sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: This guarantees that with high probability the true MDP M is in M k . In Step 5, extended value iteration (see below) is used to choose a nearoptimal policy πk on an optimistic MDP Mk ∈ M k . This policy πk is executed throughout episode k (Step 6).
Sentence 2: episode k ends when a state s is visited in which the action a = πk (s) induced by the current policy has been chosen in episode k equally often as before episode k. the total number of occurrences of any state-action pair is at most doubled during an episode.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: After taking the limit on the both sides of the inequality as ε → 0, we can prove that Equation (8) holds for p = k + 1.
Sentence 2: by induction, we can prove that Equation ( 8) holds for any non-negative integer p. The proof of Lemma 2 follows directly from the results in reference (Zhang et al., 2012).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is described by the following hypothesis class: It is known that the Rademacher complexity of H can be upper-bounded in terms of the trace of the combined kernel.
Sentence 2: most existing algorithms for learning kernels [1,4,6] add the following constraint to restrict H: (4) As we saw in the previous section, however, the tail sum of the eigenvalues of the kernel, rather than its trace, determines the local Rademacher complexity.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus, Q(t, 0) becomes Similarly, on the event p t = 1, we have that P{θ = 1} = 1 and the policy attaining the infimum in ( 3) is τ * = t+1, δ * = 1.
Sentence 2: q(t, 1) = c(t+1) + dP{D ≤ t + 1 | D > t}.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: (Without such a smoothness condition -for example, if some input x could definitively decide between some given θ 0 and θ 1 -then no such asymptotic independence statement can hold, since no more than one sample from such an x would be necessary.)
Sentence 2: we can hope that we should be able to asymptotically approximate this optimal experiment by sampling in an i.i.d.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For K ≥ 3, h(V ) is a polynomial in V of even order K − 1, whose highest order term has a positive coefficient 1/(K − 1)!.
Sentence 2: as V → ±∞, the function h(V ) goes to positive infinity and it thus has a global minimum at some value Ṽ ∈ R. At the global minimum, its derivative vanishes, .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider a k-partition P of V with k > 1, and compare P with a trivial partition {V } of V . Then, the number of clusters has increased by k − 1 and the clustering cost has increased by f [P] + c, where c is a constant.
Sentence 2: it is natural to define an average cost of P by f [P]/(|P| − 1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 14), we find that the smallest error is obtained when Matrix A corresponds to the Hessian of the merit function for α = 1: A = H (α=1) .
Sentence 2: among the various optimization strategies based on Rényi divergences, Kullback-Leibler divergence (α = 1) has the smallest asymptotic errors.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Denoting the parts of Alice's behavior observable by Bob as x, then H(g | x) is the post-observation entropy in Bob's beliefs, and optimizing it is equivalent to optimizing I(g; x) = H(g) − H(g | x), since the pre-observation entropy H(g) is not dependent on Alice's behavior.
Sentence 2: if Bob receives reward r when identifying the right goal, and 0 otherwise, then his expected reward is (1 − p err ) r. in this simplified setup, optimizing information is directly related to optimizing reward.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: With multiple bits of information at each of the parties, the existence of a differentially private protocol with a fixed accuracy for any non-trivial functionality implies the existence of a protocol with the same level of privacy and same level of accuracy for a specific functionality that only depends on one bit of each of the parties (as in [22]).
Sentence 2: if we can obtain lower bounds on accuracy for functionalities involving only a single bit at each of the parties, we obtain lower bounds on accuracy for all non-trivial general functionalities.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: But in order to have lower upper bounds for both objective regret and the clipped long-term constraint T t=1 [g i (x t )] + compared with Proposition 3 in next section, we need to use time-varying stepsize as the one used in [9].
Sentence 2: we modify the update rule of x t , λ t to have time-varying stepsize as below: If we replace the update rule in Algorithm 1 with Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To solve (4) suppose the problem has been solved to within its statistical accuracy for a set S m ⇢ S n with m = n/↵ samples where ↵ > 1.
Sentence 2: we have found a variable w m for which Our goal is to update w m using the Newton step in a way that the updated variable w n estimates w ⇤ n with accuracy V n . To do so compute the gradient of the risk as well as the Hessian H n of R n evaluated at w m and update w m with the Newton step of the regularized risk R n to compute Note that the stepsize of the Newton update in ( 7) is 1, which avoids line search algorithms requiring extra computation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Similarly, if we made this decision after n + q samples our conditional expected loss would be min x∈A μ(n+q) 1 (x).
Sentence 2: we define the d-KG factor for a given set of q candidate points z (1:q) as d-KG(z where E n [•] is the expectation taken with respect to the posterior distribution after n evaluations, and the distribution of μ(n+q) 1 (•) under this posterior marginalizes over the observations y(z (1:q) ), ∇y(z (1:q) ) = y(z (i) ), ∇y(z (i) ) : i = 1, . . . , q upon which it depends.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, Finally, for any b ∈ ∆(∼ Γn ) and s ∈ S with b(s) = 1, and any other state ŝ with b(ŝ) = 1, it must be the case that s ∼ Γn ŝ and T n (0)(s, ŝ) = 0.
Sentence 2: γn is the optimal value function for the MDP model over ∆(∼ n ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Second, we bound the minimum discrimination error in terms of the Jensen-Shannon information.
Sentence 2: in order to obtain a relationship between I JS and Fisher Information, we use an expression already derived in [7], where p(r|θ + ∆θ) is expanded up to second order in ∆θ, which yields: Fisher Information provides a good approximation of the Jensen-Shannon Information for sufficiently small ∆θ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Instead of solving this optimization problem directly, we equivalently solve the penalized objective function maximize where λ i ≥ 0 is a complexity parameter which has a one-to-one correspondence with the tuning parameter δ i .
Sentence 2: in the Bayesian point of view, the constraint terms could be viewed as the logarithm of the prior distributions over the states S. the objective can be viewed as a log posterior distribution over S. Now the Viterbi algorithm is not applicable directly since at any time t, the state S it depends on all the states at all time steps, because of the regularization terms which are non-Markovian inherently.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, if A i is random, finding A −1/2 i would be infeasible for large N . Additionally, the constraint R T i A i R i = I would lack any intuition.
Sentence 2: we restrict A i = αI + βΦ T i Φ i with α > 0 and β ≥ 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since g j is a function of h j , it suffices to check whether the minimal square ℓ 2 -norm of g j is less than 1.
Sentence 2: we solve the minimization problem of g j 2 2 w.r.t h j , which gives the optimal h j as, where is less than 1, then β j is zero and no update is needed; otherwise, we continue to the next step of checking whether β k j =0, ∀k, as follows.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The key intuition is that we stack L GNN modules and learn to assign nodes to clusters at layer l in an end-to-end fashion, using embeddings generated from a GNN at layer l − 1.
Sentence 2: we are using GNNs to both extract node embeddings that are useful for graph classification, as well to extract node embeddings that are useful for hierarchical pooling.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, H(ρ 1 ) = ρ 2 and H(ρ 2 ) = ρ 1 , i.e., {ρ 1 , ρ 2 } is a periodic orbit.
Sentence 2: all but a measure zero set S of initial conditions converge to the limit cycle {ρ 1 , ρ 2 }.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, 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.
Sentence 2: as a starting point, we first consider a simpler scenario assuming h is fixed (or observed) on the positive training examples.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If we independently solve the Play subgame, we have no way of knowing that P 1 's value for Sell is lower than the blueprint suggested, so we may still assume there is a gift of 0.5 from the Heads state based on the blueprint.
Sentence 2: in order to guarantee a theoretical result on exploitability that is as strong as possible, we use in our theory and experiments a lower bound on what gifts could be after subgame solving was applied to all other subgames.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First note that we can rewrite the convolution operator K using DFT matrices as, where B 2 R d⇥d is the DFT matrix and (•) H denotes conjugate transpose operation.
Sentence 2: note that f  Hence, we can rewrite (16) in the frequency domain as, Since B is invertible, the uniqueness of the original C decomposition is equivalent to the uniqueness of e C decomposition.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that unlike in (3), the regularization is quite important here (i.e., the case of estimators of the form n i=1 β i k(x i , •)) without which the the linear system is under determined.
Sentence 2: we propose the following class of estimators: where g λ (•) is a filter function and λ is referred to as a shrinkage parameter.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Related to this issue is the question of estimating accurately the 2 error θ − θ 0 2 2 .
Sentence 2: the above oracle bounds have the form θ − θ 0 2 2 ≤ C kλ 2 , with k = θ 0 0 the number of nonzero entries in θ 0 , as long as λ ≥ cσ √ log p. minimizing the bound does not yield a recipe for setting λ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The following lemma gives some useful properties about W n and W n . Lemma 1 Both random variables W n and W n are bounded by Haus( M n , M ).
Sentence 2: namely, The cumulative distribution function (CDF) for W n and W n are P(W n ≤ r| R n ) is the ratio of R being covered by padding the regions around R n at distance r. This lemma follows trivially by definition so that we omit its proof.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Integer programming is an NP-hard problem.
Sentence 2: the first step toward solving this problem in practice is to relax it into a continuous optimization by replacing the integer constraints (6) with continuous constraints 0 ≤ µ ≤ 1, yielding the relaxed formulation max µ (1 − µ j e)H(y|x j , w t+1 ) (10) µ j e ≤ 1, ∀j If we can solve this continuous optimization problem, a greedy strategy can then be used to recover the integer solution by iteratively setting the largest non-integer µ value to 1 with respect to the constraints.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus, the Dirichlet process has probability mass only on the K spike words, and is described by a (finite dimensional) Dirichlet distribution, In the absence of data, the parametric base measure controls the mean of this nonparametric model, regardless of ↵.
Sentence 2: we loosely say that ⇡ is "centered" around G ✓ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Without the loss of generality, we assume that the (M l + 1)-th neuron in the l-th layer is the augmented exponential neuron.
Sentence 2: the output of the network f is expressed by where σl : R M l−1 +1 → R M l +1 is a vector-valued activation function with the first M l components being the activation functions σ in the network f and with the last component being the exponential function, i.e., σl (z) = (σ(z), ..., σ(z), exp(z)).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: random variables 6, ... ,em is FO m , where F(•) is the cdf of ei.
Sentence 2: in order to find a column to add to P that is with probability 0.95 among the best 0.05 of all such columns, a random subsample of size ilogO.05/log0.951 = 59 will suffice.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: α f (y A f ) when one of its incoming message m jf or m if is 0 (see details in Appendix C-2).
Sentence 2: we can find the greedy factor (18) by maintaining an ordered multimap for the scores of outcome y A f in each factor {v(y A f , x f )} f ∈F .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To apply such a framework as a model for cortical processing, then, we take the sensory cortical activity to represent the inferred values of the latent variables.
Sentence 2: perceptual inference in this framework involves estimating the values of the causal variables that gave rise to the sensory input, while developmental (unsupervised) learning involves discovering the correct causal structure from sensory experience.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus the Equal Spacing CTC deformation (EsCTC) guides the end-to-end model training by optimizing the loss function: We consider the EsCTC as complementary to EnCTC that explicitly rules out all unreasonable alignments.
Sentence 2: we also combined the Equal Spacing CTC deformation with maximum conditional entropy regularization defined in section 3.3.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For a composite entity that consists of two entities of the same type, we'd like to capture its correlation with its twinthe composite entity made of the same basic entities but in reversed order.
Sentence 2: we add the T win relation between all pairs of twin entities: e.g., from ⟨p1, p2⟩ to ⟨p2, p1⟩, and vice versa.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 7) and 1 = (1, 1, . . . ) ∈ R N . Note that the Wiener series is represented only implicitly since we are using the RKHS representation as a sum of scalar products with the training points.
Sentence 2: we can avoid the "curse of dimensionality", i.e., there is no need to compute the possibly large number of coefficients explicitly.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The policy π instead uses updated belief-mean m t|t (smoother and better-informed than z t ) to decide the action: u t = π(m t|t , ψ).
Sentence 2: the joint distribution over the updated (random) belief and the (non-random) action is Next, the filtering 'prediction step' computes the predictive-distribution of b t+1|t = p(x t+1 |z 1:t , u 1:t ) from the output of dynamics model f given random input bt|t .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Choice of Lag Index Set L. Unlike most approaches mentioned in Section 2.2, the choice of L in TRMF is more flexible.
Sentence 2: tRMF can provide important advantages: First, because there is no need to specify the weight parameters W, L can be chosen to be larger to account for long range dependencies, which also yields more accurate and robust forecasts.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider first f . Due to memoization, the algorithm executes at most two basic operations (one comparison and one multiplication) per triplet (S, R, C), with S and R disjoint, S ⊂ C ⊆ S ∪ R, and |C| ≤ w. Subject to these constraints, a set C of size i can be chosen in n i ways, the set S ⊂ C in at most 2 i ways, and the set R \ C in 2 n−i ways.
Sentence 2: the number of basic operations needed is at most Similarly, for h the algorithm executes at most two basic operations per triplet (C, R, S), with now C and R disjoint, |C| ≤ w, and S ⊂ C. A calculation gives the same bound as for f . Finally consider g. Now the algorithm executes at most two basic operations per triplet (C, U, R), with C and U disjoint, |C| ≤ w, and ∅ = R ⊆ U .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , t N } such that |t i − t j | = |i − j|τ for all i and j, we consider a draw from the multivariate normal distribution: where each observation is generated from the cluster indicated by z (ar) w (described in Section 2.2), and each cluster is represented uniquely by a covariance matrix, Σ , whose elements are defined through the covariance kernel k(τ ; θ , γ).
Sentence 2: the parameters θ z (ar) w describe the autocorrelation content associated with each y (ar) w . We address two concerns with this formulation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We can easily extend our methods to handle this case for appropriate h. Recall that the auxiliary function that we used, depended upon the convexity of ϕ.
Sentence 2: if (ϕ •h) is a convex function, whose derivative ∇(ϕ •h) is "factorizable," then we can easily derive algorithms for this problem with link functions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, variable A does little in the PGM in (b) since W and Y are nearly independent.
Sentence 2: a similar argument can be made about variable B. variable C in the PGM in (a) can be used to explain the weak interactions between variables, whereas in the PGM in (b), variable C essentially has the task of explaining the interaction between all the variables.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then C is a margin maximizing loss function in the sense that any convergence point of the normalized solutions β(λ) β(λ) p to the regularized problems (1) as λ → 0 is an l p marginmaximizing separating hyper-plane.
Sentence 2: if this margin-maximizing hyper-plane is unique, then the solutions converge to it: Proof We prove the result separately for T = ∞ and T < ∞.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Our first key observation (already briefly mentioned in [26]) is that the critical step of rewriting the loss (1) in terms of the Laplacian (2) does not depend on any special properties of the rotation group other than the fact (a) rotation matrices are unitary (in fact, orthogonal) (b) if we follow one rotation by another, their matrices simply multiply.
Sentence 2: in general, for any group G, a complex valued function ρ : G → C dρ×dρ which satisfies ρ(g 2 g 1 ) = ρ(g 2 )ρ(g 1 ) is called a representation of G. The representation is unitary if ρ(g −1 ) = (ρ(g)) −1 = ρ † , where M † denotes the Hermitian conjugate (conjugate transpose) of M . we have the following proposition.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: After observing a transition (s, a) → s , the posterior distribution is also Dirichlet and can simply be updated as ρ s,a,s ← ρ s,a,s + 1.
Sentence 2: to model the distribution of X s,π and X s,a,π we only need to maintain a set of hyperparameters µ s,0 , λ s , α s , β s and ρ s,a for each state s and action a encountered in the MCTS search tree and update them by using Bayes' rules.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Also, note that the joint distribution p(h, s, y) depends on the transformation defined by the encoder.
Sentence 2: we can equivalently rewrite objective (1) as To analyze the equilibrium condition of the new objective (2), we first deduce the optimal discriminator D and the optimal predictor M for a given encoder E and then prove the global optimality of the minimax game.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In order to do so, it updates V such that where r t is the reward at time t and γ the discount factor.
Sentence 2: a simple way to achieve that is to transform the problem into an optimization problem where the goal is to minimize: It is also useful at this point, introduce the TD effective reinforcement signal, equivalent to the dopaminergic signal [5]: ( ) ( ) A learning rule for the weights v of V can then be devised by finding the gradient of E with respect to the weights v. Here, V is the weighted sum of the activity of I and H. the gradient is given by [ ] Adding a learning rate and negating the gradient for minimization gives the update: Developing a learning rule for the actor units and their weights W using a cost function is a bit more complex.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Instead, consider absorbing this term into the diagonal component to preserve the bound.
Sentence 2: formally, we look for a diagonal matrix F such that Σ ′′ i = Σ ′ i + F which also maintains x ⊤ Σ ′′ i x ≥ x ⊤ Σ i x for all x ∈ R d . we want to satisfy: where, for ease of notation, we take F(i) = F(i, i).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First, we assume that the distribution P (X) of the input feature x ∈ X is a product measure of P k (X) on each X k . That is, P X (dX This is typically assumed in the analysis of linear tensor estimation methods [13,6,3,21,1,36,27,37].
Sentence 2: the Hereafter, with a slight abuse of notations, we denote by f L2 = f L2(P k ) for a function f : X k → R. The inner product in the space L 2 is denoted by f, g L2 := f (X)g(X)dP X (X).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Unless f is a polynomial itself, there exists an x 0 (usually uncountably many) for which f (x 0 ) = p n (x 0 ), so if A has an eigenvalue at x 0 we have Σ f (A) = Σ pn (A).
Sentence 2: one cannot hope that the estimator (3), let alone the gradient estimator (5), to be unbiased for all matrices A.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, there is a deeper connection.
Sentence 2: if we let L(Y, Xw) denote the unregularized loss over the training set where X is the n × d matrix of instances and Y is the n-vector of class labels, we can express equation ( 1) in matrix form as Defining X to be XU (αΛ + β I) −1/2 where U and Λ are from the eigen-decomposition above, it is not hard to show that equation ( 3) is equivalent to the alternative ridge regularized learning problem That is, the two minimizers, w and w, yield the same predictions: Ŷ = Xw = X w. we can view feature network regularization as: 1) finding an embedding for the features using LLE in which all of the eigenvectors are used and scaled by the inverse square-roots of their eigenvalues (plus a smoothing term, βI, that makes the inverse well-defined); 2) projecting the data instances onto these coordinates; and 3) learning a ridge-penalized model for the new representation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Suppose there exists a central controller that collects the rewards of, and assigns the action to each individual agent, the problem reduces to the classical MDP with action space A and global reward function R c (s, a) = N −1 N i=1 R i (s, a).
Sentence 2: without such a central controller, it is essential for the agents to collaborate with each other so as to solve the multi-agent problem based solely on local information.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Here, the random variables Y i (t) are conditionally independent when X(t) is given, and the conditional probabilities W [y i (t)|x(t)] are identical for all i and t. In this paper, we impose the binary assumptions to the problem, i.e., the data sequence {X(t)} and its noisy observations {Y i (t)} are all assumed to be binary sequences.
Sentence 2: the stochastic matrix can be parameterized as where p ∈ [0, 1] represents the observation noise.
Relationship: reasoning