sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Therefore, problem (2) can be reformulated as where p x,0 and p x,1 denote the elements of the vector p corresponding to the probability values P(X = x, Y = 0) and P(X = x, Y = 1), respectively.
Sentence 2: the simple application of the minimax theorem [9] implies that the saddle point of the above optimization problem exists and moreover, the optimal decision rule is a MAP rule for a certain probability distribution P * ∈ C. there exists a pair (δ * , P * ) for which Although the above observation characterizes the optimal decision rule to some extent, it does not provide a computationally efficient approach for finding the optimal decision rule.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper we follow the formulation in [4], where the tasks' structure is summarized by a positive definite matrix D which is linked to the covariance matrix between the tasks, W W . Here, W denotes the d × T matrix whose t-th column is given by the vector w t (we have assumed for simplicity that the mean task is zero).
Sentence 2: we learn W and D by minimizing the function where γ is a positive parameter which balances the importance between the error and the penalty.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This memory term, which allows nodes to remember previous values and incorporate them into the next update, directly relates the Min-Sum Splitting algorithm to accelerated multi-step first order methods in convex optimization.
Sentence 2: we show that a proper choice of the splitting parameters recovers the same matrix that support the evolution of shift-register methods used in numerical analysis for linear solvers, and, as a consequence, we recover the same accelerated rate of convergence for consensus [45,4,24].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Alternatively, maximum belief propagation (max-BP) allows a joint maximum a posteriori (MAP) set of state variables to be determined.
Sentence 2: given a song, we generate the observations Y i , i = 1 . . . F , (where F is the number of frames in the song) and seek a set of states X i that maximize the joint product Our smoothness function ψ is the product of tempo and phase smoothness components ψ T and ψ P . For the tempo component, we use a Gaussian on the log of tempo.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the following sections we will consider several examples of applications of LDOMD to online combinatorial decision problems involving both 0-1 polytopes and general polytopes in R d . Consider an online decision problem in which the decision space C contains (not necessarily all) independent sets in a matroid M = (E, I).
Sentence 2: on each trial t, one selects an independent set I t ∈ C, and receives a loss vector t ∈ [0, 1] |E| , with t e specifying the loss for including element e ∈ E; the loss for the independent set I t is given by e∈I t t e . Here it is natural to define a mapping φ : C→{0, 1} |E| that maps each independent set I ∈ C to its characteristic vector, defined as φ e (I) = 1(e ∈ I); the loss on selecting I t ∈ C is then φ(I t ) • t . Thus here d = |E|, and φ(C) ⊆ {0, 1} |E| .
Relationship: entailment
Domain: NeurIPS
Sentence 1: As for the comparison to the setting of discounted losses of [9], we note that the latter can be cast as a special case of our setting (since the discounting weights take the special form β t,T = γ t . . . γ T −1 therein, for some sequence γ s of positive numbers).
Sentence 2: the fixed-share forecaster can satisfy the bound stated in [9, Theorem 2], for instance, by using the online tuning techniques of Section 7.3.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We conclude this section with a further relevant example of our proposed approach to the setting of applications to relational knowledge, which we plan to investigate in future work.
Sentence 2: we consider settings where the output space M corresponds to the Hyperbolic space (or Poincaré disk), which has recently been adopted by the knowledge representation community to learn embeddings of relational data to encode discrete semantic/hierarchical information [34,33].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The earliest theoretical analyses of HMC were the asymptotic "optimal scaling" results of [27], for the special case when the target distribution is a multivariate Gaussian.
Sentence 2: they showed that the Metropolis-adjusted implementation of HMC with leapfrog integrator requires a numerical stepsize of O * (d − 1 4 ) to maintain an Ω(1) Metropolis acceptance probability in the limit as the dimension d → ∞.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The form of the optimization problem (Q) is similar to that of the COSSO for smoothing spline ANOVA models (Lin and Zhang, 2006); however, our method differs significantly from the COSSO, as discussed below.
Sentence 2: our method is scalable and easy to implement even when p is much larger than n. We now derive a coordinate descent algorithm for fitting a sparse additive model.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Motivated by the above applications and the increasing need to satisfy multiple objectives, we study a wide class of multi-objective online optimization problems, and present both hardness results and (almost tight) bi-objective approximation algorithms for them.
Sentence 2: we study resource allocation problems in which a sequence of items (also referred to as impressions) i from an unknown set I arrive one by one, and we have to allocate each item to one agent (for example, one advertiser) a in a given set of agents A.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , u r } ⊆ S d , such that We expect this assumption to hold when the target function U is of the form U (x) = ∑ r i=1 f i (u ⊺ i x) for functions f i ∶ R → R with uniformly bounded third derivatives.
Sentence 2: this class includes the target functions used in logistic regression.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We are especially interested in cases where each y n is a high dimensional vector and, therefore, we assume that there exists a low dimensional manifold that governs the generation of the data.
Sentence 2: a temporal latent function x(t) ∈ R Q (with Q D), governs an intermediate hidden layer when generating the data, and the dth feature from the data vector y n is then produced from x n = x(t n ) according to where f d (x) is a latent mapping from the low dimensional space to dth dimension of the observation space and β is the inverse variance of the white Gaussian noise.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Using the notation bS := {b r | r ∈ S}, the following relationships between the continuous and fixed price versions of Υ, S W , and the expected utility are a consequence of the convexity of C. Their main purpose is to highlight the relationship between wealth and liquidity in MEU demands.
Sentence 2: they show that scaling up of liquidity is equivalent to a scaling down of wealth and that the continuously priced constraints and wealth functions monotonically approach the fixed priced versions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Hidden Parameter Markov Decision Process (HiP-MDP) [14] was developed to address this type of transfer learning, where optimal policies are adapted to subtle variations within tasks in an efficient and robust manner.
Sentence 2: the HiP-MDP paradigm introduced a low-dimensional latent task parameterization w b that, combined with a state and action, completely describes the system's dynamics T (s |s, a, w b ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The relationship between Gaussian processes and Bayesian linear regression suggests that we can define a single model that exploits both similarity and rules in forming predictions.
Sentence 2: we can do this by taking a prior that covers a broad class of functions -including those consistent with a radial basis kernel -or, equivalently, modeling y as being produced by a Gaussian process with a kernel corresponding to one of a small number of types.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the theorem statement below, "for almost all C" means potentially troublesome cost matrices form at most a (Lebesgue) zero-measure subset of S n×n , in the same way that almost all square matrices are invertible.
Sentence 2: given any matrix C ∈ S n×n , perturbing C to C + σW where W is a Wigner random matrix results in an acceptable cost matrix with probability 1, for arbitrarily small σ > 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Although Algorithm 1 is Algorithm 1 Approximate gradient descent algorithm for tuning λ Initialize the tuning parameter λ 0 , choose a step-size selection rule, and set t = 0 for t = 0, 1, 2, . . . do calculate the approximate gradient g λ t (z n ) set λ t+1 = λ t − α t g λ t (z n ) end for more computationally efficient compared to LOOCV (saving a factor of n), it might still be computationally expensive for large values of n as it still scales linearly with n. Hence, we also present an online version of the algorithm using the stochastic gradient descent idea; see Algorithm 2.
Sentence 2: initialize the tuning parameter λ 0 and set t = 0 for t = 0, 1, 2, . . . do choose a random index i t ∈ {1, . . . , n} calculate the stochastic gradient g Ridge regression with diagonal regressors: We consider the following regularized loss function:  we consider one regularization parameter per each model parameter.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As RDD is usually hard to verify, we consider a stronger sufficient condition inspired by Corollary 1. , then for any δ > 0, if the sample size satisfies where K is defined in Lemma 1, then with probability at least 1 − δ, Φ = A SIS X − 2τ −1 A SIS ∞ I p is restricted diagonally dominant with sparsity s and C 0 ≥ ρ.
Sentence 2: sIS is screening consistent for any β ∈ B τ (s, ρ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The use of such a shrinking strategy reduces the overall complexity from O(d) to O(|A|).
Sentence 2: we (a) update the gradients just on the working set, (b) update Q ( Q) just on the rows(columns) corresponding to the working set, and (c) compute the latest entries in D t , γ t , L t and S T t S t by just using the corresponding working set rather than the whole set.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that E[r k (S k , x k+1 , f k+1 )] is simply the expected improvement given, for all x ∈ X , by: where Φ is the standard Gaussian CDF and φ is the standard Gaussian PDF.
Sentence 2: the GP is used to simulate possible scenarios, and the next design to evaluate is chosen to maximize the decrease of the objective function, over the remaining iterations, averaged over all possible simulated scenarios.
Relationship: entailment
Domain: NeurIPS
Sentence 1: When some groups are included in others, the canonical graph can be simplified to yield a graph with a smaller number of edges.
Sentence 2: if h and g are groups with h ⊂ g, the edges (g, j) for j ∈ h carrying a flow ξ g j can be removed and replaced by a single edge (g, h) of infinite capacity and zero cost, carrying the flow j∈h ξ g j .
Relationship: entailment
Domain: NeurIPS
Sentence 1: 4 Simple DP-UMP tests when δ = 0 In this section, we derive the DP-UMP test when δ = 0 for simple hypotheses.
Sentence 2: given n, > 0, α > 0, θ 0 < θ 1 , and X ∼ Binom(n, θ), we find the UMP test at level α among D n ,0 for testing Before developing these tests, we introduce the Truncated-Uniform-Laplace (Tulap) distribution, defined in Definition 4.1, which is central to all of our main results.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To incorporate non-maxima suppression in the framework of RRSVM, we replaced the region ranking procedure Γ(B; w) of RRSVM by Ψ(B i ; w, α), a procedure that ranks and subsequently returns the list of regions that do not significantly overlap with one another.
Sentence 2: we use intersection over union to measure overlap, where α is a threshold for tolerable overlap (we set α = 0.5 in our experiments).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To sample test queries, we sample from the original graph (i.e., the complete graph without any removed edges), but we ensure that the test query examples are not directly answerable in the training graph.
Sentence 2: we ensure that every test query relies on at least one deleted edge (i.e., that for every test query example (q, v * ), v * / ∈ q train ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let V denote all the available views of an captured action.
Sentence 2: the encoder, parameterized by θ e , takes as input a sequence of frames for view i ∈ V , denoted as and encodes them into a sequence of low-dimensionality feature embeddings E(X i ; θe) = for each frame, we first use a downsampling CNN (denoted as "Conv") to extract a low-dimensionality feature of size h × w × k. Then, a bi-directional convolutional LSTM (denoted as "BiLSTM") runs through the sequence of extracted Conv features.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Closest to our approach are penalty methods built around the idea of mixed ℓ 1 − ℓ 2 norms.
Sentence 2: the group Lasso method [16] assumes that the components of the underlying regression vector β * can be partitioned into prescribed groups, such that the restriction of β * to a group is equal to zero for most of the groups.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Equation 1 has a similar form to that of Generalized Elastic Net (GEN) [11], which has shown good performance in nonrigid image registration [12]; note that there we directly penalized Y, while here we penalize the transformation v. The φ function represents our prior knowledge about the motion, which should be smooth.
Sentence 2: we want the velocity field v generated by template point set displacement to be smooth.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For these cases, the general strategy is to first get upper bounds and lower bounds, and then check whether any gap between the lower and upper bound can be reduced.
Sentence 2: in general, one always has the simple upperbound: the total capacity of a newtowrk is always upperbounded by the sum of the capacities of all the individual neurons (this remains true even when the circuit contains threshold gates of different degrees and different fan-ins).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Obviously, R will not be z-transported individually from these domains, but it turns out that taking both sets of experiments into account, R = z2 P (a) (y|do(z 2 ))P (b) (z 2 |x, do(Z 1 )), which fully uses all pieces of experimental data available.
Sentence 2: we were able to decompose R into subrelations such that each one is separately z-transportable from the source domains, and so is the desired target quantity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We say that Π is privacy-preserving if for every 1 ≤ i ≤ k, anything that party i can compute in time polynomial in n following the execution of Π, they could also compute in polynomial time given only their private input x i and the value y.
Sentence 2: whatever information party i is able to obtain from their view of the execution of protocol Π, it does not let them efficiently compute anything they couldn't efficiently compute just from being told the final output y of Π (and their private input x i ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: While segregated graphs preserve conditional independence structure on the observed marginal of a CG for any H [20], we chose to further restrict the set H in order to ensure that the directed edges in the segregated projection retain an intuitive causal interpretation of edges in a latent projection [28].
Sentence 2: whenever A → B in a segregated projection, A is a causal ancestor of B in the underlying causal CG.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To mitigate the cumbersome and costly process of collecting and annotating human-machine dialogs as the system learns, we utilize a model-based RL approach by training a user simulator based on a corpus of human-written relative descriptions.
Sentence 2: to emulate a single dialog turn, where the user provides feedback regarding a candidate image relative to what the user has in mind, the user simulator generates a relative caption describing the differences between the candidate image and the user's desired image.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The idea is to approximate the infinite-dimensional GP function f by a linear combination over basis functions φ j parameterized by a weight vector α ∈ R N with a Gaussian prior distribution: Now, defining the matrix Φ ij (x, ) := φ j (x i , i ), the relationship y = Φ(x, )α gives a correspondence between and α (for fixed x and y), which we assume to be one-to-one.
Sentence 2: α = Φ(x, ) −1 y.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We emphasize that the simulations do not require the use of a specific algorithm, such as the algorithms associated with the proofs of the worst-case bound (Theorem 3.2) and the average-case bound (Theorem 4.1).
Sentence 2: the simulations show that the null experiment applied in conjunction with the forcing operations (F 1 )-(F 4 ) is typically sufficient to discover most of the causal graph.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Similarly, the model evidence in (7) can be approximated by Z, the normalization constant of q: Since all the factors in (12) are Gaussian, log Z can be readily computed and maximized with respect to σ 2 , ρ 2 , γ 2 and P to find good values for these hyper-parameters.
Sentence 2: once EP has converged, the gradient of the natural parameters of the gj 's with respect to these hyper-parameters is zero [21].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Next, we show how to compute the Jacobian matrix J d . Recall that at the optimum of the smooth unconstrained optimization problem OPT1, the partial derivative of the objective with respect to any parameter must vanish.
Sentence 2: the partial derivative of 1 2 w T Cw + ℓ T (w) with respect to w i vanishes when w = w ⋆ , so where C T i denotes the ith row of the C matrix.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If G is a union of disjoint cliques, Ω (min[log log n, log ω(U)]) experiments are necessary in expectation to construct G. Observe that Lemma 3.4 explains why attempting to recursively partition the runs (used in Theorem 3.1) in sub-runs will not improve worst-case performance.
Sentence 2: a recursive procedure may produce a large number of sub-runs and, with high probability, the trick will fail on one of them.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Formally, let us consider the iterative updates for the policy parameter θ and the discriminator parameter φ, where the point estimate of θ is obtained using the distribution over φ in each iteration.
Sentence 2: given θ prev from the previous iteration, we want to utilize p posterior (φ) := p(φ|0 A , 1 E , θ prev ) that satisfies By using Monte-Carlo estimations for the expectations over trajectories in (18), the log posterior in (18) can be approximated as where E,t )).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To compare the convergence error of two costs C and C in the next section, we require that their liquidities b and b be set so that they have (approximately) the same bias, i. Theorem 5.6 also captures an intuitive relationship which can guide the market maker in adjusting the market liquidity b as the number of traders N and their risk aversion coefficients a i vary.
Sentence 2: holding μ and the cost function fixed, we can maintain the same amount of bias by setting b ∝ N/ā.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The policy is doing so through adjusting the exploration rate of each arm based on a virtual time index τ k,t associated with that arm, rather than based on the actual time period t (which is appropriate in the absence of information flows).
Sentence 2: the adaptive exploration policy explores arm k at a rate that would have been appropriate without auxiliary information flows at a future time step τ k,t .
Relationship: entailment
Domain: NeurIPS
Sentence 1: To find a principle to choose the y jl , assume that we have access to the row and column measurements on the true distributions π * ij .
Sentence 2: assume that we know the true values of the column measurements φ jk = i π * ijk and row measurements ϕ ikl = j y jl π * ijk , for a chosen set of y jl values.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, when there is prior knowledge about the unknown parameters A 0 , B 0 , e.g., when A 0 , B 0 are sparse, accurate estimation can be feasible.
Sentence 2: [3] proved that under suitable conditions the unknown parameters of a noise driven system (i.e., no control) whose dynamics are modeled by linear stochastic differential equations can be estimated accurately with as few as O(log(p)) samples.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition to the existing linear constraints ((2a), (2c) and (2b)), the non-linear constraints are approximated by sets of linear constraints representing the non-linearities' convex hull.
Sentence 2: reLUs with input of unknown sign are replaced by the set of equations: where x i[j] corresponds to the value of the j-th coordinate of x i .
Relationship: entailment
Domain: NeurIPS
Sentence 1: For our experiments, we probabilistically select which ✓ m to advance from a discrete probability distribution as a function of ✓ m 's novelty.
Sentence 2: at every iteration, for a set of agent parameter vectors ⇧ = {✓ 1 , ✓ 2 , ..., ✓ M }, we calculate each ✓ m 's probability of being selected P (✓ m ) as its novelty normalized by the sum of novelty across all policies: Having multiple, separate agents represented as independent Gaussians is a simple choice for the meta-population distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A simple choice is the uniform distribution over F , but such a distribution is not continuous over the latent space.
Sentence 2: instead, we construct a mixture of Gaussians, each of which is centered at a vector f (y i ) in F . we generate a latent vector z 2 R k in two steps: We first sample a vector µ 2 F uniformly at random, and then draw a vector z from the Gaussian distribution N (µ, 2 ), where is a smoothing parameter that controls the smoothness of the distribution of the latent space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The feasibility of these rules can be seen from Proposition 1 as well as the D.C. programing problem [11].
Sentence 2: (DR1) follows from Proposition 1 that in this case T k ∩ D = ∅, i.e., the prism T k is infeasible, and (DR2) from Proposition 1 and from the definition of µ that the current best feasible solution cannot be improved in T . We first provide illustrations of the proposed algorithm and its solution on toy examples from feature selection in Section 5.1, and then apply the algorithm to an application of discriminative structure learning using the UCI repository data in Section 5.2.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Again, this result is formulated for the realizable case for simplicity, but can easily be extended to the Tsybakov noise model as in the previous section.
Sentence 2: by reasoning quite similar to that above, a cost-adaptive variant of the Robust CAL algorithm of [14] achieves error rate er( ĥ) − er(h * ) ≤ ǫ with probability at least 1 − δ using a total cost We omit the technical details for brevity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we take additional steps towards addressing the above problems and show how the generic notion of the curvature -the deviation from modularity-of a submodular function determines both upper and lower bounds on approximation factors for many learning and constrained optimization problems.
Sentence 2: our quantification tightens the generic, function-independent bounds in [8,2,27,7,10] for many practically relevant functions.
Relationship: entailment