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arXiv · 2605.20996

Beyond the Bellman Recursion: A Pontryagin-Guided Framework for Non-Exponential Discounting

Hojin Ko·Jeonggyu Huh
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Abstract

Most value-based and actor--critic reinforcement learning methods rely on Bellman-style recursions, yet these recursions collapse under non-exponential discounting common in human preferences and survival processes. We show the breakdown is structural: exponential discounting sits at a fragile intersection of multiplicativity and time homogeneity, and violating either property breaks standard dynamic programming. To overcome this, we propose Pontryagin-Guided Direct Policy Optimization (PG-DPO), a variational framework that abandons recursion and couples the Pontryagin Maximum Principle with Monte Carlo rollouts via an Adjoint-MC projection enforcing pointwise Hamiltonian maximization. Across multi-dimensional hyperbolic and survival-discount benchmarks, PG-DPO improves accuracy and stability where equation-driven solvers and critic-based baselines diverge.

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Hojin Ko, Jeonggyu Huh. 2026-05-20. Beyond the Bellman Recursion: A Pontryagin-Guided Framework for Non-Exponential Discounting. https://arxiv.org/abs/2605.20996

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