sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: . . , x n }, let S denote the n × n similarity matrix, which is both symmetric and non-negative.
Sentence 2: s i,j = S j,i and S i,j ≥ 0, where S i,j is the element of S in the i th row and the j th column (i, j = 1, . . . , n).
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2) where agent i with type θ i arrives in period t , and with the expectation taken over future types given current reports θ ≤t .
Sentence 2: when truth-telling is an -BNE we say that the mechanism is -BNE incentive compatible and no agent can improve its expected utility by more than > 0, for any type, as long as other agents are bidding truthfully.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We measure the algorithm's performance on 3 by its cumulative loss 3 $ S 3d fe F6 X ` gD 6 $ b § " $ b9 In our analysis, we will write h and i j ` § @9 A9 @9 § #i 3 when we want to stress the fact that the cumulative loss and the hypotheses of the on-line algorithm are functions of the random sample ( 3 .
Sentence 2: throughout the paper i will denote the (deterministic) initial hypothesis of an arbitrary on-line algorithm and, for each k ml U nl £ , i o will be a random variable denoting the -th hypothesis of the on-line algorithm and such that the value of i j # f( S6 8 § A9 A9 @9 B § C( 3 $ does not change upon changes in the values of ( p fq F6 ¤ § @9 A9 @9 § C( 3 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Even for small t the curves are close to linear, showing that the local linear convergence rate kicks in essentially from the start of trade in our simulations.
Sentence 2: there exist some ĉ and γ such that, empirically, we have Ê[F (r t )] − F ≈ ĉγ t , or equivalently, log( Ê[F (r t )] − F ) ≈ log ĉ + t log γ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we show that in many cases a powerful multiplicative low-rank approximation in Frobenius norm can be obtained by applying a simple truncated SVD procedure on a crude, easyto-find spectral norm approximate.
Sentence 2: given the spectral norm approximation condition , where A k is the best rank-k approximation of A in Frobenius and spectral norm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In what follows, we let Q ⊂ m 4 be the set of all quadruples {p, q, r, s} ∈ m 4 such that only one of the three path pairs in (2.4) gives an empty set when the union of its two component paths is taken.
Sentence 2: q contains all S ∈ m 4 such that T |S is one of the configurations in Figure 2.2(a)-(c).
Relationship: entailment
Domain: NeurIPS
Sentence 1: When using the smoothed weighted max norm, we need to choose the amount of smoothing to apply to the marginals, that is, we need to choose ζ in our definition of the smoothed row and column weights, as given in (3).
Sentence 2: alternately, we could regularize simultaneously over all possible amounts of smoothing by considering the local max norm with and same for C. R and C are line segments in the simplex-they are larger than any single point as for the uniform or weighted trace norm (or smoothed weighted trace norm for a fixed amount of smoothing), but smaller than the entire simplex as for the max norm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Linear averaging weighs each iteration σ t in the average strategy by t. It does not affect regret or influence the selection of strategies on an iteration.
Sentence 2: with linear averaging the new definition for average strategy becomes σT i (I) = t∈T (tπ σ t i i σ t i ) t∈T (tπ .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In our own experiments, shown in Section 6, we use an implementation of TextonBoost [20] and augment it with quadratic higher order potentials.
Sentence 2: we use TextonBoost to generate per-pixel scores c, and then minimize f @AA a c¡e A C j jAR j jjR j nAj, where the regions R j are regions of pixels that we expect to be of the same class (e.g., by running a cheap region-growing heuristic).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To accommodate for all kinds of observed random variables described above, we introduce an auxiliary Gaussian variable y d n , such that when conditioned on the auxiliary variables, the latent variable model behaves as a standard IBP with Gaussian observations.
Sentence 2: we assume y d n is Gaussian distributed with mean z n B d and variance σ 2 y , i.e., and assume that there exists a transformation function over the variables y d n to obtain the observations x d n , mapping the real line into the observation space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: An important property of differential privacy is that the privacy guarantees degrade gracefully if the same sensitive data is used in multiple private computations.
Sentence 2: if we apply an αdifferentially private procedure k times on the same data, the result is kα-differential private as well as (α , δ)-differentially private for α = kα(e α − 1) + 2k log(1/δ)α [7,8].
Relationship: entailment
Domain: NeurIPS
Sentence 1: We measure the distance between two distributions D and D ′ using a hypothesis class-specific distance measure.
Sentence 2: let H be a hypothesis class for instance space X , and A H be the set of subsets of X that are the support of some hypothesis in H. for every hypothesis h ∈ H, {x : x ∈ X , h(x) = 1} ∈ A H . We define the distance between two distributions as: For our purposes, the distance d H has an important advantage over more common means for comparing distributions such as L 1 distance or the KL divergence: we can compute d H from finite unlabeled samples of the distributions D and D ′ when H has finite VC dimension [4].
Relationship: entailment
Domain: NeurIPS
Sentence 1: After this, in the fine-tuning stage, we build a big batch using all the data to minimize the loss L(Θ) defined in (4) with a gradient descent method.
Sentence 2: we use Adam [18], an adaptive momentum based gradient descent method, to minimize the loss, where we set the learning rate to 1.0 × 10 −3 in all our experiments.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A function g is L-Lipschitz when kg(x) g(y)k  L kx yk for all x, y, and it is H-smooth when it is differentiable and its gradient is H-Lipschitz.
Sentence 2: we consider optimization algorithms that use either a stochastic gradient or stochastic prox oracle (O grad and O prox respectively): where prox The prox oracle is quite powerful and provides global rather than local information about f . querying the prox oracle with = 0 fully optimizes f (•; z).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We assume that the models may be mis-specified.
Sentence 2: let k * be the best single model defined as: then f k * = g. We are interested in an estimator f of g that achieves a small regret This paper considers a special class of model combination methods which we refer to as model averaging, with combined estimators of the form where ŵk ≥ 0 and k ŵk = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Recent work on Boltzmann machine models and their generalizations to exponential family distributions have allowed these models to be successfully used in many application domains.
Sentence 2: the Replicated Softmax model [6] has been shown to be effective in modeling sparse word count vectors, whereas Gaussian RBMs have been used for modeling real-valued inputs for speech and vision tasks.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The first set P uB contains all sub-Gaussian distributions with unbounded support, and bounded mean and variance.
Sentence 2: we assume that there exist τ > 0 and Γ uB > 0 such that for any probability measure µ ∈ P uB and any vector a ∈ R h , The second class P B of distributions contains all sub-Gaussian distributions with bounded support X = {x : x ≤ Γ B } ⊂ R h (note that this set in fact includes all distributions with bounded support).
Relationship: entailment
Domain: NeurIPS
Sentence 1: There exist ζ > 0 and R > 0, such that The above assumption is fairly standard [7,19] in non-parametric regression.
Sentence 2: the bigger ζ is, the more stringent the assumption is, since for ζ = 0, we are assuming f H ρ < ∞, while for ζ = 1/2, we are requiring f H ∈ H ρ , since [25,19] Finally, the last assumption relates to the capacity of the hypothesis space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If this limit exists, then it equals the intrinsic dimension of M . Alternatively, Kégl [4] suggests using the easily proved inequality N M (ǫ) ≤ P M (ǫ) ≤ N M (ǫ/2) to express the capacity dimension in terms of packing numbers as .
Sentence 2: now, a simple geometric argument shows that, for any µ supported on there exists a decreasing sequence {ǫ k }, such that for sufficiently large values of k (i.e., in the high-rate regime) the ratio − log k/ log e * ∞ (k|µ) can be approximated increasingly finely both from below and from above by quantities involving the packing numbers P M (ǫ k ) and P M (2ǫ k ) and converging to the common value d cap (M ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: At high level, the algorithm proceeds through several rounds, in each performing the following operations (see also Figure 1 Concept space localization: Solve for hypothesis vector w k+1 by minimizing hinge loss subject to the constraint that w k+1 lie within a radius r k from w k ; that is, [1, 10,16] show that by setting the parameters appropriately (in particular, b k = Θ(1/2 k ) and r k = Θ(1/2 k )), the algorithm will achieve error using only k = O(log 1/ ) rounds, with O(d) label requests per round.
Sentence 2: a key idea of their analysis is to decompose, in round k, the error of a candidate classifier w as its error outside margin b k of the current separator plus its error inside margin b k , and to prove that for these parameters, a small constant error inside the margin suffices to reduce overall error by a constant factor.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For each trial, the network was simulated for 100 ms and its estimates of s 1 and s 2 were read out using a suboptimal decoding strategy.
Sentence 2: to get an estimate of s 1 , we considered only the row of neurons in the output layer whose preferred vertical locations were closest to the vertical location of the first stimulus and then we fit a Gaussian function (with amplitude, peak location and width parameters) to the activity profile of this row of neurons and considered the estimated peak location as the model's estimate of s 1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, the sampling procedure used in MCTS algorithms is not well-suited for other kinds of problems [Nakhost, 2013], such as deterministic single-agent problems where the objective is to find any solution at all.
Sentence 2: if the reward is very sparse-for example the agent is rewarded only at the end of the task-MCTS algorithms revert to uniform search.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Compared to the state of the art [5,10], the polynomial code provides order-wise improvement in terms of the recovery threshold.
Sentence 2: the recovery thresholds achieved by 1D MDS code [5,10] and product code [10] scale linearly with N and √ N respectively, while the proposed polynomial code actually achieves a recovery threshold that does not scale with N . Furthermore, polynomial code achieves the optimal recovery threshold.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section, we show that the simple 1 regularized logistic regression classifier attains fast classification error rates.
Sentence 2: we are interested in the M -estimator [21] below: which maximizes the penalized log-likelihood of the logistic regression model, which also corresponds to the conditional probability of the response given the covariates P(Y |X) for the conditional Gaussian model.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Motivated in part by the spike-sorting problem, we consider the case where we are given a signal that is the sum of known waveforms whose timing and amplitude we seek to recover.
Sentence 2: we suppose our signal can be modeled as: where the waveforms f n are known, and we seek to estimate positive amplitudes a n,j and event times τ n,j .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let Q(•) denote the marginal probability defined by The mutual information between X and Z is the expected Kullback-Leibler (KL) divergence between Q(• | X) and Q(•): We view the problem of privacy as a game between the adversary controlling P and the data owners, who use Q to obscure the samples X.
Sentence 2: we say a distribution Q guarantees a level of privacy I * if and only if sup P I(P, Q) ≤ I * .
Relationship: entailment
Domain: NeurIPS
Sentence 1: So a partition {S 1 , S 2 } of S that minimizes the description length (DL) is in We will denote by 2 S the set of all subsets of S. A set function f : 2 S → R assigns a (real) number to every subset of S. We say that f is In [1], Queyranne gives a polynomial time algorithm that finds a set A ∈ 2 S \ {S, φ} that minimizes any symmetric submodular set function (specified in the form of an oracle).
Sentence 2: queyranne's algorithm finds a non-trivial partition {S 1 , S \ S 1 } of S so that f (S 1 ) (= f (S \ S 1 )) minimizes f over all non-trivial subsets of S. The problem of finding non-trivial minimizers of a symmetric submodular function can be thought of a a generalization of the graph-cut problem.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As this data set is difficult to visualize without taking up large amounts of space, we have included the results in the supplementary material which accompanies our NIPS submission.
Sentence 2: we provide a first figure which shows the data mapped to b = 2 dimensions, with certain authors (or groups of authors) color coded-the choice of authors and their corresponding color codes follows precisely those of (Song et al., 2008).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Assume in (1) each f i is L-g-smooth, and f is µ-strongly g-convex, then if we run Algorithm 1 with Option I and parameters that satisfy then with S outer loops, the Riemannian SVRG algorithm produces an iterate x a that satisfies Algorithm 1: RSVRG (x 0 , m, ⌘, S) Parameters: update frequency m, learning rate ⌘, number of epochs S initialize x0 = x 0 ; for s = 0, 1, . . . , S 1 do The proof of Theorem 1 is in the appendix, and takes a different route compared with the original SVRG proof [18].
Sentence 2: due to the nonlinear Riemannian metric, we are not able to bound the squared norm of the variance reduced gradient by f (x) f (x ⇤ ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Loosely speaking, we first show (Lemma 10) that the Bayes error R * can be well approximated using 1-NN classifiers defined by the true (as opposed to empirical) majority-vote labels over fine partitions of X .
Sentence 2: this holds for any partition induced by a γ-net of X with a sufficiently small γ > 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The MRF also contains pairwise potentials {φ ij } that can encode dependencies between the landmark assignments.
Sentence 2: we may want to encode geometric potentials, which enforce a preference for preservation of distance or orientation for pairs of markers x i , x j and their assigned targets x k , x l . Finally, as the goal is to find a 1-to-1 mapping between landmarks in the source and target images, we also encode a set of mutual exclusion potentials over pairs of variables, enforcing the constraint that no two markers are assigned to the same candidate x k . Our task is to find the MAP solution in this MRF.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To further reduce the regret, we can break the ties according to estimated regret.
Sentence 2: with samples θ (1) ij (t), the normalized Copeland score for each arm i can be estimated as ζi (t) = 1 Then the maximum normalized Copeland score is ζ * (t) = max i ζi (t), and the loss of comparing arm i and arm j is rij (t) = ζ * (t) − 1 2 ζi (t) + ζj (t) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We assume that the cumulative intensity of the counting process is modeled by a time-varying coverage function, i.e., the observation pair in the time window [0, T ] for some T > 0, and df (S, t) = a(S, t)dt.
Sentence 2: the timevarying coverage function controls the propensity of occurring events over time.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Model scores can then be used as approximate surrogates for probabilities, obviating the computation normalizer computation.
Sentence 2: given a model of the form p η (y | x) = e η T T (y, x)−A(η, x) (1) with A (η, x) = log y∈Y e η T T (y, x) , we seek a setting of η such that A(x, η) is close enough to zero (with high probability under p(x)) to be ignored.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to estimate the value of policy µ = µ i , we follow the temporal difference learning (TD) approach.
Sentence 2: given a new estimate of the expected future returns R t at time t after taking action a t in state s t according to some policy µ, Q(s t , a t ) (dropping the dependence on µ) is updated as follows where α > 0 is a problem-dependent learning rate parameter.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition, the OLS estimation for tensor regression (2) can be rewritten as the following nonconvex problem in terms of low-rank tensor parameters o .
Sentence 2: the number of parameters for a general tensor Nevertheless, solving the tensor regression problem (5) is still expensive in terms of both computation and memory requirements, for typical settings, when n R • P D d=1 p d . the per iteration complexity is at least linear in n for popular algorithms such as block alternating minimization and block gradient descent [27,28].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since we do not have access to the oracle editor p * edit , we train the editor p edit to directly maximize L(p edit , pret ).
Sentence 2: we solve the optimization problem: In our experiments, we let p edit be a standard sequence-to-sequence model with attention and copying [12,36] (see Appendix B for details), but any model architecture can be used for the editor.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For instance, consider a single-hidden layer S-H-F MLP (with S-input H-hidden F -output nodes); there are n A =F (H + 1) terminal parameters " A (including threshold parameters) on direct connections to F terminal nodes, each of which has CA(=H + 1) direct connections, and the rest of n B =H(S + 1) parameters are not directly connected to any terminal node; hence, n B hidden parameters " B .
Sentence 2: model's parameters " (n=F CA + nB in total) can separate as: where " A k is a vector of the kth subset of CA terminal parameters directly linked to terminal node k (k = 1, • • • ,F ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: This variant, described in Figure 1, queries all labels (and stores all examples) during an initial stage of length at least (16d)/λ 2 , where λ denotes the smallest nonzero eigenvalue of the process covariance matrix E[X 1 X ⊤ 1 ].
Sentence 2: when this transient regime is over, the sampler issues a query at time t based on both the query counter N t−1 and the margin ∆ t . if evidence is collected that the number N t−1 of stored examples is smaller than our current estimate of 1/∆ 2 t , that is if ∆ 2 t ≤ (128 ln t)/(λN t−1 ), then we query (and store) the label of the next instance x t+1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: A common identifiability issue, also shared by the BIP in (1), is the scaling ambiguity.
Sentence 2: if (w \ , x \ ) solves a BIP, then so does (cw \ , c 1 x \ ) for any nonzero c 2 R. In this paper, we resolve this scaling ambiguity by finding the point in the solution set closest to the origin with respect to the `1 norm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: When coupled with kernel ridge regression, the comparison between PLST and kernel-CPLST in terms of the Hamming loss is shown in Table 5. kernel-CPLST performs well for LSDR and outperforms the feature-unaware PLST in most cases.
Sentence 2: in five out of the eight datasets, kernel-CPLST is significantly better than PLST regardless of the number of dimensions used.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To further boost the computational efficiency at each stage of the convex relaxation, we apply the pathwise optimization [10] for all algorithms in practice.
Sentence 2: at each stage, we use a geometrically decreasing sequence of regularization parameters { , where [0] is the smallest value such that the corresponding solution is zero, ↵ 2 (0, 1) is a shrin- .
Relationship: entailment
Domain: NeurIPS
Sentence 1: For Lemma 5.1 to be practically useful, it is essential that f1 and f2 be efficiently optimizable over C. We discuss two general curvature-dependent approximations that work for a large class of combinatorial constraints.
Sentence 2: we use Theorem 3.1: we decompose f into f κ and a modular part f m , and then approximate f κ while retaining f m , i.e., f = f κ + f m . The first approach uses a simple modular upper bound (MUB) and the second relies on the Ellipsoidal approximation (EA) we used in Section 3.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This paper studies the problem of learning a low-dimensional Euclidean metric from comparative judgments.
Sentence 2: consider a set of n items with high-dimensional features x i 2 R p and suppose we are given a set of (possibly noisy) distance comparisons of the form sign(dist(x i , x j ) dist(x i , x k )), for a subset of all possible triplets of the items.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Now, what happens when P [X 1 = 1|X 2 = 1] = 0.99?
Sentence 2: what can we say about P [X 1 = 1|X 2 = 1, . . . , X n = x n ] given information about probabilities P [X i = x i , X j = x j ].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Typical results study R T (w) against all w with a bounded norm in some geometry.
Sentence 2: for an invariant update, we relax this requirement and only put bounds on the predictions w x t . for some pre-chosen constant C we define K t def = w : |w x t | ≤ C . We seek to minimize regret to all comparators that generate bounded predictions on every data point, that is: Under this setup, if the data are transformed to M x t for all t and some invertible matrix M ∈ R d×d , the optimal w * simply moves to (M −1 ) w * , which still has bounded predictions but might have significantly larger norm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Conversely Nelson and Nguy ễn [NN13] showed that this is almost asymptotically optimal.
Sentence 2: every distribution over n × m matrices satisfying (1) with m = Θ(ε −2 lg(1/δ)), and such that every column has at most s non-zero entries must satisfy s = Ω((ε lg(1/ε)) −1 lg(1/δ)).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our characterization is radically different from existing results, such as in [18,8,19,9,10,11,12,13].
Sentence 2: we unify primal-dual methods [20,21], smoothing (both for Bregman distances and for augmented Lagrangian functions) [22,21], and the excessive gap function technique [1] in one.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We use the independent-sites parent-independent mutation model over multinomial vectors in [1] as our likelihood model.
Sentence 2: this model assumes that each point on the tree is associated with a D dimensional multinomial vector, and each entry of this vector on each branch of the tree evolves independently (thus independent-sites), forward in time, and with mutations occurring at rate λ d on entry d. When a mutation occurs, a new value for the entry is drawn from a distribution φ d , independently of the previous value at that entry (thus parentindependent).
Relationship: entailment