sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: In recent work on the NormalHedge algorithm [4] the authors use a potential function which is very similar to our function g(x) (see (2) below), getting strong regret guarantees to the -quantile of best experts.
Sentence 2: the use of the function g(x) seems to be quite different from ours, as is the focus of the paper [4].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Without loss of generality assume that Ψr⊗I + I⊗Ψc = RR for some matrix R. We then qualify that p(Z|Ψr, for all Z and E. For fixed values of Ψ r , Ψ c , and Γ we optimize this quadratic bound to obtain revised estimates for Z and E, noting that exact equality in (15) is possible via the closed-form solution In large practical problems, ( 16) may become expensive to compute directly because of the high dimensional inverse involved.
Sentence 2: we may still find the optimum efficiently by an ADMM procedure described in [23].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: the distortion measure d. The bound can also be applied to a generator G ′ obtained by minimizing, e.g., some fdivergence [26] between P X and P G(Z) .
Sentence 2: if W d (P X , P G ′ (Z) ) > W d (P X , P G ⋆ (Z) ) (which will generally be the case in practice) then the distortion obtained by using G ′ will asymptotically be larger than that obtained for G ⋆ .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Although we approximate Z i by a Gaussian Zi , we are still free to choose a method of computing its mean μi and covariance matrix Σi , which also influences the computation of J = ∂( μi )/∂µ \i .
Sentence 2: even if μi and Σi are general functions of µ \i and Σ \i , the derivatives ∂ μi /∂µ \i and ∂ Σi /∂Σ \i must equal the corresponding partial derivatives in (11), and ∂ μi /∂Σ \i and ∂ Σi /∂µ \i must be set to 0.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Now note that if X is an optimal solution to B then d(j, X) ≤ 1 for all points j ∈ C. So there exists a (deterministic) distribution achieving r j = 1.
Sentence 2: suppose that A generates a solution X ∈ F k where E[d(j, X)] ≤ cr j ; the set X can also be regarded as a solution to the set cover instance.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Typically, significant discriminative itemset mining aims to find all feature subsets S for which a statistical association test rejects the null hypothesis, namely Z S ⊥ ⊥ Y , after a rigorous correction for multiple hypothesis testing.
Sentence 2: for any feature subset such that Z S ⊥ ⊥ Y but Z S ⊥ ⊥ Y | C, the association between Z S and Y is exclusively mediated by the covariate C, which acts in this case as a confounder creating spurious associations.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Such methods enjoy a linear convergence rate when the cost function F is µ-strongly-convex and each f i has L-Lipschitz continuous gradients, that is, to achieve an output x which satisfies F (x) − F (x ) ≤ δ, the total number of stochastic gradient evaluations needed is O n + L/µ log 1 δ . Nesterov's acceleration [15,16,17] has also been successfully applied to construct variance-reduced methods which have an accelerated linear-convergence rate [18,19,20,21,22,23,24,25]: It is worth noting that this convergence rate has been shown to be worst-case optimal [21].
Sentence 2: all of these algorithms need explicit knowledge of the strong-convexity parameter µ.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Environment design [Zhang et al., 2009, Zhang andParkes, 2008] is a related problem, where a first agent seeks to modulate the behavior of a second agent.
Sentence 2: the interaction between agents occurs through finding a good modification of the second agent's reward function: the AI observes a human performing a task, and uses inverse reinforcement learning [Ng et al., 2000] to estimate the human's reward function.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The rationale behind this scheme is that the initial model Â(0) is obtained essentially from an uninformed, ad hoc characterizer, for which one has to expect a large model variation and thus (on the average) a poor Â(0) .
Sentence 2: the characterizer Ĉr(1) obtained from the reversed Â(0) is not uninformed any longer but shaped by a reasonable reverse model.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In fact, it is known that this affinity is efficient to compute for mixtures of Gaussians, multinomials and even hidden Markov models (Jebara et al., 2004).
Sentence 2: this permits the affinity metric to efficiently pull together parameters θ m and θ n . for mixture models, there is the presence of hidden variables h in addition to observed variables.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As we will see below, the latent count vectors {m n } N n=1 (which are functions of V and U) provide sufficient statistics for the model parameters.
Sentence 2: each element of m n (if the corresponding element in y n is one) is drawn from a truncated Poisson distribution V l,: denotes the l th row of V and Thus we can also write if y ln = 0 then m ln = 0 with probability one (Eq.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This technique is quite general, in the sense that we can always augment Ω A with a no-op action and use this technique based on r t to push the no-op actions to the end of a T -length action sequence for a pre-chosen upper bound T .  Executing a plan in a noisy environment may not always result in the goal state being reached.
Sentence 2: in the instances where a goal state is indeed reached, the executed action sequence can be used to bootstrap the learning of an optimal policy π(a | s, g), which represents the probability for action a in state s when the goal state to be reached is g. We define optimality in terms of reaching the goal using the shortest path.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In [10] it was mentioned that the computation of Ψ, which makes use of the random matrix A = (A p,i ) p≤P,i≤F , has a complexity O(F P N ).
Sentence 2: in the multi-resolution schemes described here, provided that the mother function has compact support (such as the hat functions or the Daubechie wavelets), we can significantly speed up the computation of the matrix Ψ by using a tree-based lazy expansion, i.e.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: To simplify our notations, we define the parameter µ = µ/τ , which scales the strong convexity parameter µ by 1 τ , where τ is the quadratic growth constant.
Sentence 2: in general, the constant Γ which defines the step-size parameter γ t is set to L. we allow Γ to be an arbitrary constant greater than or equal to L to facilitate the introduction of the multi-stage ORDA in the later section.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As with many other complex hierarchical Bayesian models, exact inference of the latent variables Í and Î in the BMF model is intractable (ie there is no efficient way to sample exactly from the posterior nor to compute its exact marginals).
Sentence 2: as with many other non-parametric Bayesian models, we can employ Markov Chain Monte Carlo (MCMC) methods to create an iterative procedure which, if run for sufficiently long, will produce correct posterior samples.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Then, F (U (t) ) − min In [20], single-output PNs were trained using GECO [26], a greedy algorithm with similar O τ 2 β ǫν 2 guarantees.
Sentence 2: gECO is limited to learning infinite vectors (not matrices) and it does not constrain its iterates like we do.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Then, θlinear would correspond to different power means on ŝk , depending on the choice of parameterization, e.g., where only the linear average of ŝk (when θ = σ 2 ) matches the overall empirical variance ŝ and equals the global MLE.
Sentence 2: θKL always corresponds to a linear average of ŝk , equaling the global MLE, regardless of the parameterization.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Compared to running n independent bandit algorithms (which corresponds to A ⊗ being the identity matrix), the bound in the above theorem has an extra term (i,j)∈E u i − u j 2 , which we assume small according to our working assumption.
Sentence 2: the bound has also a significantly smaller log determinant ln |M T | on the resulting matrix M T , due to the construction of φ t,k via A −1/2 ⊗ .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Our method shows the highest ARI (0.754) when the number of HMM states is 3, which shows that important dimensions are successfully discovered from the "sea" of nuisance dimensions.
Sentence 2: the ARI of K-Means decreases monotonically as the number of HMM states increases, which shows the K-Means is not robust against nuisance dimensions.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Intuitively, model selection of importance estimation algorithms (such as KMM) is straightforward by cross validation (CV) over the performance of subsequent learning algorithms.
Sentence 2: this is highly unreliable since the ordinary CV score is heavily biased under covariate shift-for unbiased estimation of the prediction performance of subsequent learning algorithms, the CV procedure itself needs to be importance-weighted [9].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The gradient is given in an explicit form: 4λW (W T W − I), and can be directly appended to the original gradient w.r.t.
Sentence 2: the current weight W . SO (1) is flawed for an obvious reason: the columns of W could possibly be mutually orthogonal, if and only if W is undercomplete (m ≥ n).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The empirical risk/loss associated with the binary space partitioned classifiers is given by: Our goal is to minimize the empirical error jointly over the family of functions g(•) ∈ G and f i (•) ∈ F . From the above equation, when the partitioning function g(•) is fixed, it is clear how one can view choice of classifiers f 0 (•) and f 1 (•) as ERM problems.
Sentence 2: even when f 0 , f 1 are fixed, it is unclear how to view minimization over g ∈ G as an ERM.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Furthermore, we notice that 1/µ o (s) ≥ τ o (s) ≥ |S o |, suggesting that "long" or "big" options are indeed more difficult to estimate.
Sentence 2: ∆ µ becomes smaller whenever the transition probabilities under policy π o are supported over a few states (B small) and the rewards are similar within the option (sp(r o ) small).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: To modify the MCU formulation to include these new variables, we replace all Euclidean distance measurements for the data points in the first dataset in (2) with the weighted distance This adds a linear function of the new weight variables to the existing distance constraints of (2).
Sentence 2: if we had to define the neighborhood of a data point itself using this weighted distance, the formulation would become non-convex.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Since W (x, s) is quadratic in the Y c (x) values, it is often the case, especially for smaller ρ-values in later AL iterations, that D W (t; x, s) is zero over most of [0, w n min (x, s)], simplifying numerical integration.
Sentence 2: this has deleterious impacts on search over (x, s), as we discuss in our supplement.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In that work a solution to the dynamics of the on-line algorithm was obtained in closed form for ¢¡ ¤£ " § learning iterations and a simple solution to the asymptotic dynamics under the optimal learning rate decay was obtained.
Sentence 2: it was noted there that modelling the dynamics on an ¢¡ ©£ " § timescale is not always appropriate, because the algorithm typically requires much longer in order to escape from a class of metastable states close to the initial conditions.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In particular, the choice of hypothesis class H and learning objective has received a lot of attention (Calders & Verwer, 2010;Zemel et al., 2013;Fish et al., 2016).
Sentence 2: data collection-the chosen set of predictive variables X, the sampling distribution p(X, A, Y ), and the training set size n-is an equally integral part of deploying fair machine learning systems in practice, and it should be guided to promote fairness.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The cycle-consistency property has been previously leveraged in CycleGAN [12], DiscoGAN [13] and DualGAN [14].
Sentence 2: in [12,13,14], cycle-consistency, L Cycle (θ, φ), is implemented via k losses, for k = 1, 2, and real-valued data such as images.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Conceptually, parameter optimization for the adversarial method forces the adversary to "behave like" certain properties of the training data sample, making labels easier to predict within the minimax prediction game.
Sentence 2: a key bottleneck for these methods has been their reliance on zero-sum game solvers for inference, which are computationally expensive relative to inference in other prediction methods, such as SVMs.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: (We note that the lower bound from [7] is proved using a construction in which the class probability of positive examples is less than the noise rate; the essence of that proof is to show that in that situation it is impossible to balance the distribution given access to noisy examples.)
Sentence 2: having a weak ranker provides enough leverage to yield a two-sided weak classifier without needing any rebalancing.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Among them are techniques that utilize a 3-D CAD model for supervision [44], employ fully convolutional feature learning [5], learn filters with geometrically consistent responses across different object instances [28], learn networks using dense equivariant image labelling [37], exploit local self-similarity within a fully convolutional network [18,19], and estimate correspondences using object proposals [7,8,38].
Sentence 2: none of these methods is able to handle non-rigid geometric variations, and most of them are formulated with handcrafted regularization.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: So, the preprocessed data is If we define E 2 (X) = median(X) and E(P flat ) = E 2 (X), then the breakdown point of E 2 is β 2 = 0.5, and the breakdown points of E is β = β 1 β 2 = 0.225.
Sentence 2: if we use another E 2 , then E can be more robust.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: If used for PBVI, an efficient direct backup computation of the optimal α-function for belief state b i should be used in line 17 that is linear in the number of observations [5,11] and which obviates the need for lines 23-25.
Sentence 2: for an alternate version of approximate value iteration that will often produce more accurate values for belief states other than those in B, one may instead retain the full cross-sum backup of line 17, but omit lines 23-25 -this yields an approximate VI approach (using discretized observations relevant only to a chosen set of belief states B if continuous observations are present) that is not restricted to alpha-functions only optimal at B, hence allowing greater flexibility in approximating the value function over all belief states.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Let us first consider the naive approach of concatenating (q, d) into a single vector and using f (q, d) = w [q, d] as a linear ranking model.
Sentence 2: this clearly does not learn anything useful as it would result in the same document ordering for any query, given fixed parameters w. considering a polynomial model: where Φ k (•) is a feature map that considers all possible k-degree terms: does render a useful discriminative model.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: . . , S p , the backfitting algorithm is a first order algorithm that guarantees to converge [5] and achieves the minimax rate of convergence as if only estimating a univariate function.
Sentence 2: for sparse learning problems, since the true index set is unknown, the backfitting algorithm no longer works due to the uncontrolled estimation variance.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: More specifically, they measured that cortical pyramidal neurons are the most efficient with a SER 1.2 while pyramidal neurons from the CA1 hippocampus region have a SER 1.6.
Sentence 2: inhibitory neurons were found to have less efficient action potentials with cerebellar Purkinje neurons having a SER 2 and cortical basket cell interneurons having a SER 2.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: (2010) adapted the greedy coordinate descent (GCD) to solve this convolutional sparse coding problem.
Sentence 2: for long signals, these techniques can be quite slow due the computation of the gradient (FISTA, ADMM, L-BFGS) or the choice of the best coordinate to update in GCD, which are operations that scale linearly in T . A way to alleviate this limitation is to use a locally greedy coordinate descent (LGCD) strategy, presented recently in Moreau et al.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The most similar to our work, in terms of goals, is [29], where the agent is given multiple suboptimal trajectories, and infers a hidden desired trajectory using Expectation Maximization and Kalman Filtering.
Sentence 2: their approach is less general, as it assumes a particular noise model in the expert data, whereas APID is able to handle demonstrations that are sub-optimal non-uniformly along the trajectory.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Alternatively, a straightforward independence chain Metropolis Hastings algorithm (see Supplementary Material) is defined by iteratively proposing A ′ ∼ q which is accepted with probability min{r(A ′ | A), 1} where A is a previous sample of a hierarchical partition and The tailoring of the proposal distribution q to this application based on normalized cuts dramatically aids in improving the acceptance rate relative to more naive tree proposals.
Sentence 2: the acceptance rate tends to decrease as higher posterior probability partitions A are discovered, especially for trees with many levels and large input spaces X for which the search space is larger.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Augmented Lagrangian Method (ALM) was invented early in 1969, and since then there have been several works developed Linear Program solver based on ALM [15,16,17].
Sentence 2: the challenge of ALM is that it produces a series of bound-constrained quadratic problems that, in the traditional sense, are harder to solve than linear system produced by IPM or Simplex methods [17].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In order to label a table, we need a draw from P k , which is obtained by inserting a customer into the corresponding restaurant.
Sentence 2: adopting the traditional hierarchical Pitman-Yor process is not straightforward in our case since the size of the context differs between levels of hierarchy, that is, a child restaurant in the hierarchy can have more than one parent restaurant to request a label from.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As a final point of comparison, the actual weight estimate obtained from solving (15) when Φ T Φ = I is equivalent to the non-negative garrote estimator that has been advocated for wavelet shrinkage [5,18].
Sentence 2: pSfrag replacements  the benefits the typically non-factorial nature of h * (x 2 ) are most pronounced when m > n, meaning there are more features than the signal dimension y.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Moreover, pruning techniques cannot be applied directly to the resulting hypersurfaces, and even second order measures do not exhibit standard quadratic forms to apply quadratic programming.
Sentence 2: convex functions can be efficiently approximated by piecewise linear functions, making it possible to apply the techniques described in Section 3.3 with a bounded error, as long as the approximation of ρ is bounded.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Then, β 1 X 1 + β 2 X 2 would be a subprogram, but β 1 X 1 + β 3 X 3 would not be even though Ŷ could have been computed equivalently as We might attempt to address this by allowing any subset of the terms used in the model to define a subprogram, thus capturing the commutativity and associativity of addition.
Sentence 2: this definition still excludes expressions such as β 1 X 1 + 0.5β 3 X 3 , which may be a stronger proxy than either β 1 X 1 or β 1 X 1 + β 3 X 3 .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Whereas the use of the Hessian matrix for second-order methods often provides dramatic speed-ups, a Hessian of (N K) 2 parameters is usually too large to fit in memory (let alone invert) for N > 1000 trials.
Sentence 2: we observe that the Hessian of our log-posterior is sparse: where C −1 is a sparse (banded) matrix, and ∂ 2 L/∂w 2 is a block-diagonal matrix.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: It has been shown that adapting a dictionary of basis functions to the statistics of natural images so as to maximize sparsity in the coefficients results in a set of dictionary elements whose spatial properties match those of V1 (primary visual cortex) receptive fields [24].
Sentence 2: the basis functions are learned under a probabilistic model where the probability density over the basis functions coefficients is factorial, whereas the sparse coefficients exhibit statistical dependencies [15,16].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: 2 It has repeatedly been shown that deep neural networks can achieve human-or super-human-level performance on certain supervised learning problems by leveraging large collections of labeled data.
Sentence 2: these successes come at a cost: Creating these large datasets typically requires a great deal of human effort (to manually label examples), pain and/or risk (for medical datasets involving invasive tests) or financial expense (to hire labelers or build the infrastructure needed for domain-specific data collection).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In general, M M F is not necessarily smooth in h since the topological structure of steady-states may change as h varies.
Sentence 2: in the next section we show that if there exists a stable and entry-wise non-negative steady-state, and if J is strongly-connected, then M M F (b + h) is both smooth and concave in h, regardless of the interaction strength.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Although we have not come across a previous study of one-sided Huber estimator in this context, we should note that it is related to the technique in [11], where samples are assumed to be nonnegative, and in the sample mean estimator summands are shrunk when they are above a certain threshold (this technique is called winsorizing).
Sentence 2: their model and application are quite different than what we consider in this paper.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: A plausible approach is to observe that at t = 1, the observations are generated by a tree graphical model; thus in principle one could learn the parameters of the underlying tree using existing algorithms [22,21,25].
Sentence 2: this approach does not directly produce the HMM parameters; it also does not work for biological sequences because we do not have multiple independent samples at t = 1; instead we have a single long sequence at the steady state, and the steady state distribution of observations is not generated by a latent tree.
Relationship: contrasting