arXiv subjects

Chiara Schindler

Publications and source records attributed to Chiara Schindler.

3 recordsLinked to original sources

arXiv2025

Asymptotic behaviour of stochastic inertial dynamics incorporating a Tikhonov regularization term

In a separable Hilbert space, we study the minimization problem of a convex smooth function with Lipschitz continuous gradient whose evaluations are corrupted by random noise. To this end, we associate a stochastic inertial system that incorporates Tikhonov regularization with the optimization problem. We establish existence and uniqueness of a solution trajectory for this system. Then, we derive an upper bound on the expected value of an appropriate associated energy function given square-integrability of the diffusion $σ_X$ before focusing on the particular case where the parameter function multiplied by the Tikhonov term is given by $\frac{1}{t^r}$ for $0<r<2$. For this setting, we show a.s. convergence rates as well as convergence rates in expectation for the function values along the trajectory to an infimal value, the trajectory process to an optimal solution and its time derivative to zero under a stronger integrability condition on $σ_X$.

math.OC↗

arXiv2025

Long-Time Analysis of Stochastic Heavy Ball Dynamics for Convex Optimization and Monotone Equations

In a separable real Hilbert space, we study the problem of minimizing a convex function with Lipschitz continuous gradient in the presence of noisy evaluations. To this end, we associate a stochastic Heavy Ball system, incorporating a friction coefficient, with the optimization problem. We establish existence and uniqueness of trajectory solutions for this system. Under a square integrability condition for the diffusion term, we prove almost sure convergence of the trajectory process to an optimal solution, as well as almost sure convergence of its time derivative to zero. Moreover, we derive almost sure and expected convergence rates for the function values along the trajectory towards the infimal value. Finally, we show that the stochastic Heavy Ball system is equivalent to a Su-Boyd-Candès-type system for a suitable choice of the parameter function, and we provide corresponding convergence rate results for the latter. In the second part of this paper, we extend our analysis beyond the optimization framework and investigate a monotone equation induced by a monotone and Lipschitz continuous operator, whose evaluations are assumed to be corrupted by noise. As before, we consider a stochastic Heavy Ball system with a friction coefficient and a correction term, now augmented by an additional component that accounts for the time derivative of the operator. We establish analogous convergence results for both the trajectory process and its time derivative, and derive almost sure as well as expected convergence rates for the decay of the residual and the gap function along the trajectory. As a final result, we show that a particular instance of the stochastic Heavy Ball system for monotone equations is equivalent to a stochastic second-order dynamical system with a vanishing damping term. Remarkably, this system exhibits fast convergence rates for both the residual and gap functions.

math.OC↗

arXiv2024

On a Stochastic Differential Equation with Correction Term Governed by a Monotone and Lipschitz Continuous Operator

In our pursuit of finding a zero for a monotone and Lipschitz continuous operator $M : \R^n \rightarrow \R^n$ amidst noisy evaluations, we explore an associated differential equation within a stochastic framework, incorporating a correction term. We present a result establishing the existence and uniqueness of solutions for the stochastic differential equations under examination. Additionally, assuming that the diffusion term is square-integrable, we demonstrate the almost sure convergence of the trajectory process $X(t)$ to a zero of $M$ and of $\|M(X(t))\|$ to $0$ as $t \rightarrow +\infty$. Furthermore, we provide ergodic upper bounds and ergodic convergence rates in expectation for $\|M(X(t))\|^2$ and $\langle M(X(t), X(t)-x^*\rangle$, where $x^*$ is an arbitrary zero of the monotone operator. Subsequently, we apply these findings to a minimax problem. Finally, we analyze two temporal discretizations of the continuous-time models, resulting in stochastic variants of the Optimistic Gradient Descent Ascent and Extragradient methods, respectively, and assess their convergence properties.

math.OC↗