sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Intuitively, limiting action selection only to temporally-extended options might hamper the exploration of the environment by restricting the policy space.
Sentence 2: we argue that in addition to the exciting work being done in heuristic and algorithmic approaches that leverage and/or dynamically discover options, it is important to build a formal understanding of how and when options may help or hurt reinforcement learning performance, and that such insights may also help inform empirically motivated options-RL research.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Nonetheless, for many of the SDPs that appear in machine learning, feasible points are easy to identify, and equality constraints are either absent or fortuitously pose no difficulty.
Sentence 2: we can gain further intuition by observing that Random Conic Pursuit's iterates, X t , are positive weighted sums of random rank one matrices and so lie in the random polyhedral cones Random Conic Pursuit optimizes the SDP (1) by greedily optimizing f w.r.t.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is well known that REINFORCE may learn slowly than other methods that use a learned action-value function (Sutton et al., 1999).
Sentence 2: we next present a function approximator for Q π t , and show the computation of policy gradient by directly sampling counts n. One can approximate the action-value function Q π t (s t , a t ) in several different ways.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since the theory contains no independent subtheories, there must be an atom a in C n which unifies with an atom a c in C c n , or C c n is empty.
Sentence 2: after shattering, all unifications bind one variable from a to a single variable from a c . Because a contains exactly n variables from B, a c must also contain exactly n (Definition 4), and because B is a root binding class, the clause of a c also contains exactly n, which contradicts the definition of C c n . C c n is empty, and because the variables in B are root, they also appear in all atoms.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Importantly, the algorithm we present is based on iterative reweighting and ADMM and is fast enough-linear in the input size when k is fixed, and highly parallelizable-to allow for computing a regularization path of features by varying the pointer cost.
Sentence 2: we may adapt the compression to the data at hand and select features that better elucidate its structure.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Similarly, the exact factors ψ i , with i = 1, . . . , n, corresponding to the prior for the latent variables z, (3), only depend on ρ and z i .
Sentence 2: the Gaussian part of the corresponding approximate factors can be removed and the multivariate Bernoulli distribution simplifies to a univariate Bernoulli.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: There are N x possible ways of choosing the positions of the x misclassified negative examples, and similarly N x possible ways of choosing the positions of the x = k − x misclassified positive examples.
Sentence 2: in view of Lemma 2, the average AUC is given by:  Each curve corresponds to a fixed ratio of r = n/(n + m).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In its simplest form, their result shows that when m D K (µx 0 ) then with high probability, ) Here, D K (µx 0 ) is the Gaussian width, a specific measure of complexity of the constrained set K when viewed from µx 0 .
Sentence 2: for our purposes, it suffices to remark that if K is properly chosen, and, if µx 0 is on the boundary of K, then D K (µx 0 ) is less than n. estimation is in principle is possible with m < n measurements.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In fact, our lower bound holds even if the algorithm is allowed to output a rank-k approximation for any 2 ≤ k = o(min(m, n)) whose cost is at most that of the best rank-2 approximation to A.
Sentence 2: we call the latter a bicriteria algorithm, since its output rank k may be larger than the desired rank k. in some sense obtaining additive error (1.1) is the best we can hope for.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is a weak solution, however, since evasion tends to be rather quick, and re-learning is a costly task, since it requires one to label a large number of instances (in crowdsourced labeling, one also exposes the system to deliberate corruption of the training data).
Sentence 2: several efforts have focused on proactive approaches of modeling the learner and adversary as players in a game in which the learner chooses a classifier or a learning algorithm, and the attacker modifies either the training or test data [13,14,15,16,8,17,18].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let η = min σ∈Sn η σ , and let p be an arbitrary member of B(p, η).
Sentence 2: by the definition of ε, we have: ∀σ ∈ A (p) , L (p , σ ) > L (p) + ε. Additionally, the definition of η gives ∀σ ∈ A (p) , L (p , σ) < L (p) + ε.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The NMDA channel has the following equation: For simpler notation, in general we use inverse time-constants In addition, the term a 1 − b 1 in the denominator is required for later easier integration in the Laplace domain.
Sentence 2: we adjust for this by defining ḡ = 12 mS/ms which represents the peak conductance (4 nS) multiplied by b 1 − a 1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In order to avoid overly aggressive pruning in the early stage of the value iterations, we need to take into account the size of the value function at every iteration.
Sentence 2: we use a sliding pruning strength specified as J E~=o j3i extent(R) where R is the initial reward diagram, j3 is the discount factor introduced earlier and n is the iteration number.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Notice that VCN using ε = 1 is equivalent to VCN as described previously, so the points for ε = 1 in Figure 3(b) correspond exactly to the VCN data in Figure 3(a).
Sentence 2: if the variance in gradient estimates precludes learning, we suggest making the policies of the modules more deterministic by decreasing exploration and increasing exploitation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Finding a radius: Remember that @i P rns, f i is differentiable with a 1{γ-Lipschitz gradient.
Sentence 2: @i P rns, f i is γ-strongly convex [14, Theorem 4.2.2, p. 83] and so D λ is γλ 2 -strongly concave: Specifying the previous inequality for Θ 1 " p Θ pλq , Θ 2 " Θ P ∆ X , one has By definition, p Θ pλq maximizes D λ on ∆ X , so we have: x∇D λ p p Θ pλq q, Θ ´p Θ pλq y ď 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If, instead, we minimize f 2 for v def = w/γ and remove the ∞ constraint, we recover exactly ridge regression for the quadratic loss case and 2 -regularized boosting for the exponential loss case.
Sentence 2: we can obtain a 1 -regularized version of Equation 17 by repeating the above steps and using 4n q − 1 2n 2 ≤ 2 q − 1 2n ∀q ∈ [0, 1/n] since, in that case, we find that KL(Q To sum up, the KL-regularized objective function f immediately follows from PAC-Bayes theory and p regularization is obtained from a relaxation of f . PAC-Bayes theory favors the use of KL regularization if the goal of the learner is to produce a weighted majority vote with good generalization.
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 (a) Ser-2g   The coordination-free approach attempts to reduce the need to coordinate guarantees and logical ordering.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In our approach, we would partition the domain, say as {{1, 3}, {2, 4, 5}}, and search over only the following 4 assignments: all groundings in each part can be either all true or all false.
Sentence 2: if we are lucky and the MAP solution is one of the 4 assignments, our approach will yield significant reductions in complexity even though no identifiable symmetries exist in the problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: o(N −1/2 ), whenever H ≥ log N d . Thus, using P = O( √ N ) random features, we deduce that the complexity of building the matrix Then in order to solve the least squares system, one has to compute Ψ T Ψ, that has numerical cost O(P 2 N ), and then solve the system by inversion, which has numerical cost O(P 2.376 ) by [5].
Sentence 2: the overall cost of the algorithm is O(2 d N 3/2 log N + N 2 ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: sample of size n from a distribution P ∈ N d . There exists a universal (numerical) constant c such that Given centralized access to the full mn-sized sample, a reasonable procedure would be to compute the sample mean, leading to an estimate with mean-squared error σ 2 d mn , which is minimax optimal.
Sentence 2: theorem 2 shows that to achieve an order-optimal mean-squared error, the total number of bits communicated must (nearly) scale with the product of the dimension d and number of machines m, that is, as dm/ log m. If we ignore logarithmic factors, this lower bound is achievable by a simple procedure: each machine computes the sample mean of its local data and quantizes each coordinate to precision σ 2 /n using O(d log(n/σ 2 )) bits.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore the solution of the (s, ρ)-trajectory-based short-sighted SSP only chooses the action a c to move the car.
Sentence 2: also, since m is even, every state s used by SSiPP for generating (s, ρ)-trajectory-based short-sighted SSPs has l s ∈ N. for even values of m, i.e., for ρ ∈ (0.5 i+1 , 0.5 i ] and i ∈ {1, 3, 5, . . . }, trajectory-based SSiPP always chooses the actions a c to move the car to l c , thus avoiding the all dead-ends.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By Lemma 15, it follows that is η -mixable, and from Theorem 11, is (η −1 S)-mixable.
Sentence 2: substituting Φ for η −1 S in Theorem 14 implies that (η /η − 1) S is convex on ri ∆ k . η ≤ η , and since from Theorem 7 η ≤ η , we conclude that η = η .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, notice that if we tighten the nonnegativity constraint to X kl ≥ , the objective function of (12) will always be finite.
Sentence 2: we can simply modify the optimization of ( 12) the objective function to become infinity, as: The following result shows that with a sufficiently small , the constrained optimization problem in (13) has a unique global optimal solution that affords a closed-form and intuitive interpretation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This would then imply that X ∈ X c . To prove the claim, observe that for i = m + 1, . . . , n, both sides of ( 24) are equal to 0.
Sentence 2: moreover, since Xk ∈ X c , Propositions 4.2 and 4.3 give it suffices to show that for ǫ > 0 sufficiently small, Towards that end, fix an index i = 1, . . . , m and consider the three cases defined in Section 4.1.1: Case (A).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let N denote the set of all pairs, and let N k denote the set of first k pairs in the random order.
Sentence 2: since each pair in N is treated equally, N k is a random subset of N of size k. k i=1 (a γ(i) /σ x ) 2 is equivalent to take the total sum of (a i /σ x ) 2 with i from 1 to d and divide it by d/k.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then, setting D L D R equal to DAB, we obtain the solution , where the subscript f denotes the retained singular value and vectors.
Sentence 2: the only modification to the method in the previous section is to replace the estimation of A and B in ( 18)-( 21) with the global solution V f Λ −1 f Λ F V T f . this method allows us to avoid the alternating optimization for finding A and B, which potentially can get stuck in undesired local minima.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If f d (z) is that estimate, an equivalent representation of the data term is g 1 (w) = ED ∇ w Eq w (Z) f D (Z) where D is uniform over subsets of data.
Sentence 2: one can define an unbiased estimator by using one of the techniques above (to cope with Eq w (Z) ) on a random minibatch D (to cope with ED).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: While the generator benefits from being able to blend pixels at object boundaries, multiplying real images by these fractional values causes the discriminator to learn that mid gray is 'real' (i.e., we push the answer towards the midpoint 0 of the normalized [−1,1] pixel space).
Sentence 2: we threshold the learned attention map for the discriminator: where t new and s new are masked versions of target sample t and translated source sample s , which only contain pixels exceeding a user-defined attention threshold τ , which we set to 0.1 (Figure 3 in the supplemental material justifies such choice).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Using dynamic programming, one can compute the optimal value function of M, but the time and space required to do so grow fast with the number of states n [7,8].
Sentence 2: the use of KBRL leads to a dilemma: on the one hand, one wants to use as many transitions as possible to capture the dynamics of M, but on the other hand one wants to have an MDP M of manageable size.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Laplacian regularization encourages nearby data points to have similar latent representations (e.g., assignments) and is widely used in spectral clustering [32,26] as well as in semi-supervised and/or representation learning [2].
Sentence 2: the model can handle non-convex (or manifold-structured) clusters, unlike standard prototype-based clustering techniques such as K-means.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the example of Düsseldorf and Hamburg, the car-license-plate cue would yield that Düsseldorf (D) is larger than Hamburg (HH), whereas the soccer-team cue would correctly favor Hamburg.
Sentence 2: how successful a lexicographic strategy is in a comparison task consisting of a partial ordering of cue profiles depends on how well the cue ranking minimizes the number of incorrect comparisons.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The directed edge from node (i; j) to node (p; q) will be assigned a weight w 1 ((i; j); (p; q)) , d((i; j)jj(p; q)) F (i p; j q), where is a free parameter of our algorithm 2 .
Sentence 2: the weight of the edge from node (i; j) to node (p; q) is proportional to their dissimilarity and to their closeness in the domain of M . Note that the edge in the opposite direction has exactly the same weight.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Partially dichotomized Gaussian model Another simple possibility to obtain a joint distribution of continuous and binary variables is to take multivariate (latent) Gaussian distribution for all variables and then dichotomize those components which should represent the binary variables.
Sentence 2: a binary variable b i is set to 1 if the underlying Gaussian variables is greater than 0 and it is set to 0 if the Gaussian variable is smaller than 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In contrast, we are interested in the "more human-like" setting where i) the number of tasks is large, ii) the number of training examples per task is small, iii) the learner observes the examples concerning each task only once, and iv) we report metrics that measure both transfer and forgetting.
Sentence 2: at training time we provide the learner with only one example at the time (or a small mini-batch), in the form of a triplet (x i , t i , y i ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This separation implies that the partition function can be computed in lower dimensional pieces As a result, the computation is exponential only in the size of the subsets α ∈ A.
Sentence 2: we can also estimate the partition function with lower dimensional random MAP perturbations, E γ [max xα\x β {θ α (x α ) + γ α (x α )}].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Inserting (2) into (5) results in Since an appropriate cost function used in ( 1) is non-negative and monotonically decreasing, C [y i F (x i )] is always negative and R(i) is non-negative according to its definition in (6).
Sentence 2: we can define our empirical data distribution as D(i) is always non-negative based on definitions of cost function in (1) and R(i) in (6).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Recall that δ is the depth (distance from a hidden node to its closest observed nodes), θ min is the minimum (absolute) edge potential and η = dmax dmin is the ratio of distance bounds.
Sentence 2: theorem 1 (Structural Consistency and Sample Requirements) Under (A1)-(A5), the probability that the proposed method is structurally consistent tends to one, when the number of samples scales as for learning Ising models on locally tree-like graphs, the sample complexity is dependent both on the minimum edge potential θ min and on the depth δ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Even the realizable case, while commonly described as "solved" via the Perceptron algorithm or linear programming (LP), is not straightforward: The Perceptron's runtime is quadratic in the inverse-margin, while solving the consistent hyperplane problem in strongly polynomial time is equivalent to solving the general LP problem in strongly polynomial time [Nikolov, 2018, Chvátal], a question that has been open for decades [Bárász and Vempala, 2010].
Sentence 2: an unconditional (i.e., infinite-precision and independent of data configuration in space) polynomial-time solution for the consistent hyperplane problem hinges on the strongly polynomial LP conjecture.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let V l be a vector field on the manifold M l . For each point x l i , let V x l i denote the value of the vector field V l at x l i .
Sentence 2: recall the definition of vector field, V x l i should be a vector in the tangent space T x l i M l . we can represent it by the coordinates of the tangent space T x l i M l as Let f l be a function on the manifold M l . By abusing the notation without confusion, we also use f l to denote the vector is, V l is a d l n l -dimensional big column vector which concatenates all the v l i 's for a fixed l. Then for each task, we aim to compute the vector f l and the vector V l . In this section, we introduce multi-task vector field learning (MTVFL).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Suppose we have a set of base kernel functions {k m } M m=1 (or base kernel matrices {K m } M m=1 ).
Sentence 2: an ensemble kernel function k (or an ensemble kernel matrix K) is then defined by the learned model from binary-class data {(x i , y i ∈ ±1)} will be of the form: Optimizing both the coefficients {α i } N i=1 and {β m } M m=1 is one particular form of the MKL problems.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From here we can see that if we let L go to infinity and set k(n) = λ −n , 0 ≤ λ ≤ 1, then we obtain the λ-return estimator in equation ( 4), since .
Sentence 2: λ-return (as used in the TD(λ) family of algorithms) is the maximum-likelihood estimator of return under the following assumptions: 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We now replace each occurence of Pu-+ c in P(d) by 1 -LaEA\{ c }Pu-+ a .
Sentence 2: since uc never occurs in d and Pu-+x ~ pv-+x, the u --+ x Fisher score entry for a document d becomes tf( ux, d) The Fisher kernel is subsequently defined to be k(d,d') = UJrIUd', where Ud is the Fisher score vector with ux-component a~n P( d) and I = Ed[UdUdTJ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ImageNet-5k, JFT [16], OpenImages [18]), different structures in the underlying network (e.g multi-scale features [20,26], deformable convolutions [8], heavier heads [28], anchor sizes, path aggregation [22]), test time augmentations like flipping, mask tightening, iterative bounding box regression etc.
Sentence 2: we compare our results with SNIP [33], which is a recent method for training object detectors on an image pyramid.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then, for any positive integer k, the β-mixing coefficient of the stochastic process Z is defined as Z is said to be β-mixing if β(k) → 0.
Sentence 2: it is said to be algebraically β-mixing if there exist real numbers β 0 > 0 and r > 0 such that β(k) ≤ β 0 /k r for all k, and exponentially mixing if there exist real numbers β 0 and a sequence of random variables is mixing when the dependence of an event on those occurring k units of time in the past weakens as a function of k. Our generalization bounds will be based on the following measure of the complexity of a class of functions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then, by construction, for any y ∈ ∆ l , we have To see how C can be used to compute Q w (z, s), we note first that the states of C can be identified with pairs (q A , q B ) where q A is a state of A, q B is a state of B, and the transitions are obtained by matching a transition in A with one in B.
Sentence 2: for any z ∈ ∆ q and s ∈ [l], let E z,s be the set of transitions of C constructed by pairing the transition in A ((z 1:q−1 , s − 1), z q , ω(z, s), (z 2:q , s)) with a transition in B: Note that, since C is deterministic, there can be only one transition leaving a state labeled with z q .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In [5], it was shown that the eigenfunctions and eigenvalues of (4) can be approximated as φ j (x i ) ≈ √ N e ji , λ j ≈ λj N , where e ji denotes the ith element of the jth eigenvector.
Sentence 2: hence, the mapping (4), can be approximated as the mapping is based on the eigenspectrum of K. The feature space data set may be represented in matrix form as Φ N ×N = [Φ(x 1 ), .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Following standard variational inference methodology, [1] we obtain the evidence lower bound of a GP-SSM augmented with inducing points In order to achieve tractability, we use a variational distribution that factorizes as where q(u) and q(x) can take any form but the terms relating to f are taken to match those of the prior (3).
Sentence 2: the difficult p(f t |...) terms inside the log cancel out and lead to the following lower bound L(q(u), q(x),θ) = −KL(q(u) p(u)) + H(q(x)) + x q(x) log p(x 0 ) where KL denotes the Kullback-Leibler divergence and H the entropy.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We denote the solution obtained in the th iteration as w ( ) .
Sentence 2: in the ( + 1)th iteration, due to the concavity property of we can bound the second term in problem (5) as follows in the ( + 1)th iteration, we need to solve a weighted version of problem (1): According to [14], the MM algorithm is guaranteed to converge to a local optimum.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: That is, Self-infolTIlation characterizes the raw likelihood of the specific n-dimensional vector of ROB values given by X . p(X) in this case is based on observing a number of n-dimensional feature vectors based on patches drawn from the area surrounding X .
Sentence 2: p( X) characterizes the raw likelihood of observing X based on its surround and -l og(p(X)) becomes closer to a measure of local contrast whereas entropy as defined in the usual manner is closer to a measure of local activity.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: There is a natural logarithmic dependence on p and a linear dependence on the average degree parameter c. Finally, the dependence on α can be explained as follows: any α-walk-summable model is also β-walk-summable for all β > α.
Sentence 2: the class of β-walk-summable models contains the class of α-walk-summable models.
Relationship: reasoning