sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Repeating this procedure results in a chain of skills leading from any state in which the agent may start to the task's goal region as depicted in Figure 1 Note that although the options lie on a chain, the decision to execute each option is part of the agent's overall learning problem.
Sentence 2: they may not necessarily be executed sequentially; in particular, if an agent has learned a better policy for some parts of the chain, it may learn to skip some options.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First, it receives and interprets the current goal given by the environment, and second, it encapsulates and memorises the features and structure of a single city region.
Sentence 2: rather than using a map, or an external memory, we propose an architecture with two recurrent pathways that can effectively address a challenging navigation task in a single city as well as transfer to new cities or regions by training only a new locale-specific pathway.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A symmetric observation system is equivalent to G t being an undirected graph or, more precisely, to a directed graph having, for every pair of nodes i� j ∈ V , either no arcs or length-two directed cycles.
Sentence 2: from the point of view of the symmetry of the observation system, we also distinguish between the directed case (G t is a general directed graph) and the symmetric case (G t is an undirected graph for all t).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, the previous variables are closely related to the new ones: when β = 0 the constraint associated with b disappears, and we recover q(x|µ); that is, p(x|µ, 0) = q(x|µ).
Sentence 2: λ i (µ, 0) = λ i (µ), and Z(µ, 0) = Z(µ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For the GGM this procedure reduces to PCA and the noise threshold depends on the eigenvalues of the covariance matrix, as expected.
Sentence 2: we show that for simple graph structures, such as hierarchical or lattice graphs, as well as the random Erdös-Rényi graph, the noise threshold can possibly grow in the network size p. leveraging the structure of network interactions can enable extraction of high-dimensional patterns embedded in heavy noise.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 8) and corresponds to the projection onto with a(x) to the solution of the standard vector-valued kernel ridge regression without constraints (we denoted Y ∈ R n×T the matrix with rows y i ).
Sentence 2: nonlinear VVR consists in: 1) computing the unconstrained kernel ridge regression estimator b(x), 2) normalizing it by a(x) and 3) projecting it onto C. The example above shows that for specific loss functions the estimation of f (x) can be significantly simplified.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For a vector y, let y m:n = (y m , y m+1 , ..., y n ) denote the sequence of entries from m to n. Initialize λ (k) , r (k) = (0, 0) at the beginning of first period.
Sentence 2: then, at each period t ≥ 1, the pair λ (k) , r (k) can be updated recursively as follows: where overall average payoff function rt (x) can be expressed as a sum of average payoff functions of individual goods.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, according to the definition of information gain, those instances with larger gradients 1 (i.e., under-trained instances) will contribute more to the information gain.
Sentence 2: when down sampling the data instances, in order to retain the accuracy of information gain estimation, we should better keep those instances with large gradients (e.g., larger than a pre-defined threshold, or among the top percentiles), and only randomly drop those instances with small gradients.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: O(1/ǫ 2 ), in the number of calls to LMO; see [14] for the optimal complexity bounds for general non-smooth optimization with LMO.
Sentence 2: 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.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Equivalently, (Vaf(a,x)}x -.J y Ef,:l Vaf (a,yi) = 0 or Because this equation holds for all a, we obtain px(a) = .J y Ef':l 8(yi -a), which is the ML solution.
Sentence 2: the sample-based method can be viewed as an approximation to ML which gets more exact as the number of samples goes to infinity.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Conditional Entropy (CE) is an information-theoretic measure that quantifies the uncertainty of random variable x when conditioned on z (or the other way around), under joint distribution π(x, z): The uncertainty of x given z is linked with Intuitively, by controlling the uncertainty of q φ (z|x) and p θ (z|x), we can restrict the solutions of the ALI objective to joint distributions whose mappings result in better reconstruction ability.
Sentence 2: we propose to use the CE in (4), denoted as L π CE (θ, φ) = H π (x|z) or H π (z|x) (depending on the task; see below), as a regularization term in our framework, termed ALI with Conditional Entropy (ALICE), and defined as the following minimax objective: L π CE (θ, φ) is dependent on the underlying distributions for the random variables, parametrized by (θ, φ), as made clearer below.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Now, since 7 © acts transitively on W , for all ¥ $ ¥ 21 ¢ W there is 5 ¢ F7 © such that ¥ 45 i@ ¥ 61 .
Sentence 2: we thus have that 7 £ we can write 7 £ I U¥ " w@ 7 £ I " since the normalizing constant for ¥ ¢ W does not in fact depend on ¥ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Unfortunately, in our model, the coupling of the conditional and marginal distribution in the likelihood makes it difficult to derive analytical formulas for the posterior distribution.
Sentence 2: note that if (X, Y ) ∼ N (µ, Σ) we introduce an approximation distribution for the datasets which decouples the marginal and conditional distributions as follows: where W is a p x × p y projection matrix and Λ is the precision matrix.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The key idea is to apply the pseudo-inverse expectation formula for standard volume sampling (see e.g., [11]) first on the subsampled estimator w ⇤ ⇡ , and then again on the full estimator w ⇤ .
Sentence 2: in the first step, this formula states: where [k] d def = {S ✓ {1, . . . , k} : |S| = d} and ⇡ S denotes a subsequence of ⇡ indexed by the elements of set S. Note that since S is of size d, we can decompose the determinant: Whenever this determinant is non-zero, w ⇤ ⇡ S is the exact solution of a system of d linear equations: the rescaling of each equation by 1 p q⇡ i cancels out, and we can simply write w ⇤ ⇡ S = (X ⇡ S ) + y ⇡ S . (Note that this is not the case for sets larger than d whenever the optimum solution incurs positive loss.)
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that the side information matrix W = (w i,j ) is provided where the w i,j s are independent random variables.
Sentence 2: now assume the scenario when we use an algorithm ALG to assign x to one of the k clusters, V u , u = 1, . . . , k. given x, ALG takes as input the random variables w i,x s where i ∈ t V t , makes some queries involving x and outputs a cluster index, which is an assignment for x.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: But a was taken at time t, so it must have been in Γ 2 t which means it maximized the index.
Sentence 2: for all optimal actions a * ∈ O(i; P, A), we have, on the event A t (i, a; ), U t (i, a * ) ≤ U t (i, a) < L * (i; P, A) − 2 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The prior distribution over a n is a product of Bernoulli distributions with parameters ρ n = {ρ(u m , u n , θ)} N m=1 .
Sentence 2: the conditional distribution is proportional to the ratio of the prior and posterior partition functions, we perform a joint update of a n and w n by collapsing out the weights to directly sample the binary entries of a n . We iterate over each entry, a m→n , and sample it from its conditional distribution given {a m →n } m =m .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: An imminent question now is how to make use of a combination of observational and interventional data (of the order of 100,000 cells / samples on 20,000 genes / variables) in the causal discovery process.
Sentence 2: the development of efficient and consistent algorithms using both observational and interventional data that are implementable within genomics is now a crucial goal.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By this, we mean that n ∀i ∈ {1, . . . , n}, i.e., the total weight assigned to each boolean complement pair of classifiers is fixed to 1/n.
Sentence 2: let For any quasi uniform Q, the output B Q (x) of the majority vote on any example x is given by the set of majority votes B Q over quasi uniform posteriors is isomorphic to the set of linear separators with real weights.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Suppose a two-armed Bernoulli bandit has mean rewards 1 /2 + ε and 1 /2 respectively and the second arm is chosen at most F < ∞ times with probability at least 1 − δ, then one can easily show that in an alternative bandit with mean rewards 1 /2 + ε and 1 /2 + 2ε there is a non-zero probability that the second arm is played finitely often and in this bandit the expected regret will be linear.
Sentence 2: sub-linear expected regret is only possible if each arm is pulled infinitely often almost surely.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For a state q i , and given observation label l, we calculate the average number of visits (at q i ) whose corresponding label is l, as E [S t = q i | l t = l, O, λ] = 1≤t≤T,lt=l γ t (i).
Sentence 2: l qi is chosen to be an l that maximized this quantity.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [2] For a closed convex set C and a point z 0 ∈ C, the normal cone to C at z 0 is Theorem 7 implies that θ * (λ) is known with λ ≥ λ max .
Sentence 2: we can estimate θ * (λ) in terms of a known θ * (λ 0 ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Take w x for example, we remove the irrelevant terms and get the following objective: where It is easy to realize that the objective function (2) can be expressed as a difference of two convex functions in different cases.
Sentence 2: we can use CCCP to solve the nonconvex optimization problem iteratively with each iteration minimizing a convex upper bound of the original objective function.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then for some constants C, c > 0.
Sentence 2: recall that we have assumed that m > n, so the probability converges to 1 with increasing m or n. While Ã is not positive definite, the triangle inequality implies a bound on the positive definite projection Ã+ (11): similarly to the earlier result for B(t), the estimator (12) formed by substituting the positive semidefinite ρ( Ã+ ) into the GLasso objective enjoys the following error bound (Zhou et al., 2011).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In [8] it is shown that dim(F, P ) = O(kd).
Sentence 2: by Lemma 2, when |S| ≥ O 1 2 (kd + log 1 δ ) , the weighted cost of S ∪ B approximates the k-median cost of P for any set of centers, then (S ∪ B, w) becomes an -coreset for P . The total communication cost is bounded by O(mn), since even in the most general case that every node only knows its neighbors, we can broadcast the local costs with O(mn) communication (see Algorithm 3).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For details on our asymptotic notation and proofs for this section, see Appendix D. Lemma 5.1.
Sentence 2: the number of vertices and edges for both the multi-and binary graphs satisfy we can examine the asymptotic behavior of the random numbers of edges and vertices by examining the asymptotic behavior of their expectations, which are provided by Lemma 5.2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Indeed, if we marginalize the latent variable z, we get that the probability of y given x is a linear combination of softmax functions of linear functions of the input variables x.
Sentence 2: the only difference with a two-layer neural network with softmax functions for the last layer is the fact that our last layer considers linear parameterization in the mean parameters rather than in the natural parameters of the multinomial variable.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First note that in the statements of Theorem 2.5.A and Theorem 2.5.B the only parameters that are assumed to grow are p and whichever other parameter is implied to grow because of ( 2) and (4).
Sentence 2: importantly, n does not necessarily grow to infinity, if for example N, 1 σ grow appropriately with p. That means that Theorem 2.5.A and Theorem 2.5.B imply non-trivial guarantees for arbitrary sample size n. The proposition below shows that if σ is at most exponential in −(1 + ) (p+2n) 2 2n + (2 + p n ) log (RQ) for some > 0, then for appropriately chosen truncation level N the LBR algorithm recovers exactly the vector β * with n samples.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Inputs: • A concept class C and a probability measure ν defined over C. The algorithm: QBC assumes a prior ν over the class of linear classifiers.
Sentence 2: the sample S induces a posterior over the class of linear classifiers which is the restriction of ν to V . the probability that QBC will query for the label of an instance x is exactly where ν|V is the restriction of ν to V . From (2) we see that there is no need to explicitly select two random hypotheses.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 4), which can be performed efficiently due to the fact that sides of the same length along one dimension (horizontal or vertical) in the coordinates of face i will also be constant along the corresponding line when projected to the coordinate system of face j.
Sentence 2: computing the side length ratios of two faces is not quadratic in the number of pixels but only in the number of horizontal or vertical lines.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For such graphs, the input to F would not contain all z i,j pairs but rather only features corresponding to valid edges of the graph, and we are only interested in invariances that preserve the graph structure, namely, the automorphisms of the graph.
Sentence 2: the desired invariance is that σ(F(z)) = F(σ(z)), where σ is not an arbitrary permutation but an automorphism.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, for any x ∈ X , for which g(x) = 1, all the hypotheses in G agree, and in particular h * and ĥ agree.
Sentence 2: applying Theorem 5.4 and the union bound we therefore know that with probability of at least 1−δ, Hanneke introduced, in his original work [5], an alternative definition of the disagreement coefficient θ, for which the supermum in ( 4) is taken with respect to any fixed ϵ > 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Both variational-EM and the wake-sleep algorithm work well when Q closely approximates P , but may fail to converge to a maximum of the likelihood if there is a significant mismatch.
Sentence 2: the efficiency of these methods depends on choosing a good approximating distribution Q(Z|Y, φ) -one that closely matches P (Z|Y, θ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We describe how to use the shadow Dirichlet to model a random pmf that has bounded variation such that |θ k − θ l | ≤ k,l for any k, ∈ {1, 2, . . . , d} and k,l > 0.
Sentence 2: to construct specified bounds on the variation, we first analyze the variation for a given M . For any d × d left stochastic matrix , so the difference between any two entries is to obtain a distribution over pmfs with bounded |θ k − θ | ≤ k,l for any k, components, it is sufficient to choose components of the matrix M such that |M kj − M lj | ≤ k,l for all j = 1, . . . , d because θ in (3) sums to 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A given node (h, i) is always picked at most once, but since its descendants may be picked afterwards, subsequent paths in the tree can go through it.
Sentence 2: n) be the empirical average of the rewards received for the time-points when the path followed by the algorithm went through (h, i): where 0 < ρ < 1 and ν 1 > 0 are parameters of the algorithm (to be chosen later by the decision maker, see Assumption 1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We show a new result relating the CVaR objective in (3) to the expected discounted-cost in presence of worst-case perturbations of the MDP parameters, where the perturbations are budgeted according to the "number of things that can go wrong".
Sentence 2: by minimizing CVaR, the decision maker also guarantees robustness of the policy.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In fact, the improvement in error compared to Loopy BP was not correlated with sub-model agreement for any K (maximum ρ = 0.0185, p < 0.05).
Sentence 2: sub-model agreement does not explain the improvement over Loopy BP, indicating that submodel disagreement is not related to the difficulty in inference problems that causes Loopy BP to underperform relative to the ensembles (e.g., due to convergence failure.)
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A pair (ā, B) is optimal for (13) if and only if the optimality condition can be written as Substituting ā = a ⋆ , B = B ⋆ , and noting that a ⋆T w ⋆ /w ⋆T B ⋆ w ⋆ = 1, the optimality condition reduces to which is precisely (12).
Sentence 2: we have shown that (a ⋆ , B ⋆ ) is optimal for (13), which in turn establishes (11).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let F j , j ∈ [d], denote the j th column of F . The multivariate distribution of F is O(m)-invariant in the sense that for a subset A ⊂ R m×d , P r (RA) = P r (A) for all R ∈ O(m).
Sentence 2: p r (M , where the inequality is due to the fact that a positive semidefinite matrix necessarily has nonnegative diagonal, and the equality is due to the independence of the random variables F T j M F j , j ∈ [d].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We first fix a value z \i for Z \i , and a value of R such that E holds, and consider the ratio of probabilities: Observe that this ratio of probabilities is equal to: which is in turn equal to: Observe that from Equation 13, Thus, the ratio of the probabilities is at most the ratio Pr(Z i ≥ γ)/ Pr(Z i ≥ γ + 2β) for γ = sup j =i z j + q(F (T, θ j , α 1 , r j ), V ) − q(F (T, θ i , α 1 , r i ), V ), which is at most e α2 by properties of the exponential distribution.
Sentence 2: we have established that when R ∈ Σ k , for all j, Thus for any such R, we can write: Equation 11 now follows by integrating R over E. PROOF:(Of Theorem 1) The proof of Theorem 1 follows from a combination of Lemmas 1 and 2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In order to find the causal direction, the authors of [GSKZ17] consider the particular case that a (i) = a (j) and either .
Sentence 2: in this case, E|C , and we have C|E if and only if E . comparing the coefficients in both directions, the causal direction is identifiable if E . Note that if either of the variances does not change, this condition always holds.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When i is large, the noise in λ i (A)( ŨA ) :,i "outweighs" the signal.
Sentence 2: we need to choose a d such that only the first d eigenvectors/eigenvalues of A are used to approximate Φ d . Here, we need to address the truncation error: the tail { √ λ i ψ i (x j )} i>d is thrown away.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Each iteration of CCCP approximates the convex part (negative joint entropy) by its tangent, that is ∂h(x 0 ), x in (6).
Sentence 2: taking derivatives of the joint entropy with respect to p(y|x i ) and evaluating at parameters at iteration t − 1, denoted as θ t−1 and θ t−1 , yields Define similarly g y (x i ), g y (x i ), and g y (x i ) for the derivative with respect to p(y|x i ), p(y |x i ) and p(y |x i ), respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Roughgarden [11], however, observed that there is often no degradation; specifically, the very same smoothness property that he identified as implying good welfare in Nash equilibrium also proves good welfare of coarse correlated equilibrium (equivalently: for outcomes from no-regret learners).
Sentence 2: for a large family of static games, we can expect strategic behavior to lead to good outcomes.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It brings us not only robustness to overfitting but also numerous benefits such as improved prediction accuracy by Bayesian ensembling (Balan et al., 2015), active learning (Gal et al., 2016), and principled/safe exploration in reinforcement learning (Houthooft et al., 2016).
Sentence 2: developing a Bayesian few-shot learning method is an important step towards robust meta-learning.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then, the number of time steps that a 1 (t) = a 2 (t + 1) = 1 does not happen is upper bounded by , which is given by regret analysis.
Sentence 2: the compensation we need to pay on arm 1 is upper bounded by P  In this section, we present experimental results for the three algorithms, i.e., the UCB policy, the modified "-greedy policy and the modified TS policy.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This also allows some other interesting adaptations.
Sentence 2: we note: is the leave-one-out template for subject k, which is indepedendent of g k . we replace (5) by: From ( 5) and ( 8) we observe that the multi-subject alignment problem reduces to a sequence of pairwise registrations, each of which registers one subject to an average of connectivity matrices.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We first contract the importance matrix of W with M along all dimensions except the D-th dimension, i.e.
Sentence 2: the result S is a square matrix whose size is the same as the current dimension D. The element S ij represents the total value of masked elements wherein we mask the i-th slice in W with the j-th slice in M . the decrease of the objective function that we can gain from swapping the i-th slice and the j-th slice is G ij as shown in Equation (10), which can also be written in the matrix format (11), where L is the diagonal vector of S Then, we only need to find the maximum elements G ij in G and swap the i-th slice and j-th slice in D dimension of W . The algorithm is described as in Algorithm 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If violations occur, we propose to project the proposed gradient g to the closest gradient g (in squared 2 norm) satisfying all the constraints (7).
Sentence 2: we are interested in: To solve (8) efficiently, recall the primal of a Quadratic Program (QP) with inequality constraints: where C ∈ R p×p , p ∈ R p , A ∈ R (t−1)×p , and b ∈ R t−1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For fixed step size γ k = γ for all k ≥ 1, Theorem 2 shows that the bias is of order O(γ 2 + 1/(Kγ)).
Sentence 2: if the number of iterations K is fixed then the choice of γ which minimizes this bound is Choosing this value for γ leads to the optimal rate for the bias of order O(K −2/3 ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Ultimately, we obtain a new factor set F := F u ∪ {F ∩ V u : F ∈ F 0 }, and probability distribution Note that during the process, not only the number of variables is reduced, but also the number of factors and the size of each factor can never grow.
Sentence 2: if sampling was feasible for the original distribution p, it will also be feasible for q, and potentially more efficient.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Obviously, every segmentation provides such a lower (albeit less tight) bound on the total evidence score.
Sentence 2: if we find large enough contiguous regions in Q, with supporting regions in ref (i.e., high enough local evidence scores), and define R 0 to be the remainder of Q, then S = R 0 , R 1 , ..., R k can provide a reasonable lower bound on GES(Q|H ref ).
Relationship: reasoning