sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: To train the deep learning network h(y; θ) with respect to θ, a gradient-based optimization algorithm is used such as the stochastic gradient descent (SGD) [31], momentum, Nesterov momentum [32], or the Adam optimization algorithm [33].
Sentence 2: for any gradient-based optimization method, it is essential to calculate the gradient of ( 7) with respect to θ as follows, it is sufficient to calculate the gradient of the empirical risk (8) to approximate (9) for any gradient-based optimization.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider a variational distribution q that lies between the input p m and p n . The log Bhattacharyya affinity with β = 1/2 can be written as follows: log B(p m , p n ) = log q(x) p m (x)p n (x) q(x) dx ≥ −D(q p m )/2 − D(q p n )/2.
Sentence 2: b(p m , p n ) ≥ exp(−D(q p m )/2 − D(q p n )/2).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider the case where the population contains many liberal users with very low anonymity levels yet one single paranoid user (user i) wants to have a maximal anonymity with δ i = n. In the k-anonymity framework, that user will require n − 1 identical copies of his data in the released database.
Sentence 2: a single paranoid user will destroy all the information of the database which will merely contain completely redundant vectors.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: (2009) is because their fully-trained (non-random) networks were overfitting more on the smaller Caltech-101 dataset than ours on the larger ImageNet informative, however, because the performance at each layer is based on different random draws of the upper layer initialization weights.
Sentence 2: the fact that layers 5, 6, and 7 result in almost identical performance across random draws suggests that multiple runs at a given layer would result in similar performance.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , p n ) T . Each parameter p i of Q, with 1 ≤ i ≤ n, approximates P(z i = 1|y, X), i.e., the posterior probability that the i-th training instance is an outlier.
Sentence 2: these parameters can be used to identify the data instances that are more likely to be outliers.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The optimality result for the vector-valued graphical model [4] implies that the optimal rate for estimating , which matches our result in (3.6).
Sentence 2: the rate in (3.6) obtained by our Tlasso estimator is minimax-optimal since it is the best rate one can obtain even when ⌦ ⇤ j (j 6 = k) are known.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Clearly, for 1 − p > p, R(steak) > R(salmon), and the diner would be rational in switching his preference.
Sentence 2: through our inferential machinery, we retrieve the anecdotal explanation for the diner's behavior: if he believes that he is more likely to be in a good restaurant (with probability (1 − p)) than not, he will prefer steak.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To create a more economical representation, we propose replacing the link attributes by a smaller set of attributes that "summarize" the information from link matrix L , possibly in combination with the term matrix T . The most obvious attributes of a document are what terms it contains.
Sentence 2: one simple way to represent the "attributes" of a document's inlinks is to aggregate the terms in the documents that link to it.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Each observation of x i in context x 1:i−1 corresponds to a customer in restaurant x 1:i−1 who is served dish x i , and each table in each restaurant u, being a draw from the base distribution G σ(u) , corresponds to a customer in the parent restaurant σ(u).
Sentence 2: the numbers of customers and tables have to satisfy the constraints where c x us = 1 if s = x i and u = x 1:i−1 for some i, and 0 otherwise.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this figure, the layer labeled "Composite" corresponds to the time-tempo variables, (tn, sn)t, for the composite rhythm, while the layer labeled "Update" corresponds to the interpretation variables ( Tn, 0" n) t. The directed arrows of this graph indicate the conditional dependency structure of our model.
Sentence 2: given all variables "upstream" of a variable, x, in the graph, the conditional distribution of x depends only on the parent variables.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 4)) of the ith component, and zi 1:K,P is parameterized by the initial pose z i 0,P and the transition variables β i 1:(T +K) .
Sentence 2: in our model, we treat z 1:N 0,P , β 1:N 1:(T +K) ,   [31] 341.2 -Brabandere et al.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We emphasize that just as in the non-adaptive setting, at no time can an algorithm stop and declare that it is TPR-, ⌧ or FWPD-, ⌧ for any finite ⌧ 2 N. This is because there may be an arm in H 1 with a mean infinitesimally close to µ 0 but distinct such that no algorithm can determine whether it is in H 0 or H 1 .
Sentence 2: the algorithm must run indefinitely or until it is stopped externally.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The hypotheses are linear classifiers X I-t (w,ifJ(x))/C =: (w,x)/C in some fixed feature space K ~ £~ where we assume that a mapping ¢ : X -+ K is chosen a priori 1 .
Sentence 2: since all we need for learning is the real-valued output (w, Xi) /C of the classifier w at the m training points in Xl, ... , Xm we can assume that w can be expressed as (see [9]) it suffices to learn the m expansion coefficients a E IR m rather than the n components of w E K. This is particularly useful if the dimensionality dim (K) = n of the feature space K is much greater (or possibly infinite) than the number m of training points.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The obvious choice is the so-called defining representation, whose elements are the familiar permutation matrices ρ def (σ) = P (σ) [P (σ)] p,q = 1 σ(q) = p 0 otherwise, since the corresponding loss function is The squared Frobenius norm in this expression simply counts the number of mismatches between the observed but noisy permutations τ ji and the inferred permutations σ j σ −1 i .
Sentence 2: furthermore, by the results of the previous section, letting P i ≡ P (σ(i)) and P ji ≡ P (τ ji ) for notational simplicity, (4) can be written in the form V ⊤ LV with similarly to the rotation case, synchronization over S n can be solved by forming V from the first d ρdef = n lowest eigenvectors of L, and extracting each P σi from its i'th n × n block.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, each corrupted row/column item has perturbed feature x i = u i + ∆u i and y i = v i + ∆v i , where ∥∆u∥ ∞ ≤ ξ 1 and Theorem 3 suggests that, if features have good quality in the sense that items with corrupted features are not too many, for example g(n), h(n) = O(log n), then sample complexity of DirtyIMC can be O(n log n √ log n) = o(n 3/2 ) as well.
Sentence 2: both Theorem 2 and 3 provide concrete examples showing that given imperfect yet informative features, the sample complexity of our model can be asymptotically lower than the lower bound of pure matrix completion (which is Ω(n 3/2 )).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By the triangle inequality, Now, using (4) and (5), Thus, ( 6) is bounded as Next, a bound on (7) follows simply by the norm comparison inequality (5): The statement follows by rearranging the terms.
Sentence 2: , where η = (e 2 − 1)η, then for all i, for all i Proof.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The interval defines constraints for the parameters because the latter quantifies the deviation of © ¡ from the benchmark 1 ¡ .
Sentence 2: the return maximization task can be stated as a constraint optimization problem with ¤ ¡ f as the actual return of asset This problem can be solved as a penalized maximization task with © is defined as a type of -insensitive error function: Summarizing, the construction of the allocation scheme consists of the following two steps: (i.)
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A person who signs the petition forwards the petition, on average, to 20.37 other addressees.
Sentence 2: by linearity of expectation, we can estimate that the ≈ 172,832.38 signers in the unobserved tree sent a total of ≈ 3,520,595.58 chain-letter e-mails in the Iraq-War petition.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We explain why Neon2 online weak works as follows.
Sentence 2: starting from a randomly perturbed point x 1 = x 0 + ξ, it keeps updating x t+1 ← x t − η (∇f i (x t ) − ∇f i (x 0 )) for some random index i ∈ [n], and stops either when T iterations are reached, or when x t+1 −x 0 2 > r. we have x t − x 0 2 ≤ r throughout the iterations, and thus can approximate .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, previous functional approaches to dendritic processing were limited because they studied dendritic computations in a firing rate-based framework [3,4], essentially requiring both the inputs and the output of a cell to have stationary firing rates for hundreds of milliseconds.
Sentence 2: they ignored the effects and consequences of temporal variations in neural activities at the time scale of inter-spike intervals characteristic of in vivo states [5].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since it is enough to show that the following holds an arbitrary 1 ≤ j ≤ m − k to prove Theorem 3: Then, (8) follows from the following inequalities: Pr The inequality (9) holds since ∥d j ∥ = 1 by the definition and Assumption 4.
Sentence 2: all we have to do is to show (10).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When facing the large-scale data but with sparse labels, these models might have computational complexity issues or parameter estimating issues.
Sentence 2: in machine learning community, numerous studies have been focusing on optimizing the parameters of these models [15][16], designing efficient algorithms [17] [18], providing sharp minimax bounds [19] and proposing novel aggregation models [9][20] [21].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Using the form of ( 9), observe that min s∈R m + y(x, s) is equivalent to min s∈R m + m j=1 2λ j ρs j + s 2 j + 2s j c j (x).
Sentence 2: for fixed x, this is strictly convex in s. its unconstrained minimum can only be its stationary point, which satisfies 0 = 2λ j ρ + 2s * j (x) + 2c j (x), for j = 1, . . . , m. Accounting for the nonnegativity constraint, we obtain the following optimal slack as a function of x: Above we write s * as a function of x to convey that x remains a "free" quantity in y(x, s * (x)).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Each term inside the summation depends on ✓S t and ✓at t , which in turn depend on all the model parameters in ⇥ via the computational graph defined by Algorithm 2 (see Appendix E for a diagram of the graph).
Sentence 2: we can learn the model parameters in ⇥ by sampling an episode in the data, computing the corresponding stochastic gradient in (7) by back-propagation on the computational graph given in Algorithm 2, and updating ⇥ by stochastic gradient/mirror descent.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The following observation will also be useful to prove error bounds for DPDA-D iterate sequence.
Sentence 2: for each i ∈ N , the definition of R k i in (17) implies that R k i (x) ∈ B 0 for all x; hence, from (19), we trivially get the following bound on µ k : Moreover, for any µ and λ we have that Theorem 3.2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is easy to see that for K * = K + this condition is satisfied provided that K((x 1 , x 1 ), (x 2 , x 2 )) ≥ 0 for all x 1 , x 1 , x 2 , x 2 ∈ X .
Sentence 2: in this case to solve (1) it is sufficient to optimize the variables (α ij ) m i,j=1 , introducing the additional constraints α ij ≥ 0 to enforce k ∈ K + .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A standard approach is to represent the collection of objects as a set of abstract points, and define distances among the points to represent similarities -the closer the points, the more similar they are.
Sentence 2: clustering is centered around an intuitively compelling but vaguely defined goal: given an underlying set of points, partition them into a collection of clusters so that points in the same cluster are close together, while points in different clusters are far apart.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is worth to mention that although it seems that the layer-wise error is scaled by a quite large product factor, S l = L k=l+1 Θk F when it propagates to the final layer, this scaling is still tractable in practice because ultimate network output is also scaled by the same product factor compared with the output of layer l. For example, we can easily estimate the norm of ultimate network output via, Y L F ≈ S 1 Y 1 F . If one pruning operation in the 1st layer causes the layer-wise error √ δE 1 , then the relative ultimate output error is .
Sentence 2: we can see that even S 1 may be quite large, the relative ultimate output error would still be about F which is controllable in practice especially when most of modern deep networks adopt maxout layer [23] as ultimate output.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 3.5 [19]) or moments accounting [2] can be used to relate the privacy guarantee of SGD to that of gradient mean estimate at each instance t. We also note that in SGD, we often sample the clients, standard privacy amplification results via sampling [2], can be used to get tighter bounds in this case.
Sentence 2: akin to [33], in the rest of the paper we just focus on the MSE and privacy guarantees of DME.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, Srebro [1] showed that finding a maximum-likelihood chordal (or triangulated or decomposable) Markov network is NP-hard even for networks of treewidth at most 2, in sharp contrast to the treewidth-1 case [2].
Sentence 2: various approximative approaches and local search heuristics have been proposed [3,1,4,5,6,7,8,9,10,11].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, for the quadratic function and the Hessian ∇ 2 F (x) = A T A.
Sentence 2: hence the parameter σ and L i satisfy (choose y = x + te i , where t ∈ R, e i ∈ R n is the unit vector), xA T Ax ≥ σ x 2 and tA T i Ae i ≤ L i |t|, ∀x, t. the σ is the smallest eigenvalue of A T A and L i = A i 2 , where A i is the ith column of the matrix A.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Eqns 9 and 10 are generalized Sylvester matrix equations of the form AXB ⊤ + CXD ⊤ = E whose unique solution X under certain regularity conditions can be exactly obtained by an extended version of the classical Bartels-Stewart method [9] whose complexity is O((p+q) 3 ) for p×q-sized matrix variable X. Alternatively, one can solve the linear system [10]: where ⊗ denotes Kronecker product and vec(X) vectorizes X in a column oriented way (it behaves as the matlab operator X(:)).
Sentence 2: the solution to Eqns (9,10) are as follows, These linear systems are of size nm r × nm r and dm c × dm c respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 2 Lower Bound on Sample Requirements: The above result states that roughly samples are required for structural consistency under any estimation method.
Sentence 2: when ρ = Θ(1) (constant fraction of observed nodes), polylogarithmic number of samples are necessary (n = Ω(poly log p)), while when ρ = Θ(m −γ ) for some γ > 0 (i.e., a vanishing fraction of observed nodes), polynomial number of samples are necessary for reconstruction (n = Ω(poly(p)).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From the training data, we learn the relative spatial distribution map of the presence of the k th 2 object, given the k th 1 object in the center.
Sentence 2: we then find out the top M highest response regions on the map, each of which has a relative location ∆  and co-occuring response w. the parameters of the k th 2 object that satisfy these relative locations, have semantic edges with θ k1,l1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In order to deal with non-uniform error corruption, our technical proof introduces a new weighted norm denoted by l w(1) , which involves the information of both localized µ 0 and µ 1 and is hence different from the weighted norms introduced in [4] for matrix completion.
Sentence 2: our proof necessarily involves new technical developments associated with such a new norm.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is well-known [4,7] that Γ is both elicitable and identifiable as the expectation of a vector-valued random variable X(ω) = (ω, ω 2 ), using for example L(r, ω) = r −X(ω) 2 and V (r, ω) = r −X(ω).
Sentence 2: we can recover σ 2 as a link of the elicitable and identifiable Γ : P → R 2 , and as no such Γ : P → R exists, we have iden(σ 2 ) = elic(σ 2 ) = 2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Defining potential function φ t (v, u) = exp w T f (v, u, x, t) , the recurrence of the normalized forward/backward score is written as, where It is easy to show that Z(x) = t s(t), and now the marginal has a simpler form of p(y t |x, w) = α(t, y t )β(t, y t ).
Sentence 2: the gradient of the marginal is, where G α and G β are the gradients of the normalized forward and backward score, respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Although S-CODE and LDC [15] achieve essentially the same level of performance on taggings that induce 17, 45, or 50 labels (Table 1), S-CODE proves superior for the induction of very fine-grained taggings.
Sentence 2: we compared the performances of S-CODE and LDC on the task of inducing 300 labels.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Decision theory recommends, after observing x A , to choose the decision d * that minimizes the risk, i.e., the expected loss, namely A natural goal in Bayesian active learning is thus to adaptively pick observations, until we are guaranteed to make the same decision (and thus incur the same expected loss) that we would have made had we run all tests.
Sentence 2: we can reduce the noisy Bayesian active learning problem to the ECD problem by defining the equivalence classes over all test outcomes that lead to the same minimum risk decision.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In our triangular model, the main property of interest is the distribution over ∆ 3 and ∆ 2 , analogous to how latent factor models and MMSB model distributions over 0-and 1-edges.
Sentence 2: subsampling strategies that preserve ∆ 3 /∆ 2 distributions (e.g.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The necessity of the more complex error term in Theorem 2 (b) (as opposed to that in part (a)) will become apparent below.
Sentence 2: when d is large, σ 2 is small, and θ, η, τ, γ ∈ C, for some compact subset C ⊆ (0, ∞), Theorem 2 suggests that consistency of ŷpcr and ŷbc in the one-shot regime hinges on asymptotic properties of The following proposition is proved in the Supplementary Material.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By arguments similar to those in Theorem 4, it can be shown that any stable non-positive steady-state is unique, attains the minimum total opinion among steady-states, and is smooth and convex for decreasing h. The above result paints a significantly simplified picture of the MF-IIM problem when J is stronglyconnected and there exists a stable non-negative steady-state m(b).
Sentence 2: given a budget H, for any feasible marketing strategy h ∈ F H , m(b + h) is the unique stable non-negative steady-state, attains the maximum total opinion among steady-states, and is smooth in h. the objective Algorithm 1: An -approximation to MF-IIM Input: System (N, J, b, β) for which there exists a stable non-negative steady-state, budget H, accuracy parameter > 0 Output: External field h that approximates a MF optimal external field h * t = 0; h(0) ∈ F H ; α ∈ (0, 1 L ) ; repeat allowing us to write down a gradient ascent algorithm that approximates a local maximum.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: According to the above theoretical analysis, the local Rademacher complexity (the tail sum of the eigenvalues of the kernel) leads to tighter generalization bounds than the global Rademacher complexity (the trace).
Sentence 2: we add the local Rademacher complexity to restrict H mkl : where λ j (K µ ) is the j-th eigenvalues of K µ and ζ is free parameter removing the ζ largest eigenvalues to control the tail sum.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is given by 1 The variance of u due to the external input can be obtained by noting that ui(t+dt) = ui(t) exp( dt/⌧ )+ R t+dt t ds exp((s t dt)/⌧ )(bi + P j Wi,jXj(t))/⌧ .
Sentence 2: in the weak coupling regime The weak coupling approximation amounts to replacing spikes of the latent neurons by intensities plus Gaussian noise.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: is a sequence of actions from the root of the search tree, and we also use p to denote the state reached after executing the path p from the root.
Sentence 2: d p is the observation distribution of the state at the end of the path p. We use p a to denote a path formed by executing all actions in p and then executing action a, and we use |p| to denote the length of the path.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Commonly, we need n convolution filters {f i } n i=1 of size d × d × c. The space complexity for storing these filters is O(d 2 cn), and the computational complexity for generating y is In contrast, the proposed spatial versatile convolution operation can extract s = d i /2 sets of feature maps simultaneously.
Sentence 2: for generating N feature maps, the space complexity for storing the proposed spatial versatile convolution filters is The computational complexity for generating feature maps using the proposed spatial versatile filters in different scales is various, which affects by the size of convolution filters, i.e., the number of non-zero elements in each M i .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, when k-means++ initialization is used, the expected number of iterations required for Lloyd's algorithm to converge is only logarithmic [1].
Sentence 2: ignoring the time required to embed the points, we can find a solution in O(n √ nk) time and space instead of the O(n 2 k) cost required by the exact method, with a strong O( √ n) improvement.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: There are two differences between DSTs and our model: (1) In DSTs, the parent chain has its own Markovian dynamics, while the team state of our model is not directly influenced by the previous team state.
Sentence 2: our model captures the emergent phenomena in which the group action is "nothing more" than the aggregate behaviors of individuals, yet it provides a useful level of representation beyond individual actions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the case where x ⋆ not set variables X i and X j to the same state, the value c-map ⋆ gives at least an upper bound that is no worse than the bound given by the relaxation alone.
Sentence 2: in particular: at least in the case where a single equivalence constraint is relaxed, the compensations implied by Equations 1 and 2 do indeed tighten a relaxation (see the auxiliary Appendix for further details).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: update the set S ui ← S ui ∪ {x oi,ui } 5: obtain the feedback let k i (x) be a vector-valued function that outputs an i-dimensional column vector with j-th entry k((x oj ,uj , S oj −1,uj , φ uj ), (x, S oi−1,ui , φ ui )) 7: let K i be an i×i matrix with (j, j )-entry k((x oj ,uj , S oj −1,uj , φ uj ), (x o j ,u j , S o j −1,u j , φ u j )) 8: update Suppose that by the end of the T -th round, an algorithm has selected T j items for the payoff function f φj ; therefore the cardinality of S j by the end of the T -th round is T j .
Sentence 2: we have T j = t≤T 1{u t = j} and T = m j=1 T j .
Relationship: reasoning