Penalized Proximal Policy Optimization (P3O)

• P3O is a reinforcement learning framework that extends PPO by incorporating explicit penalty terms to balance reward maximization with constraint enforcement.
• It utilizes diverse penalty mechanisms—such as pointwise penalties, barrier functions, and divergence metrics—to regulate policy updates and expand the solution manifold.
• Empirical results on tasks like Atari and Mujoco demonstrate that P3O improves performance, sample efficiency, and safety compliance compared to standard PPO.

Penalized Proximal Policy Optimization (P3O) refers collectively to a family of reinforcement learning algorithms that modify the Proximal Policy Optimization (PPO) framework by introducing explicit penalization terms to regulate policy updates. These penalization mechanisms serve to enforce trust-region constraints, enhance exploration, enforce constraints (such as safety), or regularize policy search, often resulting in more robust optimization and improved empirical performance compared to standard PPO. P3O terminology encompasses pointwise penalties (e.g., POP3D), barrier methods (e.g., PPO-B), KL-based regularization, pairwise update rules for relative feedback, and unconstrained reformulations with exact penalty functions.

1. Penalization Objectives and Mathematical Formulation

P3O algorithms augment the PPO surrogate objective with a penalty term designed to regularize the step between the current policy $\pi_{\text{old}}$ and proposed update $\pi_\theta$. In standard PPO, the surrogate objective is defined as

$$L^{\text{PPO}}(\theta) = \mathbb{E}_t \left[ \min \left( r(\theta) \hat{A}_t, \text{clip}(r(\theta), 1-\epsilon, 1+\epsilon) \hat{A}_t \right) \right], \quad r(\theta) = \frac{\pi_\theta(a_t|s_t)}{\pi_{\text{old}}(a_t|s_t)}$$

P3O generalizes this via a penalized surrogate:

$$L^{\text{P3O}}(\theta) = \mathbb{E}_t \left[ r(\theta) \hat{A}_t - \beta \cdot \mathcal{D}(\pi_{\text{old}}, \pi_\theta) \right]$$

where $\mathcal{D}$ can be a pointwise metric (POP3D), KL divergence (KL-PPO), barrier function, relative divergence (PPO-RPE), correntropy-induced metric (CIM-PPO), or a soft-clipping scalar (Scopic-PPO). The penalty coefficient $\beta$ regulates the trade-off between reward maximization and update regularization.

Specific penalty terms include:

• Penalized Point Probability Distance (POP3D):

$$\mathcal{D}_{\text{pp}}(\pi_{\text{old}}, \pi_\theta) = \left[ \pi_{\text{old}}(a|s) - \pi_\theta(a|s) \right]^2$$

• Logarithmic Barrier (PPO-B):

$$J^{\text{ADBAR}} = \mathbb{E} \left[ r(\theta) \hat{A}_t + \mu \cdot \log \left( \delta - \left( \sqrt{\pi_\theta(a_t|s_t)} - \sqrt{\pi_{\text{old}}(a_t|s_t)} \right)^2 \right) \right]$$

• Relative Pearson Divergence (PPO-RPE):

$$\Omega^{\text{RPE}} = C \cdot \frac{(\rho_\beta - 1)^2}{\rho_\beta}, \quad \rho_\beta = \frac{\rho}{1-\beta+\beta\rho}$$

• Correntropy-Induced Metric (CIM-PPO):

$$\text{CIM}_\sigma(\pi_{\text{old}}, \pi_\theta) = \| \Phi(\pi_{\text{old}}(\cdot|s)) - \Phi(\pi_\theta(\cdot|s)) \|_2$$

• Scopic (Sigmoid Soft-Clipping, P3O (Chen et al., 2022)):

$$L^{\text{sc}}(\theta) = \mathbb{E}_t \left[ \sigma(\tau(r(\theta)-1)) \frac{4}{\tau} \hat{A}_t \right]$$

2. Trust-Region Enforcement, Exploration, and Solution Manifold Expansion

P3O variants address limitations of PPO and TRPO by refining the enforcement of trust-region constraints and by expanding the feasible policy update region while avoiding over-penalization and restrictive update clipping.

• POP3D applies a targeted penalty on the probability assigned to the actual action taken, enabling the solution manifold (the set of parameters yielding near-optimal policies) to expand. This avoids unnecessary penalization of irrelevant action probabilities, increases robustness to parameter variations, and supports better exploration in deep neural policy spaces (Chu, 2018).
• PPO-B introduces a logarithmic barrier penalty, which "explodes" near the constraint boundary, ensuring step-wise strict feasibility and improved sampling efficiency compared to external penalty methods where feasibility is obtained only in the infinite penalty limit (Zeng et al., 2018).
• PPO-RPE and CIM-PPO utilize symmetric divergence metrics (relative Pearson divergence or kernel-based correntropy distance) to avoid directional bias and unbalanced regularization that can arise in asymmetric penalty functions such as KL divergence, particularly in high-dimensional or continuous action spaces (Kobayashi, 2020, Guo et al., 2021).

3. Penalty Design: Symmetric, Adaptive, and Exact Penalties

Penalty terms in P3O are carefully constructed to provide desirable properties:

• Symmetry: Pointwise, CIM, and relative-divergence penalties treat $\pi_{\text{old}}$ and $\pi_\theta$ equally, reducing spurious bias and exploration asymmetry.
• Adaptivity: Several P3O implementations adapt penalty coefficients or clipping thresholds online, e.g., via the effective sample size (ESS) measurements in off-policy updates (Fakoor et al., 2019), or via auto-tuning threshold-based gains in PPO-RPE.
• Exactness: For safe RL applications, P3O with ReLU penalties replaces hard constraints with exact penalty terms. The equivalence theorem guarantees, for sufficiently large penalty factor $\kappa$, the unconstrained penalized problem has the same solution set as the original constrained problem for finite $\kappa$ (Zhang et al., 2022). This ensures hard cost constraints are strictly enforced even with first-order optimization.

4. Off-Policy and Relative Feedback Extensions

P3O methodology extends PPO by combining on-policy and off-policy data, and by supporting trajectory-wise, preference-based RL objectives.

• Off-policy P3O interleaves on-policy and off-policy updates, using importance sampling and KL regularization. The ESS statistic adaptively tunes the IS ratio clipping and KL penalty coefficients to balance bias and variance (Fakoor et al., 2019).
• Pairwise P3O leverages comparative feedback in RLHF (Reinforcement Learning from Human Feedback) and is invariant to constant reward offsets (reward equivalence). Its pairwise policy gradient update computes trajectory-wise reward differences, facilitating direct alignment with relative losses (e.g., Bradley–Terry) and enabling more robust and consistent convergence in LLM alignment tasks (Wu et al., 2023).

5. Constrained Reinforcement Learning, Barrier/Central Path Methods

For safe RL and CMDPs, P3O has been adapted to handle cost and safety constraints via barrier penalties and central-path geometries:

• PPO-B and C3PO introduce a barrier function that blows up at the constraint boundary, forcing iterates to follow the central path of the feasible set (rather than oscillating or sticking at the edge). Linear (ReLU-type) penalties "move" as training progresses, enabling strict constraint satisfaction without permanent bias on the reward objective (Zeng et al., 2018, Milosevic et al., 31 May 2025).
• C3PO recasts the penalized policy optimization by coupling the reward advantage with an exact penalty-controlled cost advantage. This architecture avoids oscillatory updates and ensures convergence to the constrained optimum with high sample efficiency (Milosevic et al., 31 May 2025).

6. Empirical Performance and Implementation Considerations

P3O algorithms have demonstrated enhanced empirical performance across a range of settings:

• In Atari and Mujoco tasks, POP3D won 32/49 Atari games versus 11 for PPO, and outperformed PPO in several continuous domains (Chu, 2018).
• PPO-B achieved superior sampling efficiency, winning a greater number of Atari and Mujoco tasks compared to standard PPO (Zeng et al., 2018).
• Off-policy P3O reduced sample complexity and increased returns relative to A2C, PPO, ACER, and other strong baselines, with automatic adaptation of off-policy update factors (Fakoor et al., 2019).
• In safe RL, exact penalty P3O and central path C3PO consistently enforce constraints with minimal reward degradation, outperforming PPO-Lagrangian and trust-region algorithms in Safety Gymnasium tasks (Zhang et al., 2022, Milosevic et al., 31 May 2025).
• Pairwise P3O established superior KL-reward tradeoffs in RLHF benchmarks, confirming both theoretical and practical invariance properties under reward equivalence (Wu et al., 2023).

Typical implementation involves:

• Standard actor-critic architectures.
• Experience replay (for off-policy).
• Penalty coefficients chosen via theoretical bounds, adaptively, or via domain-driven tuning.
• Barrier function parameters calibrated to environment-specific cost budgets.
• Optionally, policy parameterizations (e.g., Beta distributions) for bounded domains or high variance mitigation (Hsu et al., 2020).
• Use of GAE for variance reduction, or omission of value function estimation in trajectory-wise variants.

7. Broader Impact, Extensions, and Open Directions

P3O provides a flexible framework for robust, stable RL in domains requiring adaptive constraint handling, reliable generalization, and preference-based alignment. Notable extensions include:

• Multi-constraint and multi-agent support (via additive penalty terms) (Zhang et al., 2022).
• Continuous optimal control via occupation measures and KL penalty regularization (Zhao et al., 2023).
• Minimax estimation in partially observable, confounded offline RL settings (Lu et al., 2022).
• Kernel-based penalty metrics (CIM) for trust-region enforcement in high-dimensional function spaces (Guo et al., 2021).

Future avenues include:

• Application of central path and receding penalty strategies to broader classes of constrained RL and safe RL tasks.
• Adaptive or automated penalty tuning for improved generalization.
• Unification of pairwise and penalized RLHF for LLMs within generic trust-region policy optimization frameworks.
• Theoretical analysis of solution manifold expansion and its implications for deep policy learning stability.

P3O thus serves as a technically rigorous, empirically validated, and extensible framework for penalized policy optimization, providing a suite of methods and insights for addressing the challenges of modern reinforcement learning.