indicated by l d ) is present in the data (indicated by i ng ) and the associated formula f g . The complete generative process first involves assigning dimensions d to groups, choosing the formula f g associated with each group, and deciding whether each group g is important: where DP is the Dirichlet process.
Sentence 2: there are an infinite number of potential groups; however, given a finite number of dimensions, only a finite number of groups can be present in the data.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Structures with the same edge sets but slightly different transition times will probably have similar likelihoods.
Sentence 2: we can add a new move that proposes a local shift to one of the transition times: let d be some small positive integer and let the new time t i be drawn from a discrete uniform distribution t i ∼ DU (t i − d, t i + d) with the constraint that t i−1 < t i < t i+1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The only property of the prediction strategy that we used was that it plays in the eigensystem of the past sufficient statistic.
Sentence 2: using the same line of argument, we can show that if for some classical prediction strategy we can obtain a meaningful regret bound by bounding each term in the regret δ t independently, we can obtain the same bound for the corresponding matrix strategy, i.e.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that the first column of T contains the impulse response curve of the closed-loop system, while the rest of the columns are obtained shifting that impulse response down.
Sentence 2: we can build the transfer matrix T either in a model-based manner, deriving the state-space characterization of the closed-loop system, or in measurement-based manner, measuring the impulse response curve.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Applying this bound and (1.8) we obtain that if η > 2 is fixed, 1/η < < 1 2 and N ≥ D η , then (2.1) Since > 1/η the first term in the RHS of (2.1) decays exponentially as a function of D (or, equivalently, as a function of N ≥ D η ).
Sentence 2: similarly, since 0 < γ < 1 and η > 1 the second term in the RHS of (2.1) decays exponentially as a function of D. Furthermore, since < 1 2 it follows that the error term for the minimizer, i.e., N − 1 2 + ≤ D η( − 1 2 ) , decays polynomially in D. in order to achieve low error estimation with high probability it is sufficient to take N = Ω(D η ) samples for any η > 2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As the input x can have highly complicated structure, we employ a dedicated model or algorithm to extract an expressive representation h from x.
Sentence 2: when we extract the representation h from x, we want the representation h to preserve variations that are necessary to predict y while eliminating information of s. To achieve the aforementioned goal, we employ a deterministic encoder E to obtain the representation by encoding x and s into h, namely, h = E(x, s).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In addition to the O(nd) cost for calculating , our algorithm requires an O(n log n) cost for sorting the entries of , as well as an additional running time of O(k ⇤ ), where k ⇤ is the number of non-zero elements at the optimum.
Sentence 2: the running time of our method is independent of any accuracy ✏, and may be significantly better compared to any off-the-shelf optimization method.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Even if available, such learning is often overkill.
Sentence 2: instead of trying to learn h everywhere, we only seek to learn the parameters w of a function f w that lives within some family, parameterized by w, so that if B ∈ argmax A⊆V :|A|≤k f w (A), then h(B) ≥ αh(A * ) for some α ∈ [0, 1] where A * ∈ argmax A⊆V :|A|≤k h(A).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Recall that r G (p, q) = v p − v q 2 G (see equation ( 7)) and use n q=1 v q = 0 to obtain that n n q=1 v q Gv q which implies the left inequality in (8).
Sentence 2: next, by Corollary 3.1, if T is the Laplacian of a tree T ⊆ G then r G (p, q) ≤ r T (p, q) for p, q ∈ V . from Lemma 3.2 we conclude that r G (p, q) ≤ d T (p, q).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Without loss of generality, let the work area (that fully contains the movement range) be an axis-aligned square The endpoint of the limb (wrist) is assumed to be connected to one end of four imaginary springs, the other end of which is sliding with no friction along rails forming the boundaries of the "work area".
Sentence 2: at every time instance each spring is parallel to one of the axes.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Features replay denotes that immediate feature is input into the module k for the first time in the forward pass at iteration t + k − K, and it is input into the module k for the second time in the backward pass at iteration t. If t + k − K < 0, we set h t+k−K L k−1 = 0 .
Sentence 2: the new problem can be written as: where denotes the gradient of the loss f (w t ) regarding ht L k with input ht L k into the module k + 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For this distribution, the equivalent covariance matrix can be given by: Now, the eigenvalues λ 1 , ..., λ n of the overall covariance matrix Σ * characterize the variance of the distribution across n dimensions.
Sentence 2: since Σ * is positive-definite, all eigenvalues are positive (this can be shown using the fact that each covariance matrix is itself positive-definite, and diag(µ to describe the variance of the feature distribution we define Diversity.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A policy, At each stage k, a stage-reward function r k : Z k × U k × W k → R, quantifies the benefits of applying a control u k to a state z k , subject to a disturbance w k . A final reward function r N +1 : Z N +1 → R, similarly quantifies the benefits of ending at a state z N +1 .
Sentence 2: the expected reward starting from state z 1 and using policy π is: where the expectation is taken with respect to the disturbances.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Recall that the truncation function ψ(•) satisfies Then, we have where the last step is due to the basic inequalities 1 + x ≤ e x and (a + b) 2 ≤ 2a 2 + 2b 2 .
Sentence 2: by Chernoff's method [Lugosi, 2009], we have with probability at least 1 − δ, we have We complete the proof by taking the union bound over all w ∈ N (W, ε).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . achieves the same risk as any other model selection criterion defined with respect to the same estimators, up to lower order terms; in other words, model averaging based on the switch-distribution achieves at least the same rate of convergence as model selection based on any model selection criterion whatsoever (the issue of averaging vs selection will be discussed at length in the full paper).
Sentence 2: the theorem requires that the prior π in ( 4) is of the form (7), and satisfies ) π M , the prior on the total number of switch points, is allowed to decrease either polynomially or exponentially (as required for Theorem 1); π T and π K must decrease polynomially.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , u(x n ) and u(x * ) are jointly Gaussian, it is easy to see that the predictive (posterior) distribution P (u * |D), u * = u(x * ) is given by where .
Sentence 2: if p = M −1 y, the optimal prediction under the model is û * = k T * p, and the predictive variance (of P (u * |D)) can be used to quantify our uncertainty in the prediction.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 1b, we start oscillatory activity and we observe the following sequence of events: (1) the activity in the bottom WTA stage ramps up due to recurrent excitation, and when it is high enough it begins activating the middle WTA stage; (2) activity in the middle WTA stage ramps up and as activity in the inhibitory population of this stage rises, it shuts down the bottom stage activity; activity in the middle WTA stage keeps on increasing until it activates the top stage; (3) activity in the top WTA stage increases, shuts down the middle stage, and provides input back into the bottom stage via the plastic connections.
Sentence 2: a bump of activity continuously jumps from one WTA stage to the next.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If φ is a smooth map φ : R M → R M and this system is observed via function, y, such that where y : R M → R, then if φ is invertible, φ −1 exists, and φ, φ −1 , and y are continuously differentiable we may apply Takens' Theorem [24] to reconstruct the complete state-space of the observed system.
Sentence 2: for each s(t), we can construct a vector s E (t), such that s E lies on a subset of R E which is an embedding of s. Because embeddings preserve the connectivity of the original vector-space, in the context of RL the mapping ψ, may be substituted for f (Eqn.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Finally, we construct our estimator by averaging over the block matrices: We note that within k 2 entries of the diagonal, each entry is effectively the sum of k 2 estimates, and as we move from k 2 to k from the diagonal, each entry is progressively the sum of one fewer entry.
Sentence 2: within k 2 of the diagonal, the entries are not tapered; and from k 2 to k of the diagonal, the entries are linearly tapered to zero.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , hk −1 that leads to error less than /2 on the problem i k. In other words: Now, by way of contradiction, suppose that .
Sentence 2: by the definition of discrepancy and the assumption on the marginal distributions it follows that for all j: Therefore by Lemma 2: by using the triangle inequality: which is in contradiction with (15).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Here we represent the transition density between m states in the following form (Fig.
Sentence 2: 1): Here, χ(x) ⊤ = [χ 1 (x), ..., χ m (x)] represent the probability of configuration x to be in a metastable (long-lived) state i these functions are nonnegative (χ i (x) ≥ 0∀x) and sum up to one ( i χ i (x) = 1∀x).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Additional static information τ includes (a) terrain features, a H × W × C T tensor that includes elements such as walkability, buildability and ground height, and (b) the faction, f me and f op , that each player picks.
Sentence 2: each input x t = (o t , τ, f me , f op ) Additionally, we pick a temporal resolution of at least s = 5 seconds between consecutive states, again aiming to model high-level dynamics rather than small but often irrelevant changes.
Relationship: reasoning