Actor-Critic Methods in RL

• Actor-critic methods are reinforcement learning algorithms that combine explicit policy optimization with value-function approximation to enable efficient learning and robust control.
• They integrate actor updates via policy gradients with critic evaluations, incorporating innovations like critic-gradient action generation and actorless frameworks for improved performance.
• These methods are extended to diverse domains such as simulation optimization and PDEs, enhanced by regularization and bias correction to ensure stability and rigorous theoretical guarantees.

Actor-critic methods are a foundational class of algorithms in reinforcement learning (RL) that combine explicit parametric policy optimization ("actor") with value-function approximation ("critic"). This two-network architecture enables both sample-efficient learning and principled policy improvement, but introduces significant challenges related to architectural choices, stability, and scaling. Recent innovations—including critic-gradient action generation, regularization, confidence bounds, and meta-learning—have further expanded the actor-critic paradigm for continuous and discrete control, simulation-based optimization, PDEs, and large-scale domains.

1. Core Principles and Standard Actor-Critic Workflows

Classical actor-critic algorithms alternate between two phases:

• Policy evaluation (critic update): Estimate either the state value $V^\pi(s)$ or state-action value $Q^\pi(s,a)$ using Bellman backup targets. Variants employ temporal-difference (TD) learning, Monte Carlo rollouts, or off-policy corrections.
• Policy improvement (actor update): The policy (typically denoted $\pi_\theta(a|s)$) is updated in the direction that increases expected return according to the critic's latest estimates, using policy-gradient methods (REINFORCE, deterministic policy gradient, advantage-weighted updates).

The critic stabilizes policy updates and enables off-policy learning via experience replay. However, architectural choices (e.g., the relative width of actor and critic networks), choice of update targets, and coupling in entropy regularization profoundly affect performance and sample efficiency (Mastikhina et al., 1 Jun 2025).

Recent work has highlighted the interplay between the actor's capacity and the critic's generalization behavior, with small or under-parameterized actors leading to degraded data coverage and overfitted critics (Mastikhina et al., 1 Jun 2025). The modular nature of actor-critic allows extensions such as regularization penalties, meta-critics, and Stackelberg game-theoretic corrections.

2. Critic-Gradient Action Generation and the Actor-Critic Without Actor Paradigm

The ACA framework ("Actor-Critic without Actor") (Ki et al., 25 Sep 2025) represents a fundamental departure by entirely eliminating the explicit actor network:

• Critic-guided sampling: Actions are produced by iterative denoising using the gradient field of a noise-level critic $Q_\phi(s,a_t,t)$, which is trained over both clean ($t=0$) and noisy ($t>0$) actions generated via a forward diffusion process.
• Policy improvement: No separate actor network is maintained; instead, policy improvement is achieved directly and immediately via the critic's current gradients. Entropy regularization is embedded by constructing Boltzmann-like target distributions: $\hat\pi_t(a_t|s)\propto\exp(wQ_\phi(s,a_t,t))$.
• Training algorithm: Data collection proceeds by critic-gradient denoising (see paper Eq. (4)), with TD-regression at $t=0$ and value-transport at $t>0$ (see paper Eq. (6)). Hyperparameters such as guidance weight $w$, number of denoising steps $T$, and the critic network architecture are crucial for robust performance.

This approach ensures tight coupling between policy improvement and the most current value estimates, eliminates actor-induced lag and instability, and retains the ability to represent complex multi-modal action distributions without diffusion-based actors (demonstrated on multi-modal bandit and MuJoCo tasks) (Ki et al., 25 Sep 2025).

3. Regularization, Bias Correction, and Stability Enhancements

Several methods address instability and bias arising from inaccurate or slow-moving critics:

• TD-Regularization: Penalizing the actor update by the critic’s squared Bellman error (TD error), $$J_\text{reg}(\theta)=J(\theta)-\lambda\mathbb{E}[\,\delta^2\,]$$, dampens large policy changes when the critic is inaccurate, improving stability and convergence, especially in high-variance or difficult domains (Parisi et al., 2018).
• Residual and Stackelberg Actor-Critic: The gap between standard actor-critic and true policy gradient is precisely characterized as $$\Delta(\theta,w)=\partial_\theta [d_\theta^\top\delta_{\theta,w}]$$, leading to practical correction schemes (e.g., Res-AC maintains a residual value network to learn this correction; Stack-AC uses implicit differentiation) (Wen et al., 2021).
• Meta-Critic Learning: An auxiliary meta-critic network is meta-trained online to directly produce additional loss signals for the actor, shaping updates to accelerate convergence and improve asymptotic policy quality (Zhou et al., 2020). Meta-critic approaches have achieved robust performance gains without test-time overhead.

These mechanisms can be implemented as plug-and-play additions to actor-critic workflows, complementing techniques such as double-critic targets, target smoothing, and advantage estimation.

4. Architectural Design, Exploration, and Asymmetric Schemes

Actor-critic architecture design—specifically the relative capacity of the actor versus critic—has substantial implications for exploration, overfitting, and sample efficiency:

• Symmetric vs. Asymmetric Architectures: Large critics with small actors tend to suffer from data sparsity, exploration collapse, and critic overfitting. The metric $o_\phi$ (ratio of validation to training TD error) quantitatively increases with actor shrinkage (Mastikhina et al., 1 Jun 2025).
• Optimistic Critics: Modifying the critic's update target to use the mean or max of an ensemble (instead of the minimum in SAC) alleviates underestimation and empowers smaller actors to achieve near full-size performance (Mastikhina et al., 1 Jun 2025). These modifications restore critic plasticity and promote exploration even under partial observability.
• Multi-modal Action Coverage: ACA achieves near-uniform mode coverage via critic-gradient denoising, outperforming diffusion-based actor methods in maintaining diversity of explored actions (Ki et al., 25 Sep 2025).

Advanced actor-critic frameworks now systematically decouple entropy regularization in the actor and critic, as in discrete-action DSAC (Asad et al., 11 Sep 2025), enhancing exploration and avoiding pitfalls of overly conservative joint regularization.

5. Extensions to Complex Domains: Simulation-Based Optimization and PDEs

Actor-critic methodology supports diverse applications beyond standard RL:

• Simulation-Based Optimization: The critic serves as a surrogate for expensive black-box objectives, and the actor adapts a sampling distribution to maximize expected simulated performance plus entropy (Li et al., 2021). This paradigm handles both continuous and discrete design spaces, supports adversarial tasks, and provides alternatives to Bayesian optimization for high-dimensional domains.
• Hamilton-Jacobi-Bellman PDEs: Neural actor-critic architectures have been successfully adapted to solve high-dimensional HJB PDEs. These employ boundary-respecting critic constructions, policy-gradient actor updates via simulated trajectories, and variance-reduced temporal-difference training for robust estimation (Zhou et al., 2021, Cohen et al., 8 Jul 2025). Infinite-width analysis yields ODE convergence guarantees and verification theorems for the learned control policies.

The modularity and expressiveness of actor-critic, including the ability to integrate advanced surrogates and domain-specific losses, make it well-suited for these settings.

6. Theoretical Analysis: Sample Complexity, Convergence, and Offline RL

Rigorous analysis quantifies the end-to-end convergence, bias, and efficiency of actor-critic algorithms:

• Sample Complexity: The two time-scale actor-critic method achieves $\mathcal{\tilde O}(\epsilon^{-2.5})$ sample complexity to reach an $\epsilon$-stationary point, balancing critic drift and actor stochasticity under general mixing and function-approximation conditions (Wu et al., 2020). Actor-critic matches SGD rates under fast critic convergence and exhibits slower rates when critic error persists (Kumar et al., 2019).
• Offline Actor-Critic: Pessimistic actor-critic procedures achieve minimax-optimal regret bounds in tabular and linear-function approximation settings by leveraging second-order cone programming and uncertainty sets, outperforming optimism-based methods that expand function class complexity (Zanette et al., 2021).
• Monotonic Policy Improvement and Decision-Aware Training: Joint actor-critic objectives based on mirror-descent lower bounds ensure monotonic policy improvement even under limited critic capacity, provided critic errors are controlled below explicit geometric thresholds (Vaswani et al., 2023).

For discrete control, off-policy actor-critic frameworks now guarantee theoretical convergence to optimal regularized value functions by decoupling entropy terms and incorporating m-step Bellman updates (Asad et al., 11 Sep 2025).

7. Empirical Benchmarks and Comparative Results

Actor-critic methods have been evaluated extensively on continuous and discrete RL benchmarks, simulation tasks, and high-dimensional control problems:

Algorithm	Performance/Return	Robustness	Memory/Compute
ACA (Ki et al., 25 Sep 2025)	Superior sample efficiency, fast early learning, competitive asymptotic performance	Robust multi-modal coverage, fewer parameters than SAC/diffusion actors	~30% fewer parameters, higher per-step compute due to denoising
OPAC (Roy et al., 2020)	State-of-the-art MuJoCo returns, stable learning	Three-critic ensemble improves robustness, exploration	Comparable runtime to SAC, moderate complexity
GreedyAC (Neumann et al., 2018)	Environment-tuned performance exceeds SAC	High robustness to entropy tuning	O(N) additional Q-eval. per state
Meta-Critic (Zhou et al., 2020)	Accelerated convergence, higher peak returns	No test-time penalty, lightweight auxiliary network	Extra meta-critic compute only during training
TD-REG (Parisi et al., 2018)	10–20% improved return, reduced variance	Plug-and-play stabilization	Minor extra backprop cost
Asymmetric/Optimistic critic (Mastikhina et al., 1 Jun 2025)	Restores returns in small-actor settings	Reduces overfitting, generalizes across tasks	No extra parameters, mean/max critic cost negligible

Across MuJoCo (Ant, HalfCheetah, Hopper, Walker2d, Humanoid), Atari, and simulation-based benchmarks, these innovations yield improved sample efficiency, robust multi-modal exploration, and reduced architectural or hyperparameter complexity.

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Actor-critic methods have evolved from two-network, gradient-coupled RL solvers to a rich family of flexible, scalable algorithms with rigorous theoretical foundations and domain-adaptive implementations. The current frontier includes actorless architectures based on critic gradients, decision-aware learning objectives, regularization for bias and stability, and principled exploration mechanisms. Open directions concern automating guidance weights, extending to non-linear function classes, adaptive critic corrections, and further theoretical analysis of scaling and bias.