sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: We can store X t as a decomposition into at most rank(X t ) ≤ kt rank-1 components.
Sentence 2: 6 for If we always use the first running time in Theorem 4.3, then Fact 4.4 implies: which roughly equals k-times the 1-SVD running time Õ nnz(∇) ∇ 2 /ε) of FW and Garber [6].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Running the observational learning algorithms like PC/IC after an intervention on a set S of variables, we can learn the new skeleton after the intervention, which allows us to identify the immediate children and immediate parents of the intervened variables.
Sentence 2: if we perform a randomized experiment on a set S of vertices in the causal graph, we can learn the direction of all the edges cut between S and V \ S. This approach has been heavily used in the literature [13,10,31].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Because H is closed under union and finite, H 0 must be in H. If ∃z ∈ H 0 ∩ Z, then z ∈ h ∩ Z for some h ∈ M(Z).
Sentence 2: this is a contradiction, because all h ∈ M(Z) are consistent with Z. H 0 is consistent with Z, and is in M(Z).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Designating by an overbar the average over randomly chosen states x and x , the mean C and variance δC 2 of the inner product are This proves that in the limit of large n and M , different columns of the random perceptron measurement matrix have inner products that concentrate around 0.
Sentence 2: the matrix of inner products is thus orthonormal almost surely, Â Â → I. with enough measurements the random perceptrons asymptotically provide an isometry.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Unlike in the convex case, in the non-convex case one can not expect a gradient-based algorithm to converge to the global minimum if only smoothness is assumed.
Sentence 2: instead of measuring functionvalue suboptimality Ef (x) − inf x f (x) as in the convex case, convergence is generally measured in terms of the squared norm of the gradient; i.e., E ∇f (x) 2 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We consider several types of tv-embedding learning task trained on unlabeled data; e.g., one task is to predict the presence of the concepts relevant to the intended task (e.g., 'desire to recommend the product') in the context, and we indirectly use labeled data to set up this task.
Sentence 2: we seek to learn tv-embeddings useful specifically for the task of interest.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For BIC we use the G-Wishart posterior mode ΩG as the plug-in estimate, since the MLE is undefined for n < d. But we use a vague and proper prior (δ 0 = 3).
Sentence 2: all three approximations will require finding the mode of a G-Wishart (for the posterior and/or the prior).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [30], such that where p T (x k ) and p T (x k , x l ) are the marginals of p T (x).
Sentence 2: after running TRW-BP, we can calculate p T (x) via Because TRW provides a convex upper bound, it is often well-suited to the inner loop of learning algorithms [e.g., 28].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For α to be closed, it must satisfy the closure condition 2π 0 e jθ(s) ds = 0.
Sentence 2: we represent curves by direction functions satisfying the average-π and closure conditions; we call this space of direction functions D. Summarizing, D is the subspace of L 2 consisting of all (direction) functions satisfying the constraints It is still possible to have multiple continuous functions in D representing the same shape.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As to the definition of B −1 z , the feasible path of each segment must start with several blanks, followed by the element corresponding to that segment more than once.
Sentence 2: we can find the maximum of entropy for every segment According to Equation 2 and the definition of segmentation, the entropy of feasible paths satisfying a specific segmentation sequence can be disassembled into the summation of the entropy of segments.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The two endpoints of any edge in the graph form a vertex cover of size k = 2, but C could conceivably be any edge.
Sentence 2: under these constraints, the only set guaranteed to contain C is the entire node set V . In this negative example, once we put v 1 into a 2-node vertex cover, the other node can be arbitrary, since it is superfluous.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since Theorem 3.4 ensures that x n visits B(x * , ε) infinitively often (almost surely), the hypothesis is guaranteed to be true.
Sentence 2: this leads to the following claim: ∀ε > 0, ∃τ 0 (a positive integer), such that: Figure 3: SMD run on the objective function of Fig.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We observe in passing, that for any K ∈ K 1 , and for any independent set I in the graph, taking since for each term in the second sum, either (i, j) is an edge, in which case either α i or α j is zero, or (i, j) is a non-edge in which case K ij ≤ 0.
Sentence 2: like ϑ(G), the Delsarte version ϑ 1 (G) is also an upper bound on the stability or independence number α(G).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Theorem 4 and Lemma 12 give an O( √ T log T )-upper bound of the regret of function optimization in Algorithm 1 under the same conditions as Theorem 4.
Sentence 2: given the convexity of f , the average of the chosen actions of any dueling bandit algorithm āT : if an optimization algorithm has a sub-linear regret bound, the above online-to-batch conversion guarantees convergence to the optimal point.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: F P R = P (C = 1|T ) and 1 − F N R = P (C = 2|T ).
Sentence 2: we have: Similarly, when log R(T ) reaches the target absent threshold (Θ 1 ), we have: given desired FPR and FNR, we can analytically compute the upper and lower thresholds for the Full Bayesian model using Eqn.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In PSRL, each agent k samples an MDP M k,0 from the common prior at time t k,0 and computes the optimal policy π k,0 (•) with respect to M k,0 , which does not change throughout the agent's interaction with the environment.
Sentence 2: the PSRL agents do not adapt to the new information in real-time.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Equivalence with the ECOC decision rule thus follows from the fact that 3 is the Hamming distance between the appropriate row of the coding matrix and the concatenation of the bits returned from the binary classifiers.
Sentence 2: with the appropriate definitions of W 2 and o , the conditional model on the ranking poset is a probabilistic formulation of ECOC that yields the same classification decisions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As discussed in Section 1.1, these methods can be viewed as different ways to optimize (5).
Sentence 2: in addition to the normalized mean square error (nMSE) [19] in prediction, we report KUKA1 -Variational Lower Bound (10 the performance in the variational lower bound (VLB) (5), which also captures the quality of the predictive variance and hyper-parameter learning.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For Gaussian designs, which we also consider, their results guarantee that for n = O (s log d), where s is the sparsity index of the regressor w * .
Sentence 2: note that for k = Θ(n), the right hand side behaves as Ω σ √ log n . the result is unable to ensure We have excluded some classical approaches to the RLSR problem from the table such as [18,1,2] which use the Least Median of Squares (LMS) and Least Trimmed Squares (LTS) methods that guaranteed consistency but may require an exponential running time.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [1] We consider a setting in which an online learning algorithm (also called an adaptive control algorithm) is used to design a controller.
Sentence 2: on each time step t, an online algorithm may (based on the observed states and action sequence so far) propose some new gain matrix K t . If we follow the learning algorithm's recommendation, then we will start choosing controls according to u = K t x.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The monotonicity of ≤ allows us to detect suboptimal strategies during variable elimination.
Sentence 2: to illustrate this, consider the variable elimination scheme in Figure 2 for two different strategies s and s , and let ψ s 1 , ψ s 2 , ψ s 3 be the valuations produced in the propagation step for strategy s and we can abort variable elimination for s after the second iteration.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In addition, this method causes less noise during training, and the noise that is caused is more gentle as the law of large numbers ensures that µ[t] and µ[∇ t] are approximately normally distributed.
Sentence 2: the added noise has much lighter tails than the highly kurtotic noise caused by the minibatch estimate of the variance used in full batch normalization.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We have proved that this can be accomplished in hardware in the address domain by presenting the network with stimulus patterns containing a set of correlated inputs and a set of uncorrelated inputs: neurons x 1 . . . x 17 are all stimulated independently with a probability of 0.05 per unit of time, while neurons x 18 . . . x 20 have the same likelihood of stimulation but are always activated together.
Sentence 2: over a sufficiently long period of time each neuron in the input layer will receive the same amount of activation, but the correlated group will fire synchronous spikes more frequently than any other combination of neurons.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The safe set after n such expansions can be found by repeatedly applying the operator in (4): Ultimately, the size of the safe set is bounded by surrounding unsafe states or the number of states in S. As a result, the biggest set that any algorithm may classify as safe without visiting unsafe states is given by taking the limit, R ✏ (S) = lim n!1 R n ✏ (S).
Sentence 2: given a tolerance level ✏ and an initial safe seed set S 0 , R ✏ (S 0 ) is the set of states that any algorithm may hope to classify as safe.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For k ∈ {1, . . . , n}, define the U-statistics where the sum is over all S ⊂ {1, . . . , n} such that |S| = k. By Hoeffding's strong law of large numbers for U-statistics [19], for any k ∈ {1, 2, . . . }.
Sentence 2: using Equations 3.1 and 3.2 we have that for any K ∈ {1, 2, . . . } and any n > K, R 2 (X 1:n ) = n 3/2 where H K is the K th harmonic number, and the last inequality follows from the standard bounds [20] on harmonic numbers: log K < H K ≤ log K + 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, since P * k X i , i = k, will be small and D * k is not trained to reconstruct X i , the residual F . In the testing phase, if the query sample y is from class k, its projective coding vector by P * k (i.e., P * k y ) will be more likely to be significant, while its projective coding vectors by P * i , i = k, tend to be small.
Sentence 2: the reconstruction residual y − D * k P * k y 2 2 tends to be much smaller than the residuals y − D * i P * i y 2 2 , i = k. Let us use the Extended YaleB face dataset [22] to illustrate this.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By measure concentration, it can be shown that holds with high probability whenever m ≥ Ω (n poly log n).
Sentence 2: to show iterative contraction of both methods, based on ( 12) and ( 13), it is sufficient to show that for some constant η ∈ (0, 1), where θ = arg min θ∈[0,2π) z − xe iθ such that e iθ = x * z/ |x * z|.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then, the probability of deleting the solution with the best f value in Q when implementing lines 8-15 is at most Taking the logarithm, we get where the inequality is since log j−2δ j+1 is increasing with j, and the last equality is since the derivative of log (j−2δ) j−2δ (j+1) j+1 with respect to j is log j−2δ j+1 .
Sentence 2: we have where the last inequality is by 0 < 1 − 2δ ≤ 1 and (1 − 2δ) 1−2δ ≥ e −1/e .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It turns out that each w (k) (η) forms a univariate polynomial whose degree is at most k. Furthermore, since f η (w) are L-smooth µ-strongly convex for any η ∈ [µ, L], standard convergence analysis for GD (e.g., [13], Theorem 2.1.14) guarantees that |w , where κ denotes the condition number.
Sentence 2: substituting Equation (6) for w (k)  we see that the faster the convergence rate of a given optimization algorithm is, the better the induced sequence of polynomials (w (k) (η)) k≥0 approximate 1/η w.r.t.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For the minimization problem in Theorem 1, the gradient descent algorithm with random initialization can converge to a global optimum only if the step size satisfies where ⇢(R) is the largest singular value of R. Remark 2.
Sentence 2: corollary 1 shows that, unlike the optimization of the ordinary least squares problem, the step size required for the convergence of the algorithm depends on the parameter to be estimated, R. estimating linear mappings with larger singular values requires the use of a smaller step size.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: There, the occupancy of regulatory binding sites corresponds to the state of neurons, the weights are set by the cooperative interaction among transcription factors, and the thresholds are the effective dissociation constants at a binding site.
Sentence 2: implementing a general N -unit neural network requires only O(N ) biochemical species, but up to O(N 2 ) significant binding interactions must be encoded in the molecular sequences.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The average numbers of agents appearing simultaneously in different steps positioned along the directions k + ±k − = constant and k ± = constant are, respectively, ∆/8 and ∆/4, and 0 for other directions.
Sentence 2: the average number of agents common in the pairs of steps {PQ, QQ 1 }, {QQ k , QP}, {QP, QR}, {PQ, QP} are ∆/8, ∆/8, ∆/8 and ∆/4 respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If ρ and V 0 are convex functions over ∆, then the value function V n of the belief MDP is convex over ∆ at any time step n. [Proof in [17,Appendix]] This last theorem is based on ρ(b, a) being a convex function over b, which is a natural property for uncertainty (or information) measures, because the objective is to avoid belief distributions that do not give much information on which state the system is in, and to assign higher rewards to those beliefs that give higher probabilities of being in a specific state.
Sentence 2: a reward function meant to reduce the uncertainty must provide high payloads near the corners of the simplex, and low payloads near its center.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Define ρ n := min i∈S |θ * S |.
Sentence 2: from equation (14b), we have in order to establish that | θ s,λ i | > 0 for all i ∈ S, and moreover that sign( θ s,λ S ) = sign(θ * S ), it suffices to show that Using our eigenvalue bounds, we have In fact, the righthand side tends to zero from our earlier results on W and R, and the assumption that λ n d → 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The spatial size of the final output, outputting the heatmaps, is set to 16×16.
Sentence 2: due to downsampling, for a network with n − 3, n ≥ 4 blocks, the resolution of the input image is H×W = 2 n ×2 n , resulting in 16×16×(32 • 2 n−3 ) tensor.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider a product a i for i  k. In order for a i to be chosen at position s after a k+1 is inserted, one of the following events has to occur: i) a i was already chosen at position s before a k+1 is inserted, and a k+1 is inserted at a position `> s, or ii) a i was chosen at position s 1, and a k+1 is inserted at a position ` s 1.
Sentence 2: consequently, we have for all i  k Consider a product a i with i  k. This product is chosen at position s only if it was already chosen at position s and a k+1 is inserted at a position `> s. for all i  k, ⇡(i, s, k + 1) = ( 1k+1,s ) • ⇡(i, s, k).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The first K − 1 stick-breaking proportions are drawn from a Beta distribution with parameters (1, α), but we assume that the final stick-breaking proportion is unity (thus implying β * is non-zero only from 1 . . . K).
Sentence 2: we only optimize the first K − 1 positions and implicitly take This constrained optimization is performed using the barrier method [17].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This extension can be defined for any set function, however it is convex if and only if the set function is submodular [17].
Sentence 2: we call a permutation ⇡ over [n] consistent 1 1 the Lovász extension can also be written as where ⇡ is a permutation consistent with x and F (;) = 0 by assumption.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given quadratic models of V (x k+1 , k + 1), F (x k , u k ) and r(x k , u k ), we can approximate the unmaximised value function, or Q-function, as a quadratic model around the present state-action pair (x k , u k ): Where the coefficients Q are computed by equating coefficients of similar powers in the secondorder expansion of ( 2) Once the local model of Q is obtained, the maximizing δu is solved for and plugged back into (3) to obtain a quadratic approximation of V k : This quadratic model can now serve to propagate the approximation to V k−1 .
Sentence 2: equations ( 4), ( 5) and ( 6) iterate in the backward sweep, computing a local model of the Value function along with a modification to the policy in the form of an open-loop term −Q −1 uu Q u and a feedback term −Q −1 uu Q ux δx, essentially solving a local linear-quadratic problem in each step.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As it turns out, the latter approach can be naturally formulated as a constrained matrix completion problem.
Sentence 2: when applying the spectral method, the (approximate) values of the target on B are arranged in a matrix H. the main difference between the two settings can be restated as follows: when learning a weighted automaton representing a distribution, unknown entries of H can be filled in with zeros, while in the general setting there is a priori no straightforward method to fill in the missing values.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The total cost T (X L , X U ) associated with the data is the total misclassification risk, plus the cost of obtaining all labeled data thus far: where the function C(•) returns the cost of obtaining an annotation for its input, and will be defined in more detail below.
Sentence 2: to measure the expected utility of obtaining any particular new annotation, we want to predict the change in total cost that would result from its addition to X L . the value of obtaining an annotation for input z is: where z (t) denotes that the input z has been merged into the labeled set with its true label t, and X U z denotes that it has been removed from the set of unlabeled data.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The basic inequality (1) for that update becomes: As in the proof of Theorem 2 of this paper, the Generalized Pythagorean theorem applies and dropping one term we get the following inequality: Adding this to the previous inequality we get: By summing over t, multiplying by n − k, and dividing by 1 − exp(−η), the bound follows.
Sentence 2: , then this is further bounded by k log n k . the r.h.s.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This drastically reduces sample complexity and vastly increases generalization capabilities of the classifier since one only need to observe one test sample to achieve invariance Lemma 2.6 also helps us in saving computation, allowing us to apply the computationally expensive Ψ (group integration) operation only once on he classifier and not the test sample.
Sentence 2: the kernel in the Invariant SVM formulation can be replaced by the form k Ψ (x, y) = φ(x), Ψ H φ(y) .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: • Y has necessarily one of the following form: • for any y ∈ Y, from the exp-concavity assumption, the function y : y → (y, y ) is convex on the interval on which it is finite 1 .
Sentence 2: the risk R is also a convex function (on the convex set of prediction functions for which it is finite).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The first term represents the amount invested in security by player , and is either 0 (if The second term is the expected cost to due to internal or direct risk of the bad event, and is either Ô Ä (which is the expected cost of internally generated bad events in the case ¼ ), or is 0 (in the case of investment, ½ ).
Sentence 2: there is a natural tension between the first two terms: players can either invest in security, which costs money but reduces risk, or gamble by not investing.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that 2(e ε + 1) 2 ≤ 32 for ε ≤ 1, and ε ≤ 1 is a common regime of interest in differential privacy.
Sentence 2: we can always use the simple binary mechanism in this regime and the resulting divergence is at most a constant factor away from the optimal one.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Taking the previous conditions into consideration, we want to find a mask M that is smooth and performs well at both SSR and SDR; examples of such masks can be seen in figure 1.
Sentence 2: more formally, given class c of interest, and an input image X, to find a saliency map M for class c, our objective function L is given by: (4) where f c is a softmax probability of the class c of the black box image classifier and TV(M ) is the total variation of the mask defined simply as: AV(M ) is the average of the mask elements, taking value between 0 and 1, and i are regularisers.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the multivariate Gaussian case, the entropy can be computed in closed form: For the contextual case, our regret bound comes also in terms of the quantity γ T , redefined so that the information gain I(y A ; f ) now depends on the observations y A = [y(x)] x∈A of the joint contextaction pairs x = (s, z), and f : S × Z → R is the payoff function over the context-action space.
Sentence 2: the kernel matrix x ∈A is defined over context-action pairs.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We first fix a value z \i for Z \i , and a value of R such that R 1 , .
Sentence 2: . . , R k all lie in Σ, and consider the ratio of probabilities: Observe that this ratio of probabilities is equal to: which is in turn equal to: Observe that from the stability condition, the ratio of the probabilities is at most the ratio Pr( , which is at most e α2 by properties of the exponential distribution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The stochastic SLQC optimization Setup: Consider the problem of minimizing a function f : R d → R, and assume there exists a distribution over functions D, such that: We assume that we may access f by randomly sampling minibatches of size b, and querying the gradients of these minibatches.
Sentence 2: upon querying a point x t ∈ R d , a random minibatch {ψ i } b i=1 ∼ D b is sampled, and we receive ∇f t (x t ), where f t (x) = 1 b b i=1 ψ i (x).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: But one could imagine requiring something less: perhaps changing distances this way should be allowed to create additional sub-structure, leading to a new partition in which each cluster is a subset of one of the original clusters.
Sentence 2: we can define Refinement-Consistency, a relaxation of Consistency, to require that if d is an f (d)-transformation of d, then f (d ) should be a refinement of f (d).
Relationship: reasoning