sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: State-of-the-art deep neural networks have recently been shown to be very unstable to worst-case perturbations of the data (or equivalently, adversarial perturbations) [17].
Sentence 2: despite the excellent classification performances of these classifiers, well-sought perturbations of the data can easily cause misclassification, since data points often lie very close to the decision boundary of the classifier.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If an algorithm always finds a hypothesis ĥ with err(D, ĥ) ≤ err(D, H)+ for > 0, then for any η ∈ (0, 1) there is a distribution D with η = err (D, H) such that the auditing complexity of this algorithm for D is Ω(dη 2 / 2 ).
Sentence 2: when η is fixed while → 0, the auditing complexity scales as Ω(d/ 2 ), similar to the passive sample complexity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Four main terms constitute our bound: the approximation error due to the limited hypothesis space (first term), the variance (second and third terms), the distance to the prior (fourth term), and a constant term decaying as O(N 2 ).
Sentence 2: as we might have expected, the only difference between the bounds for the two versions of Algorithm 1 is in the term ϕ(W s ), i.e., the distance between the optimal weights w * and the source weights W s . for the mixture version we have the (smoothened) minimum distance to the source tasks' weights (Equation ( 5)), while for the Gaussian one we have the distance to the mean of such weights.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Recently, Ramaswamy & Agarwal [39] introduce the notion of convex calibrated dimension, as the minimal dimensionality of the score vector that is required for consistency.
Sentence 2: they showed that for the Hamming loss on T binary variables, this dimension is at most T . In our analysis, we use scores of rank (T + 1), see (35) in App.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To bound the sample complexity of the DFS routine, since there are M states per level and the consensus test returns true once elimination has been performed, we know that the DFS does not visit a large fraction of the search tree.
Sentence 2: this means DFS is invoked on at most MH nodes in total, so we run elimination at most MH times, and we perform at most MKH consensus tests.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Terms (s 1 + λ * α )1 and s 2 ψ can be viewed as cost shaping.
Sentence 2: they effectively transform the costs g(x, a) to g(x, a) + s 1 + λ * α + s 2 ψ(x), so that the constraint T α Φr − Φr − λ * α 1 ≥ 0 can be met.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Suppose now that each player randomizes their play according to the sequence of probability distributions on S i generated by DA with regularizer h i .
Sentence 2: suppose that each σ i t is a random variable with the following distribution: Example Consider a zero-sum game G 1 between two players on the unit interval with payoff function u(s 1 , s 2 ) = s 1 s 2 − a 1 s 1 − a 2 s 2 , where a 1 = e−2 e−1 and a 2 = 1 e−1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: They demonstrated that the relationships between these notions of sample complexity are substantially more delicate even in the most well-studied settings of SCO.
Sentence 2: let K be a unit 2 ball and F be the set of all convex sub-differentiable functions with Lipschitz constant relative to 2 bounded by 1 or, equivalently, ∇f (x) 2 ≤ 1 for all x ∈ K. Then, known algorithm for SCO imply that sample complexity of this problem is O(1/ 2 ) and often expressed as 1/ √ n rate of convergence (e.g.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We now move to discussing safe subgame-solving techniques, that is, ones that ensure that the exploitability of the strategy is no higher than that of the blueprint strategy.
Sentence 2: in subgame Resolving [6], a safe strategy is computed for P 2 in the subgame by solving the augmented subgame shown in Figure 3b, producing an equilibrium strategy σ S . This augmented subgame differs from Unsafe subgame solving by giving P 1 the option to "opt out" from entering S and instead receive the EV of playing optimally against P 2 's blueprint strategy in S. the augmented subgame for Resolving differs from unsafe subgame solving as follows.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , F k . We require that the product of the potentials in these subnetworks maintain the same information as the original MRF.
Sentence 2: if we originally have a factor φ i ∈ F and associated factors One method of partitioning that achieves this equality is simply to select, for each potential φ i , one subgraph in which it appears unchanged, and set all of the other φ (l) i to be 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To set these parameters, we used the following parameter optimization procedure: we randomly partitioned each dataset into 4 folds and, for each tuple (λ, β, K) in the set of possible parameters (described below), we ran MDeepBoostSum, with a different assignment of folds to the training set, validation set and test set for each run.
Sentence 2: for each run i ∈ {0, 1, 2, 3}, fold i was used for testing, fold i + 1 (mod 4) was used for validation, and the remaining folds were used for training.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To find the ROC curve for our meta-classifier, we plot P D against P F for all 0 ≤ α ≤ 1.
Sentence 2: for each y ∈ {0, 1} n we can compute Pr[ℓ(Y) > ℓ(y)|H 0 ], which gives us one value for τ * and a point in ROC space (P F and P D follow directly from L and P ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, under Assumption 3.1, it attains r SR α (6) in the sense of Definition 3.1.
Sentence 2: (i) lim inf n→∞ βtp (n) − max {0, max a∈A β tp (q n ; a) − α} ≥ 0 , and (ii) lim sup n→∞ βfp (n) ≤ 0, for every strategy of the opponent, almost surely.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To generate the embedding q for a query q using Algorithm 1, we (i) represent the query using its DAG dependency graph, (ii) start with the embeddings z v1 , ..., z vn of its anchor nodes, and then (iii) we apply geometric operators, P and I (defined below) to these embeddings to obtain an embedding q of the query.
Sentence 2: we introduce two key geometric operators, both of which can be interpreted as manipulating the denotation set associated with a query in the embedding space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: There are a number of possible sign-function approximations, including hyperbolic and arc tangent, and shifted sigmoid function, the latter of which is the most flexible because of its free parameter β.
Sentence 2: 3 the sigmoid function has the form f (z) = 1 1+e −βz , where the free parameter β (an inverse temperature) determines how well the curve will approximate the discontinuous function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given training data {x (q) , y (q) } N q=1 , where each n (q) } is a set of documents of query q, and each y (q) = {y } is a set of ranking scores associated with the documents of query q, we employ Maximum Likelihood Estimation to estimate the parameters {α, β} of C-CRF.
Sentence 2: we calculate the conditional log likelihood of the training data with respect to the C-CRF model, log Pr(y (q) |x (q) ; α, β).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we study the limits of MSE recovery of high-dimensional, graph-structured noisy patterns.
Sentence 2: we assume that the patterns x are generated from a probabilistic model, either Gaussian graphical model (GGM) or Ising (binary), based on a general graph structure G(V, E), where V denotes the p vertices and E denotes the edges.
Relationship: entailment
Domain: NeurIPS
Sentence 1: When using weighted entropy approximations, it makes sense to optimize over both the linearization g and the weights ρ jointly.
Sentence 2: let T be some set of weights that yield an entropy approximation H that upper bounds H. Then, if we expand H in problem (3), we obtain minimize Note that inside the supremum, both g and ρ appear only linearly, and there is no summand that has terms from both of them.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In our implementation, we changed the auxiliary reward f in the terminal state.
Sentence 2: when the agent encounters a terminal state S ⊥ , it receives an auxiliary reward f (S, S ⊥ ) = γϕ(S ⊥ ) instead of f (S, S ⊥ ) = γϕ(S ⊥ ) − ϕ(S).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Retrace(λ) is an off-policy correction method, so it learns from n-step off-policy data by multiplying each term of the loss by a correction coefficient, the trace, in order to re-weight the data distribution to look more like the on-policy distribution.
Sentence 2: at each timestep, Retrace(λ) updates the coefficient for that term by multiplying it by λmin (1, π(as|xs) µ(as|xs) ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Another implication is that influence at larger time window T is harder to estimate, since σ(A, T ) will generally be larger and hence require more random labels.
Sentence 2: once we know how to estimate the influence σ(A, T ) for any A ⊆ V and time window T efficiently, we can use them in finding the optimal set of C source nodes A * ⊆ V such that the expected number of infected nodes in G is maximized at T . we seek to solve, where set A is the variable.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To formulate a family of objective functions which depend on both p(s), and the tuning curves, we first rely on Fisher information, I f (s), which can be written as a function of the tuning curves [1,18]: The Fisher information can be used to express lower bounds on mutual information [16], the variance of an unbiased estimator [18], and perceptual discriminability [19].
Sentence 2: the mutual information, I( r; s), is bounded by: where H(s) is the entropy, or amount of information inherent in p(s), which is independent of the neural population.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A well known example is Dantzig Selector, which minimizes a sparsity-inducing `1 norm with an `1 norm constraint.
Sentence 2: let X 2 R n⇥d be a design matrix, y 2 R n be a response vector, and ✓ 2 R d be the model parameter.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The performance of algorithm A is measured by the number of matches found by A within the T rounds.
Sentence 2: if M t (A) is the number of matches uncovered by A after t rounds of a given run, we would like to obtain lower bounds on M T (A) that hold with high probability over the random generation of boys and girls that log into the system as well as the internal randomization of A.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let F m,δ,n represent the set of those Boolean functions on {0, 1} n that can be computed by a thresholded RBM network of size m with margin δ.
Sentence 2: then, there exists a fixed number K such that, when m 2 ≤ δ2 αn , for any constant α < 1/2, the ratio of the size of the set F m,δ,n to the total number of Boolean functions on {0, 1} n (which is 2 2 n ), rapidly converges to zero with n. While interesting, existential results such as the one above does not give us a clear picture of what a particular hard-to-compute function for RBM networks might look like.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our Contribution (II): We provide new analysis for more general optimization problems beyond general convexity and strong convexity, as well as more general algorithms.
Sentence 2: we provide several concrete examples: (1) VGD achieves linear convergence under the Polyak-Łojasiewicz (PL) condition (possibly nonconvex), which matches the state-of-art result in [4]; (2) NAG achieves accelerated linear convergence (with a better constant term) under both general convex and quadratic growth conditions, which matches the state-of-art result in [19]; (3) Coordinate-gradient-type algorithms share the same ODE approximation with gradient-type algorithms, and our analysis involves a more refined infinitesimal analysis; (4) Newton's algorithm achieves linear convergence under the strongly convex and self-concordance conditions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, in general this would be a complex matrix optimization problem.
Sentence 2: it may require sophisticated optimization tools such as semidefinite programming, which may not scale well with the size of W . Fortunately, as we shall show, problem (2.4) can be efficiently solved by alternately minimizing over D and W . in Section 4 we shall show that Ω f is a function of the singular values of W only.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We compare the NN set-up to random Fourier basis expansions [34], which have been successfully applied to BO [35,36].
Sentence 2: let U ∈ R D×P and b ∈ R D be such that U ∼ N (0, I) and {b j } D j=1 ∼ U([0, 2π]).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To make inference tractable, we focus on the large-graph, small-neighborhood limit as n → ∞ and ε n (x) → 0.
Sentence 2: we will suppose that there exist scaling constants g n and a deterministic continuous function ε : D → R >0 so that where the final convergence is uniform in x and a.s. in the draw of X .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We consider learning our model with experience replay [17], a widely used technique in recent stateof-the-art systems [18].
Sentence 2: the learning process consists of multiple stages, and at each stage, the agent interacts with the environment using a fixed exploration policy ⇡ b (a t |x S 1:t , x A 1:t , a 1:t 1 ) to collect M episodes of data {w S 1:Ti , w A 1:Ti , a 1:Ti , r 1:Ti } M i=1 and saves them into a replay memory D. (We will discuss the choice of ⇡ b in section 4.)
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we propose to discover a large initial set of data-driven attributes using a dictionary learning method.
Sentence 2: assume that we have a set of N videos in a n-dimensional feature space X = [x 1 , ..., x N ], x i ∈ R n , then a data-driven dictionary is learned by solving the following problem: where are the sparse codes of X, and T specifies the sparsity that each video has fewer than T items in its decomposition.
Relationship: entailment
Domain: NeurIPS
Sentence 1: When the spike input is on (0 < t < t pw ), the dynamics of the circuit for a step input is governed by Equation ( 7) can be changed into a linear ordinary differential equation for i syn (t): In terms of the general solution expressed in (3), we have When the spike input is off (t > t pw ) and we neglect the leakage current from M 2 , then i syn (t) will exponentially decay with the same time constant defined by (11).
Sentence 2: 4 Results We have fabricated a chip containing the basic synapse circuit as shown in Figure 2 through MOSIS in a commercially-available 1.5 µm, double poly fabrication process.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is computationally convenient to compute z U (h) recursively as importance weights, just as in a sequential importance sampling particle filter [11]; this only assumes we have access to the likelihood of the observations (i.e., state transitions).
Sentence 2: the weights are initialized as z U m (h t ) = 1 M ∀m and are then updated recursively using the likelihood of the dynamics model for that particle of observations as z U m (has ) ∝ z U m (h)P (s |a, s, P m ) = z U m (h) P m (s, a, s ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Solving an MDP means to find a policy π * that maximizes the expected long-term reward: π * ∈ arg max π∈Π J π D . For any MDP there exists at least one deterministic optimal policy that simultaneously maximizes V π (s), ∀s ∈ S. For control purposes, it is better to consider action values Q π (s, a), i.e., the value of taking action a in state s and following a policy π thereafter: Furthermore, we define the advantage function: that quantifies the advantage (or disadvantage) of taking action a in state s instead of following policy π.
Sentence 2: for each state s, we define the advantage of a policy π over policy π as A π π (s) = A π (a|s)A π (s, a)da and, following [14], we define its expected value w.r.t.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This number is overall bounded (over all external loop cycles) by the number of times we can make a cell of matrix m turn from false to true (which is O(n 2 )) plus the number of times we ignore an element because the matrix cell was already set to true (which is at most O(n) per each V j , as this is the maximum number of descendants of V j and each of them can fall into this category only once, so again there are O(n 2 ) times in total).
Sentence 2: each element being removed from a todo list is either ignored (matrix already set to true) or an entry in the matrix of descendants is changed from false to true, and this can only happen O(n 2 ) times.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Background subtracted images play a fundamental role in making inferences about objects and activities in a scene and, by nature, they have structured spatial sparsity corresponding to the foreground innovations.
Sentence 2: compared to the scale of the scene, the foreground innovations are usually not only sparse but also clustered in a distinct way, e.g., corresponding to the silhouettes of humans and vehicles.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The reward distributions are assumed to be exactly generalized extreme value distributions.
Sentence 2: [21] assumes that the distributions are exactly Gumbel, P (x) = exp(−(x − m)/s)), and [9], that the distributions are exactly of Gumbel or Fréchet P (x) = exp(−(x − m) α /(sα))).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Most of the research has focused on formulating robust classification problems that can be efficiently solved via convex optimization.
Sentence 2: the robust classification method developed in [6] is based on the criterion which is similar to the Fisher discriminant ratio f . With a specific uncertainty model on the means and covariances, the robust classification problem with discrimination criterion g can be cast as a second-order cone program, a special type of convex optimization problem [5].
Relationship: entailment
Domain: NeurIPS
Sentence 1: 1 and ignoring constants, finding the MAP assignment is seen to be equivalent to the following combinatorial optimization problem: where the indicator variable x it = I[z i = t] denotes the assignment of word i to topic t. 2.1 Exact maximization for small number of topics Suppose a document only uses ⌧ ⌧ T topics.
Sentence 2: t could be large, but we are guaranteed that the MAP assignment for a document uses at most ⌧ di↵erent topics.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Because stage s depends on the historical path only through the state variables (x, x, i, j, s), C(x, x, i, j, s) is well defined and (x, x, i, j, s) transits in a Markov chain with the next state being (x, x, i, j + 1, s + 1) if stage s does not succeed and being (x + , x+ , i+1, j, s+1) if stage s succeeds, where x+ =SVRG HEB (x, T s , R s , 0.5) and x+ = arg min x∈Ω ∇f In the next, we will use backward induction to derive an upper bound for C(x, x, i, j, s) that only depends on i and j but not on s, x and x.
Sentence 2: we want to show that where A i := We start with the base case where i = S 1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: 6 The sign of t i ( ) in positive proximity 7 of 0 is the same as the first (i.e., relative to the lowest degree monomial) non-zero coefficient of the expansion of t i ( ) around 0.
Sentence 2: there exists a ¯ > 0 such that t i ( ) has the same sign as the first non-zero derivative of t i evaluated in 0 for all 0 < < ¯ .
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . u L ) ∈ U n denotes the current state of the parents (L = |pa(n)|).
Sentence 2: we can express the probability of finding node n in state x after some small time-step h, given that it was in state x at time t with x, x ∈ X n as The CIMs are connected to the joint intensity matrix R of the CTMC via amalgamation -see, for example, [13].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the ambiguous learning setting [9,10], which is a generalization of semi-supervised learning, the labeler reveals a label set Ŷi ⊆ Y for each training example xi such that ŷi ∈ Ŷi .
Sentence 2: for each training example, the learning algorithm is given a set of possibile labels the example can have (semi-supervised learning is the special case where each label set has size 1 or k).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our high-level idea is to measure the probability density of test sample on feature spaces of DNNs utilizing the concept of a "generative" (distance-based) classifier.
Sentence 2: we assume that pre-trained features can be fitted well by a class-conditional Gaussian distribution since its posterior distribution can be shown to be equivalent to the softmax classifier under Gaussian discriminant analysis (see Section 2.1 for our justification).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To date, most research in this direction has focused on approximation algorithms for finding the set with the highest probability, sometimes called the maximum a posteriori (MAP) set [Gillenwater et al., 2012, Kathuria and Deshpande, 2017, Zhang and Ou, 2016, Nikolov and Singh, 2016.
Sentence 2: the MAP objective has recently 32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montréal, Canada.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For hypothesis spaces H, H ′ , we say H essentially contains H ′ if for any h ′ ∈ H ′ , there is an h ∈ H satisfying e(h) ≤ e(h ′ ) for all D, where e(h) = E D l(y, h(x)).
Sentence 2: for any distribution D, the approximation error of H is at most the error of H ′ .
Relationship: entailment
Domain: NeurIPS
Sentence 1: If we denote by L the MAP estimate of L, then it follows that L+ is the solution to the program minimize Note the similarity with Mahoney-Orecchia regularized SDP of (2).
Sentence 2: if X = {X : Tr(X) = 1} ∩ X , then the two programs are identical except for the factor of log |X| in the optimization criterion.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Consider two sets of solutions ✓ and ✓ such that Lt1 (✓ sh , ✓ t1 ) < Lt1 ( ✓sh , ✓t1 ) and Lt2 (✓ sh , ✓ t2 ) > Lt2 ( ✓sh , ✓t2 ), for some tasks t 1 and t 2 .
Sentence 2: solution ✓ is better for task t 1 whereas ✓ is better for t 2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let G = (V, E) be a pdimensional graph and let G = (V, E) be an estimated graph.
Sentence 2: we define precision Precision is the number of correctly estimated edges divided by the total number of edges in the estimated graph; recall is the number of correctly estimated edges divided by the total number of edges in the true graph; the F 1 -score can be viewed as a weighted average of the precision and recall, where an F 1 -score reaches its best value at 1 and worst score at 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The bound O n − α 1+α log 3+αβ 2 n δ coincides with O(n − α 1+α log n) (up to a logarithmic factor) in expectation for convex and smooth loss functions [21], and largely improves the bound O(n − α 1+2α log n) in expectation for convex and non-smooth loss functions [21].
Sentence 2: if α = 1 we derive the optimal bound O(n − 1 2 log 3+β 2 n δ ) in a general case with neither Bernstein conditions on variances nor capacity assumptions on hypothesis spaces (up to a logarithmic factor).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We also provide the first general runtime proofs for dual-tree algorithms on bichromatic problems.
Sentence 2: we give the first proofs of worst-case O( ) runtimes for the following all-query problems: • All Nearest-neighbors: For all queries ∈ , find * ( ) ∈ ℛ such that * ( ) = arg min ∈ℛ ( , ).
Relationship: entailment