sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: We use a soft-max with a temperature parameter that is increased slowly through training iterations.
Sentence 2: specifically, we learn a vector w(n) ∈ R C for each location n of the multiplexing pattern, with the corresponding I(n) given during training as: where α t is a scalar factor that we increase with iteration number t. in addition to changes due to the SGD updates to w(n), the effective distribution of I(n) become "peakier" at every iteration because of the increasing α t , and as α t → ∞, I(n) becomes a zero-one vector.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore the network has N = 3 identical units: two of which will be used for the inputs and one for the output, and K = 4 memories ⇠ µ i , which are the four lines of the truth table.
Sentence 2: the energy ( 2) is equal to where the energy function is chosen to be a polynomial of degree n. For odd n, energy ( 7) is an odd function of each of its arguments, E n (x, y, z) = E n (x, y, z).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Now, all leaves in T δ are nodes that are exposed (i.e., made public) by the δ-sampling process, so in expectation T δ has at most δn leaves (and we can bound the probability that the actual number exceeds this by more than a small factor).
Sentence 2: there will also be O(δn) branching nodes, and all other nodes in T δ must be single-child nodes.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let α t = t. Algorithm 5 with update by option (A) is the case when the y-player uses OPTIMISTICFTL and the x-player adopts MIRRORDESCENT with γ t = 1 4L in Fenchel game.
Sentence 2: w T is an O( 1T 2 )-approximate optimal solution of min x∈K f (x).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, we achieve a new "per-point" guarantee E[d(j, S)] ≤ (1 + 2/e)T ≈ 1.736T , while respecting the usual guarantee d(j, S) ≤ 3T with probability one.
Sentence 2: while maintaining good global quality with probability one, we also provide superior stochastic guarantees for each user.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , G N G }, and define the (1,α)-group norm by x 1,α := N G t=1 x Gt α .
Sentence 2: if S G denotes the subset of groups where x Gt = 0, then the subgradient has the following form: where m t * α ≤ 1 for all t / ∈ S G . the group sparse norm is also decomposable with • Low-rank matrices: for the nuclear norm regularizer • * , which is defined to be the sum of singular values, the subgradient can be written as , where • 2 is the matrix 2 norm and U, V are the left/right singular vectors of X corresponding to non-zero singular values.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The proof of this theorem is provided in Appendix F. Note that for monotone rules, a larger alphawealth reward at each rejection yields a possibly higher power, but never lower power, immediately implying statement (b).
Sentence 2: we provide only a proof for statement (a) in Appendix F. For the reader interested in technical details, a key super-uniformity Lemma 1 and associated intuition for online FDR algorithms is provided in Section 3.2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For simplicity, let I (N ) (Z) be the estimate based on original samples {Z 1 , Z 2 , .
Sentence 2: . . , Z N }, where Then a certain version of Efron-Stein inequality states that: Var we have sup Z1,...,Z N To upper bound the difference | ξ i (Z) − ξ i (Z \j ) | created by eliminating sample Z j for different i 's we consider three different cases: (1) i = j; (2) ρ k,i = 0; (3) ρ k,i > 0, and conclude that The detail of the case study is in Section.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By Theorem 1.19 -1.23 of Durrett [2016], it can be proved that, for arbitrary η > 0, there exists t 0 , such that ∀t > t 0 max That is to say, when the size of the stored samples is larger than t 0 , the mixing time of the new data process with experience replay is 0.
Sentence 2: the experience replay trick improves the mixing property of the data process, and hence improves the convergence rate.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By "general" we mean the reductions are not restricted to particular classes of graphs or algorithmic approaches, but show that any computational progress on the target problem immediately transfers to progress on the source problem.
Sentence 2: for example, one of our main results establishes that the problem of determining whether given marginals, whose induced graph (the "data graph") falls within some class G, are consistent with any joint distribution reduces to the problem of MAP inference in Markov networks falling in the same class G. for instance, we immediately obtain that the tractability of MAP inference in trees or tree-like graphs yields an efficient algorithm for marginal consistency in tree data graphs; and any future progress in MAP inference for other classes G will similarly transfer.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We shall denote by S the K max -selected variables: The computational complexity of the K max -long regularization path of LASSO solutions is Most of the time, we can see that the Lasso effectively catches the true changepoint but also irrelevant change-points at the vicinity of the true ones.
Sentence 2: we propose to refine the set of change-points caught by the Lasso by performing a post-selection.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consequently, hidden state transitions depend explicitly on the elapsed time since the last observation, i.e., on the IEIs ∆t e . The transition probability a i j (∆t e ) from hidden state i to hidden state j is then given by where g i j is the conditional probability of making a transition from state i into a new state j, given that j = i.
Sentence 2: g i j has to satisfy the constraint ∑ j g i j = 1, with g ii = 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that the penalization on the margin errors of unlabeled data can be included if needed.
Sentence 2: we have the following form of TSVM that can be derived through the duality: In order to adjust the strength of the regularization raised from the unlabeled examples, we introduce a coefficient ρ ≥ 0, and modify the above problem (3) as below: Obviously, it is the standard TSVM for ρ = 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the case of hard clustering we also find thatI(Tj ; U j ) = H(Tj ) (where H(P) is Shannon's entropy).
Sentence 2: roughly speaking, we may say that H(P) is decreasing for less balanced probability distributions p. increasing (3-1 will result with a tendency to look for less balanced "hard" partitions and vice verse.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Kolter and Wong [15], which used the same network and hyperparameters as ours, reported that they could defend from perturbations with its ∞ -norm bounded by 0.1 for more than 94% examples.
Sentence 2: in the ∞ -norm, our work is inferior, if we ignore their limited applicability and massive computational demands.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If (SDP) has a compact search space, then it admits a global optimum of rank at most r, where r(r+1) 2 ≤ m [Pataki, 1998, Barvinok, 1995.
Sentence 2: if one restricts the search space of (SDP) to matrices of rank at most p with p(p+1) 2 ≥ m, then the globally optimal value remains unchanged.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Further, notice that the offset function w(x) = f ta (x) − f so (x) = x, is just a linear function.
Sentence 2: instead of directly using T ta to estimate f ta , we can use the target domain samples to find an estimate of w(x), denoted by ŵ(x), and our estimator for the target domain is just: Figure 1b shows this technique gives improved fitting for f ta .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As noted above, in a natural environment an individual is confronted with the task of discriminating between stimuli important for survival (" targets") and stimuli that are irrelevant (" distractors") .
Sentence 2: an organism must not only keep up a high rate of detecting targets but, at the same time, must strive to minimize " false alarms" to irrelevant stimuli.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As seen in, e.g., the Maximum Variance Unfolding (MVU) method (Weinberger et al., 2004) for nonlinear dimensionality reduction (see (So, 2007) for related discussion) and Pairwise Constraint Propagation (PCP) (Li et al., 2008) for constrained clustering, a nice feature of such a framework is that the problem formulation often becomes straightforward.
Sentence 2: the major challenge in optimizationbased kernel learning lies in the second component, where the key is to find an efficient procedure to obtain a positive semidefinite kernel matrix that satisfies certain properties.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [6], we can implement this by maximizing a lower bound on mutual information between generator ids and generated images: where q(c|x) is the distribution approximating p(c|x), p g (x|c) is induced by each generator G c , KL is the Kullback Leibler divergence, and the last equality is a consequence of Lemma 5.1 in [6].
Sentence 2: by modeling q(c|x) with a neural network Q(x; γ), the encoder network, maximizing I(c; x) boils down to minimizing a cross entropy loss: Utilizing the Wasserstein GAN [2] objectives, discriminator (critic) and generator maximize the following, where D(x; w) : X → R is the critic function: We call this model Disconnected Manifold Learning WGAN (DMWGAN) in our experiments.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, the increase in average contribution to the postsynaptic membrane potential for the correlated group of neurons renders this population slightly more likely to be active during the causal regime than any single member of the uncorrelated group.
Sentence 2: the synaptic strengths for this group of neurons will increase with respect to the uncorrelated group, further augmenting their likelihood of causing a postsynaptic spike.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The estimated dimension can be used to estimate the noise level present in the data set.
Sentence 2: the idea is to measure the error between the projected label vector Ĝ = π d (Y ), which approximates the true label information G. The resulting number êrr = A Note on Consistency Both the estimate of Ĝ and the noise level are consistent if the estimated dimension d scales sub-linearly with n. The argument can be sketched as follows: since the kernel PCA components do not depend on Y , the noise N contained in Y is projected to a random subspace of dimension d. Empirically, d was found to be rather stable, but in principle, the condition on d could even be enforced by adding a small sub-linear term (for example, √ n, or log n, to the estimated dimension d).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consequently, the marginals in observed and latent space are also KDEs, and the dimensionality reduction and reconstruction mappings are given by kernel regression (the conditional means E {y|x}, E {x|y}): We allow the bandwidths to be different in the latent and ambient spaces: They may be tuned to control the smoothness of the mappings and densities [1].
Sentence 2: lELVM naturally extends a LE embedding (efficiently obtained as a sparse eigenvalue problem with a cost O(N 2 )) to global, continuous, differentiable mappings (NW estimators) and potentially multimodal densities having the form of a Gaussian KDE.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The TD error δ t is a random quantity that is defined according to δ t = r t+1 − Ĵt+1 + V (s t+1 )− V (s t ), where V and Ĵ are consistent estimates of the state-value function and the average reward, respectively.
Sentence 2: these estimates satisfy E[ V (s t )|s t , π] = V π (s t ) and E[ Ĵt+1 |s t , π] = J(π), for any t ≥ 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Next observe that due to the Boolean nature of ϕ, (4) admits a simple closed form: where in general we use (•) to denote the probability of satisfying a specification using uniformly random actions.
Sentence 2: we denote by {ξ} the probability of randomly generating demonstration ξ within M . Further, note that by the law of the excluded middle, for any specification: ¬ϕ = 1 − ϕ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The online updates are summarized as: Unless r n = 0, the sum in (1) only contains one term since the only possibility is that The indexing convention is such that if r n = 0 then y n+1 is the first observation sampled from the new parameters θ.
Sentence 2: the marginal posterior predictive on the next data point is easily calculated as: the predictions from BOCPD fully integrate out any uncertainty in θ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, the SE analysis in [24] is restricted separable Lipschitz denoisers that can be described as follows: Let g 1n (r 1 , γ 1 ) be the n-th component of the output of g 1 (r 1 , γ 1 ).
Sentence 2: then, it is assumed that, for some function scalar-output function φ(•) that does not depend on the component index n. the estimator is separable in the sense that the n-th component of the estimate, x 1n depends only on the n-th component of the input r 1n as well as the precision level γ 1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Finally, note that when [31] showed that a favourable trade-off was possible for kernel ridge regression (KRR), they strongly leveraged the fact that KRR is a γ-regularized problem.
Sentence 2: all eigenvalues and eigenvectors in the Φ n Φ T n covariance matrix smaller than the γ regularization do not influence significantly the solution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Otherwise, it selects the actions with the number of observations below the minimum requirement for making the empirical divergence of each suboptimal point q ∈ C c 1 larger than log t. Unlike the N -armed bandit problem in which a reward is associated with an action, in the partial monitoring problem, actions, outcomes, and feedback signals can be intricately related.
Sentence 2: we need to solve a non-trivial optimization to run PM-DMED.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A standard calculation shows that where Note that P c ⇢ c (x, y) = 1 for all (x, y).
Sentence 2: we can rewrite, @L(W ) @Wq,r = 1 m P (x,y) x r (⇢ q (x, y) 1[q = y]) .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: These transformations, namely the diffeomorphic component deformations and the time component reparametrizations, characterize respectively the spatial and the temporal variability of propagation among the population.
Sentence 2: individual trajectories may write in the form of where the functions γ i are obtained from γ 0 through the applications of the two transformations φ i and ψ i described below.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let (h, j) be the deepest node in T n that has been expanded by the algorithm up to round n. Thus h ≥ h * n . Now, from the definition of the algorithm, we only expand a node when its value is larger than the value of all the leaves of equal or lower depths.
Sentence 2: since the node (h, j) has been expanded, its value is at least as high as that of the optimal node (h Remark 2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since log P (D) is constant for all hypotheses, its value does not enter into the minimization problem, and we can state that the hypothesis of choice ought to be such as to minimize the quantity -log P(D I H)log P(H) (3) If we follow Rissanen and regard the quantity -log P(x) as the description length of x, D L (x ), then Equation 3 instructs us to select the hypothesis that minimizes the total description length What this means is that the hypothesis that is optimal from the standpoint of the Bayesian decision maker is the same hypothesis that yields the most compact two-part code in Equation 4.
Sentence 2: besides the merits ofbrevity for its own sake, we see that maximal descriptive compactness also corresponds to maximal inferential power.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As shown in Table 1, both the IFO complexity and hard thresholding complexity of SVRG-HT are O (n + κ s ) log 1 .
Sentence 2: although the IFO complexity of SVRG-HT substantially improves over FG-HT, the overall complexity still scale linearly with respect to the sample size n. when the data-scale is huge (e.g., n κ s ) and the per-iteration hard thresholding operation is expensive, SVRG-HT could still be computationally inefficient in practice.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The physical interpretation is that of a sum of potential and kinetic energy terms, where the total energy of the system is conserved.
Sentence 2: in a system where this Hamiltonian function is constant, we can exactly compute its evolution through the pair of differential equations: These are solved exactly by x(t) = µ + a sin(t) + b cos(t) , where a and b can be identified at initial conditions (t = 0) : the exact HMC algorithm can be summarised as follows: • Initialise the allowed travel time T and some initial position x 0 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The process is non-Markovian and generates a sequence of states X t as follows.
Sentence 2: after arriving at state X t , the walker promptly "spaces out" and forgets the state X t−1 , yet it still wants to transition according to the higher-order transitions P . it invents a state Y t by drawing a random state from its history and then transitions to state X t+1 with probability P Xt+1,Xt,Yt .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Figure 1(a) shows how information flows between agents and the environment, and how Q-values are processed by the action selector in order to produce the action, u a t , and message m a t . Since this approach treats agents as independent networks, the learning phase is not centralised, even though our problem setting allows it to be.
Sentence 2: the agents are treated exactly the same way during decentralised execution as during learning.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For brevity, we will use z ∈ K to denote a global mimimizer of f which belongs to K. Recall that Algorithm 2 outputs a weighted average of the queries.
Sentence 2: we may employ Jensen's inequality to bound its error as follow, Combining this with T −1 t=0 α t ≥ Ω(T 2 ), implies that in order to substantiate the proof it is sufficient to show that, , is bounded by a constant.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As the reward gap 1 becomes larger, more attack is needed to reduce the reward of arm 1, and the slope increases.
Sentence 2: furthermore, note that Ignoring log log T terms, we have for some constant C > 0 and large enough T . We thus expect the log-cost curve as a function of log log T to behave like the maximum of two lines, one with slope 1/2 and the other with slope 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Indeed several authors have shown that favorable empirical/theoretical performance in mean square error for the preference matrix often does not translate to better performance when performance is measured using ranking metrics [11; 33; 23].
Sentence 2: collaborative preference estimation may be better posed as a collection of coupled learning to rank (LETOR) problems [25], where we seek to jointly learn the preference rankings of a set of entities, by exploiting the low dimensional latent structure of the underlying preference values.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: An immediate consequence is that given error tolerance 2 (0, 1) and with choice 2 (0, 1 4 2 ), then upper bound (10) is at most (by having each of the two terms on the right-hand side be  2 ) if n > m log 2m (i.e., = 1 + log 2 / log m), and This means that if we have access to a large enough pool of labeled time series, i.e., the pool has ⌦(m log m ) time series, then we can subsample n = ⇥(m log m ) of them to use as training data.
Sentence 2: then with choice = 1 8 2 , generalized weighted majority voting (8) correctly classifies a new time series S with probability at least 1 if the gap between sets R + and R needs to grow logarithmic in the number of latent sources m in order for weighted majority voting to classify correctly with high probability.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 12 20 While the above reveals the structure of V, we don't know V. However, we recover its column space U 2 R s(r)⇥K from the moment matrix M r (y ⇤ ), for example with an SVD.
Sentence 2: we can relate U and V by a linear transformation: V = UQ, where Q 2 R K⇥K is some unknown invertible matrix.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First, the function computing the output pose vectors of one layer is left-equivariant regarding applications of the group law if Second, the function computing activations of one layer is invariant under applications of the group law Since equivariance is transitive, it can be deducted that stacking layers that fulfill these properties preserves both properties for the combined operation.
Sentence 2: if we apply a transformation from G on the input of a sequence of those layers (e.g.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The another is that the iHMMs have one hidden state sequence s 1:t to be inferred, while the dIRM needs to estimate multiple hidden state sequences z 1:t,i given one time sequence observation.
Sentence 2: we may interpret the dIRM as an extension of the iHMM, which has N (= a number of objects) hidden sequences to handle relational data.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We can then differentiate Equation 3 to produce As described below, we will assume that the synapse strengths to each neuron are balanced between positive and negative values and that each neuron receives 256 inputs, so we can expect µ to be close to zero, and µ, ñi and cij to be much less than σ.
Sentence 2: the right term of Equation 4containing the denominator σ 3 j , can be expected to be much smaller than the left term containing the denominator σ j .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By contrast, the brain structures that seem to be associated with the evaluation of reinforcement, such as the ventral tegmental area or nucleus basalis, make only sparse projections to early sensory cortex; and these projections are frequently modulatory in character, rather than synaptic.
Sentence 2: exact computation of It might seem at first that the former of these two problems would also apply to the weight α(V i ) (in the first term of (3c )), in that execution of this portion of the update would also need to be delayed until this value-dependent weight could be calculated.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 5) and see Algorithm 3.2): ; =1 2Nonlinear conjugate gradient methods, such as Polak-Ribiere's CG (see Mizutani and Jang [13]) and Moller's scaled CG [14], are also widely-employed for training MLPs, but those nonlinear versions attempt to approximate the entire Hessian matrix by generating the solution sequence {Ih} directly as the outer nonlinear algorithm.
Sentence 2: they ignore the special structure of the nonlinear least squares problem; so does Pearlmutter's method [15] to the Newton formula, although its modification may be possible.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 8) is not convex with respect to all three unknowns (B, S and Θ) together.
Sentence 2: a good algorithm in general exhibits convergence behavior to a stationary point and we can use Block Coordinate Descent algorithm [1] to iteratively update B, S and Θ as follows: while not convergent do For i = 1, . . . , N , solve min Solve min Solve min end do Since p(x|η) is in the exponential family, the subproblem (10) with respect to B is convex and smooth with simple constraints, for which quasi-Newton methods such as projected L-BFGS [22] are among the most efficient methods.
Relationship: reasoning