sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: The simplest special case is the univariate regression problem (K = 1), in which case the function ζ(β * ) outputs an s-dimensional sign vector with elements z * i = sign(β * i ), so that ψ(β * ) = z * T (Σ SS ) −1 z * = Θ(s).
Sentence 2: the order parameter of block 1 / 2 -regression for univariate regresion is given by Θ(n/(2s log(p − s)), which matches the scaling established in previous work on the Lasso [11].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Motivated by the challenge of computing a Lyapunov function L * such that its induced set of policies contains π * , in this section, we approximate * with an auxiliary constraint cost , which is the largest auxiliary cost that satisfies the Lyapunov condition: L (x) ≥ T π B ,d [L ](x), ∀x ∈ X , and the safety condition L (x 0 ) ≤ d 0 .
Sentence 2: the larger the , the larger the set of policies F L . by choosing the largest such auxiliary cost, we hope to have a better chance of including the optimal policy π * in the set of feasible policies.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, spikes are known to consume a significant fraction of the metabolic budget [30,32] placing a premium on their total number.
Sentence 2: we can postulate that neuronal spike trains find a trade-off between the mean squared error in the decoded spike train relative to the input signal and the total number of spikes, as expressed by the following cost function over a time interval T: where x is the analog input signal, s is the binary spike sequence composed of zeros and ones, and is the linear filter.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 19), and the diagonal elements of D are determined by κ.
Sentence 2: substituting this into the definitions of μj , γi , ηj , ¯ i , we have that after preconditioning using JR if the conditions of Lemma 1 hold for X, β * , they will continue to hold after preconditioning using JR.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We eliminate A and consider , where g(X) := ψ 0 (X) + δ(X|≥ 0), and where each f t (X) for 1 ≤ t ≤ T is defined as where g t (a) := ψ t (a) + δ(a|≥ 0).
Sentence 2: for simplicity, assume that (35) attains its unique 1 minimum, say a * , then f t (X) is differentiable and we have ∇ X f t (X) = (Xa * − y t )(a * ) T . we can instantiate (28), and all we need is a subroutine for solving (35).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The Part-Detector (Section 3.1) performance on our validation set predicts heat-maps that contain many false positives and poses that are anatomically incorrect; for instance when a peak for face detection is unusually far from a peak in the corresponding shoulder detection.
Sentence 2: in spite of the improved Part-Detector context, the feed forward network still has difficulty learning an implicit model of the constraints of the body parts for the full range of body poses.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is because in the proof of Theorem 1, the set D φ has the property that exactly one action V t ∈ D φ is awake in any round t, and t (V t ) = 1[y t = y t ].
Sentence 2: if we consider a ranking where the actions in D φ are ranked at the top positions (in arbitrary order), the loss of this ranking exactly equals the number of errors made by the disjunction φ on the input sequence.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Metric-based methods for model selection are complementary to our approach in that they are desgined to prefer models/algorithms that behave similarly on the labeled and unlabeled data [Sch97, SS02, BC03], while disagreement is a measure of self-consistency on the same dataset (in this paper, unlabeled data only).
Sentence 2: our method is also applicable to scenarios in which the test and training distributions are different.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This Markov chain has as limit operator H (0) aa , as the granularity of the job market increases along with the number of observations.
Sentence 2: in trying to recover the geometry, distribution and vector field, we are actually interested in estimating the full advective effect of the diffusion process generated by H We obtain an embedding of the job market that describes the relative position of jobs, their distribution, and the natural time progression from each job.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Finally, notice that, in view of the algorithm's definition, every b i corresponds to a rejected price.
Sentence 2: by Proposition 3, there exist nodes n i (not necessarily distinct) such that p ni = b i and It is immediate that δ r n ≤ 1/2 and δ l n ≤ 1/2 for any node n, thus, we can write T. The last inequality holds since at most T prices are offered by our algorithm.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To prove our theorem, we construct a joint distribution over pairs (A, P) such that when M is given too small a sample X, and A asks k − 1 statistical queries, then either M does not answer all the queries accurately or A outputs a k-th query q * such that q * (P) − q * (X) > ε.
Sentence 2: m cannot be both accurate and satisfy post hoc generalization.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: More formally, assume that the data elements lie in an unknown ultrametric space pA,distq and so exhibit a natural hierarchical clustering defined by this ultrametric.
Sentence 2: the input is a random graph generated as follows: an edge is added between nodes u,v P A with probability p " f pdistpu,vqq, where f is an (unknown) non-increasing function with range p0,1q.vertices that are very close in the ultrametric (and so very similar) have a higher probability to have an edge between them than vertices that are further apart.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Furthermore, it is easy to construct a value oracle for Φ t simply by using the value oracles for the f τ .
Sentence 2: the optimization in step 3 is poly-time solvable given oracle access to Φ t . While the algorithm itself is a simple extension of Hannan's [5] follow-the-perturbed-leader algorithm, previous analysis (such as Kalai and Vempala [9]), which rely on linearity of the cost functions, cannot be made to work here.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Importantly, the hippocampus can also encode these conjunctions rapidly, and therefore it shares some of the same functionality as the phonological loop mechanism (i.e., rapidly encoding arbitrary conjunctions required for tasks).
Sentence 2: it is likely that the hippocampus and the prefrontal-mediated working memory system (including the phonological loop) are partially redundant with each other, and work together in many tasks .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, we minimize the following normalized cut objective value [10]: In (11), d(V c ) is the volume of the vertex set V c for balancing cluster sizes, and the numerator is the sum of off-diagonal entries, which corresponds to E 2 F . As shown in [10,3], minimizing the normalized cut is equivalent to finding cluster indicators x 1 , .
Sentence 2: . . , x c to maximize where D is a diagonal matrix with a common way for getting cluster indicators is to compute the leading k eigenvectors of D −1/2 S λ D −1/2 and then conduct kmeans on these eigenvectors.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 2), as given by The sufficient condition for the GGN approximation to be exact is that the network makes a perfect prediction for every given sample, that is, J L = 0, or [N ] i stays in the linear region for all i, that is, G L•N has less rank than KT and is positive semidefinite provided that H L is.
Sentence 2: l is chosen to be a convex function so that H L is positive semidefinite.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This implies that for any θ and for all non-zero ϑ, If AdEFGM has minimal representation at θ * , then there must exist sufficiently small constants δ > 0 and β > 0 such that for any θ ∈ {∥θ − θ * ∥ ≤ δ}, ϑ ⊤ ∇ 2 L(θ; X n )ϑ ≥ β∥ϑ∥ 2 .
Sentence 2: assumption 2 holds true when AdEFGM has minimal representation at θ * .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, the learning problem becomes convex once the latent variable h is fixed for positive examples.
Sentence 2: we can train the LSVM by an iterative algorithm that alternates between inferring h on positive examples and optimizing the model parameter w. Dual form with fixed h on positive examples : Due to its nature of non-convexity, it is not straightforward to derive the dual form for the general LSVM.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Rather than relying on learners to discover both sparse features and a visual hierarchy fully automatically, we use external "real-world" knowledge expressed in hierarchical structures to bias which sparsity patterns we want the learned discriminative feature representations to exhibit.
Sentence 2: our end-goal is not any sparsity pattern returned by learners, but the patterns that are in concert with rich high-level semantics.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From the time change property [10] of point processes, if the sequences come from the point process λ(t) , then the integral Λ = tt+1 ti λ(s)ds between consecutive events should be exponential distribution with parameter 1.
Sentence 2: the QQ plot of Λ against exponential distribution with rate 1 should fall approximately along a 45-degree reference line.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A more complex instance of this kind of approach is the Population of Linear Experts (POLE) model [7], in which hidden units each represent different linear functions, but the weights from input to hidden nodes indicate which linear function should be used to make predictions for particular input values.
Sentence 2: the model can learn non-linear functions by identifying a series of local linear approximations, and can even model situations in which people seem to learn different functions in different parts of the input space.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let v⇡ 2 R |S|⇥d be the matrix of gradients (with respect to the policy parameters ✓) of v ⇡ for each state s, and G 2 R |S|⇥d the matrix where each row corresponding to state s is the vector g(s).
Sentence 2: then we obtain We can prove a similar result for the deterministic policy gradient objective, for a deterministic policy, ⇡ : S !
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A useful property of CVaR, which we exploit in this paper, is its alternative dual representation [1]: where E ξ [Z] denotes the ξ-weighted expectation of Z, and the risk envelope U CVaR is given by U CVaR (α, P) = ξ : ξ(ω) ∈ 0, 1 α , ω∈Ω ξ(ω)P(ω)dω = 1 .
Sentence 2: the CVaR of a random variable Z may be interpreted as the worst-case expectation of Z, under a perturbed distribution ξP.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: While all previously mentioned approaches retain certain stability properties of the original representation with respect to common metrics in TDA (such as the Wasserstein or Bottleneck distances), they also share one common drawback: the mapping of topological signatures to a representation that is compatible with existing learning techniques is pre-defined.
Sentence 2: it is fixed and therefore agnostic to any specific learning task.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In addition, the standard of error of E-C Bandit, hLinUCB, Logistic Bandit and PBMUCB's relative CTR performance among 100 trials are 0.032, 0.031, 0.030, 0.041, respectively.
Sentence 2: the variance of our offline evaluation is small and the improvement from our solution to the baselines are statistically significant.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, this method suffers from 1) greatly increased time complexity and computational burden, 2) inability to exhaust all possible dimensionalities and 3) lack of consensus between different functionality tests as their results can differ.
Sentence 2: we need a universal criterion that can reflect the relationship between the dimensionality and quality of word embeddings in order to establish a dimensionality selection procedure for embedding methods.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , R |R| by noting that in moving between instances k and k + 1, only r k is added to the conditioning set.
Sentence 2: by reassigning r k as the root for the BP instance associated with r k+1 (i.e., A ∪ R k as the conditioning set), only the second half of the message passing schedule (disseminating messages from the root to the leaves) is necessary.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For the case of on-policy sampling, the answer here is an affirmative one, as formalized in the following theorem.
Sentence 2: [17], Lemma 6) Let w ⋆ D be the unique solution to where D is the stationary distribution of P . Then for on-policy sampling with linear function approximation, not only does TD converge to its fixed point, but we can also bound the error of its approximation relative to Π D V − V D , the lowest possible approximation error for the class of function approximators.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The input X t has a spatial size of W × H with C in channels, and ConvLSTM has a convolutional kernel size of K × K with C out channels.
Sentence 2: the parameter size of ConvLSTM can be calculated as 3 : The parameter size of ConvLSTM is very large, partly due to the convolutional structures.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This also means that it may not be possible to design efficient online algorithm with sublinear regret without making any assumptions on either A or X because an efficient online algorithm with sublinear regret would imply an efficient algorithm for solving (1) in the offline case.
Sentence 2: we focus on obtaining regret bounds for the dictionary update, assuming that the at each timestep the sparse codes given to the batch and online algorithms are "close".
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let {x(t)} be a set of measurements lying in the moving hyperplane Notice that the main difference between linear regression and hyperplane estimation is that in the latter case the parameter vector b(t) is constrained to lie in the unit sphere S D−1 .
Sentence 2: instead of applying standard gradient descent as in (2), we must follow the negative gradient direction along the geodesic curve in S D−1 passing through b(t).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For instance, when π m is implemented according to x m starting from time τ m , the intensity x m+1 := f (x m , π m (x m )) at time τ m+1 depends on x m and the control π m (x m ), but is also random due to the stochasticity of the process during time [τ m , τ m+1 ).
Sentence 2: the design of π needs to take future uncertainties into considerations.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 2(a) and (b), each solid edge e can only be adjacent to a set of dashed edges, say {e 1 , e 2 , • • • , e K }, in a solid-dashed graph, and no two solid edges are adjacent to each other.
Sentence 2: we can directly merge the solid-edge weight and length to its adjacent dashed edges by shares one vertex with e as in Fig.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Over time, skills represented internally as options and their models also become available to the agent as action choices.
Sentence 2: q B maps states s and actions a (both primitive and options) to the expected long-term utility of taking that action a in state s. Salient Events In our current implementation we assume that the agent has intrinsic or hardwired notions of interesting or "salient" events in its environment.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In our experiments, we observe that even with arguably the state-of-the-art implementation [Geng et   2017], PL without screening still has a significantly larger memory footprint compared with that of NW.
Sentence 2: the experiments for PL without screening are not fully conducted in Figure 1b,1c, and 1d for networks with thousands of nodes.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: That is, if a basis function is learned at one position and scale, then it is assumed to be repeated at all positions (spaced apart by two positions horizontally and vertically) and scales (in octave increments) .
Sentence 2: the entire set of basis functions for tiling a large image may be learned by adapting only a handful of parametersi.e., the wavelet filters and the scaling function that is used to expand them across scale.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: considered as members of another class which represents all classes of D u , making the total number of classes for the purpose of this evaluation C + 1.
Sentence 2: this measure can be highly sensitive to the dataset bias since the number of samples belonging to D u may be many times higher than those belonging to D c . One may argue that a simplified weighted accuracy might solve this issue but it still would not be able to provide a measure of the number of samples from D u that were classified as belonging to a known class c ∈ C. Algorithms are Incomparable: Since confidences across algorithms cannot be meaningfully related, comparing them based on their individual confidences becomes tricky.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When a ij = 1, no matter q jz is 1 or 0, we have Suppose the groups found by the base models have balanced size, i.e., n i=1 a ij = γ where γ is a constant for ∀j.
Sentence 2: then the objective function can be approximated as: when the classification and clustering algorithms generate balanced groups, with the same set of constraints in Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The leading contribution to µ is then given by µ(x; J) ∼ J • tanh(βx) as β goes to zero; The expression is valid in the asymptotic region L 1 for a fixed R. Then, the formula for the delta function yields [8]  where we have used ρ(x; x) = x − tanh(βx).
Sentence 2: we have for given β 2 C. Inserting (12), ( 13) into (11), we get where we rewrite x = βx.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: One can verify that the following equality holds: In view of that, it follows that ( 7) is equal to for any coordinate j 2 [1, N].
Sentence 2: for any coordinate j 2 [1, N], by Hölder's inequality, we have ) 2 is the empirical variance of the j-th coordinate and where r = sup y (y) 2 is the empirical `1-norm radius of the data.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, for F (X) = log det X and A(θ) ∈ R d×d , the exact computation of ∇ θ F (A(θ)) can take as much as O(d 3 k), where k is the number of parameters in θ.
Sentence 2: it is desirable to avoid computing, or at the very least reduce the number of times we compute, the gradient of F (A(θ)) exactly.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider sets S and S 0 , and assume g(•), b(•), and p(•) are such that Since s P  t P for S ✓ T , we obtain f S (e) f T (e) 0.
Sentence 2: it is also easy to see that we get f T (e) f S (e) f T (e) (1 c)f S (e) 0 for all S ✓ T and e 6 2 T . Finally, for poor elements,  We consider several benchmarks.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let v k (s, a) be the number of times action a has been chosen in episode k when being in state s, and denote the total number of episodes by m. Then setting ∆ k := s,a v k (s, a)(ρ * − r(s, a)), with probability at least 1 − δ 12T 5/4 the regret of UCCRL after T steps is upper bounded by (cf.
Sentence 2: section 4.1 of [11]), Next, we consider the regret incurred when the true MDP M is not contained in the set of plausible MDPs M k . fix a state-action pair (s, a), and recall that r(s, a) and pagg (•|s, a) are the estimates for rewards and transition probabilities calculated from all samples of state-action pairs contained in the same interval I(s).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Following the KM, we use a Gaussian random matrix M for the memory, and approximate samples of M using its mean R. We use subscripts t for M t , R t and U t to distinguish the memory or parameters after the online update at the t'th step when necessary.
Sentence 2: we use a neural network encoder e(x) → z to deterministically map an external input x to embedding z.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Notably, it was found that when a rodent was moving passively, i.e., a situation where no internal motor command is available, the head-direction encoded by ADN cells was still leading the actual position of the rodent's head by around ∼ 50ms, even larger than that in a free-moving condition [5].
Sentence 2: extra anticipation strategies may exist in neural systems.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In other words, marginalizing over an observation and its associated marginal's parameter (which can be taken to be x N and θ N without loss of generality) still produces a similar isd posterior on the remaining observations X /N and parameters Θ /N .
Sentence 2: we need: We then would recover the posterior formed using the formula in Equation 1 with only N − 1 observations and N − 1 models.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First, it is easy to see that when z = 0, these polynomials attain a value of unity (i.e., Q m (0) = 1).
Sentence 2: secondly, it can be shown that for any s and z ∈ choosing the approximate solution as 7 d (m) s = −H(λ s ) −1 (I − Q m (H(λ s ))) g s guarantees an -close solution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given a bandwidth σ > 0, for a set Θ, one can smooth and normalize µ Dg as follows, where Z = Θ u∈Dg N(x; u, σI)dx, N is a Gaussian function and I is an identity matrix.
Sentence 2: each PD can be regarded as a point in a probability simplex In case, one chooses Θ as an entire Euclidean space, each PD turns into a probability distribution as in [Anirudh et al., 2016, Adams et al., 2017.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Surrogate Upper Bound of Composite System: We can get better insight for the first two terms of the objective in (OPT) if we view z ∈ {0, 1} as a latent variable and consider the composite system Pr(y|x) = z Pr(z|x; g) Pr(y|x, f z ).
Sentence 2: a standard application of Jensen's inequality reveals that, − log(Pr(y|x)) ≤ E q(z|x) (f z (x), y) + D KL (q(z|x) Pr(z|x; the conditionalentropy of the composite system is bounded by the expected value of our loss function (we overload notation and represent random-variables in lower-case format): This implies that the first two terms of our objective attempt to bound the loss of the composite system; the third term in the objective together with the constraint serve to enforce budget limits on the composite system.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To see the importance of this proxy, let us first describe a different approach to the problem at hand that in particular extends directly the approach of [8] who show that if an upper-bound on V T is known in advance, then the optimal regret is attainable.
Sentence 2: one might guess that having an unbiased estimator for V T will eliminate the need in this prior knowledge.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Superposition: Sample F ∼ DP(ν), and sample (c D , c F ) ∼ Dir(T qµ t−1 (Ω), ν(Ω)).
Sentence 2: then set D t to be the union of (θ, c D π) for all (θ, π) ∈ D t and (θ, c F π) for all (θ, π) ∈ F . D t is a random convex combination of D t and F , where If the DDP is used as a prior over a mixture model, these three operations allow new mixture components to arise over time, and old mixture components to exhibit dynamics and perhaps disappear over time.
Relationship: reasoning