arXiv research

Sharp bounds on weak convergence rate for rough volatility models.

Study approximates weak error for specific stochastic models with rough and Gaussian mean-reverting volatility.

Study on error rates for approximating rough volatility models.

Studied SGD convergence under weak conditions.

Paper proposes an algorithm to recover full supervision from weakly labeled data.

Study rough volatility models using path-dependent PDEs and fractional Brownian motions.

Improved volatility models for option pricing with weak error rates.

Establishes a microstructural foundation for a rough log-normal volatility model.

Paper proves large deviation principle for stochastic approximations.

Near-Exponential Convergence Rates for kNN Classification

Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first order schemes for constrained convex models under progressively relaxed assumptions. Stochastic proximal point is an iterative scheme born from the adaptation of proximal point algorithm to nois…

Adaptive algorithm improves convergence rate of Langevin dynamics.

Novel weak MLMC scheme for Lévy-driven SDEs, applied to financial derivatives pricing.

In this paper, we quantitative convergence in $W_2$ for a family of Langevin-like stochastic processes that includes stochastic gradient descent and related gradient-based algorithms. Under certain regularity assumptions, we show that the iterates of these stochastic processes converge to an invariant distribution at a…

Large learning rates cause oscillations in NN weights that improve generalization.

New algorithm improves convergence of gradient boosting trees.

The paper provides convergence guarantees for multicalibration gradient boosting.

Proves weak convergence equals mean convergence in GGC.

Boosting is a learning scheme that combines weak prediction rules to produce a strong composite estimator, with the underlying intuition that one can obtain accurate prediction rules by combining "rough" ones. Although boosting is proved to be consistent and overfitting-resistant, its numerical convergence rate is rela…

We present a novel approach for nonparametric regression using wavelet basis functions. Our proposal, $\texttt{waveMesh}$, can be applied to non-equispaced data with sample size not necessarily a power of 2. We develop an efficient proximal gradient descent algorithm for computing the estimator and establish adaptive m…

The stochastic gradient descent has been widely used for solving composite optimization problems in big data analyses. Many algorithms and convergence properties have been developed. The composite functions were convex primarily and gradually nonconvex composite functions have been adopted to obtain more desirable prop…

Paper generalizes extragradient methods for solving equations and inclusions with improved convergence rates.

New bounds for generative models under weaker assumptions.

Variable selection for models including interactions between explanatory variables often needs to obey certain hierarchical constraints. The weak or strong structural hierarchy requires that the existence of an interaction term implies at least one or both associated main effects to be present in the model. Lately, thi…

Sparse-penalized deep neural networks improve performance in weakly dependent processes.

Unified analysis for decentralized SGD across various topologies and updates.

In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô's formula, we get rid, in the asset price dynamics, of the stochastic i…

Paper analyzes high probability convergence of adaptive SGD with momentum.

The AdaBoost algorithm was designed to combine many "weak" hypotheses that perform slightly better than random guessing into a "strong" hypothesis that has very low error. We study the rate at which AdaBoost iteratively converges to the minimum of the "exponential loss." Unlike previous work, our proofs do not require …

Geometric analysis improves convergence of variational inference.

The significance of the study of the theoretical and practical properties of AdaBoost is unquestionable, given its simplicity, wide practical use, and effectiveness on real-world datasets. Here we present a few open problems regarding the behavior of "Optimal AdaBoost," a term coined by Rudin, Daubechies, and Schapire …

We introduce cylindrical projections to simulate infinite-dimensional occupation flows of diffusions.

The paper shows how MMD metrizes weak convergence for certain kernels.

In this paper we study utility maximization with proportional transaction costs. Assuming extended weak convergence of the underlying processes we prove the convergence of the corresponding utility maximization problems. Moreover, we establish a limit theorem for the optimal trading strategies. The proofs are based on …

We study the rate of growth of normalized Hodge numbers along a tower of abelian covers of a smooth projective variety with semismall Albanese map. These bounds are in some cases optimal. Moreover, we compute the $L^2$-Betti numbers of irregular varieties that satisfy the weak generic Nakano vanishing theorem e.g., var…

Proves curvature tensor convergence for smoothable spaces.

Weak labels can significantly speed up learning for strong tasks.

Study reveals convergence properties of SGD with random learning rate.

Discrete time analogues of ergodic stochastic differential equations (SDEs) are one of the most popular and flexible tools for sampling high-dimensional probability measures. Non-asymptotic analysis in the $L^2$ Wasserstein distance of sampling algorithms based on Euler discretisations of SDEs has been recently develop…

Study shows financial value of weak information converges in discrete vs continuous markets.

Gradient boosting is a prediction method that iteratively combines weak learners to produce a complex and accurate model. From an optimization point of view, the learning procedure of gradient boosting mimics a gradient descent on a functional variable. This paper proposes to build upon the proximal point algorithm, wh…

This paper investigates tradeoffs among optimization errors, statistical rates of convergence and the effect of heavy-tailed errors for high-dimensional robust regression with nonconvex regularization. When the additive errors in linear models have only bounded second moment, we show that iteratively reweighted $\ell_1…

Two new Frank-Wolfe algorithms improve convergence for constrained optimization.

Extends stochastic modified equations for optimization algorithms.

Improved algorithms for convex-concave min-max optimization and monotone variational inequalities.

Study improves weak error estimates for rough volatility models.

We construct a general stochastic process and prove weak convergence results. It is scaled in space and through the parameters of its distribution. We show that our simplified scaling is equivalent to time scaling used frequently. The process is constructed as an integral with respect to a Poisson random measure which …

In this paper we discuss the possibility of using multilevel Monte Carlo (MLMC) methods for weak approximation schemes. It turns out that by means of a simple coupling between consecutive time discretisation levels, one can achieve the same complexity gain as under the presence of a strong convergence. We exemplify thi…