Regularized Exploration Frameworks in RL

• Regularized exploration frameworks are methods that integrate convex entropy regularizers into the classical MDP formulation to systematically balance exploration and exploitation.
• They leverage convex duality and mirror descent to derive soft Bellman optimality equations, enabling dynamic programming with adaptive regularization strength.
• These frameworks offer robust convergence guarantees and guide the design of algorithms like TRPO and DPP, improving sample efficiency and ensuring principled policy updates.

Regularized exploration frameworks in reinforcement learning (RL) refer to a family of approaches that explicitly incorporate convex optimization-based regularization terms—most prominently, various forms of entropy—into the policy optimization process. These frameworks unify and theoretically ground modern RL algorithms that promote systematic exploration, offering robust convergence properties and performance guarantees by leveraging deep connections to convex duality, mirror descent, and dynamic programming. By embedding convex regularizers (such as Shannon or relative entropy) into the classical Markov decision process (MDP) formulation, these frameworks enable principled control over the exploration–exploitation trade-off, elucidate the convergence behaviors of a wide class of algorithms, and provide a foundation for designing novel exploration strategies in both discrete and continuous settings.

1. Convex Optimization Formulation for Regularized MDPs

Classical RL policy optimization for long-term average reward can be cast as a linear program over stationary state–action distributions: $$\mu^* = \arg\max_{\mu \in \Delta} \quad \rho(\mu) = \sum_{x, a} \mu(x, a) r(x, a)$$ subject to the flow constraints ensuring stationarity. Here, $\Delta$ is the set of feasible stationary distributions satisfying

$$\sum_a \mu(y, a) = \sum_{x, a} P(y \mid x, a) \mu(x, a), \ \forall y.$$

Regularized exploration frameworks generalize this by introducing a convex regularizer $R(\mu)$ and optimizing

$$\max_{\mu \in \Delta} \left\{ \sum_{x, a} \mu(x, a) r(x, a) - \frac{1}{\eta} R(\mu) \right\}, \quad \eta > 0$$

where $R(\mu)$ is typically chosen to be negative (conditional or relative) entropy.

Key regularizers include:

Regularizer	Mathematical Form	Notable Case
Relative entropy	$$R_S(\mu) = \sum_{x, a} \mu(x, a) \log[\mu(x, a) / \mu'(x, a)]$$	REPS, Mirror Descent
Conditional entropy	$$R_C(\mu) = \sum_x \nu_\mu(x) \sum_a \pi_\mu(a|x)\log[\pi_\mu(a|x)]$$	TRPO, DPP, Policy Iteration

Where $\nu_\mu(x) = \sum_a \mu(x, a)$ and $\pi_\mu(a|x) = \mu(x, a)/\nu_\mu(x)$. This convex regularized structure is the foundation for a unified view of regularized policy optimization, facilitating analytical connections to duality and dynamic programming.

2. Duality, BeLLMan Optimality, and Regularized BeLLMan Equations

A central technical result is that regularization by conditional entropy $R_C(\mu)$ leads to a Lagrange dual that yields a nonlinear system closely resembling the BeLLMan optimality equations: $$V^*(x) = \frac{1}{\eta} \log \sum_{a} \pi_{\mu'}(a|x) \exp\left\{ \eta \left[r(x, a) - \rho^* + \sum_y P(y|x, a)V^*(y) \right] \right\}.$$ This is the "regularized average-reward BeLLMan optimality equation," showing that entropy-regularized policy optimization can be performed by dynamic programming with "soft" maximum operators. The form of regularization determines the structure of these dual (BeLLMan-like) equations. The Legendre–Fenchel transform appears as a critical tool, enabling: $$\Omega^*(q) = \max_{\pi \in \Delta_A} \{\langle \pi, q \rangle - \Omega(\pi)\}$$ and yielding unique softmax-, sparsemax-, or composite-type greedy policies via the mapping $G_{\Omega}(v) = \nabla\Omega^*(q)$.

3. Algorithmic Schemes: Mirror Descent, Policy Iteration, and Practical Implementations

The convex–dual formulation provides an interpretative and practical framework linking algorithms to standard convex optimization methods:

• Mirror Descent (MD): Algorithms such as REPS and TRPO can be cast as approximate or exact mirror descent steps in the policy space, using relative entropy as the Bregman generating function.
• Dual Averaging (DA): Policy optimization steps that optimize surrogate objectives (as in Asynchronous Advantage Actor-Critic or A3C) correspond to DA, where regularization manages the update velocity and convergence properties.
• Regularized Greedy Policy and Policy Iteration: The softmax update underlying TRPO is a direct instantiation of entropy-regularized (soft) policy greedy steps, where the update is

$$\pi^*_\eta(a|x) \propto \pi_{\mu'}(a|x)\exp\left\{\eta A^*(x, a)\right\},$$

with $A^*$ the advantage function from the dual solution. Adaptive variants (as in TRPO, DPP) are highlighted as superior to fixed-regularization methods (RegVI) due to their improved accounting for drifting state distributions.

Algorithmic variants differ in how closely they approximate the exact regularized policy iteration; exact methods guarantee global convergence, while out-of-family approximations (such as entropy-regularized policy gradients based on non-convex parameterizations) may lack convergence guarantees.

4. Convergence, Empirical Behavior, and the Role of Regularization Strength

The theoretical analysis exploits convexity to establish global convergence for algorithms that correctly instantiate the dual update or regularized BeLLMan operator. In particular, exact TRPO (as formulated via the dual) always converges to the globally optimal policy due to the non-expansion property of the regularized policy improvement operator. In contrast, entropy-regularized policy gradients—as in A3C—optimize non-convex objectives and may thus become stuck at suboptimal points or diverge if the optimization landscape drifts erratically due to sampling and evolving policies.

Empirical evaluation (e.g., in gridworld environments) confirms three regimes depending on the regularization parameter $\eta$:

• Large $\eta$ (weak regularization): Policy becomes greedy prematurely, potentially leading to suboptimal exploitation and failure to discover high-reward strategies.
• Small $\eta$ (strong regularization): Policy remains overly stochastic ("soft"), accumulating insufficient reward.
• Intermediate $\eta$ (balanced): The trade-off between exploration and exploitation induces policies that both explore sufficiently and target optimal long-term rewards. Adaptive methods (regularized iterative algorithms with time-varying $\eta$) empirically outperform fixed-regularization schemes.

5. Theoretical and Practical Implications

The unified, convex-regularized framework offers several notable implications:

• Principled Design and Comparison of Algorithms: By mapping TRPO, DPP, and variants onto convex optimization methods (Mirror Descent, Dual Averaging), the framework explains convergence behaviors and provides a taxonomy rooted in convex analysis.
• Regularized Exploration Strategy: Entropy regularization—especially when using conditional entropy over state–action distributions—naturally induces temporally-consistent and systematic exploration. Parameter $\eta$ serves as a temperature controlling stochasticity, with adaptation mechanisms providing practical levers in continuous control and discrete domains.
• Foundation for New Methods: By aligning RL with tools from convex optimization (e.g., composite objective methods or more sophisticated Bregman divergences), the framework enables novel approaches to exploration and policy learning, with potential cross-fertilization in related fields such as robust planning or adversarial games.
• Broader Relevance: The theoretical tools, particularly the strong duality between entropy-regularized MDPs and soft BeLLMan equations, support both analysis and further algorithmic development across RL, bandit problems, and planning under uncertainty.

6. Summary

Regularized exploration frameworks, as formalized via entropy-regularized MDPs and their convex optimization-based extensions, provide a unified theoretical and practical basis for the design, analysis, and implementation of exploration strategies in reinforcement learning. The use of convex regularizers—most prominently, Shannon and relative entropies—leads to dual formulations that anchor much of modern RL within mirror descent and dual averaging principles. These insights yield exact convergence guarantees for algorithms such as TRPO, explain empirical trade-offs in regularization strength, and establish the mathematical underpinnings required for principled design of robust, convergent, and sample-efficient exploration mechanisms in both classical and deep RL.