augmented lagrangian ppo

Safety-constrained reinforcement learning often suffers from constraint violations and training instability. To address this, we propose Proactive Constraint Policy Optimization (PCPO), which introduces a predictive penalty mechanism that imposes costs before the policy approaches safety boundaries, and a boundary-aware intrinsic reward to guide safe exploration. Theoretically, we establish—for the first time—upper and lower bounds linking the duality gap to policy update performance, thereby guaranteeing convergence and stability. Methodologically, PCPO unifies Lagrangian relaxation, barrier functions, intrinsic rewards, and policy iteration into an end-to-end, safety-driven optimization framework. Experiments demonstrate that, compared to existing post-hoc correction methods, PCPO significantly reduces constraint violation rates while improving robustness and convergence stability across diverse safety-critical benchmarks.

This paper addresses constrained nonconvex optimization problems involving both equality and inequality constraints. We propose an Adaptive Proximal Augmented Lagrangian Method (AP-ALM), which introduces a novel joint adaptive update rule for the penalty parameter and proximal term: rapid growth in early iterations to accelerate convergence, followed by controlled damping in later stages to mitigate ill-conditioning. Coupled with an inexact subproblem solver, the method ensures global convergence under mild assumptions. Theoretically, AP-ALM inherits the convergence guarantees of the classical Augmented Lagrangian Method (ALM) while relaxing the requirement for exact subproblem solutions. Numerical experiments demonstrate its robustness and efficiency on both convex and nonconvex benchmarks, achieving significantly faster convergence rates and enhanced numerical stability compared to state-of-the-art alternatives.

This work addresses the challenge of balancing task performance and strict safety constraints in robotic control within safe reinforcement learning. The authors propose PPO-EAL, a novel framework that integrates the exact augmented Lagrangian method into Proximal Policy Optimization (PPO). By combining clipped policy updates with quadratic penalty terms, the approach provides theoretical guarantees for constraint satisfaction, while a momentum-based update rule for Lagrange multipliers enhances dual variable stability. PPO-EAL is the first first-order safe RL method to achieve exact constraint enforcement without requiring excessively large penalty coefficients, offering both convergence guarantees and deployment robustness. Experiments demonstrate superior performance over existing methods across multiple robotic tasks, achieving simultaneous improvements in reward and safety. Notably, the method enables zero-shot sim-to-real transfer for a gear assembly task, significantly boosting success rate and operational robustness.

This work addresses the long-standing lack of theoretical unification between the two predominant variants of Proximal Policy Optimization (PPO)—namely, PPO with a clipped surrogate objective (PPO-Clip) and PPO with KL divergence penalty. By analyzing the per-sample KL divergence, the study reveals that PPO-Clip implicitly implements a sample-wise KL regularization with a stepwise coefficient. Through closed-form derivations involving the importance sampling ratio and the advantage function, the authors construct an adaptive KL penalty coefficient that exactly reproduces the gradient updates of PPO-Clip. Empirical validation on five MuJoCo continuous control tasks demonstrates nearly identical training curves between this reformulation and the original PPO-Clip, confirming their theoretical equivalence. This insight not only provides a unified interpretation of PPO but also opens new avenues for algorithm design based on adaptive KL regularization.

For nonconvex optimization problems with nonlinear equality constraints, this paper proposes an inexact augmented Lagrangian method employing a norm penalty with exponent strictly between 1 and 2. The method constructs Hölder-smooth subproblems under convex feasibility and weak regularity assumptions, leveraging the first use of a non-integer-power Euclidean norm as the augmentation term. We establish, for the first time, accelerated first-order algorithm complexity bounds for such subproblems. Theoretically, we reveal an intrinsic trade-off: constraint violation converges faster as the exponent decreases, while dual residual decay remains controllably degraded. Numerical experiments demonstrate that the proposed method achieves superior constraint satisfaction accuracy and iteration efficiency compared to the standard squared-augmented Lagrangian method.

This study addresses the challenges of hyperparameter tuning and poorly understood interaction mechanisms in Proximal Policy Optimization (PPO) by establishing its first closed-loop, non-asymptotic convergence theory. Methodologically, we construct synchronous and asynchronous analysis frameworks to systematically characterize error propagation arising from actor-critic coupling, the clipping mechanism, and finite-batch reuse. Furthermore, we rigorously model Generalized Advantage Estimation (GAE) and Monte Carlo targets under explicit coverage assumptions. Our analysis reveals error amplification bounds and temporal dependency conditions, proving polynomial sample complexity. Ultimately, we derive an $O(T^{-2/5})$ bound on stationarity and tracking accuracy, providing solid theoretical guidance for PPO practice.

This work addresses the limitations of Proximal Policy Optimization (PPO)—notably its low sample efficiency due to gradient information loss from hard clipping—and the instability of unclipped methods like Surrogate Policy Optimization (SPO), which suffer from unbounded gradients. To overcome these issues, the authors propose Anchored Neighborhood Optimization (ANO), a unified trust-region framework that incorporates a “re-descending influence principle” to dynamically suppress the impact of outliers on policy updates, eschewing both monotonic penalties and hard thresholds. Theoretical analysis demonstrates that this mechanism is crucial for stability in high-variance stochastic optimization. Empirical results show that ANO achieves state-of-the-art performance on MuJoCo benchmarks, significantly outperforming PPO and SPO, and remains stable even with learning rates up to three times higher than standard values, effectively preventing policy collapse.

This work addresses the challenge of last-iterate convergence in policy optimization for constrained Markov decision processes (CMDPs) by proposing a general framework based on an inexact augmented Lagrangian method. It is the first to systematically apply the classical augmented Lagrangian approach to CMDPs. The framework efficiently solves subproblems via Projected Q-Ascent (PQA), circumventing the computational and storage overhead associated with mixed policies. It provides global last-iterate convergence guarantees for tabular, log-linear, and nonlinear policy classes. Empirical results demonstrate that the method achieves convergence performance on par with existing algorithms across both discrete and continuous control tasks while supporting complex policy representations.

This work addresses the long-standing lack of rigorous convergence theory for Proximal Policy Optimization (PPO) and clarifies the theoretical underpinnings of its multi-epoch minibatch update mechanism. The authors interpret PPO updates as an approximate policy gradient ascent procedure with controlled bias and, by incorporating stochastic reshuffling techniques, establish the first convergence proof framework for PPO under standard assumptions. Furthermore, they identify a weight collapse issue in truncated Generalized Advantage Estimation (GAE) at episode boundaries and propose a corrective modification. Both theoretical analysis and empirical evaluation demonstrate that the proposed correction significantly enhances PPO’s performance in environments with strong terminal signals, such as Lunar Lander.

This work addresses the limitations of traditional numerical methods—which rely heavily on gradients and initial guesses—and the slow convergence of evolutionary algorithms in high-dimensional constrained optimization. To overcome these challenges, the paper proposes embedding a population-based stochastic optimizer, such as CMA-ES, into an augmented Lagrangian (AL) framework, replacing local solvers in AL subproblems with gradient-free global search. This approach represents the first systematic integration of the AL method’s robust constraint-handling capabilities with the strong exploratory power of evolutionary algorithms, effectively balancing feasibility enforcement and global exploration. Experimental results demonstrate that the proposed method significantly outperforms both pure evolutionary algorithms and state-of-the-art solvers like IPOPT on standard benchmark problems, particularly excelling in high-dimensional nonconvex landscapes riddled with numerous local minima and saddle points.