sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Further assume that there exists a constant r > 0 such that Φ is differentiable for any ν such that ν − µ ∞ ≤ r. Then, a subset of a linear subspace.
Sentence 2: for −Θ t , −Θ t−1 ∈ V , w t+1 = w t . Hence, by Theorem 3.9, On the other hand, note that f i ∞ ≤ M . Then where the last inequality is due to Hoeffding's inequality.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The goal in many works in revenue maximization is to find a price p such that r * − r(p) ≤ , or alternatively, to bound r * /r(p).
Sentence 2: given a revenue-estimation error n , one can clearly maximize the revenue within an additive error of 2 n by simply posting a price p * n ∈ arg max p r n (p), thereby attaining revenue good revenue estimation implies good revenue maximization.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then by introducing the variable u = p 1 n 1, the objective in problem (7) satisfies hp, zi = z + hu, zi = z + hu, z zi because hu, 1i = 0.
Sentence 2: thus problem ( 7) is equivalent to solving Notably, by the Cauchy-Schwarz inequality, we have hu, z zi  p 2⇢ kz zk 2 /n = p 2⇢s 2 n /n, and equality is attained if and only if Of course, it is possible to choose such u i while satisfying the constraint if inequality (8) holds for the vector z-that is, there is enough variance in z-we have For losses `(✓, X) with enough variance relative to `(✓, X that is, those satisfying inequality (8), then, we have A slight elaboration of this argument, coupled with the application of a few concentration inequalities, yields the next theorem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As n becomes large-i.e., the spike count is obtained over a larger period of time-the binomial is well approximated by a Gaussian: Using the Gaussian approximation, Equation 1 can be rewritten in the form of a logistic function: , where and fxi = f xi /n denotes the observed spike rate for a feature detector.
Sentence 2: because of the logistic relationship, if attentional priority is given to locations in order of their target probability, P (T x |F x , ρ), then it is equivalent to rank using r x + n 2 s x .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By Lemma 3, every d-cell in the partition induced by the k objects corresponds to an equally probable ranking of those objects.
Sentence 2: the probability that a query is ambiguous is the number of cells intersected by the corresponding hyperplane divided by the total number of d-cells, and therefore A(k, d, U ) = P (k,d) Q(k,d) .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Take y = x * + ( /κ) ĝt , and observe that y − x * ≤ /κ.
Sentence 2: the ( , κ, x * )-SLQC assumption implies that ĝt , y − x t ≤ 0, and therefore Setting η = /κ, the above implies, after T iterations for which f It can be shown that strict-quasi-convexity and (G, /G, x * )-local-Lipschitzness of f implies that f is ( , G, x * )-SLQC ∀x ∈ R d , ∀ ≥ 0, and x * ∈ arg min x∈R d f (x).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For preprocessing method A, normalization generally produced a small improvement.
Sentence 2: promising results can be obtained using the average ERP in response to standard beeps, using ICA followed by normalization (fourth results column): error rates of 5-15% for some some subjects are comparable with the performance of, for example, well-trained patients in an SCP paradigm [1], and correspond to information transfer rates of 0.4-0.7 bits per trial (say, 4-7 bits per minute).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consistent with [31], the first hidden layer with L 1 or L 2 regularization is capable of keeping the interactions separated, but at the second hidden layer, nearly all hidden units entangle the interactions at every threshold when at least one of the interactions is modeled.
Sentence 2: the regularization does very little to prevent the pairwise interactions from entangling within the neural network.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Obtaining E t from A t can be achieved by the polar-decomposition, which is a cubic-time operation: currently there are no algorithms known that can update an existing polar decomposition from an updated Cholesky factor in less than cubic time.
Sentence 2: if our goal is to apply the fast Cholesky update, we have to perform the update without this correction factor This introduces some error, but we will show in the following that we can expect this error to be small and to decrease over time as the algorithm converges to the optimum.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The results of the previous section imply that even though the discriminative logistic regression algorithm has a lower asymptotic error, the generative naive Bayes classifier may also converge more quickly to its (higher) asymptotic error.
Sentence 2: as the number of training examples m is increased, one would expect generative naive Bayes to initially do better, but for discriminative logistic regression to eventually catch up to, and quite likely overtake, the performance of naive Bayes.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As a consequence, we have to use a different Rademacher complexity with the absolute value inside the supremum [35,36], whose contraction makes the coefficients of ( 15) doubled compared with Lemma 8 of [19]; moreover, we have to assume G is closed under negation to change back to the standard Rademacher complexity without the absolute value [28].
Sentence 2: the differences of ( 13) and ( 14) are mainly due to different proof techniques and cannot reflect the intrinsic differences of empirical risk minimizers.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Importantly, it can be shown that the updated one still satisfies the conditions of Theorem 2 for large n. Since the bound given in Theorem 2 is uniform, the inequality ( 8) can be recursively applied to the sequence of f (t) (t = 1, 2, . . . ).
Sentence 2: , we have that where poly(d, K) means a polynomial of d, K. if s 2 < 1 and n is sufficiently large compared with d and K, then the second term is smaller than the first term and we have S n ≤ Cn − 1 1+s with a constant C. Furthermore, we can bound the L 2 -norm from the true one as in the following theorem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Figure 1(b) illustrates how belief state B can be reached in one step from tangent belief state B , and how the backed up value function improves B 's current value.
Sentence 2: if we add a node to the controller that maximizes the value of B , its improved value can subsequently be backed up to the tangent belief state B , breaking out of the local optimum.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Next, we compute the variability in RKHS norm of a set of conditional mean embeddings as the deviance of the RKHS norms of that set.
Sentence 2: using the KCDC measure S X→Y with we compute the deviance in RKHS norm of the set {p(Y |X = x i )} n i=1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For this, we need the following result: Lemma  This result establishes that, when C t converges to a certain shape (but not necessary to a certain scaling), A t and thus E t will also converge (up to scaling).
Sentence 2: thus, as we only need the norm of p σ,t+1 , we can rotate the coordinate system and by multiplying with E t we obtain if E t E T t−1 t→∞ −→ I, the error in the norm will also vanish due to the exponential weighting in the summation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The corresponding probability bounds c d,x,y are 0.1, 0.05, 0 (no dependence on d, x, y).
Sentence 2: and in the same broad spirit as for COMPAS, decreases in income, small changes in age, and small increases in education (events (1), ( 2)) are permitted with small probabilities, while larger changes in age and education (event (3)) are not allowed at all.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A practitioner asking by how much two samples differ does not want to receive a blizzard of answers for every conceivable kernel and bandwidth setting, but a single measure that satisfies some "reasonable" notion of distance across the family of kernels considered.
Sentence 2: in Section 4, we propose a generalization of the MMD, yielding a new distance measure between P and Q defined as which is the maximal RKHS distance between P and Q over a family, K of positive definite kernels.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Initialize() sets the initial g-values of all vertices to infinity and sets their rhs-values according to Equation 1¡ 03-04¢ .
Sentence 2: initially E GF IH QP RF is the only locally inconsistent vertex and is inserted into the otherwise empty priority queue with a key calculated according to Equation 2¡ 05¢ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The motion energies from two neighboring locations are considered as the retinal images to the left and right eyes.
Sentence 2: we can construct a differential motion opponency by placing two populations of phase shift motion energy neurons at different spatial locations and the energy ( ) v E Δ Θ of the opponency is the squared modulus of the averaged phase shift motion energies over space and phase, where ( ) , , w x y Θ is the profile for differential motion opponency and v Δ is the velocity difference between the two spatial regions defined by the kernel ( ) , , w x y Θ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: YGGDRASIL, on the other hand, scales well up to D = 20, for which it runs up to 6× faster.
Sentence 2: for the Yahoo dataset, YGGDRASIL's speed-up is even greater because of the higher number of features p -recall that the communication cost for PLANET is proportional to 2 D and p. for D = 18, YGGDRASIL is up to 24× faster than Spark MLLIB.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Proof of Theorem 4.5: A and g B defined in Lemma 4.8.
Sentence 2: it follows from Lemma 4.6 and Lemma 4.7 that for V 2 R n⇥d with V T V = I we have the Fair PCA problem is equivalent to We proceed to prove the claim by contradiction.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Equation (3) defined a prior on s with parameters W . We now add a hyperprior on W under which the expected power of both tangential components at the vth voxel is determined by a hyperparameter α v . For notational convenience we collect the α v into a vector α and introduce a d s × d v indicator matrix J, with J iv = 1 if the ith source is located in the vth voxel and 0 otherwise.
Sentence 2: each column of J contains exactly two unit entries, one for each tangential component of the corresponding voxel dipole.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We thus have and where in the last line, we use the Taylor expansion again to expand f around Q k c + 1 − Q 2 k e and the bracket • denotes the average over two independent random variables c ∼ P c and e ∼ N (0, 1).
Sentence 2: we have where the function G(Q) is defined in (11).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The Optimization Problem It has been argued in [3,18,11,17] and finally shown in [5] that AdaBoost and Logistic Regression under certain condition asymptotically converge to a combined hypothesis minimizing the respective loss on the training sample, where is a linear combination of hypotheses from , i.e. .
Sentence 2: they solve the optimization problem: where we defined a matrix with .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As we mentioned before, we would like to promote low-dimensional vectors for regularization that are not strongly correlated in high-dimensional space h, therefore we propose the function exp(−|a|/β) that takes the highest values for a close to 0.
Sentence 2: we promote the pairs of vectors in the criterion L M AC for which distances are around M/2 and put less force to the pairs for which the distances in high-dimensional space are close to 0 and M . While optimizing L M AC in each iteration of gradient method we assume that b h,k are constant and calculated from the h(x) layer of discriminator by application of the sign(•) function.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Under the conditions of Theorem 4.1, for any M ∈ M \ Q, (z, η) is not M-proper.
Sentence 2: proof Since the columns of V are in the ambiguity sets and form a basis of R d , span(∪ c−1 j=0 A j ) = R d . the n-th column of any arbitrary M can be represented as m n = c j=1 α n,j w j for some w j ∈ A j and some coefficients α nj .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If ρ(r, r(A)) and ρ(c, c(A)) are both at most 1, then under the assumption that r(A) − r 1 + c(A) − c 1 > ε , our progress is at least Likewise, if either ρ(r, r(A)) or ρ(c, c(A)) is larger than 1, our progress is at least 1/2n ≥ 1 28n ε 2 .
Sentence 2: we terminate in at most 28nε −2 log(s/ ) iterations.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In Figures 2a and 2b, we cut the plot in 50% because, in these two databases, the discrete attributes present a mode value that is present for more than 80% of the instances.
Sentence 2: the SIBP and the BPMF algorithms assign probability close to one to the mode, which results in an artificial increase in the average test log-likelihood for larger percentages of missing data.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The stationary condition for q( w qm , s qm ) is where Z is a normalizing constant that does not depend on { w qm , s qm }.
Sentence 2: we have q( w qm |s qm = 0) ∝ q( w qm , s , where C = e log p(Y, wqm,sqm=0,Θ) q(Θ) is a constant that does not depend on w qm .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 2), can be written as , and shows that PCA will maximize variance, no matter whether this variance comes about through a single marginalized variance, or through mixtures thereof.
Sentence 2: we need to modify this objective function such that solutions w that do not mix variances-thereby falling along one of the axes in x-space-are favored over solutions w that fall into the center in x-space.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Previous work in CDec-POMDPs develops an actor-critic RL approach when the joint reward is additively decomposable among agents [25], and is unable to address non-decomposable team rewards.
Sentence 2: we develop multiple actor-critic approaches for the general team reward setting where some (or all) joint-reward component may be non-decomposable among agents.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The random variable K t , which denotes the index of the current theoretical peak index, is a deterministic function of its parents, such that p(K t = K t−1 + δ t |K t−1 , δ t ) = 1.
Sentence 2: δ t > 1 corresponds to the deletion of δ t − 1 theoretical peaks.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: M be the best linear predictor of i,j|S to be the entry in β Note that in general β , where Pa (k) (j) denotes the parents of node j in G (k) .
Sentence 2: when ij , then there exists a set S such that β (k) i,j|S stays invariant across k = {1, 2}.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus, synchrony between A and B occurs whenever N A * F L = N B * F R , i.e., for a specific set of filter pairs (F L , F R ).
Sentence 2: in our model, sounds presented at a given location induce specific synchrony patterns, which then activate a specific assembly of postsynaptic neurons (coincidence detection neurons), in a way that is independent of the source signal (see Goodman and Brette, in press).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that if we examine the posterior distributions of the weights we see that if each x n contains only one nonzero value then each factor in the likelihood is dependent on only one of the K weights and can be multiplied into the corresponding k th spike-and-slab.
Sentence 2: since the product of a spike-and-slab and a Gaussian is still a spike-and-slab, if we stimulate only one neuron at each trial then this posterior is also spike-and-slab, and the variational approximation becomes exact in this limit.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Using this inequality we obtain Note that L is unbiased and g does not depend on S i or S j .
Sentence 2: for every fixed setting of S i and z, E Sj ∼P g(S i ) • L(S j←z , S j ) = g(S i ) • E P [L(S j←z )] = 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Using the above procedure, at each timestep the number of bits stored in each B i h grows by: If the entries of z i,h are not close to zero, this compares favorably with the naïve cost of 32 bits per timestep.
Sentence 2: the total storage cost of TBPTT for a RevGRU model with hidden state size H on a sequence of length T will be 7 : in the idealized case, the number of bits stored equals the number of bits forgotten.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Indeed, when µ 1 > 0.9 and 2 (T ) with a probabilistic argument, where (T ) converges to 0 as T goes to infinity.
Sentence 2: for large µ 1 and small µ 2 (so are µ i for i 2), we have that E[μ 1 (t i (k))] µ i = ⌦(µ 1 µ i ) holds for any i and k 2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Also by definition of 'ljJ7r, F( ()) = L,s ,a p7r (s )n( a; s, ()) ' ljJ7r (s, a )' ljJ7r (s, a) T. Substitution leads to: Solving for w gives w = F(()) -l\71}(()), and the result follows from the definition of the natural gradient.
Sentence 2: sensible actor-critic frameworks (those using f7r(s , a; w)) are forced to use the natural gradient as the weights of a linear function approximator.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus the duality gap at the k-th iteration is given by: We define the relative duality gap as: Rel.gap := Dgap/(1 + |pobj| + |dobj|), where pobj and dobj are respectively the objective function values of the primal problem (12) at point X k , and the dual problem (2 )}, we measure the relative changes of objective function value F (X) and the iterates X and Y as follows: We terminate ALM when either Note that in (19), computing log det(X k ) is easy since the spectral decomposition of X k is already available (see ( 14) and ( 15)), but computing log det( Σ − Λ k ) requires another expensive spectral decomposition.
Sentence 2: in practice, we only check (20)(i) every N gap iterations.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given E, a normalized version of the Laplacian can be defined as L = 2(diag(1 T E) − E)/(1 T E1), where 1 is the m × 1 all-ones vector.
Sentence 2: the average squared error between adjacent lattice outputs can be compactly represented as inclusion of this term penalizes first-order differences of the function at the scale of the lattice.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is easy to see that the above lemma is still valid even when the condition of Ω-isomerism is replaced by k-isomerism.
Sentence 2: hereafter, we may say that a space is isomeric (k-isomeric, Ω-isomeric or Ω T -isomeric) as long as its basis matrix is isomeric.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As for the optimal arm, when N 1 (t) , with high probability N 1 (t) = max j N j (t).
Sentence 2: the controller does not need to pay compensation in time slots with a(t) = 1 and N  The second algorithm we propose is a modified "-greedy policy, whose details are presented in Algorithm 2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Regardless of the specific setting being considered, the main challenge in the analysis of privacy amplification by subsampling resides in the fact that the output distribution of the mechanism µ = M S (x) ∈ P(Z) is a mixture distribution.
Sentence 2: in particular, writing µ y = M(y) ∈ P(Z) for any y ∈ Y and taking ω = S(x) ∈ P(Y ) to be the (finitely supported) distribution over subsamples from x produced by the subsampling mechanism, we can write µ = y ω(y)µ y = ωM , where M denotes the Markov kernel operating on measures defined by M. proving privacy amplifications results requires reasoning about the mixtures obtained when sampling from two neighbouring datasets x X x , and how the privacy parameters are affected by the mixture.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 2, it suffices to pick For Theorem 3, it essentially asks for a well-conditioned design matrix at the sparsity level 2s + r. Note that (3) implies κ 2s+r ≥ σ max (Σ)/σ min (Σ), which in return requires a well-conditioned covariance matrix.
Sentence 2: to guarantee that κ 2s+r ≤ 1 + ǫ for some ǫ > 0, it suffices to choose Σ such that σ max (Σ)/σ min (Σ) < 1 + ǫ and pick .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since the eigenvalue decomposition is invariant under permutation, Ā and ĀP have the same eigenvalues and eigenvectors.
Sentence 2: as e → ∞, K clusters of columns appear in ( ĀP ) e . ĀP can be made block-diagonal by clustering the columns of ( ĀP ) ∞ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus, for that Player i, there exists some strategy k ∈ S i \ {j} such that x ik (t) ≥ x ik (0) + 1 2(ni−1) • x ij (0), and hence • min k∈Si x ik (0) max k∈Si x ik (0) .
Sentence 2: the regret of Player i is at least 1 εiT times the RHS of the above inequality.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We need to bound δ t = w t (η) • t + 1 η ln(w t (η) • e −η t ), which is a convex function of t by Lemma 10.
Sentence 2: its maximum is achieved when t lies on the boundary of its domain, such that the losses k t are either 0 or 1 for all k, and in the remainder of the proof we will assume (without loss of generality) that this is the case.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, previous approaches proposed algorithms in which the optimal statespace decomposition was computed based on the optimal solution in the original (non-hierarchical) representation 15,16 .
Sentence 2: the resulting state-space partition was designed for a specific (optimal) environment solution rather than the dynamics of the planning algorithm itself, and also required a priori knowledge of the optimal solution to the planning problem (which may be difficult to obtain in general and renders the resulting hierarchy obsolete).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [2015] derived that the GTD algorithms to minimize (2.3) is equivalent to the stochastic gradient algorithms to solve the following convex-concave saddle point problem with x as the parameter θ in the value function, y as the auxiliary variable used in GTD algorithms.
Sentence 2: we consider the general convex-concave stochastic saddle point problem as below where X x ⊂ R n and X y ⊂ R m are bounded closed convex sets, ξ ∈ Ξ is random variable and its distribution is Π(ξ), and the expected function φ(x, y) is convex in x and concave in y. Denote z = (x, y) ∈ X x × X y X , the gradient of φ(z) as g(z), and the gradient of Φ(z, ξ) as G(z, ξ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The Rice distribution has the property that For simplicity we perform inference using R 2 , as opposed to R, and transform accordingly: The complete likelihood ( 8) is the product of two exponential family models and is exponential family itself, parameterized with base measure h and partition factor g: By inspection we see that the natural parameters η and sufficient statistics S are the same as a Gaussian with unknown mean and variance.
Sentence 2: we apply the normal-gamma prior on (ν, τ ) as it is the conjugate prior for the complete data likelihood.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The average of these functions reaches its minimum of r , so we can view optimizing these functions as the task of discovering the vectors v reven if only v k is missing, a suboptimality better than b/(6 p k) > ✏ cannot be achieved.
Sentence 2: the deterministic algorithm must complete at least k rounds of optimization, each comprising at least ⌃ m 2 ⌥ queries to h F in order to optimize F . The key to this construction is that even though each term |hx, v r 1 i hx, v r i| appears in m/2 components, and hence has a strong effect on the average F (x), we can force a deterministic algorithm to make ⌦(m) queries during each round before it finds the next relevant term.
Relationship: reasoning