Anc-VI Sets a New Standard for Reinforcement Learning Optimization

Abstract and 1 Introduction

1.1 Notations and preliminaries

1.2 Prior works

2 Anchored Value Iteration

2.1 Accelerated rate for Bellman consistency operator

2.2 Accelerated rate for Bellman optimality opera

3 Convergence when y=1

4 Complexity lower bound

5 Approximate Anchored Value Iteration

6 Gauss–Seidel Anchored Value Iteration

7 Conclusion, Acknowledgments and Disclosure of Funding and References

A Preliminaries

B Omitted proofs in Section 2

C Omitted proofs in Section 3

D Omitted proofs in Section 4

E Omitted proofs in Section 5

F Omitted proofs in Section 6

G Broader Impacts

H Limitations

4 Complexity lower bound

We now present a complexity lower bound establishing optimality of Anc-VI.

The so-called “span condition” of Theorem 5 is arguably very natural and is satisfied by standard VI and Anc-VI. The span condition is commonly used in the construction of complexity lower bounds on first-order optimization methods [13, 14, 23, 25, 59, 65] and has been used in the prior state-ofthe-art lower bound for standard VI [37, Theorem 3]. However, designing an algorithm that breaks the lower bound of Theorem 5 by violating the span condition remains a possibility. In optimization theory, there is precedence of lower bounds being broken by violating seemingly natural and minute conditions [35, 40, 98].