| Publications about 'gradient flows' |
| Articles in journal or book chapters |
| Solutions of optimization problems, including policy optimization in reinforcement learning, typically rely upon some variant of gradient descent. There has been much recent work in the machine learning, control, and optimization communities applying the Polyak-Łojasiewicz Inequality (PLI) to such problems in order to establish an exponential rate of convergence (a.k.a. ``linear convergence'' in the local-iteration language of numerical analysis) of loss functions to their minima under the gradient flow. Often, as is the case of policy iteration for the continuous-time LQR problem, this rate vanishes for large initial conditions, resulting in a mixed globally linear / locally exponential behavior. This is in sharp contrast with the discrete-time LQR problem, where there is global exponential convergence. That gap between CT and DT behaviors motivates the search for various generalized PLI-like conditions, and this paper addresses that topic. Moreover, these generalizations are key to understanding the transient and asymptotic effects of errors in the estimation of the gradient, errors which might arise from adversarial attacks, wrong evaluation by an oracle, early stopping of a simulation, inaccurate and very approximate digital twins, stochastic computations (algorithm ``reproducibility''), or learning by sampling from limited data. We describe an ``input to state stability'' (ISS) analysis of this issue. We also discuss convergence and PLI-like properties of ``linear feedforward neural networks'' in feedback control. Much of the work described here was done in collaboration with Arthur Castello B. de Oliveira, Leilei Cui, Zhong-Ping Jiang, and Milad Siami. This is a short paper summarizing the slides presented at my keynote at the 2025 L4DC (Learning for Dynamics \& Control Conference) in Ann Arbor, Michigan, 05 June 2025. A partial bibliography has been added. |
| This paper studies the effect of perturbations on the gradient flow of a general constrained nonlinear programming problem, where the perturbation may arise from inaccurate gradient estimation in the setting of data-driven optimization. Under suitable conditions on the objective function, the perturbed gradient flow is shown to be small-disturbance input-to-state stable (ISS), which implies that, in the presence of a small-enough perturbation, the trajectory of the perturbed gradient flow must eventually enter a small neighborhood of the optimum. This work was motivated by the question of robustness of direct methods for the linear quadratic regulator problem, and specifically the analysis of the effect of perturbations caused by gradient estimation or round-off errors in policy optimization. Interestingly, we show small-disturbance ISS for three of the most common optimization algorithms: standard gradient flow, natural gradient flow, and Newton gradient flow. |
| Recent work on data-driven control and reinforcement learning has renewed interest in a relatively 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 paper 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. |
| Conference articles |
| This work explores generalizations of the Polyak-Lojasiewicz inequality (PLI) and their implications for the convergence behavior of gradient flows in optimization problems. Motivated by the continuous-time linear quadratic regulator (CT-LQR) policy optimization problem -- where only a weaker version of the PLI is characterized in the literature -- this work shows that while weaker conditions are sufficient for global convergence to, and optimality of the set of critical points of the cost function, the "profile" of the gradient flow solution can change significantly depending on which "flavor" of inequality the cost satisfies. After a general theoretical analysis, we focus on fitting the CT-LQR policy optimization problem to the proposed framework, showing that, in fact, it can never satisfy a PLI in its strongest form. We follow up our analysis with a brief discussion on the difference between continuous- and discrete-time LQR policy optimization, and end the paper with some intuition on the extension of this framework to optimization problems with L1 regularization and solved through proximal gradient flows. |
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