sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Assume m ≥ 12 and n t, we have, with a probability at least 1 − 2e −t , Remark Theorem 3 implies that if the empirical loss L (w * , S) ≤ O(log m/m), for most positive instance x + (i.e., 1 − O(log m/m)), the percentage of negative instances ranked above x + is upper bounded by O( log m/n).
Sentence 2: we observe that m and n play different roles in the bound; that is, because the empirical loss compares the positive instances to the negative instance with the largest score, it usually grows significantly slower with increasing n. For instance, the largest absolute value of Gaussian random samples grows in log n. we believe that the main effect of increasing n in our bound is to reduce δ (decrease at the rate of 1/ √ n), especially when n is large.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For any ⌘, ⌘ ′ > 0, there exists z ∈ , with z k ≠ 0 for all k ∈ [p], such that the following holds for the distribution-weighted combining rule h Lemma 6 indicates that for the solution z, h ⌘ z has essentially the same loss on all source domains.
Sentence 2: our problem consists of finding a parameter z verifying this property.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We will use the following notation: Note that q t , v t satisfy the Bellman equations underlying π t , P and r t . For reasons to be made clear later in the paper, we shall need the state distribution at time step t given that we start from the state-action pair (x, a) at time t − N , conditioned on the policies used between time steps t − N and t: It will be useful to view µ N t as a matrix of dimensions |X × A| × |X |.
Sentence 2: µ N t,x,a (•) will be viewed as one row of this matrix.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Indeed, since all of the squared MMD estimates calculated on the training data using each of the base kernels are negative, it is unlikely the statistic computed on the data used for the test will exceed the (always positive) threshold.
Sentence 2: when no entries in η are positive, we (arbitrarily) select a single base kernel k u with largest ηu /σ u,λ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given a point (x, y) = (µ 1 (H), µ 2 (H)) on the graph in Figure 2, where µ i (H) is the probability by which player i plays Heads, we know that our expected reward is  We wish to show that We consider each part separately.
Sentence 2: in the losing section, we let g(t) = x = t and h(t Similarly, we can show that we receive 1/4 reward over C w . C R 1 (x, y)dt = 1/3 > 0, and we have shown that we receive a payoff greater than the Nash equilibrium payoff of zero over every cycle.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, we define the set of all local neighborhoods as N B {(i, j)| i,j = 1, ρ(x i , x j ) ≤ γ} .
Sentence 2: we minimize the following objective function Note that we do not need to have the same metric in both the input and the feature space, they do not even have in general the same dimension.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since in general normalized cuts are thought to be superior to mincuts, the transductive inference approach that we used in the later experiments is built on the above spectral hypergraph clustering method.
Sentence 2: as shown in [20], with the least square loss function, the classification function is finally given by f = (I − ξΘ) −1 y, where the elements of y denote the initial labels, and ξ is a parameter in (0, 1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Re-arranging, we have: L + I = mI − B.
Sentence 2: since λ n (B/m) < 1, we can apply the Neumann series representation [Suhubi, 2003, page 390] to this expression: we can re-write f (S) as an infinite sum of traces of matrix powers: Note that B i S here means (B S ) i and not (B i ) S . Submodularity.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Nevertheless, its application to continuous distributions requires discretization of these distributions, leading to computational intractability when one desires good accuracy and, hence, has to use fine discretization with large n, which leads to the necessity of solving an optimization problem of large dimension.
Sentence 2: this approach is not directly applicable in our setting of continuous distributions, and it is not clear whether it is applicable in the decentralized distributed setting with arbitrary networks.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We now provide an unbiased estimator of the expected generalization error of θn , which generalized the Akaike information criterion [22] (for a proof, see [23]): Proposition 2 In addition to the assumptions of Proposition 1, we assume that where The criterion G is a sum of two terms: the second term corresponds to a variance term and will converge to zero in probability at rate O(n −1 ); the first term, however, which corresponds to a selection bias induced by a specific choice of sampling distributions, will not always converge to the minimum possible value L u (θ 0 ).
Sentence 2: in order to ensure that our active learning method are consistent, we have to ensure that this first term is going to its minimum value.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that the bound (10) consists of two main terms: the first term captures the geometric convergence of the population EM operator from Theorem 1, while the second term characterizes the radius of convergence in terms of sample complexity, which is O( 1/n).
Sentence 2: with probability at least 1 − δ, we have where c, c are universal constants.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is wellknown that a Markov process on variables x will converge to a desired equilibrium distribution 7r(x) if the transition probabilities obey the detailed balance conditions, w(x'lx)7r(x) = w(xlx')7r(x'), (10) where w(x'lx) is the probability of generating x' from x.
Sentence 2: any search algorithm searching on a flat space should have dynamics which obeys, and PEIL does not obey this.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It follows from a result of Cesa-Bianchi and Lugosi [1] that for some constant B.
Sentence 2: by Markov's inequality we conclude that If |||M ||| * = ∥D∥ * = o (n/(q + log(pq))) 1/4 , then returning to (6.1), this gives us a bound on R c (M )− Rc (M ) that is o P (1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In fact, we find that these features are better face detectors than a linear filter trained from labeled data, achieving up to 86% AUC compared to 77% on labeled validation data.
Sentence 2: our results emphasize that not only can unsupervised learning algorithms discover object-selective features with no labeled data, but that such features can potentially perform better than basic supervised detectors due to their deep architecture.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Further, on the duration problem in the CAP data sets, PBCMC also achieved the reported accuracy using λ = 5000 and φ = 10 −2 , and on the location problem it acheived negligibly better accuracy using λ = 5000 and φ = 10 −3 or φ = 10 −4 .
Sentence 2: simply using an even more stable parameterization of PBCMC did not improve performance on these problems either.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We use a probabilistic argument to switch from the randomization over F 0 to the randomization over S ⇠ D and show that there exists a deterministic F ✓ F such that f i (S) = f j (S) for all f i , f j 2 F w.h.p.
Sentence 2: over S ⇠ D. By a union bound this holds for all samples S. for such a family of functions F = {f 1 , .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This observation is based on an important property of Newton-like methods, namely that there is much more freedom in choosing a Hessian approximation than in computing the gradient [7,3].
Sentence 2: a smaller sample O k can be employed for updating the inverse Hessian approximation H k than for computing the batch gradient g S k k in the search direction −H k g S k k . In summary, by ensuring that unresponsive nodes do not constitute the vast majority of all working nodes in a fault-tolerant parallel implementation, or by exerting a small degree of control over the creation of the samples S k in the multi-batch method, one can design a robust method that naturally builds upon the fundamental properties of BFGS updating.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We used the following conjugate priors (Dirichlet distributions) over θ l and α l,m as: Here, ξ and ζ are hyperparameters and in this paper we set ξ = 2 and ζ = 2, each of which is equivalent to Laplace smoothing for θ l,i and α l,m , respectively.
Sentence 2: the objective function to find Θmap is given by Of course, the third term on the RHS of Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since the predictor might lose information with respect to these features in order to yield the predictions, only the features from the encoder go through the decoder to be merged back into the main path (as illustrated in Figure 2 and formalized in Equation ( 1)), so that more information is kept for use in the primary task.
Sentence 2: all parameters learned in the predictor are to be thrown away after training, i.e.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 1 CF-2g closely resembles the serial Seq-2g, but the elements e 2 V are no longer processed in a fixed order.
Sentence 2: the sets A, B are replaced by potentially stale "bounds" Â, B, where Â is a subset of the "true" A and B is a superset Intuitively, the bounds hold because A e , B e are stale versions of A ι(e)−1 , B ι(e)−1 , which are monotonically non-decreasing and non-increasing sets.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Suppose now that the induction holds for some t 1.
Sentence 2: since by definition of v t , subtracting ⌘t v t from x t can only decrease positive entries in x t (see proof of Observation 2), and both v t , v + t are vertices of P (and thus in {0, 1} n ), and ⌘t = 2 t , it follows that each entry i in x t+1 is given by: x t+1 can also be written in the form 2 t st+1 for some st+1 2 N n . By definition of t and the monotonicity of {⌘ t } t 1 , we have that 2 t 2 t+1 is a positive integer.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In addition to p i -denoting the fact that wise man i wears a red hat -to model each agent's individual knowledge, we need to use a modality K j , j ∈ {1, 2, 3}, which represents the relative notion of knowledge at each time point t 1 , t 2 , t 3 .
Sentence 2: k j p i denotes the fact that agent j knows that agent i wears a red hat.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We start from an existing probabilistic logic programming language, ProbLog [De Raedt et al., 2007], and extend it with the capability to process neural predicates.
Sentence 2: the idea is simple: in a probabilistic logic, atomic expressions of the form q(t 1 , ..., t n ) (aka tuples in a relational database) have a probability p. the output of neural network components can be encapsulated in the form of "neural" predicates as long as the output of the neural network on an atomic expression can be interpreted as a probability.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , x M are conditionally independent given y, then their covariance matrix is diagonal, i.e., cov(x, x|y) = diag(y).
Sentence 2: by the law of total covariance, the covariance in ( 6) has the form where the last equality follows by the multilinearity property of cumulants (see Appendix C.1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Specifically, if we have K observations of x, denoted x (k) , k = 1, ..., K, then the estimate of µ i , denoted μi , is given by We can still use the maximum entropy formulation described above; the only difference is that we replace µ by μ.
Sentence 2: the maximum entropy distribution is given by q(x| μ) (Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As can be seen from the proof, this constant has no special meaning and can be replaced by any constant strictly larger than 1 at the cost of increasing the constant C. Note that the O P (1/ √ n) rate obtained by Biau and Bleakley (2006) has a factor 1 in front of the bias term.
Sentence 2: theorem 1 provides a significant improvement unless the bias overly dominates the second term of the bound (i.e., the complexity of G is too small).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Benchmark dataset: Except for synthesizing natural looking profile view face images, the proposed DA-GAN also aims to generate identity preserving face images for accurate face-centric analysis with state-of-the-art deep learning models.
Sentence 2: we evaluate the possibility of "recognition via generation" of DA-GAN on the most challenging unconstrained face recognition benchmark dataset IJB-A (15).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since at least half of the Q measure of the classifiers err on an example incorrectly classified by the majority vote, it follows that the error rate of the majority vote is at most twice the error rate of the Gibbs classifier.
Sentence 2: given enough training data, the PAC-Bayes theorem will give a small risk bound on the majority vote classifier only when the risk of the Gibbs classifier is small.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To express such negative costs in a graph cut, we point out an equivalent formulation with positive weights: since m(V) is constant, minimizing m(S) = i∈S m(i) is equivalent to minimizing the shifted function m(S) − m(V) = −m(V \ S).
Sentence 2: we instead minimize the sum of positive weights on the complement of the solution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Clearly, F is defined up to a scale factor: if F 33 = 0, one can assume that F 33 = 1.
Sentence 2: each pair x ↔ x of matching points in images I and I yields a linear constraint on the eight remaining coefficients of F. Because of the quantification and the presence of noise in images, these linear relations are satisfied up to some error.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since such beliefs are modeled as Gaussians, this probability has a unique global maximum at µ b = µ c and Σ N → 0 + .
Sentence 2: a final cost J final = w 0 µ b − µ c 2 2 + w 1 tr Σ N can approximate the negated log-probability of successfully catching the ball while rendering the optimal control problem solvable.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By conditional expectation, Thus, we can see that the key idea of decoupling L(a, w) into Q L dec (a, a ′ , w), is essentially "recoupling" Q L dec (g 1 , g 2 , w) via conditional expectation -the "recoupled" term L can be viewed as an approximation of L(a, w).
Sentence 2: notice that by Fact (ii), for any convex function φ, by choosing φ(t) = |t| p , we can control moments of sup w∈S L(a, w) via sup w∈S L(a, w) For tetrahedral polynomials, L = L, so the approximation is exact.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Such a measure could assign absolute numbers to each of the possibilities, but any such static assignment would make it impossible for the propositions u(x 1 , X ) > u(x 2 , X ) and u(x 1 , X ) < u(x 2 , X ) to hold simultaneously, as is desired of a context-sensitive utility measure.
Sentence 2: we see that it is impossible to design a utility function u such that u : X × C → R. If we wish to incorporate the effects of context variation on the desirability of a particular world possibility, we must abandon a foundational premise of existing statistical decision theory -the representational validity of absolute utility.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: More precisely, there exists π * ∈ Π HR , (p * , r * ) ∈ s∈S Ps such that  This leads to sup π∈Π HR w π, µ * , (s ini ) = w π * , µ * , (s ini ) = inf µ∈CS w π * , µ, (s ini ) .
Sentence 2: part (ii) of Theorem 1 holds.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The results shown in figure (1) demonstrate that with a linear model the likelihood function will display a peak corresponding to each source location, but with the multiplicative model only a spurious location that is consistent among the kernels remains and information about the two sources is lost.
Sentence 2: we use a model in which time difference and level difference information is first combined multiplicatively within frequency channels and is then summed across frequency.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since dj measures how "connected" point j is to other points in the same cluster, it will be dj = O(n) in this case, so the sum, which is over 0(n 2 ) terms, is in turn divided by djdk = O(n 2 ) .
Sentence 2: as long as the individual Ajk's are small, the sum will also be small, and the assumption will hold with small fl.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For the full breaking, adjacent breaking, and any top-k breaking, the three statements in Theorem 2 are equivalent for any data D. For any position-k (with k ≥ 2) and any bottom-k (with k ≤ m − 1), 1 and 2 are not equivalent to 3 for some data D. Ford, Jr. [6] identified a necessary and sufficient condition on data D for the MLE under PL to be unique, which is equivalent to condition 1 in Theorem 2.
Sentence 2: we have the following corollary.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If an internet user clicks uniformly at random over the outgoing links, then the probability that the user will click on page i given she is currently on page j is where the denominator is the out-degree or number of web pages that page j links to.
Sentence 2: m is the transition matrix of a Markov chain and under mild conditions, the probability that a "random surfer" will be on any page regardless of where she starts is given by the vector p that solves p = Mp.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If H is a graph and p ≥ 1, Ω H,p reduces to the Laplacian regularization Ω G,p .
Sentence 2: note that for characteristic functions of sets, the difference between the hypergraph cut and its approximations such as clique and star expansion carries over to Ω H,p and Ω G CE ,p , respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Before seeing the preferences ¢ , the vector £ ¢ ¥ £ ¢¡ forms a joint prior Gaussian derived from the GP.
Sentence 2: after incorporating the ¢ information, the posterior mean of £ is where the user preference function for a song s, £ ¡ 0) , is a linear blend of kernels ¡ £ ¢ §¥ £ ¡ 0) § that compare the metadata vector for song ) with the metadata vectors £ £¢ for the songs that the user expressed a preference.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To track to the previous frame (t − 1), a backward dense optical flow field Since the optical flow field is typically real-valued, the sampling position p t+τ becomes fractional.
Sentence 2: the corresponding feature x( p t+τ ) is derived via interpolation with a specific sampling kernel G, written as In this paper, we will not go deeper into the usage of different choices of sampling kernels G and use the bilinear interpolation as default.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Furthermore, all X feasible for the SDP have identical trace equal to n, so that the search space of the SDP is compact.
Sentence 2: our results stated below apply: For p = √ 2n , for almost all C, even though (Max-Cut BM) is non-convex, any local optimum Y is a global optimum (and so is X = Y Y ), and all saddle points have an escape (the Hessian has a negative eigenvalue).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, in our case z is a Gaussian random variable, which means (f (z; w), y) is now itself a random variable given a label y.
Sentence 2: we use the expectation E z∼N (µ,Σ;θ) (f (z; w), y) as the overall loss between the label y and a time series S given its Gaussian representation N (µ, Σ; θ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus, in stark contrast to the setting considered here, the size of a τ -divergence ambiguity set has no impact on the corresponding optimal estimator.
Sentence 2: moreover, the least favorable prior Q = N d (µ, S ) for a τ -divergence ambiguity set always satisfies in order to harm the statistician, nature only perturbs the second moments of the signal but sets all second moments of the observation as well as all cross moments to their nominal values.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore using equation (3.9) and equation above, we have that Note that since g | α has Gaussian distribution, we have, p g|α (0) = (2π) −d/2 ( α 6 6 ) −d/2 .
Sentence 2: thus using two equations above, we can bound E [h(α)] by it suffices to control the RHS of (3.10), which is much easier than the original Kac-Rice formula.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A key observation from Theorem 2 is that the optimal multilabel classifier can be obtained by thresholding the marginal instance-conditional probability for each label P(Y m = 1|x) and, importantly, that the optimal classifiers for all the labels share the same threshold ⇤ .
Sentence 2: the effect of the joint distribution is only in the threshold parameter.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For p ij = 1/2, we need about Θ( log T D(p ij ||1/2) ) time-slots to distinguish it from 1/2 [5].
Sentence 2: when choosing i as the first candidate, the regret of comparing it with all other arms can be estimated by R( 1) ij (t)||1/2).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Also, ||x|| 2 ≤ ||x|| 2 ≤ p, so we obtain the desired bound.
Sentence 2: proof sketch of Theorem 2: Under the same notation as the previous proof, notice that Proof sketch of Corollary 1: Let * = (1 − γ) log 2 p. Since i = 2 −i(1−β) ∀i < L − * + 1 and i = 0 otherwise, we have for ≥ * and since L = log 2 p, λ ≥ 2 β(L− * ) 2 β−1 = p βγ 2 β−1 , which is increasing in p. we can pick k = 2 * and since 2 * /p = p −γ , the result follows.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, the tensor factorization algorithm returns a set of vectors {( µ h , w h ) : h = 1, . . . , k}, where each ( µ h , w h ) estimates a particular column of C c (for some µ cl ) and a particular diagonal entry of W (for some w l ).
Sentence 2: it is important to note that the tensor factorization algorithm doesn't provide a one-to-one correspondence between the recovered column and the true columns of C c . µ 1 , .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Second, the coarser time granularity of the high level process implies fewer resampling steps and fewer opportunities for particle depletion.
Sentence 2: if we can model how much drift is expected to occur over a small section of the robot's trajectory, we can maintain this extra uncertainty longer, and resolve inaccuracies or ambiguities in the map in a natural fashion.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Because of the flexibility this allows in constructing Σ s , and therefore Σ b , some additional notation is required to proceed.
Sentence 2: a new decomposition of Σ b is defined as where Also, using commutative properties of the trace operator, L(γ) only depends on the data B through the d b ×d b sample correlation matrix BB T . to reduce the computational burden, we replace B with a matrix B ∈ d b ×rank(B) such that B B T = BB T . This removes any per-iteration dependency on n, which can potentially be large, without altering that actual cost function.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If (X , H) is neighborly, then GBS requires n 0 = O(log |H|) queries in noiseless conditions [Now08].
Sentence 2: under the conditions of Theorem 2, the query complexity of the simple noise-tolerant GBS algorithm is O(log |H| log log δ ), a logarithmic factor worse than the optimal query complexity.
Relationship: reasoning