# Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning

arXiv: https://arxiv.org/abs/2108.02632 · HTML: https://mlsanity.com/paper/2108.02632/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Aug 2021.

## Built on by (most cited first, 8 of 8)

- [A Fisher-Rao gradient flow for entropy-regularised Markov decision processes in Polish spaces](https://mlsanity.com/paper/2310.02951/) (2023)
- [On the ISS Property of the Gradient Flow for Single Hidden-Layer Neural Networks with Linear Activations](https://mlsanity.com/paper/2305.09904/) (2023)
- [Fusing Multiple Algorithms for Heterogeneous Online Learning](https://mlsanity.com/paper/2312.05432/) (2023)
- [Remarks on the Polyak-Lojasiewicz inequality and the convergence of gradient systems](https://mlsanity.com/paper/2503.23641/) (2025)
- [On the Robustness of Derivative-free Methods for Linear Quadratic Regulator](https://mlsanity.com/paper/2506.12596/) (2025)
- [Some remarks on gradient dominance and LQR policy optimization](https://mlsanity.com/paper/2507.10452/) (2025)
- [Descend or Rewind? Stochastic Gradient Descent Unlearning](https://mlsanity.com/paper/2511.15983/) (2025)
- [Global linear convergence of entropy-regularized softmax policy gradient beyond tabular MDPs](https://mlsanity.com/paper/2605.24939/) (2026)

## Abstract (arXiv)

Recent work on data-driven control and reinforcement learning has renewed interest in a relative old field in control theory: model-free optimal control approaches which work directly with a cost function and do not rely upon perfect knowledge of a system model. Instead, an "oracle" returns an estimate of the cost associated to, for example, a proposed linear feedback law to solve a linear-quadratic regulator problem. This estimate, and an estimate of the gradient of the cost, might be obtained by performing experiments on the physical system being controlled. This motivates in turn the analysis of steepest descent algorithms and their associated gradient differential equations. This note studies the effect of errors in the estimation of the gradient, framed in the language of input to state stability, where the input represents a perturbation from the true gradient. Since one needs to study systems evolving on proper open subsets of Euclidean space, a self-contained review of input to state stability definitions and theorems for systems that evolve on such sets is included. The results are then applied to the study of noisy gradient systems, as well as the associated steepest descent algorithms.