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Backstepping Temporal Difference Learning

Han-Dong Lim, Donghwan Lee

TL;DR

This work introduces a control-theoretic framework based on backstepping to design convergent off-policy TD learning algorithms with linear function approximation. By constructing stabilizing continuous-time dynamics and applying backstepping, the authors derive Backstepping TD (BTD) and recover single-time-scale variants of TDC, along with generalized versions like TDC++. The approach unifies and extends established off-policy TD methods (GTD2, TDC) under a common ODE-based stability analysis, with convergence guarantees grounded in stochastic approximation theory. Empirical results on standard benchmarks demonstrate stability and competitive performance even in domains where standard TD is unstable, and the authors discuss extensions to nonlinear function approximation and broader control-theoretic settings.

Abstract

Off-policy learning ability is an important feature of reinforcement learning (RL) for practical applications. However, even one of the most elementary RL algorithms, temporal-difference (TD) learning, is known to suffer form divergence issue when the off-policy scheme is used together with linear function approximation. To overcome the divergent behavior, several off-policy TD-learning algorithms, including gradient-TD learning (GTD), and TD-learning with correction (TDC), have been developed until now. In this work, we provide a unified view of such algorithms from a purely control-theoretic perspective, and propose a new convergent algorithm. Our method relies on the backstepping technique, which is widely used in nonlinear control theory. Finally, convergence of the proposed algorithm is experimentally verified in environments where the standard TD-learning is known to be unstable.

Backstepping Temporal Difference Learning

TL;DR

This work introduces a control-theoretic framework based on backstepping to design convergent off-policy TD learning algorithms with linear function approximation. By constructing stabilizing continuous-time dynamics and applying backstepping, the authors derive Backstepping TD (BTD) and recover single-time-scale variants of TDC, along with generalized versions like TDC++. The approach unifies and extends established off-policy TD methods (GTD2, TDC) under a common ODE-based stability analysis, with convergence guarantees grounded in stochastic approximation theory. Empirical results on standard benchmarks demonstrate stability and competitive performance even in domains where standard TD is unstable, and the authors discuss extensions to nonlinear function approximation and broader control-theoretic settings.

Abstract

Off-policy learning ability is an important feature of reinforcement learning (RL) for practical applications. However, even one of the most elementary RL algorithms, temporal-difference (TD) learning, is known to suffer form divergence issue when the off-policy scheme is used together with linear function approximation. To overcome the divergent behavior, several off-policy TD-learning algorithms, including gradient-TD learning (GTD), and TD-learning with correction (TDC), have been developed until now. In this work, we provide a unified view of such algorithms from a purely control-theoretic perspective, and propose a new convergent algorithm. Our method relies on the backstepping technique, which is widely used in nonlinear control theory. Finally, convergence of the proposed algorithm is experimentally verified in environments where the standard TD-learning is known to be unstable.
Paper Structure (36 sections, 12 theorems, 77 equations, 2 figures, 15 tables, 5 algorithms)

This paper contains 36 sections, 12 theorems, 77 equations, 2 figures, 15 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonlinear system theory
  4. Stochastic approximation and O.D.E. approach
  5. Backstepping control
  6. Markov Decision Process
  7. Temporal difference learning
  8. Gradient temporal difference learning
  9. Designing TD-learning through backstepping
  10. Backstepping TD
  11. Recovering single time-scale TDC
  12. Generalizing TDC++
  13. Experiments
  14. Conclusion
  15. Acknowledgements
  16. ...and 21 more sections

Key Result

Lemma 2.1

Suppose that assm:borkar_meyn in the Appendix holds, and consider the stochastic approximation in (eq:sa). Then, for any initial $x_0 \in \mathbb{R}^n$, $\sup_{k\geq 0}||x_k|| < \infty$ with probability one. In addition , $x_k \rightarrow x^e$ as $k \rightarrow \infty$ with probability one, where $x

Figures (2)

  • Figure 1: Backstepping diagram
  • Figure 2: O.D.E. dynamics of first element of $\lambda_t$ in Baird counter example

Theorems & Definitions (26)

  • Definition 2.1: Control Lyapunov function sontag2013mathematical
  • Lemma 2.1: Borkar and Meyn theorem borkar2000ode
  • Lemma 3.1
  • proof : Proof sketch
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Lemma 3.2
  • Theorem 3.2
  • Lemma 3.3
  • ...and 16 more