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Penalized Proximal Policy Optimization for Safe Reinforcement Learning

Linrui Zhang, Li Shen, Long Yang, Shixiang Chen, Bo Yuan, Xueqian Wang, Dacheng Tao

arXiv:2205.11814v2cs.LGcs.AImath.OC

TL;DR

Safe RL must improve reward while satisfying hard constraints, yet existing methods have difficulty with efficient updates and constraint satisfaction. P3O uses an exact penalty and clipped surrogate to optimize an equivalent unconstrained objective; experiments report stronger reward improvement and constraint satisfaction than state-of-the-art methods, including in multi-constraint and multi-agent settings.

  • Problem

    Safe RL seeks high-return policies that satisfy safety constraints, while existing approaches face difficulties with efficient updates and hard constraint satisfaction.

    Source: §1 · p. 1

  • Method

    P3O combines an exact penalty for constraints with a clipped PPO surrogate to optimize an equivalent unconstrained objective, with a finite penalty factor.

    Source: §1 · pp. 1–2

  • Results

    P3O outperforms several state-of-the-art algorithms in reward improvement and constraint satisfaction, including in multi-constraint and multi-agent scenarios.

    Source: §1 · p. 2, §5 · p. 4

  • Takeaways & Limitations

    P3O uses first-order minimization and avoids high-dimensional Hessian inversion, a benefit the paper highlights for large CMDPs with deep neural networks.

    Source: §6 · p. 6

Abstract

from arXiv · show

Safe reinforcement learning aims to learn the optimal policy while satisfying safety constraints, which is essential in real-world applications. However, current algorithms still struggle for efficient policy updates with hard constraint satisfaction. In this paper, we propose Penalized Proximal Policy Optimization (P3O), which solves the cumbersome constrained policy iteration via a single minimization of an equivalent unconstrained problem. Specifically, P3O utilizes a simple-yet-effective penalty function to eliminate cost constraints and removes the trust-region constraint by the clipped surrogate objective. We theoretically prove the exactness of the proposed method with a finite penalty factor and provide a worst-case analysis for approximate error when evaluated on sample trajectories. Moreover, we extend P3O to more challenging multi-constraint and multi-agent scenarios which are less studied in previous work. Extensive experiments show that P3O outperforms state-of-the-art algorithms with respect to both reward improvement and constraint satisfaction on a set of constrained locomotive tasks.

1 Introduction

Safe RL must balance reward with hard safety constraints, but existing approaches face approximation, computation, and scalability difficulties. P3O replaces constrained policy iteration with an equivalent unconstrained objective and is reported to improve reward and constraint satisfaction across challenging settings.

1 Introduction

  • P3O uses an exact penalty and a clipped PPO surrogate to form an equivalent unconstrained objective, avoiding quadratic approximation and Hessian inversion.The introduction also states that the method supports arbitrary initial policies and multiple constraints.Source: §1 · p. 1
  • The authors prove exactness for a sufficiently large finite penalty factor and describe an adaptive penalty-tuning algorithm.They extend clipped surrogate objectives and advantage normalization to CMDPs, reporting that these enable a fixed penalty factor across tasks.Source: §1 · p. 2
  • P3O outperforms several state-of-the-art algorithms in reward improvement and constraint satisfaction, including in multi-constraint and multi-agent scenarios.The paper describes these scenarios as less studied in prior safe RL work.Source: §1 · p. 2

2 Related Work

  • Prior approaches include primal-dual methods sensitive to initialization and multiplier learning rates, quadratic-optimization methods with approximation and scalability drawbacks, and IPO’s feasible-initialization requirement.The related-work discussion contrasts P3O’s exact penalty with IPO’s log barrier, which may have numerical issues or fail to yield the primal solution exactly.Source: §2 · p. 2

3 Preliminaries

The preliminaries define the MDP and CMDP objects used to state safe policy optimization: maximize expected return while respecting cost limits.

3 Preliminaries

  • An MDP is specified by state and action spaces, a reward function, transition probabilities, and an initial-state distribution; a stationary policy maps states to action distributions.The passage also introduces the expected discounted return as the policy objective.Source: §3 · p. 2
  • A CMDP adds cost functions and upper limits, defining feasible stationary policies as those whose expected discounted costs satisfy every limit.The limits are denoted d_i for the corresponding constraints.Source: §3 · p. 2
  • Safe RL seeks a policy that maximizes expected discounted return over the CMDP’s feasible policy set.Source: §3 · p. 2

4 Methodology

P3O turns constrained policy optimization into a penalized unconstrained objective, then uses clipped surrogates for iterative updates. The method supports finite penalty exactness, approximate-error analysis, and extensions to multiple constraints and agents.

4 Methodology

  • The new policy from the constrained update is stated to yield monotonic return improvement while satisfying hard constraints.Source: §4 · p. 3
  • ReLU penalties produce an unconstrained objective equivalent to the constrained problem when the penalty factor is sufficiently large but finite.The penalized term takes effect when its corresponding constraint is violated; when all constraints are met, the objective equals standard policy optimization.Source: §4 · p. 3
  • Replacing the future state distribution with the current policy’s distribution makes the objective tractable on collected trajectories, with resulting bias analyzed for approximate error.Source: §4 · p. 3
  • P3O incorporates a trust-region constraint through clipped surrogate objectives and alternately estimates and minimizes the unconstrained loss.The first objective term matches standard PPO; clipped constraint terms prevent updates from jumping outside the trust region during penalty reduction.Source: §4 · pp. 3–4
  • The benchmark figure covers Circle, Gather, Navigation, and Simple Spread, pairing task rewards with safety constraints.Navigation has separate hazard and pillar costs with different upper limits; Simple Spread constrains collisions without shared agent observations.Source: Figure 1 · p. 4
  • The method increases the penalty factor adaptively and uses advantage normalization to support a fixed factor across tasks; reported learning is stable across a wide range of penalty values.Source: §4 · p. 4
  • The paper extends P3O to multi-constraint CMDPs by adding penalized terms and to decentralized multi-agent settings where agents cannot share observations.Source: §4 · p. 4

5 Experiments

Experiments compare P3O on single-constraint benchmarks, sensitivity and threshold tests, a two-constraint navigation task, and a decentralized multi-agent task. The reported results show reward improvements alongside constraint satisfaction across these settings.

Single-Constraint Scenario

  • P3O is reported as the best at improving policy while satisfying the constraint; CPO shows more violations, especially on SafetyGym PointGoal.PPO-Lagrangian is described as conservative and less steady during learning.Source: Figure 2 · p. 5, §5 · p. 5, Figure 7 · p. 6
  • +15%, +24%, and +9% higher reward improvement over CPO, PPO-Lagrangian, and FOCOPS on AntCircle, and +34%, +23%, and +31% on PointGather, respectively.P3O also satisfies the hard constraint and converges closer to its upper limit.Source: Table 1 · p. 5, §5 · p. 5

Sensitivity Analysis

  • On AntCircle, P3O remains stable across a wide range of κ settings and learns constraint-satisfying policies across tested cost limits d.With different d values, it converges precisely to the cost upper limit.Source: Figure 3 · p. 5, §5 · p. 5, Figure 4 · p. 5

Multi-Constraint Scenario

  • When both navigation costs are included, P3O satisfies each constraint at convergence, while single-cost variants fail to control the ignored cost.The return is lower than in unconstrained and single-constrained settings under the stricter constraints.Source: §5 · p. 6, Figure 5 · p. 6

Multi-Agent Scenario

  • MAP3O improves total reward while satisfying the mutual-collision constraint in decentralized training and execution; its centralized-critic version performs better on reward and constraint satisfaction.Both algorithms use recurrent networks to address partial observability.Source: §5 · p. 6, Figure 6 · p. 6

6 Conclusion

P3O uses an exact-penalty unconstrained objective for safe reinforcement learning, with empirical reward and constraint benefits and compatibility with multi-constraint and multi-agent settings.

6 Conclusion

  • P3O uses an exact-penalty reformulation to optimize an unconstrained objective for safe reinforcement learning.The paper presents it as effective and easy to implement.Source: §6 · p. 6
  • P3O empirically achieves better reward performance and constraint values.Source: §6 · p. 6
  • P3O is naturally compatible with multi-constraint and multi-agent scenarios.Source: §6 · p. 6
  • First-order minimization avoids high-dimensional Hessian inversion, which the paper identifies as beneficial for large CMDPs with deep neural networks.Source: §6 · p. 6
  • The paper identifies end-to-end visual tasks as future work because they are less studied in safe reinforcement learning.Source: §6 · p. 6

Supplementary Material

The supplementary material contains detailed proofs, a multi-agent algorithm description, and experimental implementation details.

Supplementary Material

1 Proof of Proposition 1.

The proposition states that the policy update improves return monotonically while satisfying hard constraints, and its proof adapts the performance difference lemma to establish the result.

1 Proof of Proposition 1.

  • The proof applies a version of the Performance Difference Lemma adapted to more general settings.Source: §1 · p. 8
  • The policy update yields monotonic return improvement and hard constraint satisfaction.Source: §1 · p. 8
  • The proof replaces the lemma’s function with the reward and cost functions, reformulates the policy-update problem, and omits a constant term from the objective.Source: §1 · p. 8

2 Proof of Theorem 2.

The theorem establishes exact equivalence between constrained and penalized optimization when the penalty factor is at least the largest component of the corresponding optimal Lagrange multiplier vector.

2 Proof of Theorem 2.

  • At this penalty threshold, an optimum of the constrained problem is also a minimizer of the penalized problem.Source: §2 · p. 8
  • Conversely, a minimizer of the penalized problem is optimal for the constrained problem, including when feasibility is considered explicitly.Source: §2 · p. 9
  • For κ ≥ ||λ̄||∞, the constrained and penalized problems share the same optimal solution set.Source: §2 · p. 9

3 Proof of Theorem 3.

The proof section sets up bounds and restates Theorem 3 for arbitrary policies. It relates problems (M) and (N) through a δ term.

3 Proof of Theorem 3.

  • The cited passages state that problem (3) is rewritten and that problem (4) is rewritten, introducing each rewrite without showing its full expression.Source: §3 · p. 9
  • Theorem 3 is restated for arbitrary policies and relates problems (M) and (N) through a δ term.The passage states this relation when problem (M) is replaced with problem (N).Source: §3 · p. 9
  • The proof invokes Lemma 1 and Lemma 4, then uses the stated bounds to complete Theorem 3.Lemma 4 is introduced for a function on S × A × S and policies π and π′.Source: §3 · pp. 9–10

4 Multi-Agent P3O

The paper extends P3O to decentralized multi-agent reinforcement learning as MAP3O. It describes both centralized-training/decentralized-execution and fully decentralized variants.

4 Multi-Agent P3O

  • MAP3O extends P3O to decentralized multi-agent refinement learning scenarios.The paper introduces this extension as inspired by MAPPO.Source: §4 · p. 10
  • The decentralized partially observed CMDP uses global reward and constraint functions, with the joint policy optimized subject to constraints.The paper defines the setting using state, action, observation, transition, reward, and constraint components.Source: §4 · p. 10
  • The paper also summarizes a fully decentralized MAP3O iteration for each agent.Its detailed scheme is assigned to Algorithm 3.Source: §4 · p. 10
  • Under CTDE, a centralized critic takes concatenated local observations, while the framework avoids joint-action inputs.The framework is described as learning a single policy.Source: §4 · p. 10
  • Algorithm 4 collects trajectories and reward and cost returns, then updates the policy with the P3O loss and regresses value functions.Its inputs include an initial policy, reward-value function, and cost-value function.Source: §4 · p. 10

5 Implementation Details.

The implementation details describe constrained environments, agent representations, and where experiment-specific and algorithm-specific parameters are listed.

5.1 Environments

  • Circle uses d = 10, xlim = 3, and a threshold of 50, restricting safe motion to a narrower region than the reward-maximizing path.Its cost is 1[x > xlim].Source: §5.1 · p. 10
  • Gather rewards apples and penalizes bombs; Navigation assigns separate thresholds of 25 for hazards and 20 for pillars.Gather's threshold is 0.5, and Simple Spread uses a collision cost with threshold 0.5.Source: §5.1 · pp. 10–11

5.2 Agents

  • The experiments use Point and Ant agents for single constraints, mass-point or wheeled-car agents for multi-constraints, and partially observed particle agents for multi-agent tasks.The multi-agent setting has five actions per particle.Source: §5.2 · p. 11

5.3 Parameters

  • Algorithm-specific and experiment-specific parameters are summarized separately in Tables 2 and 3.The supplied passages identify the tables but do not give their parameter values.Source: §5.3 · p. 11, Table 2 · p. 11, Table 3 · p. 11
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