# Dynamic Stochastic Approximation for Multi-stage Stochastic Optimization

arXiv: https://arxiv.org/abs/1707.03324 · HTML: https://mlsanity.com/paper/1707.03324/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2017.

## Built on by (most cited first, 3 of 3)

- [Neural Stochastic Dual Dynamic Programming](https://mlsanity.com/paper/2112.00874/) (2021)
- [Numerical Methods for Convex Multistage Stochastic Optimization](https://mlsanity.com/paper/2303.15672/) (2023)
- [Mission-Aligned Learning-Informed Control of Autonomous Systems: Formulation and Foundations](https://mlsanity.com/paper/2507.04356/) (2025)

## Abstract (arXiv)

In this paper, we consider multi-stage stochastic optimization problems with convex objectives and conic constraints at each stage. We present a new stochastic first-order method, namely the dynamic stochastic approximation (DSA) algorithm, for solving these types of stochastic optimization problems. We show that DSA can achieve an optimal cal O(1/ε4) rate of convergence in terms of the total number of required scenarios when applied to a three-stage stochastic optimization problem. We further show that this rate of convergence can be improved to cal O(1/ε2) when the objective function is strongly convex. We also discuss variants of DSA for solving more general multi-stage stochastic optimization problems with the number of stages T > 3. The developed DSA algorithms only need to go through the scenario tree once in order to compute an ε-solution of the multi-stage stochastic optimization problem. As a result, the memory required by DSA only grows linearly with respect to the number of stages. To the best of our knowledge, this is the first time that stochastic approximation type methods are generalized for multi-stage stochastic optimization with T ≥ 3.