Paper analyzes LSA algorithm bias and error bounds with RR extrapolation.
problem Analyzing bias and high-order error bounds of LSA with Markovian noise.
method Polyak-Ruppert averaging, linearization, Richardson-Romberg extrapolation.
result RR extrapolation effectively cancels the leading bias term.
Paper analyzes \FedAvg's convergence and introduces a new algorithm to reduce bias.
problem Analyzing convergence and bias in Federated Averaging.
method Markov property, first-order bias expansion, Richardson-Romberg extrapolation.
result Bias in \FedAvg can be decomposed into noise and client heterogeneity components.
The paper analyzes SGD with Richardson-Romberg extrapolation for convex optimization problems.
problem Solving strongly convex and smooth minimization problems efficiently.
method Combining SGD with Polyak-Ruppert averaging and Richardson-Romberg extrapolation.
result An expansion of the mean-squared error of the estimator with respect to the number of iterations.
Quantized Variational Inference improves ELBO optimization with fast convergence.
problem Maximizing Evidence Lower Bound (ELBO) for variational inference.
method Optimal Voronoi Tesselation for variance-free gradients, Richardson extrapolation for asymptotic improvement.
result Quantized Variational Inference leads to fast convergence with comparable computational cost.
Paper examines constant stepsize in LSA for Markovian data inference.
problem Improving statistical inference with constant stepsize in LSA for Markovian data.
method Established CLT, used averaged LSA iterates, applied Richardson-Romberg extrapolation.
result Constant stepsize leads to better CI coverage, especially with limited data.
Study Q-learning with constant stepsize, proving convergence and bias, and applying extrapolation.
problem Understanding and optimizing Q-learning with constant stepsize.
method Connecting Q-learning to a Markov chain, proving distributional convergence and bias, applying Richardson-Romberg extrapolation.
result Explicit expression for the linear coefficient of the asymptotic bias and improvement of RR extrapolation method.
New method reduces bias in incomplete data using deliberate missingness.
problem Systematic gradient biases in incomplete data for stochastic learning.
method Richardson-SGD debiasing procedure with deliberate missingness.
result Reduces gradient bias from O(∥p∥) to O(∥p∥2). Study on bias and extrapolation in LSA with Markovian data, showing bias reduction with Richardson-Romberg extrapolation.
problem Bias in LSA with constant stepsizes and Markovian data.
method Viewing LSA as a Markov chain, proving convergence and bias expansion, and applying Richardson-Romberg extrapolation.
result Bias is proportional to the stepsize up to higher order terms, and Richardson-Romberg extrapolation reduces the bias.
Improves numerical solution of ill-conditioned linear systems for machine learning.
problem Wastefulness and instability in solving ill-conditioned linear systems.
method autonugget combines Richardson extrapolation to determine the solution of the ill-conditioned system, improving accuracy over a single nugget.
result Improves accuracy of numerical solution of ill-conditioned linear systems.
Study on bias of constant-step stochastic approximation with Markovian noise.
problem Understanding the bias in stochastic approximation algorithms with Markovian noise.
method Infinitesimal generator comparisons to analyze bias, Lyapunov equation for time-averaged bias, Richardson-Romberg extrapolation for bias reduction.
result Bias of the algorithm is of order O(α) and time-averaged bias is αV+O(α2), where V is a constant. The coupled system, where one is a degenerate parabolic equation and the other has not a diffusion term arises in the modeling of European options with liquidity shocks. Two implicit-explicit (IMEX) schemes that preserve the positivity of the differential problem solution are constructed and analyzed. Numerical experim…
New estimators for intrinsic dimension and Wasserstein distance improve OT accuracy.
problem Intrinsic dimension estimation and Wasserstein distance estimation in large-scale OT.
method Introduces novel estimators for intrinsic dimension and Wasserstein distance.
result Simple, tuning-free estimator of OT and fast intrinsic dimension estimator.
We consider the minimization of an objective function given access to unbiased estimates of its gradient through stochastic gradient descent (SGD) with constant step-size. While the detailed analysis was only performed for quadratic functions, we provide an explicit asymptotic expansion of the moments of the averaged S…
For the numerical solution of the American option valuation problem, we provide a script written in MATLAB implementing an explicit finite difference scheme. Our main contribute is the definition of a posteriori error estimator for the American options pricing which is based on Richardson's extrapolation theory. This e…
In this paper, we treat the problem of evaluating the asymptotic error in a numerical integration scheme as one with inherent uncertainty. Adding to the growing field of probabilistic numerics, we show that Gaussian process regression (GPR) can be embedded into a numerical integration scheme to allow for (i) robust sel…
New insights into stochastic methods for solving variational inequalities.
problem Understanding convergence behaviors of stochastic algorithms in variational inequalities.
method Re-casting SEG/SGDA as Markov Chains to analyze their probabilistic structures.
result The average iterate is asymptotically normal with a unique invariant distribution for various VIPs.
Study on nonsmooth contractive SA with constant stepsize and Q-learning.
problem Understanding convergence and bias in nonsmooth contractive SA with different noise types.
method Proposed prelimit coupling technique for steady-state convergence and derived asymptotic bias.
result Asymptotic bias of nonsmooth SA is proportional to the square root of the stepsize.
Generalized Schouten, Froelicher-Nijenhuis and Froelicher-Richardson brackets are defined for an arbitrary Lie algebroid. Tangent and cotangent lifts of Lie algebroids are introduced and discussed and the behaviour of the related graded Lie brackets under these lifts is studied. In the case of the canonical Lie algebro…
Paper proposes a new algorithm to reduce derivative pricing computation time.
problem Derivative pricing computational inefficiency.
method Combines multilevel Richardson-Romberg and importance sampling.
result Reduces computational time while maintaining accuracy.
A new method for faster estimation of Wasserstein distance using Sinkhorn divergence.
problem Estimating the squared Wasserstein distance between probability distributions.
method Proposes a new estimator based on the Sinkhorn divergence with debiasing terms, and analyzes its sample complexity and computational efficiency.
result The proposed estimator allows higher regularization levels, leading to improved computational complexity and speedup in practice.
This is an expository paper in which we explain how basic, standard, results about simple Lie algebras can be obtained by geometric arguments, following ideas of Cartan, Richardson and others.
Probabilistic numerics expands numerical tasks with black box methods.
problem Difficult conditioning of random variables in numerical tasks.
method Construct probabilistic numerical methods based on final outputs, extrapolating limiting quantities.
result Higher orders of convergence achieved in various numerical tasks.
The rough Bergomi (rBergomi) model, introduced recently in [5], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet remarkably fits with empirical implied volatility surfaces. In the absence of analytical European option pricing methods f…
Study limits of adjoint orbits for Lie groups, describing nilpotent orbits.
problem Understanding limits of adjoint orbits for Lie groups.
method Systematic and topological study of limits of continuous families of adjoint orbits for non-compact simple Lie groups.
result Explicit description of nilpotent orbits in terms of Richardson orbits for hyperbolic semisimple elements.
We demonstrate the use of several tools from Algebraic Combinatorics such as Young tableaux, symmetry operators, the Littlewood-Richardson rule and discrete Fourier transforms of symmetric groups in investigations of algebraic curvature tensors.
RTE enables extrapolation to new tasks by learning task transformations.
problem Learning systems struggle to generalize to unseen tasks.
method Relational Task Extrapolator (RTE) learns task transformations to enable extrapolation.
result RTE substantially outperforms existing approaches on extrapolation tasks.
We extend nonparametric models to handle extrapolation, providing bounds for inference.
problem Challenges in nonparametric statistical inference when evaluating outside the conditioning variable's support.
method Introduced a class of extrapolation assumptions and a consistent estimation procedure to handle extrapolation.
result Validated extrapolation-aware conclusions through various applications and real-world data.
Study convex hulls of orbits for compact groups, defining new invariants related to polynomial degrees.
problem Understanding properties of convex hulls of coadjoint orbits of compact groups.
method Introduce partial convex hulls and use them to define numerical invariants.
result Orbits with new invariants form rational convex polyhedral cones related to Littlewood-Richardson cones.
Binary encoding enables neural networks to extrapolate periodic functions.
problem Extrapolating periodic functions without prior knowledge of their form.
method Normalized Base-2 Encoding (NB2E) for continuous numerical values.
result MLPs using NB2E can successfully extrapolate diverse periodic signals.
Method controls extrapolation in prediction profiles for statistical and machine learning models.
problem Avoiding invalid predictions due to extrapolation in prediction profiles.
method Genetic algorithm optimization over constrained factor regions.
result Optimal factor settings without constraint are often invalid and extrapolated.
The paper extends IPC framework to stationary physical systems and validates it with a photonic system.
problem Characterizing the computational capabilities of stationary physical systems in a principled, data-efficient way.
method Extended IPC framework, established fundamental results, derived asymptotic bias, introduced data-efficient estimation methods.
result IPC strongly correlates with machine-learning performance and provides a reliable estimate of system dimensionality.
For the first time in mathematical finance field, we propose the local weak form meshless methods for option pricing; especially in this paper we select and analysis two schemes of them named local boundary integral equation method (LBIE) based on moving least squares approximation (MLS) and local radial point interpol…
Neural networks struggle with extrapolation, but a new framework allows them to learn counterfactual invariances.
problem Neural networks' inability to extrapolate beyond training data distribution.
method Introduces a learning framework that allows neural networks to extrapolate over group transformations based on counterfactual invariances.
result Neural networks can learn counterfactual invariances from a single environment, overcoming their limitations in extrapolation.
Study shows LLMs can extrapolate rules from out-of-distribution prompts.
problem Understanding LLMs' ability to generalize from unexpected inputs.
method Formal languages and rule-based scenarios to evaluate LLMs' OOD behavior.
result LLMs can extrapolate rules from out-of-distribution prompts, even in complex scenarios.
New methods reduce extrapolation errors in feature importance.
problem Flawed feature importance methods using unrestricted permutations lead to extrapolation errors.
method Three new approaches: conditional model reliance, Knockoffs with Gaussian transformation, and restricted ALE plot designs.
result Theoretical and numerical results show our strategies reduce/eliminate extrapolation.
New method smooths integrands for efficient option pricing.
problem Improving numerical performance of option pricing methods.
method Combining hierarchical adaptive sparse grids, quasi-Monte Carlo, and numerical smoothing.
result Improved efficiency of ASGQ and QMC methods for high-dimensional problems.
New models extrapolate false alarms in ASV without new data.
problem Reliable extrapolation of false alarm rates in ASV without new speaker data.
method Generative models in ASV score space for arbitrary systems.
result Models accurately extrapolate false alarm rates for large speaker populations.
Neural networks extrapolate poorly in simple tasks but succeed in complex ones.
problem Understanding neural networks' extrapolation capabilities and conditions for success.
method Analyzing ReLU MLPs and GNNs, connecting to neural tangent kernel.
result ReLU MLPs learn linear functions but not most nonlinear ones, while GNNs succeed in complex tasks due to task-specific non-linearities.
A new principle for extrapolating regression outside training data.
problem Regression extrapolation when predictions are outside training data range.
method Data-adaptive marginal transformation and simple relationship assumption.
result Progression method offers guarantees on approximation error beyond training data range.
The problem of classifying, upto isometry (or similarity), the orientable spherical, Euclidean and hyperbolic 3-manifolds that arise by identifying the faces of a Platonic solid is formulated in the language of Coxeter groups. In the spherical and hyperbolic cases, this allows us to complete the classification begun by…
We extend return extrapolation to incorporate asymmetry and saturation, finding that asymmetric nonlinear extrapolation leads to lower welfare loss.
problem Optimal portfolio choice under stochastic volatility
method Smooth, nonlinear extrapolation function with sentiment and variance hedging
result Lower welfare loss with asymmetric nonlinear extrapolation
Logical neural networks solve mazes by filling dead ends, but not all methods generalize well.
problem Understanding how logical neural networks extrapolate solutions to mazes.
method Examined recurrent and implicit neural networks trained on maze-solving tasks.
result Models fail to generalize well to diverse maze sizes, suggesting limitations in learning scalable algorithms.
Accelerates Riemannian gradient methods with extrapolation.
problem Optimizing functions on manifolds efficiently.
method Extrapolating iterates in Riemannian gradient descent.
result Achieves optimal convergence rate and computational advantage.
Improves reinforcement learning extrapolation in Gridworlds.
problem Generalizing to unseen states in reinforcement learning.
method Avoiding deterministic action choice, using ego-centric representation, incorporating symmetry, and adding an entropy term.
result Significant improvement in extrapolation performance in Gridworld environments.
New methods improve neural network extrapolation to long sequences.
problem Neural networks struggle with extrapolation to very long or adversarial sequences.
method Activation binning and localized differentiable memory architecture.
result No extrapolation errors detected within memory constraints.
Concept modulation models unify identifiability and extrapolation in conditional latent variable models.
problem Reliable generalization in conditional latent variable models
method Concept modulation models (CMMs) with structure AoΛoCoX result Lifts identifiability to conditional settings and controls extrapolation through attribute potentials.
New framework shows ERM is optimal for both interpolation and extrapolation in domain generalization.
problem Formalizing and solving the challenges of domain generalization.
method Reformulated domain generalization as an online game between a risk-minimizing player and an adversary.
result ERM is minimax-optimal for both interpolation and extrapolation in domain generalization.
NSR enables neural networks to reason with continuous numbers and extrapolate.
problem Quantitative reasoning and extrapolation in neural networks.
method Proposes Neural Status Register (NSR) for continuous number reasoning.
result NSR achieves extrapolation to numbers many orders of magnitude larger than training data.