sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: On the other hand, most methods for reducing dimension will also bring H 0 and H 1 "closer together," making them harder to distinguish.
Sentence 2: mindful of the fact that the Hotelling test measures the separation of H 0 and H 1 in terms of δ Σ −1 δ, we see that the statistical distance is driven by the Euclidean length of δ. we seek to transform the data in such a way that the dimension is reduced, while the length of the shift δ is mostly preserved upon passing to the transformed distributions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Finally, for a partition P , let Z be the indicator variable of a partition, Z = s iff X ∈ P s . With this notation, one can write where H are the full and conditional Shannon entropies.
Sentence 2: dER algorithm can be interpreted as seeking a partition that maximizes the information between current known state (Z), and the next step from it (Y ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This implies that lim n→∞ nE[f ( θn ) − f (θ * )] has a finite limit.
Sentence 2: this interpretation explains why the averaging produces a sequence of estimators which converges to the solution θ * pointwise, and that the rate of convergence of Note that (a) our result is stronger since it is independent of the lowest eigenvalue of H, and (b) for other losses than quadratic, the same properties hold except that the mean under the stationary distribution does not coincide with θ * and its distance to θ * is typically of order γ 2 (see Section 3).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: While the information-theoretic limits of the Stochastic Block Model have been heavily studied (in slightly different ways), none of the other models considered in this paper have been studied before.
Sentence 2: we provide new information-theoretic results for the Exponential Random Graph Model (ERGM), the Latent Space Model (LSM), the Directed Preferential Attachment Model (DPAM), and the Directed Small-world Model (DSWM).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: • For q = 2: When we fix i, j, s, t which are all different, and take expectation over ⇡, then all products of off-diagonal elements of A appear the same number of times in E ⇡ A ⇡(i),⇡(j) A ⇡(s),⇡(t) .
Sentence 2: this quantity is proportional to P p6 =r v p v r , where v is the vector of all non-diagonal entries of A.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then by Proposition 1, there exists at least one i ∈ {1, • • • , n} such that lim inf k∈K,k→∞ We next consider the following two cases: (a) There exist a subsequence K of K and an integer k > 0 such that for all k ∈ K, k ≥ k, the GD-step is adopted.
Sentence 2: then for all k ∈ K, k ≥ k, we have by the optimality condition of the above problem and properties of the Clarke subdifferential (Proposition 4 in Supplement A), we have Taking limits with k ∈ K for Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For local consistency, we enforce the predicted value for close locations to be similar via spatial Laplacian regularization.
Sentence 2: we can formulate the forecasting task as the following optimization problem over the model coefficient tensor W: Though cokriging and forecasting are two different tasks, we can easily see that both formulations follow the global and local consistency principles and can capture the inter-correlations from spatialtemporal data.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, for each FoE model, the Hessian of the energy function log(1 + 1 2 (J T i x r ) 2 ) is lower bounded by − J T i Ji 8 (proof in the supplementary material).
Sentence 2: we simply add an extra term γx T r x r with γ > J T i Ji 8 to obtain the concave-convex decomposition log(1+ 1 2 (J T i x r ) 2 ) = {log(1+ 1 2 (J T i x r ) 2 )+ γx T r x r } − γx T r x r . We utilize a pre-trained 5 × 5 filterbank with 24 filters, and conduct experiments on the BM3D benchmark 1 with noise level σ = 15.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As there are exactly (2;;) distinct choices of dE {I, ... , ko} training examples from the 2m examples Z there are no more than (2emlko)kO different subsamples to be used in w. For each particular subsample z ~ Z the weight vector w is a member of the class of linear classifiers in a ko (or less) dimensional space.
Sentence 2: from (7) it follows that for the given subsample z there are no more (2emlko)kO different dichotomies induced on the double sample Z E z2m.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It follows from Lemma 3 and Lemma 1 that, if u i = 0, then x * i = 0.
Sentence 2: by applying the above iterative procedure, we can find the minimizer of ( 5) by solving a reduced problem that has p ′ ≤ p variables and g ′ ≤ g groups.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, Tσ 2 (p) = IE(X 2 ) − IE(X) 2 is the familiar plug-in statistic for the variance, while T unbiased Obviously, a plug-in statistic yields an unbiased estimate concerning the distribution that is plugged in.
Sentence 2: when the empirical distribution is plugged in, a plug-in statistic typically does not give an unbiased estimate concerning the (unknown) true distribution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: and leave through one scalar r.v.
Sentence 2: let ζ i (u, v|C) ∈ (−∞, 0] denote the warped MI between the enter and exit r.v.s of interstitial vectoral node i on Pu:v , with conditioning set C ⊆ V \ {u, v}.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The equivalence between the two problems can be verified using convex analysis [8] by noting that the Lagrangian for the constrained optimization ( 4) is exactly the objective in the optimization (5) (plus a constant that does not depend on θ), where κ is the Lagrange multiplier.
Sentence 2: (4) can be solved by solving either (5) or ( 6) for an appropriate κ or γ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The log likelihood is computed by where S = 1 k k i=1 (y i − µ)(y i − µ) T . To facilitate the following analysis we first assume V projects Y to a 1-D space, i.e., p = 1 and V = v T , and thus Note that v T Cv is the variance of the object samples in the projected space, and we need to impose a constraint, e.g., v t Cv = 1, to ensure that the minimum likelihood solution of v does not increase the variance in the projected space.
Sentence 2: let v T Cv = 1, the optimization problem becomes we obtain an equation exactly like the Fisher discriminant analysis for a binary classification problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To model this, we follow the Bayes-Adaptive MDP framework, and include the φ and ψ vectors in the state of the BAPOMDP.
Sentence 2: the state space S ′ of the BAPOMDP is defined as S ′ = S × T × O, where T = {φ ∈ N |S| 2 |A| |∀(s, a), s ′ ∈S φ a ss ′ > 0} represents the space in which φ lies and O = {ψ ∈ N |S||A||Z| |∀(s, a), z∈Z ψ a sz > 0} represents the space in which ψ lies.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Theorem 2 guarantees that to get an ε guarantee for the accuracy of the optimization it suffices to perform n(ε) = O(1/ε) evaluations of f , where we are now treating log(δ 2 ) − log(δ 1 ) as a constant.
Sentence 2: the number of functions evaluations is O(1/ε), and, since the time complexity of every evaluation is O(h|S| 2 |A|), the total time complexity of the algorithm is O(h|S| 2 |A|/ε).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This takes the same values as X i,t , but its distribution is given by a different exponential parameter ( θi instead of θt but for all θ ∈ [ θt , θt+1 ], var(X i,t, θ ) ≤ e ηB var(X i,t ).
Sentence 2: hence, for η < 0.567/(B + λ), Q(ᾱ t ) is non-increasing in t. The proof is in [1].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the parallel setting, we cannot investigate L * directly and we may only analyze it through L N . However, calculating the total loss function directly and exactly requires excessive inter-processor communication, which carries a huge computational burden as the number of processors increase.
Sentence 2: we must approximate the true parameter θ * = arg min θ∈Θ L * (θ) by an empirical risk minimizer θ = arg min θ∈Θ L N (θ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: One may imagine, however, that training data for the handling of a load can be obtained along a fixed reference trajectory T r for calibration purposes, and also along a trajectory typical for that load, say T m for the m th load.
Sentence 2: for each load, 2000 random training samples are acquired at a common reference trajectory T r = (p 2 , s 3 ), and an additional 2000 random training samples are acquired at a trajectory unique to each load; Table 1 gives the combinations.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let Z denote the random variable w(x) L h (x) − R(h).
Sentence 2: then, |Z| ≤ M . By lemma 2, the variance of the random variable Z can be bounded in terms of the Rényi divergence d α+1 (P Q): by Bernstein's inequality [4], it follows that: Setting δ to match this upper bound shows that with probability at least 1−δ, the following bound holds for the importance weighting method: Using the sub-additivity of √ • leads to the simpler expression These results can be straightforwardly extended to general hypothesis sets.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We first note that, as desired, our model selects features that have large variation in their probabilities across the clusters (rows in Figure 3).
Sentence 2: it is straight-forward to read what makes each column different from the others: the mammals in the third column do not lay eggs; the insects in the fifth column are toothless and invertebrates (and therefore have no tails).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let us now consider the family of distributions P with marginal distribution PX and regular conditional distributions Q x := P (• − m(x))/σ(x) x , x ∈ X, where m : X → R and σ : X → (β, 1/β) are as in the previous example.
Sentence 2: then Q x has a τ -quantile of type αβ, because we obtain for s ∈ [0, αβ] the inequality P has a τ -quantile of p-average type αβ if and only if P does have a τ -quantile of p-average type α.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Each iteration requires one matrix-matrix multiply with K XX , and the subsequent work to derive these inference terms takes negligible additional time (Appendix B).
Sentence 2: p iterations of mBCG requires O(nt) space (see Appendix B) and O(p Ξ( K XX )) time, where Ξ( K XX ) is the time to multiply K XX by a n × t matrix.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Rearranging the product from equation 2, we achieve a likelihood in terms of recipient states x : x |p e −q x |p Tp (3) where we define p as {x}×{u}×(X \(X ×U X )) in each partition P x , and likewise: q x |p = q xx |u , M x |p = M xx |u , and T p = T x|u .
Sentence 2: cTBNs are one instance of partition-based CTBNs, with partitions corresponding to a specified donor state x and parents setting u.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The twist from the standard setup is that the losses of all the experts are not uniformly bounded in the same range.
Sentence 2: indeed, for the setup described for the oracle inequality (3), class W k will produce predictions with norm as large as 2 k . here, we assume that expert i incurs losses in the range [−c i , c i ], for some known parameter c i ≥ 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Sample-MIPS, as mentioned earlier, is essentially a sampling algorithm with replacement scheme to draw entries of Z such that (j, t) is sampled with the probability proportional to z jt .
Sentence 2: sample-MIPS can be thought as a traversal of (j, t) entries using in a stratified random sequence determined by a distribution of the values of {z jt }, while the core idea of Greedy-MIPS is to iterate (j, t) entries of Z in a greedy sequence induced by the ordering of {z jt }.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By letting , we can fit (6) to X = F Y A in Definition 2.
Sentence 2: learning {O i } i∈ [3] is reduced to selecting k rank-one matrices from {X t,i ⊗ X t,j } t∈[n] indexed by A whose conical hull covers the q matrices {Y (t) } t∈ [q] .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Similarly to Theorem 2 in (Abbasi-Yadkori, Pal, and Szepesvari, 2011) 2 , for any state history and with probability of at least 1 − δ, it holds for all t > 0 that θT t,a x(s t ) − θ * a T x(s t ) ≤ we define δ = δ/k and bound this probability for all the actions, i.e., ∀a, t > 0 Recall that any valid action a at state s satisfies E[e t (s, a)] = θ * a T x(s t ) ≤ .
Sentence 2: we can eliminate action a at state s t if This ensures that with probability 1 − δ we never eliminate any valid action.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This simple covariance calculation was much cheaper in our experiments than the moment conversion with cost O(pq 2 ) or the SVD with cost O(p 3 q 3 ), both of which are independent of the data set size N T . In contrast, each iteration of EM requires at least O(N T (p 3 + pq)) time.
Sentence 2: the computational advantage of PLDSID is expected to be especially great for large data sets.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The effectiveness of spectral clustering hinges crucially on the construction of the graph Laplacian and the resulting eigenvectors that reflect the cluster structure in the data.
Sentence 2: we construct an objective function that consists of the graph Laplacians from all the views of the data and regularize on the eigenvectors of the Laplacians such that the cluster structures resulting from each Laplacian look consistent across all the views.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From our analysis of multiplicative approxinlate greedy (see Lemma 3), given a multiplicative approximation factor (1 + f=) in the approximate nearest neighbor method, the approximate greedy coordinate descent has the convergence rate: K . 1\:1,,,3 for some constant K > O.
Sentence 2: the num-'ber of iterations required to obtain a solution with accuracy fopt is given by, T greedy = ~:~.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, the expressiveness of GPs comes at a price: solving (approximate) inference for a GP with N data instances has time and space complexities in Θ(N 3 ) and Θ(N 2 ), respectively.
Sentence 2: gPs have traditionally been viewed as a tool for problems with small-or medium-sized datasets Table 1: Comparison between SVDGP and variational GPR algorithms: SVI [10], iVSGPR [2], VSGPR [24], and GPR [19], where N is the number of observations/the size of a mini-batch, M , M α , M β are the number of basis functions, and D is the input dimension.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For dataset 'ecoli', 'wine' and 'iris', the averaged error rates of the 'EM Model' are very close to the cases when there are no distractor class labels.
Sentence 2: we can conclude that the 'EM Model' is able to reduce the noise caused not only by random label ambiguity but also by some systematic label ambiguity.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From (10), we observe that Σ i+1 Σ i for all i, so Σ n+1 Σ i+1 Σ 1 = I for all i.
Sentence 2: the conditions on Σ * in ( 18) are satisfied by where the last equality holds since we assumed that an update was made for the ith example.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Here s = 4, and This gives N S ψ (z 1 ) = � p ∈ Δ 3 : p = (q 1 + q 2 , q 3 , q 4 ) for some q ∈ Δ 4 , q 1 − q 2 − q 3 − q 4 = 0 � = � p ∈ Δ 3 : p = (q 1 + q 2 , q 3 , q 4 ) for some q ∈ Δ 4 , q A similar procedure yields N S ψ (z 2 ) = Q ord 2 and N S ψ (z 3 ) = Q ord 3 .
Sentence 2: by Theorem 7, we get that ψ is classification calibrated with respect to � ord over Δ 3 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Furthermore, for every γ in this range, this tradeoff between m 1 and m 2 is necessary, up to constant factors.
Sentence 2: our results smoothly interpolate between the known bounds of Θ(n 2/3 ) on the sample size necessary in the setting where one is given two equal-sized samples [6,9], and the bound of Θ( √ n) on the sample size in the setting in which the sample is drawn from one distribution and the other distribution is known to the algorithm [22,29].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The Bethe approximation can reflect dependence of random variables (dependence between U and V in p(U, V |A) in our problem) to some extent.
Sentence 2: one can intuitively expect that performance of the AMP algorithm is better than that of the variational Bayes matrix factorization, which treats U and V as if they were independent in p(U, V |A).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By (23), if Ẑ = CH, then On the other hand, if Γ C :,i 2 = 1 and H i,: = H i,: 2 C :,i Γ, then we have ||| Ẑ||| * = i H i,: 2 , implying the optimality of (C, H).
Sentence 2: given Γ, the recovery problem (22) has been reduced to finding a vector µ and matrix C such that µ ≥ 0, C :,i ∈ C(Γ) for all i, and C diag(µ)C Γ = Ẑ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Although both V b2 and V b1 converge to zero, the rate and the type of convergence are not the same: nV b2 converges in law to some random variable while the behaviour of nV b1 is unspecified.
Sentence 2: tests that utilize V b2 usually give lower Type II error then the ones that use V b1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Also, let λ = β P 2 for β > 0 and x 0 − x * small.
Sentence 2: we can thus approximate the right parenthesis of the bound by the bound on the precision of the extrapolation is approximately equal to Also, if we use equation ( 8), it is easy to see that where ζ is defined in (9).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The term (s) = s T ⇤s is a slack variable-when ⇢ > s T P s or s = 0 (so the second term is nonpositive), it can be made sufficiently large so that the first constraint holds regardless of s T P f ⇡ (s), but when ⇢  s T P s and s 6 = 0 (so the second term is positive), we must have s T P f ⇡ (s) < 0 since s T ⇤s 0 by the second constraint.
Sentence 2: properites (i) and (ii) hold from (1), and ( 2) verifies (iii) for all if a solution ⇢ > 0 is found, then V is a Lyapunov function with region of attraction U .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, there is no ----------------------1 a -------------------------------------------------------------- ) restriction on the range of preferred disparities when using position shifts.
Sentence 2: it has been suggested that position shifts can be used to "bias" the preferred disparity of a population around a rough estimate of the stimulus disparity, and then use a population of neurons tuned by phase shifts to obtain a more accurate estimation of the actual disparity.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is because (i) the reference predictors do not need it; and (ii) if we did include the cost of observing the loss function for the reference predictors, then the loss of each predictor would just be increased by c d+1 T , and so the regret R T would just be reduced by c d+1 T , making it substantially easier for the learner to achieve sublinear regret.
Sentence 2: we prefer the current regret definition as it promotes the study of regret when there is a price attached to observing the loss functions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: That is, if a state s is visited for multiple times, then each time the distribution (of uncertain parameters) µ s is the same.
Sentence 2: mathematically, we can adapt the augmented state space as in the non-stationary model, and requires that µ s,t does not depend on t. the set of admissible distributions is The next theorem is the main result of this section; it shows that a sequentially robust policy is distributionally robust to both stationary and non-stationary models.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Matters are only slightly more complicated here.
Sentence 2: proposition 10 If trial t is a mistake, and q maximizes entropy subject to membership in S t ∩ T t , then it is on the separating hyperplane for T t . Proof: Because q and p both satisfy S t , any convex combination of the two satisfies S t . if q was on the interior of T t , we could find a probability distribution with higher entropy that still satisfies both S t and T t by taking a tiny step from q toward p. This would contradict the assumption that q is the maximum entropy member of S t ∩ T t . This implies that the next hypothesis of a ROME algorithm is either on J t (the separating hyperplane T t ) only, or on both J t and H t (the separating hyperplane of S t ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Not only so, but every once in a generation the training model is allowed to switch its gating vector that results the best fitness-score, which comes as a crucial feature of our optimization for highly multi-modal solution spaces.
Sentence 2: among a population, an elite neural network model f (x; θ, µ elite ) is selected, where µ elite denotes the chromosome that causes an individual to yield the best fitness-score.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider what happens if we run a 1 -smooth boosting algorithm, where the initial distribution D satisfies D rel / U k = τ.
Sentence 2: at each stage we will have D rel t (x) ≤ 1 • D rel (x) for all x ∈ {−1, 1} k , and consequently we will have by Lemma 2 each distribution D t will satisfy d T V (D rel t , U k ) ≤ 1− 2 /(4τ 2 ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A principled way to achieve such a goal is to impose a boundedness constraint on 1 -like norms of these maps to achieve sparsity and simultaneously impose smoothness by penalizing their Laplacian.
Sentence 2: we propose the following penalized dictionary-learning model The ingredients in the model can be broken down as follows: 2 measures how well the current dictionary V explains data X i from subject i.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, we can construct a feasible solution Ŵ = W − G • E, where G ij = sign(W ij ) and • indicates the entrywise product of two matrices.
Sentence 2: the assumption of this theorem implies that From Lemma 1 we have log det Ŵ ≥ log det W − p γσmin( W ) E F . Since W * is the optimal solution of P λ and Ŵ is a feasible solution of P λ , log det W * ≥ log det Ŵ ≥ log det W − p γσmin( W ) E F . Also, since W is the optimal solution of P Λ and W * is a feasible solution of P Λ, we have log det W * < log det W . By the mean value theorem and some calculations, we have |f , which implies (7).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Now, for such a utility measure, if it is true that for any two possibilities {x i , x j } and any two contexts {c k , c l }, then the choice behavior of an agent maximizing u(x, c) would be equivalent to one maximizing u(x).
Sentence 2: for the inclusion of context to have any effect, there must exist at least some Note however, that the context in this operationalization is simply a collection of other possibilities, i.e.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The Bregman divergence representation for the scaled distribution is given by where the last equality follows from Lemma 3.1 and the fact that, for a Bregman divergence, D βφ (•, •) = βD φ (•, •).
Sentence 2: as β increases under the above scaling, the mean is fixed while the distribution becomes increasingly concentrated around the mean.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Lemma 2 Let us approximate (or assume) If the policy is non-degenerate for the task (so the dimension of θ, d, is equal to |S| i=1 (|A i | − 1), where |S| and |A i | are the numbers of states and the available actions at state s i , respectively) and ω * denotes the gNG(ι), then the 'near' optimal auxiliary function ρ * in the 'near' optimal estimating function g * ι,θ (s, a; ω) Proof Sketch: The covariance matrix for the criterion of the auxiliary function ρ is approximated as The |A i | degrees of freedom.
Sentence 2: there exist ρ * and ∇ θ ln{d θ (s) ι π(a|s;θ)} ω that satisfy (14).
Relationship: reasoning