Generalised Bellman recurrence and three dualities in sequential decision-making
Authors: Fernando E. Rosas, David Hyland, Daniel Polani
Organizations: Department of Informatics, University of Sussex · Department of Computer Science, University of Oxford · Department of Computer Science, University of Hertfordshire
What gives the Bellman equation its form? We show that the recursive properties of optimal value functions follow from three conditions: that the dynamics decomposes through sufficient statistics, that the return decomposes recursively, and that the aggregation of uncertainty is compatible with both. When all three conditions hold on a common state, the Bellman equation arises from their mutual consistency; when one fails, tractability can often be recovered by augmenting the state or by deforming return or dynamics. The same conditions are shown to give rise to three dualities: one between probability and return, one between return and aggregation, and one between aggregation and probability. Our framework reveals these dualities as arising from a single construction, unifying methods developed separately across reinforcement learning, control, and decision theory.
In continual reinforcement learning, carefully managing the stability-plasticity tradeoff remains a core challenge. Recent work by Abel et al. (2025) formalized this dilemma by defining plasticity as the generalized directed information from an agent's observations to its actions, and empowerment as the generalized directed information from its actions to its observations. This formulation successfully reframes the traditional stability-plasticity tradeoff as an empowerment-plasticity tradeoff. However, while extensive literature exists on optimizing for empowerment, there is currently no research addressing the optimization of plasticity under this new definition. This paper presents preliminary work toward optimizing plasticity within Markov decision processes. We show that there exists a Bellman optimality equation for optimizing plasticity similar to previous work for empowerment.
We introduce Bellman-sufficient information complexity for minimax analysis of sequential decision problems. A Bellman-sufficient state retains enough of the history to close the controlled recursion, while an index Y=χ(Ω) specifies the decision-relevant information being charged. The upper bound is a log-penalized Bellman program; the lower bound is a Bellman--Fano comparison along an algorithm-dependent reference trajectory. If the two values match at a common localization scale and the stated admissibility, calibration, and growth conditions hold, they form an information-risk sandwich. UCB, E2D, and AMS/EBO control or relax the upper Bellman bracket in different ways. For the main application, we give a negative answer to a widely studied form of the GP--UCB minimax-optimality question. For every 0<α<1/4, we construct one bounded continuous kernel whose minimax regret is Θ(T1−α) along an infinite sequence of horizons, while two globally calibrated GP--UCB rules incur linear regret under one fixed truth. An epochwise finite-marginal action-index AIR Bellman policy, implemented through robust AIR/AMS/EBO control, attains the minimax order. The construction separates realized information from the cost of uniform optimism: many low-value directions inflate the exploration multiplier and change the trajectory. Through the canonical RKHS feature map, it also yields a finite-horizon polynomial minimax separation for the specified maximal-information-calibrated LinUCB rule. A reproducible experiment illustrates the mechanism.
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.