Switching-Geometry Analysis of Deflated Q-Value Iteration
This academic paper introduces a novel joint spectral radius (JSR) framework for analyzing rank-one deflated Q-value iteration (Q-VI) within discounted Markov decision process control. The study focuses on an all-ones residual correction, interpreting the algorithm through the geometry of switching systems. It presents the first JSR-based convergence analysis for deflated Q-VI in policy optimization problems. The analysis demonstrates that while standard Q-VI has a JSR equal to the discount factor gamma, removing the invariant all-ones direction via a quotient space yields a projected system with a potentially smaller JSR. This allows for a sharper characterization of convergence rates compared to traditional bounds. Furthermore, the authors prove that this correction is equivalent to a scalar recentering of standard Q-VI, meaning the greedy-policy sequence remains unchanged when initialized from the same point. The primary benefit identified is not a change in the decision-making problem itself, but a more precise mathematical description of convergence geometry after eliminating redundant components. This research contributes to the fields of optimization, control theory, and artificial intelligence by enhancing the theoretical understanding of reinforcement learning algorithms.
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Switching-Geometry Analysis of Deflated Q-Value Iteration
This academic paper introduces a novel joint spectral radius (JSR) framework for analyzing rank-one deflated Q-value iteration (Q-VI) within discounted Markov decision process control. The study focuses on an all-ones residual correction, interpreting the algorithm through the geometry of switching systems. It presents the first JSR-based convergence analysis for deflated Q-VI in policy optimization problems. The analysis demonstrates that while standard Q-VI has a JSR equal to the discount factor gamma, removing the invariant all-ones direction via a quotient space yields a projected system with a potentially smaller JSR. This allows for a sharper characterization of convergence rates compared to traditional bounds. Furthermore, the authors prove that this correction is equivalent to a scalar recentering of standard Q-VI, meaning the greedy-policy sequence remains unchanged when initialized from the same point. The primary benefit identified is not a change in the decision-making problem itself, but a more precise mathematical description of convergence geometry after eliminating redundant components. This research contributes to the fields of optimization, control theory, and artificial intelligence by enhancing the theoretical understanding of reinforcement learning algorithms.
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