sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: (c) For any F UHPR (T, δ) with F UHPR (T, δ) → ∞ as T → ∞, there is an algorithm that satisfies that uniform high-probability regret bound on some MDP but makes infinitely many mistakes for any sufficiently small accuracy level ε > 0 for that MDP.
Sentence 2: a high-probability regret bound (uniform or not) cannot be converted to a finite (ε, δ)-PAC bound.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Two neurons A and B responding to two different sides (left and right), with receptive fields N A and N B , transform the signals N A * F L * S and N B * F R * S into spike trains.
Sentence 2: synchrony between A and B occurs whenever N A * F L = N B * F R , i.e., for a specific set of filter pairs (F L , F R ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given the training data points {(x i , ŷi )} N i=1 , a natural candidate for the center of the Wasserstein ball is the empirical distribution PN = 1 N N i=1 δ (xi,ŷi) , where δ (xi,ŷi) denotes the Dirac point measure at (x i , ŷi ).
Sentence 2: we henceforth examine the distributionally robust optimization problem equipped with a Wasserstein ambiguity set.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For node i, with probability the random walk enters the cycle, with probability 1/2 the random walk chooses any neighbor in the clique; and with probability 1/2 − the random walk stays at node i.
Sentence 2: we can show that the expected return time to node i is E i [T i ] scales linearly in n and k. Suppose we observe only the local neighborhood of constant size r around node i.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As a result, on event E, the regret of UGapEc cannot be bigger than , because then it contradicts the stopping condition of the algorithm, i.e., B J( n+1) ( n + 1) < .
Sentence 2: we have P r Ω( n+1) ≤ ≥ 1 − δ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As a comparison, note that the corresponding ODE for gradient descent with updates whose finite difference scheme has the characteristic equation det(λ − (1 − ∆t∇ 2 f )) = 0.
Sentence 2: to guarantee −I n 1 − ∆t∇ 2 f I n in worst case analysis, one can only choose ∆t ≤ 2/L for a fixed step size, which is much smaller than the step size 2/ √ L for (3.1) when ∇f is very variable, i.e., L is large.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: So the expected number of positive sampled points would be less than ( 1 2 − ǫ)(2 log n/ǫ 2 ) = (log n/ǫ 2 − 2 log n/ǫ).
Sentence 2: by Chernoff bounds, the chance of the sample having a majority of positive-predicting functions would be at most e −(2 log n/ǫ) 2 /2(log n/ǫ 2 −2 log n/ǫ) = e −2 log n/(1−ǫ) = n −2/(1−ǫ) .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Furthermore, the mean and the variance remain in the domain Ω, that is g(Ω) ⊆ Ω, where When iteratively applying the mapping g, each point within Ω converges to this fixed point.
Sentence 2: we consider activations of a neural network to be normalized, if both their mean and their variance across samples are within predefined intervals.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By (13), we have α j = λ p/(2πσ 2 ) + λ 2 p/(2πσ 2 ) + β j /σ 2 .
Sentence 2: one can pre-compute the eigenvectors once, and only update the eigenvalues as a function of θ k . This will make the computation efficient both in time and memory.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The first considers the behavior of EM when the dimension d = d n may grow with the sample size n. Our proof of Theorem 7 reveals that the parameter error of the t-th iterate (in Euclidean norm) is of the order d/n as t → ∞.
Sentence 2: we conjecture that the theorem still holds as long as d n = o(n).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This guarantees that on any finite interval [t 0 , T ), the functions η(t) and w(t)/W (t) are bounded.
Sentence 2: −η(t)∇f (X) and w(t) W (t) (∇ψ * (Z) − X) are Lipschitz functions of (X, Z), uniformly in t ∈ [t 0 , T ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this case, the perturbation matrices in the lemma can be expressed as ∆ 1 = Ξ, ∆ 2 = Ξ, P = 2(1 − ξ )Ξ.
Sentence 2: we get the expansions of the eigenvalues and eigenvectors as d k + ε χ 1k , c k + ε χ 2k , v k + εψ 1k and u k + εψ 2k , where    is satisfied, the O(ε) term χ 1k (resp.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, we could let DcjJ be an information divergence and qN some kernel estimate, that is, a filtered version of the empirical measure (see [4] for an independent approach along these lines).
Sentence 2: this doesn't quite work, however, because the kernel induces an arbitrary scale; if this scale is larger than the natural scale of f and p(< V, X » for some V but not others, our estimate will be biased away from K. DcjJ and PN have to be asymptotically scale-free in some sense.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let us write ŵt = ŵt + ŵt⊥ , where ŵt is the projection of ŵt on the rows of Π πt F , i.e.
Sentence 2: ∃α t ∈ R |S| : ŵt = F T Π T πt αt , and ŵt⊥ is orthogonal to the rows of Π πt F . Our problem now is to find a distribution on deterministic policies P that satisfies the conditions stated in Proposition 1 in addition to the feature matching conditions φ π k (s) = φk .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For a given orthonormal basis {ϕ j } of L 2 ([0, 1]), smoothness parameter β > 1/2 and radius C, the function class F β [C] is given by If we choose the trigonometric basis as our orthonormal basis, then membership in the class F β [C] corresponds to certain smoothness constraints on the derivatives of f . More precisely, for j ∈ N, consider the orthonormal basis for L 2 ([0, 1]) of trigonometric functions: Now consider a β-times almost everywhere differentiable function f for which .
Sentence 2: definition 1 (essentially) captures densities that have Lipschitz-continuous (β − 1)th derivative.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This term is convex and, therefore, the problem in (7) is convex: The objective is convex (sum of linear and convex functions) and constraint z p ∈ ∇ L is affine.
Sentence 2: one can minimize this constrained convex problem for each p by solving the Karush-Kuhn-Tucker (KKT) conditions 8 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this case, the probability of an output sequence only depends on the total number of zeros or ones in the sequence.
Sentence 2: we only get t independent measurements from windows of length t, hence windows of length O(n) instead of O(log 2 n) are necessary for identifiability (also refer to Blischke [8] for more discussions on this case).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let us assume that Q has a density h with respect to µ.
Sentence 2: then, the sets {h > ρ}, ρ > 0, describe the concentration of Q. to define anomalies in terms of the concentration we only have to fix a threshold level ρ > 0, so that an x ∈ X is considered to be anomalous whenever x ∈ {h ≤ ρ}.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From Lemma 3 below, we know that in an open-loop POMDP, two belief trajectories can only get closer to each other, as they progress.
Sentence 2: to get a δ-cover of the space R(b 0 ) that the robot scout can reach from a given b 0 , it suffices to first compute a δ/2-cover C of the initial belief points for all possible subsequences of moves and then take the union of the δ/2-covers of the belief points traversed by the subsequences whose initial belief points lie in C. The δ-cover of R(b 0 ) then has its size bounded by O(n m (2m /δ) m (16 mn/δ) 2 m + h ), where h = log ζ (δ/4n).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The kernel of h k defined by K 2 and ( 7) is denoted by K : N × N → R and depends on β, θ.
Sentence 2: letting E[•] denote the expectation operator, the prior model on the impulse responses is given by The noise variance σ 2 will always be estimated via a preliminary step using a low-bias ARX model, as described in [24].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In light of this, the motivation to formulate the problem of multi-step ahead prediction in the online setting is quite clear: attaining regret in this setting would imply that our algorithm's performance 6: end for is comparable with the best 1-step ahead recursive AR predictor in hindsight (even if the latter is misspecified).
Sentence 2: our goal is to minimize the following regret term: where K denotes the set of all 1-step ahead recursive AR predictors, against which we want to compete.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Notice this is the same as variational inference for unsupervised LDA: since we averaged the response variable out of the right-hand side in ( 12), what remains is the standard unsupervised LDA model for Z 1:N and θ.
Sentence 2: given a new document, we first compute E q [Z 1:N ], the variational posterior distribution of the latent variables Z n . Then, we estimate the response with Up to this point, we have confined our attention to an unconstrained real-valued response variable.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For any t 2 [1, T ], define a distribution D t over the pairs (i, n), with n in {1, 2} where Z t is the normalization factor given by Z In order to derive an explicit formulation of the descent direction that is based on the weighted error of the classification function h j and the expected value of the rejection function r j , we use the distributions D  Using these definition, we show (see Appendix D) that the descent direction is given by This equation shows that Z 1,t and 2 p c(1 c)Z 2,t re-scale the weighted error and expected rejection.
Sentence 2: finding the best descent direction by minimizing this equation is equivalent to finding the best scaled trade-off between the misclassification error and the average rejection cost.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that for this dataset we report only accuracy (recall), because, unlike traditional object detection, there are no overlapping detections and every stroke is assigned to a symbol.
Sentence 2: a false positive always causes a false negative, so recall and precision are redundant: e.g., misclassifying one segment in a three-segment "H" makes it impossible to recognize the original "H" correctly.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For any given policy π (mapping belief state to action), the expected cost is At any time t, the observer can either choose to stop and declare one of the locations to be the target, or choose to continue and look at location λ t+1 .
Sentence 2: the expected cost associated with stopping and declaring location i to be the target is: And the minimum expected cost for continuing sensing at location j is: The value function V (p, i), or the expected cost incurred following the optimal policy (π * ), starting with the prior belief p 0 = p and initial observation location λ 1 = i, is: Then the value function satisfies the following recursive relation (Bellman, 1952), and the action that minimizes the right hand side is the optimal action π * (p, k): This can be solved using dynamic programming, or more specifically value iteration, whereby we guess an initial value of the value function and iterate eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As we will see in the next section, the MSPE of ridge regression can be bounded in a sharp way when the design matrix is approximately low rank, and there we will analyze mspe( β |X) for the pilot estimator.
Sentence 2: when near low-rank structure is available, the only remaining issue in controlling the right side of line ( 12) is to bound the quantity The very recent work of Bobkov and Ledoux [15] provides an in-depth study of this question, and they derive a variety bounds under different tail conditions on F 0 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: While an interesting generalization of the usual KDE, this approach assumes that the data manifold M is known in advance, and that we have access to certain geometric quantities related to this manifold such as intrinsic distances between its points and the so-called volume density function.
Sentence 2: this Riemannian KDE cannot be used directly in a case where the data lives on an unknown Riemannian submanifold of R N . Certain tools from existing nonlinear dimensionality reduction methods could perhaps be utilized to estimate the quantities needed in the estimator of [9], however, a more straightforward method that directly estimates the density of the data as measured in the subspace is desirable.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For an index tree T , let For a node G i j , the set A i j is the sum of B t k corresponding to all its descendant nodes and itself, and the set C i j the sum excluding itself.
Sentence 2: by the definitions of A i j , B i j , and C i j , we have which implies that P ∂φ(0 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First, the baseline intensity γ 0 captures users' inherent and long-term preferences to items, regardless of the history.
Sentence 2: second, the triggering kernel γ(t, t i ) quantifies how the influence from each past event evolves over time, which makes the intensity function depend on the history T . a Hawkes process is essentially a conditional Poisson process [14] in the sense that conditioned on the history T , the Hawkes process is a Poisson process formed by the superposition of a background homogeneous Poisson process with the intensity γ 0 and a set of inhomogeneous Poisson processes with the intensity γ(t, t i ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If we use the time complexity from [17], then finding the span of the top k singular vectors of an n × m matrix takes O(nmk) time.
Sentence 2: if ϑ is a constant, both simple-EVD and c-EVD need O(nαr) time, whereas, Alt-Min-RPCA needs O(nαr 2 ) time per iteration [17].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To do so, note first that h f errs on an example (x, y) iff y h f (x) ≤ 0.
Sentence 2: the exp-loss function, defined as exp(−y h f (x)), is a smooth upper bound of the zero-one error, which equals to 1 if y h f (x) ≤ 0 and to 0 otherwise.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We note that although DDPG trains a deterministic policy µ φ , its behavior policy, which is used to collect experience during training is augmented with Gaussian (or Ornstein-Uhlenbeck) noise [25].
Sentence 2: actions are collected as a t ∼ N (µ φ (s t ), σ) for fixed standard deviation σ, which we will shorten as a t ∼ µ φ (s t ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, multiplying the matrix by k x largely cancels this effect, since k x is effectively a row of a kernel matrix similar to K , and will likely concentrate most weight on the high eigenvalue eigenvectors.
Sentence 2: ultimately, it is not K itself, but the relationship between the eigenvectors of K and the data vector y that determines which part of the spectrum of K the result of GP regression is most sensitive to.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We assume the emission distribution is also normally-distributed, Pr(s t |x) = N (s t |x, σ 2 e ), which, from Bayes' rule, results in a piecewise normal posterior distribution where and the running average st = t t =1 s t /t.
Sentence 2: the posterior distribution depends only on s and t, the two sufficient statistics of the sequence s 1:t . For the case of a piecewise uniform prior, the variance σ 2 t = σ 2 e t , which decreases inversely in proportion to elapsed time.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In contrast, the and £ responses are much slower.
Sentence 2: if $ £ ¡ ¤£ $£ ¡ ¤¦ ¦£ ¡ £ V( moves at & ¨¢ upwards as in maneuver III.1, the helicopter will tend to track the ¦ -component of the trajectory much more quickly, so that it accelerates into a climb steeper than & ¢ , resulting in a "bowed-out" trajectory.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: OOPS, however, is searching in program space instead of raw solution space.
Sentence 2: in principle it should be able to solve arbitrary instances by discovering the problem's elegant recursive solution: given and three pegs G % ¨ § ¨© (source peg, auxiliary peg, destination peg), define procedure The -th task is to solve all Hanoi instances up to instance .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Here we replace the exact computation of A i • X by employing the l 2 -sampling technique used in [15] in order to estimate this quantity by viewing only a single entry of the matrix A i (line 9 ).
Sentence 2: an important property of this sampling procedure is that if we can estimate the product A i • X with constant variance, which is important for our analysis.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, the most favourable case (in terms of relative efficiency) seems to be when µ 2 = • • • = µ K : then and T * = 2( √ K − 1 + 1) 2 /∆ 2 , leading to lim sup while Lai and Robbins' lower bound yields 2(K − 1)/∆.
Sentence 2: the difference grows with K as 2 √ K − 1 log(T )/∆ , but the relative difference decreases.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let Ĥ(W ) denote the piecewise continuous function achieved when the discrete problem is solved approximately using submodular optimization, then we have Ĥ(W ) ↵ • H(W ) for every W , where ↵ 2 [0, 1] is the approximation factor.
Sentence 2: solving the max-max problem in Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As we have solved for Y * in closed form in Equation ( 8), we can plug this result into Equation ( 1) to obtain a formulation of the problem of optimizing Π * that does not require a simultaneous optimization over Y .
Sentence 2: under these conditions, Equation ( 1) is equivalent to By evaluating the first order conditions on Equation (10), we can see that the relaxed solution, Π * , to Equation (10) must lie in the principal subspace of H n M H n . 1 for the problem of simultaneously optimizing the structure matrix, Y 0, and the partition matrix, one can use the same spectral relaxation as in Equation ( 4), and use the resulting partition matrix to solve for the optimal assignment for Y using Equation (8).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If the true gradients are g(θ, w) = ∇ w L D and h(θ, w) = ∇ θ L G , then we can define g(θ, w) = g(θ, w) + M (w) and h(θ, w) = h(θ, w) + M (θ) with random variables M (w) and M (θ) .
Sentence 2: the gradients g θ, w and h θ, w are stochastic approximations to the true gradients.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider the shortest path distance between arbitrary pairs of nodes i and j (that are not eliminated.)
Sentence 2: because the corrupted nodes are removed, the whole path is around C. Also, by the uniform density assumption, walking on the shortest path in G is equivalent to walking on C with "uniform speed", i.e., each edge on the path will map to an approximately fixed distance on C. the shortest path distance scales with the latent distance, i.e., (d − 1) c , which implies Theorem 2.2 (cf.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Small values of the transition weights in the k th row of Π therefore prevent component k from being excited by the other components and by itself.
Sentence 2: specifically, because the shape parameter for the gamma prior over θ (t) k involves a linear combination of θ (t−1) and the transition weights in the k th row of Π, small transition weights will result in a small shape parameter, shrinking θ (t) k . the component weights play a critical role in the PGDS by enabling it to automatically turn off any unneeded capacity and avoid overfitting.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The minimax-optimal binwidth n is known to be of the form O n −1/(2+d) , assuming a training data of size n from a metric space of unknown dimension d, and unknown Lipschitz target function f . This setting of n would then yield a minimax error rate of O n −2/(2+d) .
Sentence 2: the dimension d is the most important problem variable entering the rate (and the tuning of n ), while other problem variables such as the Lipschitz properties of f are less crucial in comparison.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Here, ∆ G is the adjacency matrix such that and D∆ G [i, j] = 0, for i = j.
Sentence 2: defining VΛV = L G as the eigen-system of the graph Laplacian, and the projection of z t onto the feature space VΛ − as zt = Λ − V zt yields: the transportation distance can be approximated by a 2 metric through a similarity transformation of the original space.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, letting p = µ/ µ 1 for µ a random measure taken from either D or D , we note that with high probability p 2 = Θ(n −1/2 ) and p 3 = Θ(n −2/3 ).
Sentence 2: m = o(n 2/3 / 4/3 ) samples are sufficient to distinguish between p being uniform and being Ω( ) far from uniform.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let δ > 0 and v ∼ B n , the smoothed version of a function f : R n → R is defined as follows: The next lemma of [5] ties between the gradients of f and an estimate based on samples of f : Lemma 5.
Sentence 2: let u ∼ S n , and consider the smoothed version f defined in Equation (4), then the following applies: n δ f (x + δu)u is an unbiased estimator for the gradients of the smoothed version.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If we can bring these two thresholds together, we will achieve the target CA and CR rates.
Sentence 2: the first algorithm we propose involves training a series of classifiers, attempting to make classifier n+1 achieve better CA and CR rates by focusing its effort on examples from classifier n that lie between θ CA and θ CR ; the positive examples must be pushed above θ CR and the negative examples must be pushed below θ CA .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A large class of graph-based algorithms in semi-supervised learning and clustering is based either explicitly or implicitly on the cut.
Sentence 2: we discuss first in Section 2.1 the hypergraph cut and the corresponding approximations.In Section 2.2, we introduce in analogy to graphs, the total variation on hypergraphs as the Lovasz extension of the hypergraph cut.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If we focus only on the first row of the inequality system, we obtain In fact, when the contribution from E c (q) can be ignored, then applying [16, Lemma 3] shows that v(t + 1) converges linearly if −γ 1 a 0 + γ 2 1 a 1 < 0, which is possible as a 0 > 0.
Sentence 2: if E c (t) also converges linearly, then it is anticipated that E g (t) would do so as well.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Recalling that α ≥ 0, we have that sign(v j • x) = f (x) if and only if angle from v to x lies in (π/2, π/2 + α).
Sentence 2: since the margin condition implies α ≤ π/2 − γ as noted above, we have after conditioning on G, we have that v 1 , ..., v k are still independent and the events sign(v 50π , where we used Lemma 2 for the final inequality.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We show there are problem dependent constants a(Θ, ) and b(Θ, ) such that the rates of convergence of all possible procedures are lower bounded by a(Θ, )/ √ nI * and that there exist procedures achieving convergence rates of b(Θ, )/ √ nI * , where the ratio b(Θ, )/a(Θ, ) is upper bounded by a universal constant.
Sentence 2: we establish and quantify explicitly the tradeoff between statistical estimation and the amount of privacy.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Next, we examine the threshold control scheme in the equation (6), where the threshold is controlled to maintain the mean firing rate of the network at f. q(t) in equation ( 15) is equal to the mean firing rate because q(t) x i (t) under the condition x i (t) = {0, 1}.
Sentence 2: the threshold is adjusted to satisfy the following equation, Figure 3 shows the overlap m t (t) as a function of loading rate α with f = 0.1.
Relationship: reasoning