sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: In this construction, rather that drawing the ✓ ⇤ i ⇠ G 0 once as in (5), one may draw ✓ ⇤ i ⇠ G 0 for the first block of time, and then a simple Gaussian auto-regressive model is employed to allow the {✓ ⇤ i } drift a small amount between consecutive blocks.
Sentence 2: if {✓ ⇤ ji } represents the atoms for block j, then ✓ ⇤ j+1,i ⇠ N (✓ ⇤ ji , 1 0 ), where it is imposed that 0 is large.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The estimation of discrete distributions given finite data -"histogram smoothing" -is a canonical problem in statistics and is of fundamental importance in applications to language modeling, informatics, and safari organization (1)(2)(3).
Sentence 2: estimation of discrete distributions under Kullback-Leibler (KL) loss is of basic interest in the coding community, in the context of two-step universal codes (4,5).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper we assess the performance of the first player using the notion of regret.
Sentence 2: given a number of rounds T and a fixed vector u ∈ S, we define the regret of the first player as the excess loss for not consistently playing the vector u, Our main result is an algorithmic framework for the first player which guarantees low regret with respect to any vector u ∈ S. we derive regret bounds that take the following form where f : S → R and L ∈ R + .
Relationship: entailment
Domain: NeurIPS
Sentence 1: [14] for online submodular function maximization: we run n copies of some low regret online learning algorithm, E 1 , E 2 , .
Sentence 2: . . E n , each with action space A = V . We use the ith copy E i to select the ith item in each predicted sequence S t . the predictions of E i will be v 1 i , v 2 i , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: This problem is often referred to as multiclass learning.
Sentence 2: in other multiclass problem settings it is common to encode the set Y as a prefix of the integers {1, . . . , r}, however in our setting it will prove useful to assume that the labels are encoded as the set of r standard unit vectors in r . the i'th label in Y is encoded by the vector whose i'th component is set to 1, and all of its other components are set to 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: • We obtain a characterization of the optimal sample complexity for estimating the entropy rate of a stationary reversible Markov chain in terms of the sample size, state space size, and mixing time, and partially resolve one of the open questions raised in [2].
Sentence 2: we show that when the mixing is neither too fast nor too slow, the sample complexity (up to a constant) does not depend on mixing time.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, a learning algorithm is a sample compression learning algorithm (that is identified solely by a compression set z i and a message string σ) iff there exists a Reconstruction Function R : (X × Y) |i| × K −→ H, associated with A.
Sentence 2: here, H is the (data-dependent) classifier space and R outputs a classifier R(σ, z i ) when given an arbitrary compression set z i ⊆ S and message string σ chosen from the set M(z i ) of all distinct messages that can be supplied to R with the compression set z i .
Relationship: entailment
Domain: NeurIPS
Sentence 1: That is, for each destination point C , the total of the incoming G -weight has to sum to e . In fact, it is quite easy to prove that this yields unbiased estimates [12]  Crucially, this argument does not depend on how the block decomposition is chosen or how the G -weights are set, so long as they satisfy (1).
Sentence 2: one could fix any block decomposition and weighting scheme, even one that depends on the target distribution £ and random variable ¢ , without affecting the unbiasedness of the procedure.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Pearce model instead spreads graded activation over x, based on a measure of similarity between the observed stimulus compound (or element) and the compounds represented by the model's configural units.
Sentence 2: if we denote the number of stimulus elements present in an observed stimulus pattern a as size(a), and in the pattern represented by the ith configural unit as size(i), then the activation of unit i by pattern a is given by x i = size(overlap(a, i)) 2 /(size(a) • size(i)).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore, we can compute the conditional expectation of p(jlx), denoted by Pr(jlxi = z), given that Xi assumes value z, where Xi represents the ith component of x.
Sentence 2: pr(jlxi = z) = E[Pr(jlx)lxi = z] = J Pr(jlx)p(xlxi = z)dx.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We prove that for each i ∈ [d], X h i (t) converges weakly to a deterministic function X i (t) as η → 0.
Sentence 2: we rewrite (3.8) as, (3.9) Taking the limit of η → 0 at a fix time t, we have Since ∇f ( X h (t)) 2 is bounded at the time t, we have 1 η Var X h (t + h) − X h (t) X h (t) = O(h).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given lack of space, we cannot detail the kernelization of all these methods, but they follow in the same manner as in the above two examples.
Sentence 2: each of these methods may be run in kernel space, and our analysis yields new insights into these methods; for example, kernelization of LMNN [11] using Theorem 1 avoids the convex perturbation analysis in [16] that leads to suboptimal solutions in some cases.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We define the subspace H m and dictionary I as γ-preserving w.r.t.
Sentence 2: space H n if Notice that the inverse since φ i ∈ Φ n , we have that in the worst case the error is bounded as we know that in the best case the error can be reduced up to Note that the directions associated with the larger eigenvalues are the ones that occur most frequently in the data.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If we believe that P "closely approximates" P f , and know the probability p xi i ≡ P(M xi a→i = 1 for all a ∈ C(i)) that x i is in the consistent domain for variable i of f , for every variable i, clause a and possible assignment x i , we can use them to dynamically order both the variables and the values they can take during search.
Sentence 2: we first compute p 1 i = P(M u a→i = 1 for all a ∈ C − (i)) and p 0 i = P(M u a→i = 1 for all a ∈ C + (i)) for each variable i, where C + (i) and C − (i) are the sets of clauses where variable i appears as a positive and a negative literal, respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let (f, ε) be a solution to optimization problem (5), then with probability at least 1 − 2ζ, Theorem 3 suggests that if we want to control the non-coverage rate on average at the nominal α −1 or α 1 rates with high probability, we should choose the α −1 or α 1 values to be slightly smaller than the desired ones in optimization (3) in practice.
Sentence 2: we need to make The next theorem ensures that the empirical ambiguity probability from solving (5) based on a finite sample will converge to the ambiguity given by the solution on an infinite sample (under the constraints E(H −ε (Y f (X))|Y = j) ≤ α j , j = ±1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The causal support model can be most easily described in the context of causal graphical models [12] (see Figure 1(b)).
Sentence 2: we consider two graphical models, Graph 0 (G 0 ) and Graph 1 (G 1 ), and we want to determine the log posterior odds of the models given some data D (i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Multi-stage ORDA further improves this rate to be uniformly-optimal.
Sentence 2: sAGE [8] achieves a nearly optimal rate since the parameter D in the convergence rate is chosen such that E x t − x * 2 2 ≤ D for all t ≥ 0 and it could be much larger than V ≡ V (x * , x 0 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: This metric can be addressed by all models which operate by first stochastically generating a sequence of hidden states and then defining a distribution over the data-space given the state sequence.
Sentence 2: for our model we approximate the marginal by a Monte Carlo estimate of (2) Note that sampling from p(h) boils down to sampling ξ 1...T from independent standard normals and then applying F g . the non-autoregressive property of our model allows us to estimate all words in some set S using K samples each by using only K independent trajectories h overall.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We can use a special case of a more general algorithm given in the appendix (Appendix B.2, Algorithm 3).
Sentence 2: we replace line 10 with the step w ⌧ +1 = (r ) 1 (r (w ⌧ ) r ⌧ ) where (w) = 1 2 kwk 2 r . We set r = log(p 0 )/(log(p 0 ) 1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The AM criterion is used to label T-F units that do not belong to any segments generated in initial segregation; such segments, as discussed earlier, tend to miss unresolved harmonics.
Sentence 2: unit u ij is labeled as target speech if the final square error is less than half of the total energy of the corresponding signal and the AM repetition rate is close to the estimated target pitch: Psychoacoustic evidence suggests that to separate sounds with overlapping spectra requires 6-12% difference in F0 [6].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Assume that F can be parameterized as F = {f θ : θ ∈ R d }, that we are given a sufficient statistics summary function φ : X × Y → R d , and that function evaluation is given by f θ (x, y) = θ, φ(x, y) .
Sentence 2: we let we have chosen our mechanism to encode a particular exponential family model, with C x (•) chosen as the conditional log partition function over the distribution on y given x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This demonstrates the effectiveness of making FastGRNN's parameters low-rank, sparse and quantized and allows FastGRNN to fit on the Arduino  Uno having just 2 KB RAM and 32 KB flash memory.
Sentence 2: fastGRNN was able to recognize the "Hey Cortana" wakeword just as accurately as leading RNNs but with a 1 KB model.
Relationship: entailment
Domain: NeurIPS
Sentence 1: By combining the result in Corollary 5 with Theorem 4, we can infer that in the heavily loaded regime, the time taken by Q-ThS to achieve O poly(log t)/✏ 2 t scaling is, in some sense, order-wise close to the optimal in the ↵-consistent class.
Sentence 2: for any 2 (0, 1), there exists a scaling of K with ✏ such that the queue-regret under Q-ThS scales as O poly(log t)/✏ 2 t for all t > (K/✏) while the regret under any ↵-consistent policy scales as ⌦ (K log t/ log log t) for t < K/✏.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Correspondingly, the bipartite graph along which messages are passed has ⌘p(p + 1)/2 minimizer-nodes that connect to the (⌘ + 1)p equality-nodes.
Sentence 2: the equalitynode associated with the break-point variable x i (s), ⌘ > s > 0, is connected to 2(p 1) different g coll minimizer-nodes and two different g cost C minimizer-nodes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In these cases, if we want to analyze the full procedure (from a learning theoretical sense), it is desirable to have a more precise information on the selected subspace than just its reconstruction error.
Sentence 2: from a learning complexity point of view, it is important to ensure that functions used for learning stay in a set of limited complexity, which is ensured if the selected subspace is stable (which is a consequence of its convergence).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Ridge leverage scores are closely connected to the residual Π ⊥ m φ i 2 after the projection Π m discussed in Definition 2.
Sentence 2: using Lemma 2 we have that the residual can be bounded as It is easy to see that, up to a factor γ 1−ε , high-RLS points are also high-residual points.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Instead, we impose a weaker condition known as restricted strong convexity (RSC).
Sentence 2: such conditions have been discussed in previous literature [9,1], and involve a lower bound on the remainder in the first-order Taylor expansion of L n . our main statistical result is based on the following RSC condition: where the α j 's are strictly positive constants and the τ j 's are nonnegative constants.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our performance bound is based on showing that W can be sandwiched between two simplicies with a small scaling factor.
Sentence 2: consider the following simplex,  Using Hoeffding's inequality [18] (see Appendix B) with ⌧ = b q log m n , we have and a union bound over j = 1, . . . , m gives us where the last inequality follows from Bernoulli's inequality.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, we note that the quantity E c∼U (0,1) er 0-1,c D Thresh D, η,c • η that appears in Theorem 10 is also the area under the cost curve [20,22]; since this quantity is upper bounded in terms of regret rank D [f ] by virtue of Theorem 4, we also get a weak regret transfer bound from bipartite ranking to problems where the area under the cost curve is a performance measure of interest.
Sentence 2: this implies that algorithms that are statistically consistent with respect to AUC can also be used to construct algorithms that are statistically consistent w.r.t.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The expected number of edges originating in x equals the expected number of points which lie in the intersection of this ball with the other side of the hyperplane.
Sentence 2: setting we have E(N 1 |X 1 = x) = (n − 1)g(x, r n ), since the number of points in the intersection of B(x, r n ) with the other side of the hyperplane is binomially distributed with parameters n − 1 and g(x, r n ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, identifying representative eigenvectors is not very trivial because in real life depending on the separation between high density clusters the representative eigenvectors can have no sign change up to some small precision > 0.
Sentence 2: we say that a vector e = (e 1 , e 2 , ..., e n ) ∈ R n has no sign change up to precision if either ∀i e i > − or ∀i e i < .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Against such a pricing strategy, it is in the buyer's best interest to be truthful, accepting if and only if v t ≥ p * , and the seller would receive a revenue of Informally, a no-regret algorithm is able to learn D from previous interactions with the buyer, and converge to selecting a price close to p * .
Sentence 2: we therefore define regret as: Finally, we will be interested in algorithms that attain o(T ) regret (meaning the averaged regret goes to zero as T → ∞) for the worst-case D. we say A is no-regret if sup D Regret(A, D, T ) = o(T ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Unlike previous linear models which behave differently under l ∞ and l 2 attacks, nonlinear DNN models show a consistent trend of adversarial robustness with respect to the sparsity.
Sentence 2: we observe increased r ∞ and r 2 values under different attacks when continually pruning the models, until the sparsity reaches some thresholds and leads to inevitable capacity degradations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let k denote iteration count, T k denote time when iteration k starts for k 2 N. Initially, k = 0, T 0 = 0, and for t 2 [T k , T k+1 ), the policy is in iteration k. The iteration is updated from k to k + 1 when starting with t = T k , all ball-action (B i , a) pairs have been visited at least once.
Sentence 2: in the policy description, the counter N k (c i , a) records how many times the ball-action pair (B i , a) has been visited from the beginning of iteration k till the current time t; that is, N k (c i , a) = P t s=T k 1{Y s 2 B i , a s = a}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The rank of L 0 is unknown, the support set Ω (i.e., the locations of the nonzero entries of S 0 ) and its cardinality (i.e., the amount of the nonzero entries of S 0 ) are unknown either.
Sentence 2: the magnitudes of the nonzero entries in S 0 may be arbitrarily large.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In particular, when this probability is proportional to exp{η s M e (s−1)}, for a certain choice of η s , this algorithm is known to minimize regret [1,2].
Sentence 2: by letting j s (s = 1, 2, . . .) denote the observed actions of the opponent up to stage s, and letting σ X denote the strategy induced by the experts algorithm, we have The main deficiency of the regret minimization approach is that it fails to consider the influence of chosen actions of a player on the future choices of the opponent -the inequality (1) holds for any fixed sequence (j s ) of the opponent's moves, but does not account for the fact that different choices of actions by the agent may induce different sequences of the opponent.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 00 00 n=O n=O a necessary and sufficient condition is an ~ 0 for all n E N if no assumption about the dimensionality of the input space is made (for finite dimensional spaces of dimension d, the condition is that bn ~ 0).
Sentence 2: the polynomial series expansion in dot product kernels plays the role of the Fourier transform in translation invariant kernels.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , (t s , y s )} having likelihood N (Y ; x T , Λ) give rise to a posterior GP s (x; µ s , k s ) with GPs are closed under linear maps.
Sentence 2: the joint distribution over x and its derivative is A recursive algorithm analogous to RK methods can be constructed [5,6] by setting the prior mean to the constant µ(t) = x 0 , then recursively estimating xi in some form from the current posterior over x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2 Unless otherwise stated, the embedding we refer to is the aforementioned w2vNEWS embedding, a d = 300-dimensional word2vec [19,20] embedding, which has proven to be immensely useful since it is high quality, publicly available, and easy to incorporate into any application.
Sentence 2: we downloaded the pre-trained embedding on the Google News corpus, 3 and normalized each word to unit length as is common.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Rather than attempting to identify direct interactions between neurons (where the number of interactions grows with the number of neurons squared), we propose to extract a smaller number of latent variables from each population and study how these latent variables interact.
Sentence 2: we propose extensions to probabilistic canonical correlation analysis (pCCA) to capture the temporal structure of the latent variables, as well as to distinguish within-population dynamics from across-population interactions (termed Group Latent Auto-Regressive Analysis, gLARA).
Relationship: entailment
Domain: NeurIPS
Sentence 1: persistence lanscapes [Bubenik, 2015], tangent vectors from the mean of the square-root framework with principal geodesic analysis [Anirudh et al., 2016], or persistence images [Adams et al., 2017]), using points in PDs as roots of a complex polynomial for concatenated-coefficient vector representations [Di Fabio and Ferri, 2015], or using distance matrices of points in PDs for sorted-entry vector representations [Carriere et al., 2015], (ii) implicit representation via kernels such as the Persistence Scale Space (PSS) kernel, motivated by a heat diffusion problem with a Dirichlet boundary condition [Reininghaus et al., 2015], the Persistence Weighted Gaussian (PWG) kernel via kernel mean embedding [Kusano et al., 2016], or the Sliced Wasserstein (SW) kernel under Wasserstein geometry [Carriere et al., 2017].
Sentence 2: geometry on PDs plays an important role.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We find that the network with STD can generate both static and traveling bumps, and STD enhances the performance of the network in tracking external inputs.
Sentence 2: we find that STD endows the network with slow-decaying plateau behaviors, namely, the network being initially stimulated to an active state will decay to silence very slowly in the time scale of STD rather than that of neural signaling.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this case, x can also be regarded as a linear transformation of a vector s containing independent unit-variance Gaussian components, as: x = Σ −1/2 s. In fact, the Gaussian is the only density that is both elliptically symmetric and linearly decomposable into independent components [14].
Sentence 2: the Gaussian densities correspond to the intersection of the class of ESDs and the class assumed by the ICA methods.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, the conditional probability of ending up at x 1 = 1 given that the algorithm did leave x 0 is , which does not depend on α.
Sentence 2: provided α is small, SR, on average, makes the same decisions/transitions with learning rate α as it does with learning rate α/10; it just takes 10 times longer to make those decisions when α/10 is used.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Specifically, the Copeland score is defined as j =i 1(pij > 1/2), and the normalized Copeland score is defined as ζi = A dueling bandit algorithm Γ decides which pair of arms to compare depending on the historic observations.
Sentence 2: define a filtration H t−1 as the history before t, i.e., Ht−1 = {a (1) τ , a τ , wτ , τ = 1, 2, . . . , t − 1}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since L(Θ) is a separable sum over the N points, then the theorem follows, with L i (θ i ) summing those training points in S i , L j (θ j ) summing those in S j , and the remaining points being summed in the "constant" term.
Sentence 2: the respective terms are: Note that L i depends on the parameters θ k of other nodes k besides i but it does not depend on θ j .
Relationship: entailment
Domain: NeurIPS
Sentence 1: However stateof-the-art neural networks reached the hardware limits both in terms the computational power and the memory.
Sentence 2: modern networks reached the memory limit with 89% [21] or even 100% [25] memory occupied by the weights of the fully-connected layers so it is not surprising that numerous attempts have been made to make the fully-connected layers more compact.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, as mentioned in Section 1, such solutions do not fare well on the three key requirements mentioned above due to training data inefficiency (see Section 1).
Sentence 2: while the actual class-signature is a small part of the entire data point, in absence of more fine-grained labels, existing methods typically process the entire sequential data point to predict the class label which leads to expensive computation and large model sizes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Later on, in Section 7 and Section 8, we will see that the similarities just described go beyond the structure of the processes, and they extend to the acceleration mechanism itself.
Sentence 2: the choice of splitting parameters that yields subdiffusive convergence rates, matching the asymptotic rates of shift register methods, is also shown to depend on global information about G. We apply Min-Sum Splitting to solve network averaging.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For a sequence of learning rates α ∈ [0, 1] N and a constant γ ∈ [0, 1], Q-learning [26], a very important algorithm in the multi-agent systems literature, updates its Q-values for an experience e t ∈ E by 3 Interruptibility Orseau and Armstrong [16] recently introduced the notion of interruptions in a centralized context.
Sentence 2: an interruption scheme is defined by the triplet < I, θ, π For Bob, I Bob = 0 and for Adam, I Adam = 1 if his car goes fast and Bob is not too close and I Adam = 0 otherwise.
Relationship: entailment