Q-Learning with Stable Infinite-Dimensional Linear Function Approximation

• URL: http://arxiv.org/abs/2608.22636v1
• Date: Sun, 23 Aug 2026 22:40:11 GMT
• Title: Q-Learning with Stable Infinite-Dimensional Linear Function Approximation
• Abstract summary: We develop a stable infinite-dimensional linear function approximation framework for Q-learning from a single Markovian behavior-policy trajectory.<n>We illustrate the framework through Q-learning with linear density approximation and output-layer neural weight training under a frozen pretrained network.
• Score: 10.189658648290257
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Q-learning with linear function approximation can be unstable because an arbitrary approximation architecture need not preserve the Bellman contraction. We develop a stable infinite-dimensional linear function approximation framework for Q-learning from a single Markovian behavior-policy trajectory. The learning variable is a coefficient field $θ\in C(\mathbb L)$ on a compact latent metric space $(\mathbb L,ρ)$. The framework uses a reconstruction operator that maps $θ$ to a continuous Q-function and a compression operator that maps Bellman updates back to latent coordinates. Nonexpansiveness of both operators induces a contractive latent Bellman map on $C(\mathbb L)$, with a unique fixed point $θ^*$ whose reconstruction approximates the optimal Q-function up to representation error. We propose two stochastic approximation (SA) algorithms and establish their sup-norm convergence bounds with a leading term of order $\widetilde O(n^{-1/2})$. The infinite-dimensional formulation provides a powerful abstraction for identifying the structures that govern statistical difficulty. Smoothness of the compression map in $ρ$ is inherited by $θ^*$ and the SA iterates, allowing uniform estimation errors to be controlled through covering numbers of $(\mathbb L,ρ)$ rather than the dimension of $C(\mathbb L)$. Remarkably, the SA algorithms we propose are agnostic to the choice of $ρ$, and thus can automatically adapt to both the smoothness and the geometry. We further illustrate the framework through Q-measure-learning with linear density approximation and output-layer neural weight training under a frozen pretrained network.

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