Policy gradient methods enjoy strong practical performance in numerous tasks in
reinforcement learning. Their theoretical understanding in multiagent settings, however,
remains limited, especially beyond two-player competitive and potential Markov games.
In this paper, we develop a new framework to characterize optimistic policy gradient
methods in multi-player Markov games with a single controller. Specifically, under the
further assumption that the game exhibits an equilibrium collapse, in that the marginals
of coarse correlated equilibria (CCE) induce Nash equilibria (NE), we show convergence
to stationary ϵ-NE in O(1/ϵ2) iterations, where O(⋅) suppresses polynomial factors in
the natural parameters of the game. Such an equilibrium collapse is well-known to
manifest itself in two-player zero-sum Markov games, but also occurs even in a class of
multi-player Markov games with separable interactions, as established by recent work.
As a result, we bypass known complexity barriers for computing stationary NE when
either of our assumptions fails. Our approach relies on a natural generalization of
the classical Minty property that we introduce, which we anticipate to have further
applications beyond Markov games.
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A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games
Recent developments in domains such as non-local games, quantum interactive proofs, and quantum generative adversarial networks have renewed interest in quantum game theory and, specifically, quantum zero-sum games. Central to classical game theory is the efficient algorithmic computation of Nash equilibria, which represent optimal strategies for both players. In 2008, Jain and Watrous proposed the first classical algorithm for computing equilibria in quantum zerosum games using the Matrix Multiplicative Weight Updates (MMWU) method to achieve a convergence rate of O(d/ε2) iterations to ε-Nash equilibria in the 4d-dimensional spectraplex. In this work, we propose a hierarchy of quantum optimization algorithms that generalize MMWU via an extra-gradient mechanism. Notably, within this proposed hierarchy, we introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as O(d/ε) iterations to ε-Nash equilibria. This quadratic speed-up relative to Jain and Watrous’ original algorithm sets a new benchmark for computing ε-Nash equilibria in quantum zero-sum games.
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- Award ID(s):
- 2023505
- PAR ID:
- 10526085
- Publisher / Repository:
- 7th International Conference on Quantum Techniques in Machine Learning
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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