Exploring Starts Are Not Enough: Counterexamples and a Fix for Monte Carlo Exploring Starts
Authors: Octave Oliviers, Glenn Vinnicombe
Organizations: Department of Engineering · University of Cambridge · Cambridge, UK
Abstract
The asymptotic behaviour of Monte Carlo Exploring Starts (MCES) is a long-standing open question in reinforcement learning, even in the tabular setting. We investigated the convergence properties of tabular MCES by constructing examples in which the algorithm converges to suboptimal solutions. This paper presents new counterexamples for both initial-visit and first-visit MCES and gives a convergence-restoring modification for the initial-visit case. We show that stable suboptimal solutions may exist for initial-visit MCES with sample-average updates even when greedy actions are updated more often than non-greedy actions on average. However, by scaling learning rates inversely to update frequencies on a state-by-state basis, convergence to optimality is guaranteed. Unlike previous uniformisation methods, this modification is applicable to large-scale problems that require approximating the estimated value function. We then extend the example to show that sample-average first-visit MCES may also converge to suboptimal solutions. This largely settles a fundamental open problem and shows that exploring starts alone do not guarantee convergence to optimality. More broadly, these results highlight that convergence depends critically on the relative size and frequency of updates applied to different actions, making the choice of learning rates and the balance between exploration and exploitation central to the analysis of MCES and the implementation of scalable Monte Carlo control methods.
The asymptotic behaviour of Monte Carlo optimistic policy iteration (MC-O-PI) is a long-standing open question. When the model of the environment is unknown, as is common in practice, the only known condition that guarantees convergence to optimality is impractical. In its canonical form, this condition requires that the episodes used for policy evaluation be initialised uniformly over the entire state-action space. This paper strictly relaxes that requirement. Specifically, we prove that initial-visit MC-O-PI converges to optimality even when updates are uniform only over the actions within each state. This allows episodes to start in different states at arbitrary frequencies; a realistic implementation when the state space is large or unknown but the action space in each state is manageable. The proof departs from the classical analysis of Tsitsiklis whose central commutativity argument no longer applies when states are updated at different frequencies. Instead, we first show that the mean-field dynamics of MC-O-PI generate monotonically improving policies when updates are uniform over the actions in each state, and then prove that noise cannot consistently prevent this improvement by extending the lock-in argument of the combined stability-ODE method. This approach suggests a new way to study optimistic policy-iteration algorithms in general.
Tsitsiklis proved convergence of Monte Carlo optimistic policy iteration under a uniform update structure and identified nonuniform update frequencies as a delicate obstruction. We give a certified negative answer for the natural scalar-stepsize, unnormalized asynchronous state-value recursion with fixed nonuniform state-selection probabilities. In a three-state, two-action discounted MDP, the nonuniform update frequencies induce a diagonally scaled greedy-policy mean field with a certified nonconstant attracting hybrid periodic orbit. With a bounded unbiased geometric-horizon estimator and Robbins--Monro stepsizes, the original stochastic recursion remains trapped near the cycle with positive probability and therefore fails to converge. The example pinpoints a geometric obstruction: uniform sampling gives radial residual contraction, whereas scalar nonuniform sampling anisotropically distorts the residual dynamics and can generate switched attracting cycles.
This paper explores the efficacy of quasi-Monte Carlo (QMC) weight initialization for meta-reinforcement learning within modern benchmark environments. Various sampling methods are used to bound a population-based search and aggregate an optimal prior from a baseline set of tasks. The QMC meta-priors show improvements in training convergence compared to modern orthogonal (SB3) defaults when extrapolated to similar unseen continuous control environments. In dissimilar tasks, the orthogonal orientation was globally superior for an unbiased search.