Finite-time analysis of Multi-timescale Stochastic Optimization Algorithms
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| Format: | Preprint |
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2026
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| author | Kartikey, Kaustubh Bhatnagar, Shalabh |
| author_facet | Kartikey, Kaustubh Bhatnagar, Shalabh |
| contents | We present a finite-time analysis of two smoothed functional stochastic approximation algorithms for simulation-based optimization. The first is a two time-scale gradient-based method, while the second is a three time-scale Newton-based algorithm that estimates both the gradient and the Hessian of the objective function $J$. Both algorithms involve zeroth order estimates for the gradient/Hessian. Although the asymptotic convergence of these algorithms has been established in prior work, finite-time guarantees of two-timescale stochastic optimization algorithms in zeroth order settings have not been provided previously. For our Newton algorithm, we derive mean-squared error bounds for the Hessian estimator and establish a finite-time bound on $\min\limits_{0 \le m \le T} \mathbb{E}\| \nabla J(θ(m)) \|^2$, showing convergence to first-order stationary points. The analysis explicitly characterizes the interaction between multiple time-scales and the propagation of estimation errors. We further identify step-size choices that balance dominant error terms and achieve near-optimal convergence rates. We also provide corresponding finite-time guarantees for the gradient algorithm under the same framework. The theoretical results are further validated through experiments on the Continuous Mountain Car environment. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_29380 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finite-time analysis of Multi-timescale Stochastic Optimization Algorithms Kartikey, Kaustubh Bhatnagar, Shalabh Machine Learning We present a finite-time analysis of two smoothed functional stochastic approximation algorithms for simulation-based optimization. The first is a two time-scale gradient-based method, while the second is a three time-scale Newton-based algorithm that estimates both the gradient and the Hessian of the objective function $J$. Both algorithms involve zeroth order estimates for the gradient/Hessian. Although the asymptotic convergence of these algorithms has been established in prior work, finite-time guarantees of two-timescale stochastic optimization algorithms in zeroth order settings have not been provided previously. For our Newton algorithm, we derive mean-squared error bounds for the Hessian estimator and establish a finite-time bound on $\min\limits_{0 \le m \le T} \mathbb{E}\| \nabla J(θ(m)) \|^2$, showing convergence to first-order stationary points. The analysis explicitly characterizes the interaction between multiple time-scales and the propagation of estimation errors. We further identify step-size choices that balance dominant error terms and achieve near-optimal convergence rates. We also provide corresponding finite-time guarantees for the gradient algorithm under the same framework. The theoretical results are further validated through experiments on the Continuous Mountain Car environment. |
| title | Finite-time analysis of Multi-timescale Stochastic Optimization Algorithms |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2603.29380 |