analyze policy gradients

This work addresses the theoretical and practical bias in policy gradient methods for discounted reinforcement learning, arising from state-action distribution mismatch—i.e., discrepancies between the stationary distributions of the behavior and target policies. Methodologically, we first prove, for the tabular setting, that policy gradients converge to the global optimum despite distribution mismatch. We then extend the analysis to general function approximation via a biased stochastic gradient descent (Biased SGD) framework, characterizing how gradient bias affects convergence rate and optimality. Leveraging the policy gradient theorem, Markov chain stationary distribution modeling, and generalization error analysis, we quantitatively expose the gap between classical theoretical assumptions (e.g., exact gradient estimation) and practical implementations. Our contributions provide novel theoretical guarantees on the robustness of policy gradient algorithms and inform the design of more reliable approximate policy optimization methods.

Conventional policy optimization relies heavily on approximate dynamic programming or reinforcement learning (RL) frameworks, requiring explicit modeling of actions, control signals, and reward functions—limiting generalizability across diverse control and learning tasks. Method: This paper introduces a novel paradigm that embeds parameterized policies directly into autonomous dynamical systems, enabling joint optimization of policy parameters and system dynamics at the continuous-time dynamical level—bypassing RL-specific abstractions. Contribution/Results: The approach unifies behavior cloning, mechanism design, system identification, and state estimation within a single differentiable optimization framework, without requiring reward engineering or action-space specification. Theoretically, its gradient updates are shown to be equivalent to standard policy gradients, natural gradients, and PPO updates. Empirically, it achieves performance on par with state-of-the-art RL methods across diverse control benchmarks and provides a more intrinsic, fully differentiable foundation for optimizing generative AI systems.

This work addresses the low sample efficiency of traditional model-free reinforcement learning methods—such as Proximal Policy Optimization (PPO)—which rely on high-variance advantage estimates. The authors propose Analytic Policy Gradients (APG), a method that leverages differentiable environment dynamics to compute exact, end-to-end gradients of policy returns with respect to policy parameters. To mitigate gradient degradation in long-horizon tasks, APG incorporates a segment-wise backpropagation mechanism and combines Monte Carlo estimation with critic-guided bootstrapping for effective gradient guidance. Evaluated on four continuous control benchmarks under identical network architectures and training protocols, APG consistently outperforms PPO, demonstrating substantially higher sample efficiency and faster convergence.

Policy gradient (PG) methods suffer from low sample efficiency and slow convergence in continuous control due to their reliance on newly sampled on-policy trajectories. This work provides the first theoretical proof that reusing historical off-policy trajectories achieves the optimal convergence rate of Õ(ε⁻¹). To this end, we propose a power-mean-corrected multiple importance weighting estimator and design the Randomized Policy Gradient (RPG) algorithm, which integrates off-policy learning with stochastic gradient analysis. Theoretically, RPG attains the state-of-the-art sample complexity for PG methods. Empirically, it significantly outperforms existing advanced PG algorithms across benchmark continuous-control tasks. Our core contributions are: (i) establishing the first rigorous convergence guarantee for trajectory reuse in PG; and (ii) introducing a bias-correction mechanism that simultaneously achieves theoretical optimality and practical effectiveness.

Policy optimization algorithms suffer from poor interpretability and error-prone implementation due to the complexity of Markov decision process (MDP) modeling and inconsistent use of discounted versus average-reward settings. Method: This paper introduces a unified analytical framework that, for the first time, systematically integrates generalized ergodicity theory with perturbation analysis to characterize the steady-state behavior of diverse policy optimization algorithms under both discounted and average-reward criteria. Contribution/Results: The framework clarifies fundamental algorithmic principles, identifies and corrects common implementation pitfalls, and significantly enhances interpretability and robustness. Empirical validation on MDP modeling and linear quadratic regulator (LQR) benchmarks confirms the framework’s ability to capture algorithmic consistency. Quantitative analysis further demonstrates that minor adjustments to key design parameters exert decisive influence on convergence properties and performance.

This work addresses the absence of a unified principled framework in existing large language model policy optimization methods, which obscures the mechanistic roles and design motivations of diverse algorithms within their objective functions. Starting from the expected reward objective, the paper constructs a diagnostic unified framework structured around two axes: trajectory-side and reward-side factors—centered on trajectory probabilities and rewards, respectively. Through first-principles derivation, this framework systematically integrates the evolutionary logic of REINFORCE, PPO, GRPO, and their variants (e.g., Agentic RL, GRPO-OPD), revealing compound failure modes that cannot be resolved by improvements on either axis alone and delineating the failure boundaries of current approaches. The proposed framework provides a principled foundation for joint policy optimization, offering both extensibility and diagnostic capability.

This work addresses the slow convergence and low sample efficiency of Generative Flow Networks (GFlowNets) in structured discrete sampling tasks by leveraging their theoretical connection to entropy-regularized reinforcement learning. For the first time, Proximal Policy Optimization (PPO) is integrated into the GFlowNet training framework. Through a systematic design of policy gradient updates, advantage estimation, and baseline mechanisms, the proposed approach substantially enhances training stability and data efficiency. Experimental results on benchmark tasks—including synthetic energy functions and molecular graph generation—demonstrate that the method achieves faster convergence and higher sampling efficiency compared to standard GFlowNet objectives.

Standard policy gradient methods often converge to suboptimal stationary points within restricted policy classes due to their reliance on single-step Q-function updates. This work proposes a generalized k-step policy gradient method that overcomes such myopia and avoids distribution mismatch by coupling stochasticity across a k-step temporal window. Theoretically, we establish—for the first time—that this approach exponentially approaches the performance of the optimal deterministic policy, requiring only smoothness and differentiability of the value function. By integrating projected gradient and mirror descent techniques, the algorithm achieves an exponentially near-optimal solution within O(1/T) iterations, demonstrating broad applicability to complex settings such as state aggregation and partially observable multi-agent coordination.

This work addresses the lack of a unified theoretical framework for reinforcement learning, which has hindered systematic analysis of its convergence, sample complexity, and generalization. Building upon Markov decision processes and Bellman operators, the paper introduces a cohesive analytical framework that integrates tools from operator theory, stochastic approximation, convex duality, and function approximation. This framework encompasses a broad range of algorithms, including value iteration, policy iteration, temporal difference methods, off-policy learning, and constrained MDPs. By leveraging contraction mappings, monotone operators, martingale techniques, mirror/proximal optimization, concentration inequalities, and mixing process theory, the study establishes finite-sample performance bounds and asymptotic convergence guarantees for diverse reinforcement learning algorithms, thereby forging a rigorous theoretical bridge between probability theory, optimization, and statistics.

This work investigates Group Relative Policy Optimization (GRPO), revealing that its use of intra-group average return as a baseline imposes a zero-sum constraint on the advantage function, which undermines credit assignment and induces gradient sparsity, thereby hindering multi-step reasoning. Building upon the policy gradient theorem, the study is the first to demonstrate that the GRPO gradient matrix possesses an intrinsic rank-2 structure, proving that its effective rank remains approximately two regardless of group size. It further establishes that this baseline is optimal under specific conditions and quantifies the exacerbation of gradient sparsity during training. Through theoretical analysis, singular value decomposition, and empirical validation on Nemotron-4B with GSM8K, the paper identifies credit assignment as the critical bottleneck limiting multi-step reasoning performance.