Boosting Soft Q-Learning by Bounding

TL;DR

This work introduces a zero-shot framework to bound the optimal soft Q-function $Q^*(s,a)$ from any bounded value estimate in entropy-regularized RL, enabling double-sided bounds that facilitate clipping during training. By deriving bounds from $V(s)=\frac{1}{\beta}\log \mathbb{E}_{a\sim\pi_0} e^{\beta Q(s,a)}$ and $\Delta(s,a)=r(s,a)+\gamma \mathbb{E}_{s'}V(s')-Q(s,a)$, the authors design a clipping mechanism that accelerates learning in both tabular and function-approximation settings, and prove convergence of the clipped Bellman operator to $Q^*$. The approach extends to continuous spaces via batch-extrema approximations and Lipschitz-based probabilistic bounds, with empirical validation showing faster convergence and robust training in diverse tasks. Overall, the bounds provide a principled way to reuse prior value information to tighten target estimates and improve data efficiency in value-based RL. Potential applications include tighter initializations, ensemble-based bound tightening, and integration with model-based or actor-critic methods to boost performance in real-world tasks.

Abstract

An agent's ability to leverage past experience is critical for efficiently solving new tasks. Prior work has focused on using value function estimates to obtain zero-shot approximations for solutions to a new task. In soft Q-learning, we show how any value function estimate can also be used to derive double-sided bounds on the optimal value function. The derived bounds lead to new approaches for boosting training performance which we validate experimentally. Notably, we find that the proposed framework suggests an alternative method for updating the Q-function, leading to boosted performance.

Figures (7)

• Figure 1: Schematic illustration of the main contribution of this work. Given any approximation (red curve) to the optimal value function of interest (black curve), we derive double-sided bounds (blue curves) that lead to clipping approaches during training. Based solely on the current approximation for $Q(s,a)$ (red curve), we derive double-sided bounds on the unknown optimal value function $Q^*(s,a)$ (black curve). In the right panel, we show the different clipping methods, which are described further in the "Experimental Validation" section. In "Hard Clipping", the target is replaced with the exceeded bound; in "Soft Clipping", an additional loss term is appended to the Bellman loss, proportional to the magnitude of the bound violation.
• Figure 2: Here we show specific results on a representative environment, and further examples are given in the Appendix. At each step, the agent receives a small penalty if it has not reached the goal (orange diamond). The discount factor $\gamma=0.98$ and inverse temperature parameter $\beta=5$ are fixed throughout these experiments. From left to right: (1) The optimal policy is shown at the inset. The greedy policy is evaluated during training for the various methods presented. "Baseline Bounds" refers to clipping during training with $\left[\frac{r_\textrm{min}}{1-\gamma}, \frac{r_\textrm{max}}{1-\gamma}\right]$. (2,3) The mean and range of $Q$-values and the proposed bounds (Equation \ref{['eq:gen_double_sided_bound']}). Clipping during training constrains the $Q$-values to a tight range much faster than without clipping. Each method is averaged over $30$ random initializations.
• Figure 3: Speed of learning (measured as area under evaluation reward curve) with $Q$-value clipping during TD updates. Each point is the result of averaging over 30 randomly generated $7\times 7$ mazes with stochastic transitions. Further details of the experiment are given in Appendix \ref{['app:code']}.
• Figure 4: We test the proposed clipping methods (labeled None, Hard, and Soft; described below) across the classic control suite. We fine-tuned each environment's hyperparameters (details in Appendix \ref{['app:code']}). The average evaluation reward plotted is the reward achieved by following the stochastic optimal policy, averaged over 5 episodes. Each method in a given environment is averaged over 30 random initializations, with the 95% bootstrapped confidence interval shaded. To ensure the performance stems from our bounds alone, we have not included the simpler $R_\text{min,max}/(1-\gamma)$ bounds which are likely to improve the performance further.
• Figure 5: Examples of the random maps generated for the tabular experiments.

Theorems & Definitions (25)

• Theorem 1
• Proposition 1
• Theorem 2 (Informal)
• Lemma A
• proof
• Theorem 1
• proof
• Proposition 1
• proof
• Corollary 1