On a Stochastic Differential Equation with Correction Term Governed by a Monotone and Lipschitz Continuous Operator
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| Format: | Preprint |
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2024
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| _version_ | 1866929329932337152 |
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| author | Bot, Radu Ioan Schindler, Chiara |
| author_facet | Bot, Radu Ioan Schindler, Chiara |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_17986 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a Stochastic Differential Equation with Correction Term Governed by a Monotone and Lipschitz Continuous Operator Bot, Radu Ioan Schindler, Chiara Optimization and Control Probability 34F05, 47H05, 60H10, 68W20 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. |
| title | On a Stochastic Differential Equation with Correction Term Governed by a Monotone and Lipschitz Continuous Operator |
| topic | Optimization and Control Probability 34F05, 47H05, 60H10, 68W20 |
| url | https://arxiv.org/abs/2404.17986 |