sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Besides using a somewhat different hide-andseek construction as mentioned earlier, it also relies on the simple but powerful observation that any b-memory online protocol is also a (b, κ, m/κ ) protocol for arbitrary κ.
Sentence 2: we only need to prove the theorem for (b, κ, m/κ ) for some κ (chosen to equal d(d − 1) in our case) to automatically get the same result for b-memory protocols.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For an optimal policy and an unbiased estimator, this parameter gradient must, by definition, map to the zero policy gradient, i.e., to ∇ π J(θ k ) = 0.
Sentence 2: an optimal policy at a fixed point is defined solely by the zero policy gradient, making the policy equal to π(0), which is the uniformly stochastic policy.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: a' ¢ HC by where 7 ' ( TS denotes the inward branching factor of point ' ( TS .
Sentence 2: a simple induction proof can be used to show that © 6 G ©' ¢ DC e . the new G -weighting scheme provides an efficient unbiased method for implementing GIS that does not use any free parameters.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In information theory, the Kraft-McMillan inequality states that, ignoring rounding issues, for every uniquely decodable code with a code length function λ, there is a probability distribution ω such that λ i = − log ω i for all symbols i = 1, . . . , n, and vice versa.
Sentence 2: the log loss can be interpreted as the code length assigned to the observed outcome.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We have where the Stein matching set F * is the linear span of {P x k(x, Equivalently, f (x) = P x φ(x) + b and φ is in the linear span of {k(x, Extending Lemma 3.1, one can readily see that the SVGD fixed points can approximate the expectation of functions that are close to F * .
Sentence 2: specifically, let F * be the neighborhood of F * , that is, the SVGD approximation can be viewed as prioritizing the functions within, or close to, F * .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Difficult to Generalize Proof of Lemma 3.2.
Sentence 2: we compute the gradient and Hessian of g(x), rg(x) = Mx kxk 2 x, r 2 g(x) = 2xx > M + kxk 2 • I .a critical point x satisfies rg(x) = Mx kxk 2 x = 0, and thus it must be an eigenvector of M and kxk 2 is the corresponding eigenvalue.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: p∈P d(p, C) and p∈P d 2 (p, C)), where d(p, C) = min c∈C d(p, c) is the minimum distance from p to a center in C. For all the 3 objectives, given a set C ⊆ P of k centers, the best set P can be derived from P by removing the z points p ∈ P with the largest d(p, C).
Sentence 2: we shall only use a set C of k centers to denote a solution to a (k, z)-center/median/means instance.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A formal way of enforcing robustness is to require that the distribution of f (X; θ f ) conditional on Z (and possibly Y ) be invariant with the nuisance parameter Z.
Sentence 2: we wish to find a function f such that for all z, z ∈ Z and all values s ∈ S of f (X; θ f ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In addition, our probabilistic model allows dealing with missing data and, in the supervised dimensionality reduction case, can incorporate additional unlabeled data one may have access to, making our CCA algorithm work in a semi-supervised setting.
Sentence 2: apart from being a general, nonparametric, fully Bayesian solution to the CCA problem, our framework can be readily applied for learning useful predictive features from labeled (or partially labeled) data in the context of learning a set of related tasks.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Critically, it does not require either re-running MaxEnt or recomputing the Hessian, since the optimization is always done around the optimal single-task parameters, ↵ (t) .
Sentence 2: eLIRL can pay a small cost to do this optimization when it is faced with performing on a previous task, but it gains potentially improved performance on that task by benefiting from up-to-date knowledge in L, as shown in our results.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The binomial CDF yields the probability of making at most τ max q mistakes total, and the (1 − γ) term above specifies the probability of recovering the true class if a maximum of this many mistakes were made.
Sentence 2: there are three probabilistic events required for the semantic output code classifier to predict a novel class and the total (joint) probability of these events is: P (there are p feature predictors with true error ≤ τ max q /p) • P (at most τ max q mistakes made | there are p feature predictors with true error ≤ τ max q /p ) • P (recovering true class | at most τ max q mistakes made) and since τ max q = G −1 q (γ), the total probability is given by: In summary, given desired error parameters (1−γ) and (1−δ) for the two classifier stages, Equation 2 provides the total probability of correctly predicting a novel class.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the following time step t, if M receives an action A t ≤ m (i.e., a feature acquisition action) from the agent and the time step t is less than the maximum length T , the sample model generates the next state S t+1 and the reward R t+1 by The next state S t+1 contains the positive and negative symptoms in S t , and also one additional feature x At (i.e., the A th t position of x) acquired by the action A t . If M receives a feature acquisition action at the last step of an episode (t = T ), the next state S t+1 and the reward R t+1 are where S ⊥ is the terminal state.
Sentence 2: the agent is punished if it asks too many questions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, in practice the dependencies between patches are not the same everywhere, and our expected value for (d i − d j ) 2 may depend on the features of the local patches.
Sentence 2: to improve accuracy we extend the model to capture the "variance" term σ 2 2rs in the denominator of the second term as a linear function of the patches i and j's relative depth features y ijs (discussed in Section 3.2).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let x π be the set of examples selected by π.
Sentence 2: the Gibbs error of a non-adaptive policy π is simply the optimal non-adaptive policy selects a set S of examples maximizing its Gibbs error, which is defined by p0 g (S In general, the Gibbs error of a distribution P is , where the summation is over elements in the support of P . The Gibbs error is a special case of the Tsallis entropy used in nonextensive statistical mechanics [4] and is known to be monotone submodular [7].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For simplicity denote N 1 = N/2.
Sentence 2: algorithm 1 run on V VT (or equivalently, just V) can be used to find the conic combination weights M ≈ M. Since M being the product of T with a diagonal matrix where T has unit column sum, we can extract T = M D−1 , where D is a diagonal matrix with Dii = M(:, i) 1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, for a fixed ⇡ p , the constraints and objective depend only on inner products between a small number of points: the 4 attack points, the class means µ (on the clean data), and the model ✓.
Sentence 2: we can solve for the optimal attack locations with a semidefinite program on a 7 ⇥ 7 matrix.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For this general definition of diversity, the marginal gain is Since h is concave, the gain h 1 + G i ∩ S − h G i ∩ S decreases as S becomes larger.
Sentence 2: the marginal gain of an item a is proportional to how rare each group G i a is in the list S. In each step of the greedy algorithm, we maximize r(y) + λd(y|S).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: More specifically, given a general bounded convex body K and a continuous function F that is monotone, smooth, and (weakly) DR-submodular we show that • All stationary points of a DR-submodular function F over K provide a 1 2 approximation to the global maximum.
Sentence 2: projected gradient methods with sufficiently small step sizes (a.k.a.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since xc is not an optimal point of f , we have f (x new ) < f (x c ).
Sentence 2: now consider the two possible cases: (i) if xnew is consistent with the given active set and sign vector, updating x := xnew strictly decreases the objective; (ii) if xnew is not consistent with the given active set and sign vector, let xd be the first zero-crossing point (where any coefficient changes its sign) on a line segment from xc to xnew , then clearly xc = xd , and f (x d ) < f (x c ) by convexity of f , thus we finally have f ( 4 the discrete line search described in Step 3 ensures a decrease in the objective value.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Silverstein and Choi have shown that the support of ρ(λ) can be determined by the intervals between extrema of z(m ρ ) [12] and this has been used to determine the signal component of a spectrum when O(N ) equal strength symmetry-breaking directions are present [13].
Sentence 2: since C in equation ( 1) only contains a finite number of symmetry-breaking directions then in the limit N → ∞ these will have zero measure as defined by H. in this limit the eigenvalue density would appear to be identical to the isotropic case.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given KN fixed activation signals z n k ∈ R T , associated to signals X n ∈ R P ×T , the D-step aims to update the K spatial patterns u k ∈ R P and K temporal patterns v k ∈ R L , by solving: where The problem ( 8) is convex in each block of variables {u k } k and {v k } k , but not jointly convex.
Sentence 2: we optimize first {u k } k , then {v k } k , using in both cases a projected gradient descent with an Armijo backtracking line-search (Wright and Nocedal, 1999) to find a good step size.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: According to the analysis in [4], the success condition (regarding L 0 ) of RPCA is where µ(L 0 ) = max{µ 1 (L 0 ), µ 2 (L 0 ), µ 3 (L 0 )} and c r > 0 is some numerical constant.
Sentence 2: rPCA will be less successful when the coherence parameters are considerably larger.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then, for any > 0 and any SPD matrix A 0, the function b The result shows that if at each round t we sample u t ⇠ U (@B 1 (0)), define an SPD matrix A t and play the point y .
Sentence 2: we can use FTRL with these smoothed gradient estimates: , at the cost of the approximation error from f t to b f t . Furthermore, the norm of these estimate gradients can be bounded.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If we assume that the advantages A(s, a) are distributed with N (A(s, a)|m A (s), σ 2 A (s)), all normalized advantage values Ā(s, a) have the same distribution.
Sentence 2: the denominator of π 1 is constant for all states and we can use the term exp(τ Ā(s, a)) ∝ π 1 (a|s) directly as weighting for the regression defined in Equation (10).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: One crucial requirement of a MP representation is that the parameters of a single primitive are easy to acquire from demonstrations.
Sentence 2: to facilitate the estimation of the parameters, we will assume a Gaussian distribution for p(w; θ) = N (w|µ w , Σ w ) over the parameters w. the distribution of the state p(y t |θ) for time step t is given by ) and, thus, we can easily evaluate the mean and the variance for any time point t. As a ProMP represents multiple ways to execute an elemental movement, we also need multiple demonstrations to learn p(w; θ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Assume that the inputs and outputs follow a "true" joint distribution q(x, y).
Sentence 2: then any transmission method with codelength L satisfies (Mackay, 2003) the gain (per data point) between the codelength L and the trivial codelength H(y) is the mutual information between inputs and outputs (Mackay, 2003).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The modified prior P would then assign mass P (h ) = P (h)/N to each noisy copy h of h. The conditional distribution P (X T | h ) is still deterministic (obtained by flipping the outcome of one of the tests).
Sentence 2: each hypothesis h i in the original problem is now associated with a set H i of hypotheses in the modified problem instance.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since the objective value is the sum of the total cost from all point-sets {G 1 , • • • , G n }, by Markov inequality, the contribution from G l should be no more than 2 times the average cost with probability at least 1/2, i.e.,  From ( 3) and triangle inequality, if replacing each q j by p l j , the objective value becomes where ( 4) follows from triangle inequality, and (5) follows from (3).
Sentence 2: the first part of the theorem is true.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, by taking η = Θ(1) and T = Θ(1/λ), the algorithm attains an -accurate solution after O(log(1/ )/λ) stochastic iterations of without-replacement sampling, and O(log(1/ )) sequential passes over the data to compute gradients of F (•).
Sentence 2: this implies that as long as 1/λ (which stands for the condition number of the problem) is smaller than O(m/ log(1/ )), the number of withoutreplacement stochastic iterations is smaller than the data size m. assuming the data is randomly shuffled, we can get a solution using only sequential data passes, without the need to reshuffle.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The following lemma gives a convenient method of computing a tight upper bound on gain( t , y ) for any supergraph t of t. For any t ⊇ t and y ∈ {±1}, the gain of t , y is bounded by µ(t) (i.e., gain( t y ) ≤ µ(t)), where µ(t) is given by where I(•) is the indicator function.
Sentence 2: if we focus on the case y = +1, then for any t ⊇ t and y ∈ {±1}, gain( t , y ) ≤ µ(t).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: During the initial training phase, to reach a minima of "width" d the weight vector w t has to travel at least a distance d, and this takes a long time -about exp(d) iterations.
Sentence 2: to reach wide ("flat") minima we need to have the highest possible diffusion rates (which do not result in numerical instability) and a large number of training iterations.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We prove that if a surrogate loss is convex, then it cannot be calibrated with respect to any of the AP, the ERR or the PD.
Sentence 2: if one of these metrics should be optimized, the price to pay for the computational advantage of convex losses is an inconsistent learning procedure, which may converge to non-optimal predictions as the number of examples increases.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The experimental procedures are described in detail in [ 21 ] .
Sentence 2: after spikesorting, spike trains from 32 neurons were binned at 2 0ms resolution, and the response of a neuron to a stimulus at time t was defined to consist of the the spike-counts in the 1 0 bins between 40ms and 240ms after t. each population response Y t is a 3 20 dimensional vector.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The advantage of considering the estimating function is the ability 1) to characterize an entire set of consistent estimators, and 2) to find the optimal estimator with the minimum parameter estimation variance from the set of estimators [23,29].
Sentence 2: by applying this to WLPS, we can characterize an entire set of estimators, which maximizes the expected reward without identifying the state transition distribution, and find the optimal estimator with the minimum estimation variance.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Similarly, define êg (γ) := E g (1/γ); this function is concave in γ as it is a pointwise infimum (indexed by x) of functions linear in γ [see e.g., §3.2.3 in 6].
Sentence 2: its derivative ê g (γ) = 1 2 P g 1/γ (x − γ −1 y) − x 2 = q g (1/γ), is a decreasing function of γ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From the definition of H − , we can choose f : For Part 3, first note that, for any φ, ψ is continuous on [0, 1] and ψ(0) = 0 by Lemma 4, part 6, and hence θ i → 0 implies ψ(θ i ) → 0.
Sentence 2: we can replace condition (3b) by (3b') For any sequence To see that (3a) implies (3b'), let φ be classification-calibrated, and let (θ i ) be a sequence that does not converge to 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the next two subsections, we bound Let A := {x : 0 ≤ x 1 ≤ h and h ≤ x 2 ≤ 1 − h}.
Sentence 2: we have For x ∈ A, we have for x ∈ A we have Since p ∈ Σ κ (2, L) and 0 < x 1 ≤ h, we have For any x ∈ A, we can bound the bias term where the last inequality follows from the fact that ∂p(x) ∂x1 , ∂p(x) ∂x2 ≤ Lh, by the Hölder condition and the assumption that the density p(x) has vanishing partial derivatives on the boundary points.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: While it is often infeasible to store the original matrix A in the memory, it is also often infeasible to materialize all pairwise distances in A.
Sentence 2: in applications such as online learning, databases, search engines, online recommendation systems, and online market-basket analysis, it is often more efficient if we store B ∈ R n×k in the memory and estimate any pairwise distance in A on the fly only when it is necessary.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Notice that from Lemma 3.5 we know z j+1 z j 2 W for all j 2 [k].
Sentence 2: thus for all w 2 W 1 \ W k+1 there exists i 2 we must have h 2 + (W \ B(0, r)) ✓ W k+1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, M exhibits some nice properties for any Θ: M 0, M I, tr(M ) = tr((ΘΘ ) † (ΘΘ )) = rank(ΘΘ ) = rank(Θ).
Sentence 2: (10) Here I is the identity matrix, and we also use M 0 to encode M = M . tr(M ) provides a convenient proxy for controlling the rank of the latent representation, i.e.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, we show that for multilabel metrics constructed as instance-, micro-and macroaverages, the population optimal classifier can be decomposed into binary classifiers based on the marginal instance-conditional distribution of each label, with a weak association between labels via the threshold.
Sentence 2: our analysis extends the state of the art from a few known multilabel classification metrics such as Hamming loss, to a general framework applicable to many of the classification metrics in common use.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Hence the input vectors used to train the next layer would be x .
Sentence 2: if the computational cost of a single inner product x i β k (e.g., logistic regression) is one, then that for T hidden layers would be about T t=1 (K t−1 + K t + 1)K t+1 /(V + 1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consequently, when applying the algorithm to the variational reformulation of the problem of interest (3), we are able to get an ǫ-optimal solution within at most O(1/ǫ 2 ) LMO calls.
Sentence 2: semi-Proximal Mirror-Prox generalizes previously proposed approaches and improves upon them in special cases of problem (3); see Appendix D.2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is because if u ∈ B � (i 1 , r 1 ) ∩ B � (i 2 , r 2 ), then assuming r 1 ≥ r 2 (w.l.o.g.
Sentence 2: ), triangle inequality implies that d(i 1 , i 2 ) < r 1 (the strict inequality is because we defined the balls to be 'open'); in an integral solution, we will set at most one of x i1r1 and x i2r2 to 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Each dataset was divided into 10% for testing and 90% for training.
Sentence 2: the input to all of the algorithms tested was, for each point in the training set: (1) a vector in ℜ n (where n is different for each set) and (2) a value from 1 to 5 denoting the rank of the group to which it belongs.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Recall the definition that v over all unit-length vectors, it must lie in the range of because otherwise the component outside the range will not contribute.
Sentence 2: we can choose v Consider the following decomposition: The second and the third term can be bounded by Wely's inequality (Lemma C.4) and basic properties of � A (Lemma A.3).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: = inf When Z is continuous, ρ CVaR (Z; α) is well-known to be the mean of the α-tail distribution of Z, E [ Z| Z > q α ], where q α is a (1 − α)-quantile of Z.
Sentence 2: selecting a small α makes CVaR particularly sensitive to rare, but very high costs.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: samples of a variable distributed according to Bernoulli(p i ), an unbiased estimator for p k i is their product, namely the estimator which is 1 if all the tosses come up heads, and otherwise is 0.
Sentence 2: if we average over all t k subsets of size k, and then average over the population, we still derive an unbiased estimator.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The method is based on minimizing the expected reconstruction error E[g(h(θ)) − θ] of the task parameter θ onto the manifold M. Here h is the mapping from the manifold to the lower dimensional Euclidean space and g is the mapping from the lower dimensional Euclidean space to the manifold.
Sentence 2: the composition g • h maps a point belonging to manifold to the manifold, using the mapping to the Euclidean space as an intermediate step.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The linear SVM solves the following optimization problem [3,4,11]: where λ ∈ R + is the regularization parameter and l hinge the hinge loss function defined as: With a linear SVM, the feature space is the same as the input space.
Sentence 2: when the input features are the voxels of a 3D image, each element of w opt = (w opt v ) v∈V also corresponds to a voxel.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The measurement at time will be denoted ¡ £¢ , and ¤ ¢ denotes the control asserted in the time interval ¥ ¦ ¨ § © .
Sentence 2: the data is given by Following common notation in the controls literature, we use the subscript ¢ to refer to an event at time and the superscript ¢ to denote all events leading up to time .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then f h (x, y) > λ + δ/2 > f h (x, ŷ) but y ∈ [ŷ − τ, ŷ + τ ], a contradiction since ŷ is the maximizer of the KDE in τ radius neighborhood when restricted to X = x.
Sentence 2: there exists y 0 ∈ M x such that y 0 ∈ [ŷ − τ, ŷ + τ ].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Typically one step of gradient descent on data from the replay buffer is taken per action in the environment, as (with the exception of [Barth-Maron et al., 2018]) a greater reliance on replay data leads to unstable performance.
Sentence 2: we propose that the replay buffer may frequently contain information that could significantly improve the policy of an agent but never be fully integrated into the decision making of an agent.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Although different sampling techniques could be used in principle [15], our goal is to model correlations between each variable X uv and some neighboring region G uv in G. Such a neighborhood should be large enough to make ∆λ 2 (u, v, G uv ) sufficiently informative with respect to the overall network, but also small enough to keep the spectral decomposition of G uv computationally tractable.
Sentence 2: in order to sample G uv , we propose to draw V uv by performing k 'snowball waves' on G [16], using u and v as seeds, and then setting E uv to be the edge set induced by V uv in G (see Algorithm 1 for the details).
Relationship: reasoning