sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: At level n, we perform the same outer sum, but for each trajectory we sum the probabilities of the histories for agents k = j that would lead to the beliefs we are interested in.
Sentence 2: the sequence distributions at time t are sufficient for computing any desired element of the infinite belief sequence B j, * (h j 0:t ) for any agent j and history h j 0:t . Updating the distributions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To do so, we define the regularized empirical risk R n (w) := L n (w) + (cV n /2)kwk 2 and the corresponding optimal argument and attempt to minimize R n with accuracy V n . Since the regularization in ( 4) is of order V n and (3) holds, the difference between R n (w ⇤ n ) and L(w ⇤ ) is also of order V n -this is not immediate as it seems; see [16].
Sentence 2: the variable w n solves the ERM problem in (2) to within its statistical It follows that by solving the problem in (4) for n = N we find w ⇤ N that solves the expected risk minimization in (1) up to the statistical accuracy V N of the full training set T . In the following section we introduce a class of methods that solve problem (4) up to its statistical accuracy faster than traditional deterministic and stochastic descent methods.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Here, KL f t g i t is the KL divergence between two Gaussians and π i is the mode weight of g t . The final objective function is: We now show that by modeling the prior as MoG and the posterior as Gaussian, minimizing the approximation results in KL divergence minimization.
Sentence 2: let define the log-likelihood we have (see Supplementary material for full derivation): minimizing D var results in KL divergence minimization.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [5]): In what follows, CV, ∆CV and JK will generically be denoted by μ: where, slightly abusing notation, i Note that μ is the average of identically distributed (dependent) variables.
Sentence 2: it asymptotically converges to a normally distributed variable, which is completely characterized by its expectation E[μ] and its variance Var[μ].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Notice that when σ C jk is a Rademacher variable, it has the same distribution as z l C jk σ C jk since z l C jk ∈ {−1, 1}.
Sentence 2: using the first result of theorem 6 we have that with probability 1 − δ over the samples S, all h in H satisfy: Now putting in equation ( 8) the third point of theorem 6, with H = H K,B as defined in theorem 6 with Z = X , we obtain the following theorem: Where n S + , n S − are the number of positive and negative instances in S, and σ and ν also depend on S, and are such that x σ(i) is the i-th positive instance in S and ν(j) the j-th negative instance.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In addition, even if it was possible to solve them analytically, they are still subject to the limitations of the original graphical models, such as the need to use exponential family distributions and conjugate priors for tractability.
Sentence 2: the next step in the proposed method is to map the probability distributions and the corresponding fixed-point equations into a Hilbert space, where some of these limitations can be relaxed.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This theory is not in FO 2 , not in RU , and is not domain-liftable using R. However, the first two clauses are of the form described in Lemma 1, the third and fourth are in FO 2 (and also in RU ), and the third clause, which contains Assigned(v, j), has no other PRVs with more than one LV.
Sentence 2: this theory is in S 2 FO 2 (and also in S 2 RU ) and domain-liftable based on Theorem 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Formally, we aim to compute a Stackelberg equilibrium of the game in which the learner moves first by choosing a linear classifier g(x) = w T x and all the attackers simultaneously and independently respond to g by choosing x according to a query-based algorithm optimizing the cost function c(x , x A ) subject to query and cost budget constraints.
Sentence 2: we term this approach Stackelberg game multi-adversary model (SMA).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Toward this we note that Corollary 3 holds for an arbitrary loss, and Lemma 5, and Theorem 6 hold for a sufficiently smooth loss with bounded 2nd derivative w.r.t.
Sentence 2: w. The conversion to Bayes risk in Corollary 8 holds for any loss convex in p. the result of Corollary 8 holds more generally for any sufficiently smooth loss that has bounded 2nd derivative in w and that is convex in p. We provide an application of this more general result in the next section.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To accommodate prior information on the class labels, we generalize the previous framework so that the estimated label distribution jJ(y I Xi) has low relative entropy with the prior on the class labels.
Sentence 2: the objective function (1) and its EM -bound (4) can be modified to be where " i,y is the prior probability for the i-th training example to have class label y.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Here, we introduce a new approach using a near-maximum entropy model, that makes this type of analysis feasible for very high-dimensional data-the model parameters can be derived in closed form and sampling is easy.
Sentence 2: our NearMaxEnt approach can serve as a tool for testing predictions from a pairwise maximum entropy model not only for low-dimensional marginals, but also for high dimensional measurements of more than thousand units.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: So we can directly derive the convergen ce order as D,, (8) = Op (nl / 2 ) and D(f 0) = Op( nl ).
Sentence 2: by using t he rela tion (2.4) , it follows that R,, (8 ) -L* = Op( nl / 2 ) and R(f 0) -L * = Op (nl ).5 t he TOP kernel h as a much b etter convergen ce rate in R(f 0)' which is a strong motiva tion to use the TOP kernel instead of plug-in estimate.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By the central limit theorem, z i , z j / √ n d ⇒ N(0, 1).
Sentence 2: σ 2 z i , z j /n ≈ N(0, σ 4 /n), and we need to choose the (rescaled) threshold proportional to √ σ 4 = σ 2 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Randomly select M training samples indexed by ξ k,1 , ξ k,2 , ...ξ k,M ; 3: Randomly select i k ∈ {1, 2, ..., n} with uniform distribution; 4: For the shared memory platform, one can exactly follow ASYSG-CON on the computer network using software locks, which is expensive 4 .
Sentence 2: in practice the lock free asynchronous parallel implementation of SG is preferred.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The semi-smooth Newton method for the solution of ( 10) can be stated as the iteration α t+1 = α t − g(α t ) −1 p(α t ) , (13) where g is a generalized derivative of p. Proposition 12 (Generalized derivative of p).
Sentence 2: if prox h•D −1/2 is Newton differentiable with generalized derivative G, then so is the mapping p with a generalized derivative Furthermore, g is nonsingular with a uniformly bounded inverse on R. as p is Newton differentiable with nonsingular generalized derivative whose inverse is also bounded, the general semi-smooth Newton convergence theorem implies that (13) converges superlinearly to the unique root of (10).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As we optimize (7) via gradient descent, the policy changes only locally.
Sentence 2: we can achieve better data-efficiency by restricting the exploratory actions u with (x, u) ∈ S n to be close to π n , u ∈ [π n (x) − ū, π n (x) + ū] for some constant ū. Computing the region of attraction by verifying the stability condition on a discretized domain suffers from the curse of dimensionality.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Additionally, most formulations of apprenticeship learning are actually harder than reinforcement learning; apprenticeship learning algorithms typically invoke reinforcement learning algorithms as subroutines, and their performance guarantees depend strongly on the quality of these subroutines.
Sentence 2: these apprenticeship learning algorithms suffer from the same challenges of large state spaces, exploration vs. exploitation trade-offs, etc., as reinforcement learning algorithms.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Furthermore, the Gibbs empirical risk R S (G i ab ) is small when the training points are located far away from the Bayes decision surface G i ab (x) = 1/2 (with G i ab (x j ) → y j ∀j ∈ i).
Sentence 2: the Gibbs classifier with the smallest guarantee of risk should perform a non trivial margin-sparsity tradeoff.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Taking the logarithm and eliminating constant terms, we obtain the following objective: In most cases, the same prior is used for all features, so we have β k = β for all k. This objective is convex, and can be optimized efficiently using methods such as conjugate gradient or BFGS, although care needs to be taken with the discontinuity of the derivative at 0.
Sentence 2: in principle, we can simply optimize this objective to obtain its globally optimal parameter assignment.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This feature is expressive enough to differentiate whether an object just grazes over table's edge (steep change in vertical distance) versus, it first goes up and over the table and then moves down (relatively smoother change).
Sentence 2: the features obtained from object-environment interaction are φ obj−env (•) ∈ R 20 (3*4+2+6=20).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In contrast, P b is defined to Granger cause P a when: λ a (t|H a (t)) = λ a (t|H Q (t)), where Q = P a ∪ P b .
Sentence 2: given two processes P a and P b whose intensity functions follow Eq (3), Granger causality arises when α ba = 0 2 4 Learning GRANGER-BUSCA Recall from Figure 1 that GRANGER-BUSCA exhibits a cluster like behavior.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, we find that removing instance normalization from the discriminator at that stage is helpful as we do not want its final prediction to be influenced by zero values coming from the background.
Sentence 2: we update the adversarial energy L adv of Equation 2 to: Algorithm 1 summarizes the training procedure for learning F S→T ; training F T →S is similar.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By this definition, it can be seen that the entry (t, Thus, the entries of matrix A are multivariate polynomials of entries of B.
Sentence 2: furthermore, where A S and A Y are the rows corresponding to set S and Y in matrix A, respectively, and matrix ⇤ i is a diagonal matrix where the entries of vector      )), then we will reject the null hypothesis H ij 0,Y,S .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: provides an exact solution for the ground state of a 2D lattice Ising model without a ferromagnetic field [13,8,9,10] in O(N Computing a lower bound: At a given iteration, prior to adding a newly generated set of constraints we can compute the total residual constraint violation over all layers of hierarchy by ∆ = l (θ l + λ l + ω l−1 − ω l ) • z l . In the supplement we demonstrate that the value of the dual objective plus 3 2 ∆ is a lower-bound on the relaxed closest ultrametric problem in Eq 9.
Sentence 2: as the costs of the minimum-weight matchings approach zero from below, the objective of the reduced problem over Ẑ approaches an accurate lower-bound on optimization over ΩL Expanding generated cut constraints: When a given cut z l produces more than two connected components, we found it useful to add a constraint corresponding to each component, following the approach of [20].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Assume that the second moment of the rewards of each arm are bounded uniformly as E X 2 i ≤ V and let c > 0 be arbitrary.
Sentence 2: then, the regret of Boltzmann-Gumbel exploration satisfies Notably, this bound coincides with that of Theorem 3, except that σ 2 is replaced by V . by following the proof of Corollary 1, we can show a distribution-independent regret bound of order √ KT log K. Let us now present the proofs of our main results concerning Boltzmann-Gumbel exploration, Theorems 3 and 5.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Using the law of total probability, we have the following equation for the filtering distribution: We can think of the integral in (5) as a sum over a distribution of experts, where each expert corresponds to a single pose history, u 1:t−1 .
Sentence 2: based on its hypothesis about pose history, each expert has an opinion about the current pose of the object, U t , and the texture maps of the object and background, V t and B t . Each expert also has a credibility, a scalar that measures how well the expert's opinion matches the observed image y t . (5) can be interpreted as follows: The filtering distribution at time t is obtained by integrating over the entire ensemble of experts the opinion of each expert weighted by that expert's credibility.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Each (z, t) 2 Q denotes the (j, t) entry of Z where j = iters[t].current() and z = z jt = h jt w t . Note that there are most k elements in the max-heap at any time.
Sentence 2: we can implement Q by a binary heap such that 1) Q.top() returns the maximum pair (z, t) in O(1) time; 2) Q.pop() deletes the maximum pair of Q in O(log k) time; and 3) Q.push((z, t)) inserts a new pair in O(log k) time.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given that the number of inducing variables determines the amount of the approximation, or compression of the DPP model, we examine different settings for this number and see whether the corresponding trained models provide similar estimates for the overdispersion measures.
Sentence 2: we train the DPPs under different approximations having m ∈ {50, 100, 200, 400, 800, 1200} inducing variables and display the estimated overdispersion measures in Figures 3(a) and 3(b).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It contains two types of edges: 1) entity-entity edge connecting every two entity variables (y i and y j for 1 ≤ i = j ≤ n. 2) entity-relation edges connecting every relation variable y k (where k > n) to its two entity variables.
Sentence 2: our graph is not a complete graph and our goal is to design an architecture that will be invariant to any automorphism of the graph, such as permutations of the entity variables.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: by the null hypothesis H 0 , we can derive a global evidence score on the hypothesis that Q was generated from ref via the segmentation S (for simplicity of notation we use the symbol H ref also to denote the global hypothesis): Namely, the global evidence induced by S is the accumulated sum of the local evidences provided by the individual segments of S. The statistical significance of such an accumulated evidence is expressed by: Consequently, we can accumulate local evidence of non-overlapping regions within Q which have similar regions in ref for obtaining global evidence on the hypothesis that Q was generated from ref .
Sentence 2: for example, if we found 5 regions within Q with similar copies in ref , each resulting with probability less than 10% of being generated by random, then the probability that Q was generated by random is less than (10%) 5 = 0.001% (and this is despite the unfavorable assumption we made that the rest of Q was generated by random).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, in this form we see that p(v j |u σ(t) , π 1:t−1 ) is directly related to the change in the partial energy from π 1:t−1 to H(v j , u σ(t) , π 1:t−1 ) -the larger the change the bigger the resulting probability will be.
Sentence 2: the matching choices will be made solely based on the changes in the partial energy.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, based on general techniques developed in [15] and greatly extended in [1], it can be shown that online learnability is information-theoretically possible for such W. However, these techniques do not provide a computationally efficient algorithm.
Sentence 2: to the best of our knowledge, there is currently no efficient (polynomial time) online algorithm, which attain non-trivial regret.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In signal detection theory, the sensitivity index, conventionally denoted d , is a measure of how well the annotator can discriminate the two values of z i [14].
Sentence 2: it is defined as the Mahalanobis distance between µ 0 and µ 1 normalized by s, the lower σ j , the better the annotator can distinguish between classes of z i , and the more "competent" he is.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Intuitively, this occurs because the now finite spines of the prior p(w; H), which bound the variational approximation, do not allow us to place infinite prior density in any region of weight space (as occurred previously when any γ i → 0).
Sentence 2: if any γ i goes to zero with a, b > 0, the associated approximate prior mass, and therefore the approximate evidence, must also fall to zero by ( 16).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Second, it can be proved that the pairwise losses in Ranking SVM, RankBoost, and RankNet, and the listwise loss in ListMLE are all upper bounds of the essential loss.
Sentence 2: we come to the conclusion that the loss functions used in these methods can bound (1−NDCG) and (1−MAP) from above.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Each neuron that spikes at time step k will randomly evoke, with probability W ij , one recurrent excitatory post-synaptic potential (EPSP) at time step k + 1, after some network delay.
Sentence 2: we define the number of recurrent EPSPs received by neuron l in the j-th sub-population as a j l . a j l is the sum of N k independent (but not identically distributed) Bernoulli trials: where The sum a j l follows the so-called "Poisson binomial" distribution [22] and in the limit approaches the Poisson distribution: The detailed analysis of the distribution of a i l and the proof of equation 6 are provided in the supplementary materials.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, for many GPs -and ordinary kriging in particular -the correlation between two points decreases as a function of distance.
Sentence 2: the full GP model can be approximated well by a local GP, where only the k nearest neighbors of the query point are used to compute the prediction value; this reduces the computation time to O(k 3 log(n)) per prediction, since O(log(n)) time is required to find the k-nearest neighbors using spatial indexing structures such as balanced kd-trees.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, both z +ε = (y + ε, x 1 , ..., x d ) and z −ε = (y − ε, x 1 , ..., x d ) are, with probability α, on the correct side of the regression surface, defined by z = (ŷ, x 1 , ..., x d ).
Sentence 2: z +ε differs from z by at most +ε in the first dimension, and z −ε differs from z by at most −ε in the first dimension.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This root finding procedure will take no more than O@nA steps in the worst case.
Sentence 2: stepping beyond a single threshold potential, we now assume that the submodular function to be minimized can be written as a nonnegative linear combination of threshold potentials and a modular function, i.e., f @AA a c ¡ e A C j d j © wj;yj @AA: we have the smoothed Lovász extension, and its gradient: f @xA a c ¡ x C j d j © wj;yj @xA and r f @xA a c C j d j r © wj;yj @xA: We now wish to use the accelerated gradient descent algorithm of [18] to minimize this function.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Assume W has been first normalized to have unit-norm columns, w i , w j is essentially the (i, j)-the element of the Gram matrix W T W , and i = j requires us to consider off-diagonal elements only.
Sentence 2: we propose the following mutual coherence (MC) regularization term inspired by (3: Although we do not explicitly normalize the column norm of W to be one, we find experimentally that minimizing (4) often tends to implicitly encourage close-to-unit-column-norm W too, making the objective of (4) a viable approximation of mutual coherence (3) 2 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since B = XAY for positive diagonal matrices X and Y , Lemma 1 implies B is the optimal solution to min By Lemma 7, there exists a matrix P ∈ U r ,c such that P − P * 1 ≤ 2 ( r − r 1 + c − c 1 ).
Sentence 2: moreover, since B is an optimal solution of (3), we have by Hölder's inequality where we have used the fact that 0 ≤ H(B), H(P ) ≤ 2 log n. Lemma 7 implies that the output P of ROUND(B, U r,c ) satisfies the inequality B − P 1 ≤ 2 ( r − r 1 + c − c 1 ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Our model assumptions based on the PL model imply some regularity properties for the pairwise marginals, such as strong stochastic transitivity and stochastic triangle inequality (see Appendix A of [28] for the proof).
Sentence 2: the INTERLEAVED FILTER [28] and BEAT THE MEAN [29] algorithms can be directly applied in our online framework.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In each point v ∈ V, we have a probability distribution p atlas (•|v) ∈ R T ×A which informs about the tissue type and the atlas region in v. Without any loss of generality, one can assume that the tissue information is encoded in the atlas.
Sentence 2: we consider the probability p atlas (•|v) ∈ R A .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The § ¡ 43 ¢ -mismatch kernel is based on a feature map to a vector space indexed by all possible subsequences of amino acids of a fixed length ¡ ; each instance of a fixed ¡ -length subsequence in an input sequence contributes to all feature coordinates differing from it by at most ¢ mismatches.
Sentence 2: the mismatch kernel adds the biologically important idea of mismatching to the computationally simpler spectrum kernel presented in [14].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For a solution to be optimal, the Karush-Kuhn-Tucker (KKT) conditions must be satisfied [3].
Sentence 2: which can be rearranged as For being a probability measure, the variables π ijk have to satisfy Eliminating λ ij by jointly considering Equations ( 5) and ( 6), we obtain a labeling model in the exponential family: Plugging Equation ( 7) into (4) and performing some algebraic manipulations, we prove Theorem 2.1 Let π ij be the maximum entropy distributions in (3).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Our sum assumption corresponds to a situation where the feature map Φ(x) and feature space F for k is the concatenation of the feature maps Φ v (x) for each kernel k v , i.e, F = v∈V F v and Φ(x) = (Φ v (x)) v∈V .
Sentence 2: looking for a certain β ∈ F and a predictor function f (x) = β, Φ(x) is equivalent to looking jointly for β v ∈ F v , for all v ∈ V , and As mentioned earlier, we make the assumption that the set V can be embedded into a directed acyclic graph.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We first notice that, by using (4) in (1), we obtain q(h 1:S 1:M ) = i q(h i 1:M ).
Sentence 2: we then observe that we can rewrite the update for q(h i 1:M ) as proportional to the joint distribution of the following linear gaussian state-space model (LGSSM) where inference on q(h 1:S 1:M ) can be accomplished with LGSSM smoothing routines [7].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By comparison, for the greedy heuristic, a training example (X (i) , Y (i) = α k ) contributes log(λ k f k (X (i) )) to w(k, k ).
Sentence 2: the greedy estimator can be seen as a "soft" version of MV that also greedily optimizes the MLE objective.
Relationship: reasoning