R-GTD: A Geometric Analysis of Gradient Temporal-Difference Learning in Singular Regimes

Researchers propose R-GTD, a regularized gradient temporal-difference learning algorithm that maintains convergence guarantees even when the feature interaction matrix becomes singular—a practical limitation in existing GTD methods. The geometric analysis provides explicit error bounds and addresses a key stability challenge in off-policy reinforcement learning with function approximation.

Gradient temporal-difference learning has become foundational for off-policy policy evaluation in reinforcement learning, but a critical assumption undermines its real-world applicability: the nonsingularity of the feature interaction matrix (FIM). When this matrix becomes singular—a common occurrence in practical deployments with high-dimensional or redundant feature spaces—existing GTD algorithms suffer from instability and performance degradation. This paper directly addresses this gap through R-GTD, introducing regularization that reformulates the mean-square projected Bellman error minimization problem. The key contribution lies not merely in adding regularization, but in proving convergence to a unique solution under singular conditions, which prior regularization attempts failed to guarantee theoretically. The geometric analysis framework employed here provides explicit error bounds that practitioners can calculate, offering quantifiable performance expectations rather than asymptotic guarantees alone. For the reinforcement learning and AI communities, this work bridges theory and practice by removing a restrictive assumption that prevented deployment in realistic scenarios. The singular FIM condition frequently emerges when working with overparameterized neural networks or correlated feature representations, making this advancement particularly relevant to modern deep reinforcement learning applications. The empirical validation through experiments strengthens confidence in the method's practical utility. Looking forward, this theoretical foundation may accelerate adoption of GTD methods in production systems and inspire similar regularization approaches for other algorithms constrained by similar assumptions.

• →R-GTD guarantees convergence even when the feature interaction matrix is singular, removing a major practical limitation of standard GTD algorithms.
• →Geometric analysis provides explicit error bounds rather than asymptotic convergence rates, enabling practitioners to quantify expected performance.
• →The reformulated optimization objective naturally incorporates regularization without requiring external parameter tuning beyond standard approaches.
• →Results apply to off-policy policy evaluation with function approximation, a critical component in modern reinforcement learning pipelines.
• →Empirical validation confirms theoretical predictions, suggesting practical utility for high-dimensional or overparameterized learning scenarios.