arXiv subjects

Lorick Huang

Publications and source records attributed to Lorick Huang.

11 recordsLinked to original sources

arXiv2026

Continuized Nesterov Momentum Achieves the $O(\varepsilon^{-7/4})$ Complexity in Smooth Nonconvex Optimization

For first-order optimization of non-convex functions with Lipschitz-continuous gradient and Hessian, the best-known complexity for reaching an $\varepsilon$-approximation of a stationary point is $\mathcal{O}(\varepsilon^{-7/4})$. The existing algorithms achieving this bound are based on momentum, but are always complemented with safeguard mechanisms that erase the accumulated momentum if a certain condition is violated. Whether such momentum-control mechanisms are fundamentally necessary has remained an open question. We show that randomizing the parameters enables one to achieve this complexity in expectation when using momentum without any of such mechanisms, and we improve the numerical constant factor of the bound in the case of a large enough number of iterations. From an analysis perspective, we do so by leveraging the continuized method, which interprets the algorithm as a realization of a continuous-time stochastic differential equation (SDE) involving a Poisson process. We show that this SDE converges in probability to the Heavy Ball ordinary differential equation when the stepsize goes to zero, paralleling the behavior of more classical instances of Nesterov momentum.

math.OC↗

arXiv2025

Continuized Nesterov Acceleration for Non-Convex Optimization

In convex optimization, continuous-time counterparts have been a fruitful tool for analyzing momentum algorithms. Fewer such examples are available when the function to minimize is non-convex. In several cases, discrepancies arise between the existing discrete-time results, namely those obtained for momentum algorithms, and their continuous-time counterparts, with the latter typically yielding stronger guarantees. We argue that the continuized framework (Even et al., 2021), mixing continuous and discrete components, can tighten the gap between known continuous and discrete results. This framework relies on computations akin to standard Lyapunov analyses, from which are deduced convergence bounds for an algorithm that can be written as a Nesterov momentum algorithm with stochastic parameters. In this work, we extend the range of applicability of the continuized framework, e.g. by allowing it to handle non-smooth Lyapunov functions. We then strengthen its trajectory-wise guarantees for linear convergence rate, deriving finite time bounds with high probability and asymptotic almost sure bounds. We apply this framework to the non-convex class of strongly quasar convex functions. Adapting continuous-time results that have weaker discrete equivalents to the continuized method, we improve by a constant factor the known convergence rate, and relax the existing assumptions on the set of minimizers.

math.OC↗

arXiv2025

Weak uniqueness for the PDE governing the joint law of a diffusion and its running supremum

In a previous work [8], it was shown that the joint law of a diffusion process and the running supremum of its first component is absolutely continuous, and that its density satisfies a non standard weak partial differential equation (PDE). In this paper, we establish the uniqueness of the solution to this PDE, providing a more complete understanding of the system's behavior and further validating the approach introduced in [8].

math.AP↗

arXiv2024

CV@R penalized portfolio optimization with biased stochastic mirror descent

This article studies and solves the problem of optimal portfolio allocation with CV@R penalty when dealing with imperfectly simulated financial assets. We use a Stochastic biased Mirror Descent to find optimal resource allocation for a portfolio whose underlying assets cannot be generated exactly and may only be approximated with a numerical scheme that satisfies suitable error bounds, under a risk management constraint. We establish almost sure asymptotic properties as well as the rate of convergence for the averaged algorithm. We then focus on the optimal tuning of the overall procedure to obtain an optimized numerical cost. Our results are then illustrated numerically on simulated as well as real data sets.

math.OC↗

arXiv2024

Rate of Convergence in the Functional Central Limit Theorem for Stable Processes

In this article, we quantify the functional convergence of the rescaled random walk with heavy tails to a stable process.This generalizes the Generalized Central Limit Theorem for stable random variables infinite dimension. We show that provided we have a control between the randomwalk or the limiting stable process and their respective affine interpolation, we canlift the rate of convergence obtained for multivariate distributions to a rateof convergence in some functional spaces.

math.PR↗

arXiv2023

Local Limit Theorems and Strong Approximations for Robbins-Monro Procedures

The Robbins-Monro algorithm is a recursive, simulation-based stochastic procedure to approximate the zeros of a function that can be written as an expectation. It is known that under some technical assumptions, Gaussian limit distributions approximate the stochastic performance of the algorithm. Here, we are interested in strong approximations for Robbins-Monro procedures. The main tool for getting them are local limit theorems, that is, studying the convergence of the density of the algorithm. The analysis relies on a version of parametrix techniques for Markov chains converging to diffusions. The main difficulty that arises here is the fact that the drift is unbounded.

math.PR↗

arXiv2021

Convergence of the Discrete-Time Compound Hawkes Processwith Exponential or Erlang Kernel

Due to its clustering and self-exciting properties, the Hawkes process has been used extensively in numerous fields ranging from sismology to finance. Since data is often aquired on regular time intervals, we propose a piece-wise constant model based on a Discrete-Time Hawkes Process (DTHP). We prove that this discrete-time model converges to the usual continuous-time Hawkes process as the time-step tends to zero.

math.PR↗

arXiv2021

The Malliavin-Stein method for Hawkes functionals

In this paper, following Nourdin-Peccati's methodology, we combine the Malliavin calculus and Stein's method to provide general bounds on the Wasserstein distance between functionals of a compound Hawkes process and a given Gaussian density. To achieve this, we rely on the Poisson embedding representation of an Hawkes process to provide a Malliavin calculus for the Hawkes processes, and more generally for compound Hawkes processes. As an application, we close a gap in the literature by providing the first Berry-Esséen bounds associated to Central Limit Theorems for the compound Hawkes process.

math.PR↗

arXiv2018

A Local Limit Theorem for Robbins-Monro Procedure

The Robbins-Monro algorithm is a recursive, simulation-based stochastic procedure to approximate the zeros of a function that can be written as an expectation. It is known that under some technical assumptions, a Gaussian convergence can be established for the procedure. Here, we are interested in the local limit theorem, that is, quantifying this convergence on the density of the involved objects. The analysis relies on a parametrix technique for Markov chains converging to diffusions, where the drift is unbounded.

math.PR↗

arXiv2015

A Multi-Step Richardson-Romberg Extrapolation Method For Stochastic Approximation

We obtain an expansion of the implicit weak discretization error for the target of stochastic approximation algorithms introduced and studied in [Frikha2013]. This allows us to extend and develop the Richardson-Romberg extrapolation method for Monte Carlo linear estimator (introduced in [Talay & Tubaro 1990] and deeply studied in [Pag{è}s 2007]) to the framework of stochastic optimization by means of stochastic approximation algorithm. We notably apply the method to the estimation of the quantile of diffusion processes. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost.

math.PR↗

arXiv2015

A Parametrix Approach for some Degenerate Stable Driven SDEs

We consider a stable driven degenerate stochastic differential equation, whose coefficients satisfy a kind of weak H{ö}rmander condition. Under mild smoothness assumptions we prove the uniqueness of the martingale problem for the associated generator under some dimension constraints. Also, when the driving noise is scalar and tempered, we establish density bounds reflecting the multi-scale behavior of the process.

math.PR↗