This paper introduces two novel modifications to the Dynamic sAmpling Policy Optimization (DAPO) algorithm [1], approached from a mixed-policy perspective. Standard policy gradient methods can suffer from instability and sample inefficiency, particularly in sparse reward settings. To address this, we first propose a method that incorporates a pre-trained, stable guiding policy ($\piphi$) to provide off-policy experience, thereby regularizing the training of the target policy ($\pion$). This approach improves training stability and convergence speed by adaptively adjusting the learning step size. Secondly, we extend this idea to re-utilize zero-reward samples, which are often discarded by dynamic sampling strategies like DAPO's. By treating these samples as a distinct batch guided by the expert policy, we further enhance sample efficiency. We provide a theoretical analysis for both methods, demonstrating that their objective functions converge to the optimal solution within the established theoretical framework of reinforcement learning. The proposed mixed-policy framework effectively balances exploration and exploitation, promising more stable and efficient policy optimization.
Training neural networks with reinforcement learning (RL) typically relies on backpropagation (BP), necessitating storage of activations from the forward pass for subsequent backward updates. Furthermore, backpropagating error signals through multiple layers often leads to vanishing or exploding gradients, which can degrade learning performance and stability. We propose a novel approach that trains each layer of the neural network using local signals during the forward pass in RL settings. Our approach introduces local, layer-wise losses leveraging the principle of matching pairwise distances from multi-dimensional scaling, enhanced with optional reward-driven guidance. This method allows each hidden layer to be trained using local signals computed during forward propagation, thus eliminating the need for backward passes and storing intermediate activations. Our experiments, conducted with policy gradient methods across common RL benchmarks, demonstrate that this backpropagation-free method achieves competitive performance compared to their classical BP-based counterpart. Additionally, the proposed method enhances stability and consistency within and across runs, and improves performance especially in challenging environments.
We contribute the first provable guarantees of global convergence to Nash equilibria (NE) in two-player zero-sum convex Markov games (cMGs) by using independent policy gradient methods. Convex Markov games, recently defined by Gemp et al. (2024), extend Markov decision processes to multi-agent settings with preferences that are convex over occupancy measures, offering a broad framework for modeling generic strategic interactions. However, even the fundamental min-max case of cMGs presents significant challenges, including inherent nonconvexity, the absence of Bellman consistency, and the complexity of the infinite horizon. We follow a two-step approach. First, leveraging properties of hidden-convex--hidden-concave functions, we show that a simple nonconvex regularization transforms the min-max optimization problem into a nonconvex-proximal Polyak-Lojasiewicz (NC-pPL) objective. Crucially, this regularization can stabilize the iterates of independent policy gradient methods and ultimately lead them to converge to equilibria. Second, building on this reduction, we address the general constrained min-max problems under NC-pPL and two-sided pPL conditions, providing the first global convergence guarantees for stochastic nested and alternating gradient descent-ascent methods, which we believe may be of independent interest.
Ensuring convergence of policy gradient methods in federated reinforcement learning (FRL) under environment heterogeneity remains a major challenge. In this work, we first establish that heterogeneity, perhaps counter-intuitively, can necessitate optimal policies to be non-deterministic or even time-varying, even in tabular environments. Subsequently, we prove global convergence results for federated policy gradient (FedPG) algorithms employing local updates, under a {\L}ojasiewicz condition that holds only for each individual agent, in both entropy-regularized and non-regularized scenarios. Crucially, our theoretical analysis shows that FedPG attains linear speed-up with respect to the number of agents, a property central to efficient federated learning. Leveraging insights from our theoretical findings, we introduce b-RS-FedPG, a novel policy gradient method that employs a carefully constructed softmax-inspired parameterization coupled with an appropriate regularization scheme. We further demonstrate explicit convergence rates for b-RS-FedPG toward near-optimal stationary policies. Finally, we demonstrate that empirically both FedPG and b-RS-FedPG consistently outperform federated Q-learning on heterogeneous settings.
Due to their expressive capacity, diffusion models have shown great promise in offline RL and imitation learning. Diffusion Actor-Critic with Entropy Regulator (DACER) extended this capability to online RL by using the reverse diffusion process as a policy approximator, trained end-to-end with policy gradient methods, achieving strong performance. However, this comes at the cost of requiring many diffusion steps, which significantly hampers training efficiency, while directly reducing the steps leads to noticeable performance degradation. Critically, the lack of inference efficiency becomes a significant bottleneck for applying diffusion policies in real-time online RL settings. To improve training and inference efficiency while maintaining or even enhancing performance, we propose a Q-gradient field objective as an auxiliary optimization target to guide the denoising process at each diffusion step. Nonetheless, we observe that the independence of the Q-gradient field from the diffusion time step negatively impacts the performance of the diffusion policy. To address this, we introduce a temporal weighting mechanism that enables the model to efficiently eliminate large-scale noise in the early stages and refine actions in the later stages. Experimental results on MuJoCo benchmarks and several multimodal tasks demonstrate that the DACER2 algorithm achieves state-of-the-art performance in most MuJoCo control tasks with only five diffusion steps, while also exhibiting stronger multimodality compared to DACER.
Policy gradients in continuous control have been derived for both stochastic and deterministic policies. Here we study the relationship between the two. In a widely-used family of MDPs involving Gaussian control noise and quadratic control costs, we show that the stochastic and deterministic policy gradients, natural gradients, and state value functions are identical; while the state-control value functions are different. We then develop a general procedure for constructing an MDP with deterministic policy that is equivalent to a given MDP with stochastic policy. The controls of this new MDP are the sufficient statistics of the stochastic policy in the original MDP. Our results suggest that policy gradient methods can be unified by approximating state value functions rather than state-control value functions.
Boolean Satisfiability (SAT) solvers are foundational to computer science, yet their performance typically hinges on hand-crafted heuristics. This work introduces Reinforcement Learning from Algorithm Feedback (RLAF) as a paradigm for learning to guide SAT solver branching heuristics with Graph Neural Networks (GNNs). Central to our approach is a novel and generic mechanism for injecting inferred variable weights and polarities into the branching heuristics of existing SAT solvers. In a single forward pass, a GNN assigns these parameters to all variables. Casting this one-shot guidance as a reinforcement learning problem lets us train the GNN with off-the-shelf policy-gradient methods, such as GRPO, directly using the solver's computational cost as the sole reward signal. Extensive evaluations demonstrate that RLAF-trained policies significantly reduce the mean solve times of different base solvers across diverse SAT problem distributions, achieving more than a 2x speedup in some cases, while generalizing effectively to larger and harder problems after training. Notably, these policies consistently outperform expert-supervised approaches based on learning handcrafted weighting heuristics, offering a promising path towards data-driven heuristic design in combinatorial optimization.
We study the policy testing problem in discounted Markov decision processes (MDPs) under the fixed-confidence setting. The goal is to determine whether the value of a given policy exceeds a specified threshold while minimizing the number of observations. We begin by deriving an instance-specific lower bound that any algorithm must satisfy. This lower bound is characterized as the solution to an optimization problem with non-convex constraints. We propose a policy testing algorithm inspired by this optimization problem--a common approach in pure exploration problems such as best-arm identification, where asymptotically optimal algorithms often stem from such optimization-based characterizations. As for other pure exploration tasks in MDPs, however, the non-convex constraints in the lower-bound problem present significant challenges, raising doubts about whether statistically optimal and computationally tractable algorithms can be designed. To address this, we reformulate the lower-bound problem by interchanging the roles of the objective and the constraints, yielding an alternative problem with a non-convex objective but convex constraints. Strikingly, this reformulated problem admits an interpretation as a policy optimization task in a newly constructed reversed MDP. Leveraging recent advances in policy gradient methods, we efficiently solve this problem and use it to design a policy testing algorithm that is statistically optimal--matching the instance-specific lower bound on sample complexity--while remaining computationally tractable. We validate our approach with numerical experiments.
Reinforcement learning (RL) has proven effective for fine-tuning large language models (LLMs), significantly enhancing their reasoning abilities in domains such as mathematics and code generation. A crucial factor influencing RL fine-tuning success is the training curriculum: the order in which training problems are presented. While random curricula serve as common baselines, they remain suboptimal; manually designed curricula often rely heavily on heuristics, and online filtering methods can be computationally prohibitive. To address these limitations, we propose Self-Evolving Curriculum (SEC), an automatic curriculum learning method that learns a curriculum policy concurrently with the RL fine-tuning process. Our approach formulates curriculum selection as a non-stationary Multi-Armed Bandit problem, treating each problem category (e.g., difficulty level or problem type) as an individual arm. We leverage the absolute advantage from policy gradient methods as a proxy measure for immediate learning gain. At each training step, the curriculum policy selects categories to maximize this reward signal and is updated using the TD(0) method. Across three distinct reasoning domains: planning, inductive reasoning, and mathematics, our experiments demonstrate that SEC significantly improves models' reasoning capabilities, enabling better generalization to harder, out-of-distribution test problems. Additionally, our approach achieves better skill balance when fine-tuning simultaneously on multiple reasoning domains. These findings highlight SEC as a promising strategy for RL fine-tuning of LLMs.
Reinforcement Learning (RL) has made significant strides in various domains, and policy gradient methods like Proximal Policy Optimization (PPO) have gained popularity due to their balance in performance, training stability, and computational efficiency. These methods directly optimize policies through gradient-based updates. However, developing effective control policies for environments with complex and non-linear dynamics remains a challenge. High variance in gradient estimates and non-convex optimization landscapes often lead to unstable learning trajectories. Koopman Operator Theory has emerged as a powerful framework for studying non-linear systems through an infinite-dimensional linear operator that acts on a higher-dimensional space of measurement functions. In contrast with their non-linear counterparts, linear systems are simpler, more predictable, and easier to analyze. In this paper, we present Koopman-Inspired Proximal Policy Optimization (KIPPO), which learns an approximately linear latent-space representation of the underlying system's dynamics while retaining essential features for effective policy learning. This is achieved through a Koopman-approximation auxiliary network that can be added to the baseline policy optimization algorithms without altering the architecture of the core policy or value function. Extensive experimental results demonstrate consistent improvements over the PPO baseline with 6-60% increased performance while reducing variability by up to 91% when evaluated on various continuous control tasks.
Retrieval-augmented generation (RAG) integrates large language models ( LLM s) with retrievers to access external knowledge, improving the factuality of LLM generation in knowledge-grounded tasks. To optimize the RAG performance, most previous work independently fine-tunes the retriever to adapt to frozen LLM s or trains the LLMs to use documents retrieved by off-the-shelf retrievers, lacking end-to-end training supervision. Recent work addresses this limitation by jointly training these two components but relies on overly simplifying assumptions of document independence, which has been criticized for being far from real-world scenarios. Thus, effectively optimizing the overall RAG performance remains a critical challenge. We propose a direct retrieval-augmented optimization framework, named DRO, that enables end-to-end training of two key components: (i) a generative knowledge selection model and (ii) an LLM generator. DRO alternates between two phases: (i) document permutation estimation and (ii) re-weighted maximization, progressively improving RAG components through a variational approach. In the estimation step, we treat document permutation as a latent variable and directly estimate its distribution from the selection model by applying an importance sampling strategy. In the maximization step, we calibrate the optimization expectation using importance weights and jointly train the selection model and LLM generator. Our theoretical analysis reveals that DRO is analogous to policy-gradient methods in reinforcement learning. Extensive experiments conducted on five datasets illustrate that DRO outperforms the best baseline with 5%-15% improvements in EM and F1. We also provide in-depth experiments to qualitatively analyze the stability, convergence, and variance of DRO.
We propose reinforcement learning (RL) strategies tailored for reasoning in large language models (LLMs) under strict memory and compute limits, with a particular focus on compatibility with LoRA fine-tuning. Building on early policy gradient methods with baseline subtraction, we design critic-free methods that operate on a small, informative subset of output tokens to reduce memory usage and stabilize training. We introduce S-GRPO, a stochastic variant of Group Relative Policy Optimization, and T-SPMO, a token-level prefix matching approach for fine-grained credit assignment. Applied to Qwen2-1.5B, our methods raise accuracy on the SVAMP benchmark from 46% to over 70% and show strong performance on multi-digit multiplication. Surprisingly, full-token GRPO under LoRA fails to improve over the base model, suggesting that selective token-level optimization may act as an implicit regularizer in low-parameter training regimes.
A key challenge in reinforcement learning (RL) is managing the exploration-exploitation trade-off without sacrificing sample efficiency. Policy gradient (PG) methods excel in exploitation through fine-grained, gradient-based optimization but often struggle with exploration due to their focus on local search. In contrast, evolutionary computation (EC) methods excel in global exploration, but lack mechanisms for exploitation. To address these limitations, this paper proposes Evolutionary Policy Optimization (EPO), a hybrid algorithm that integrates neuroevolution with policy gradient methods for policy optimization. EPO leverages the exploration capabilities of EC and the exploitation strengths of PG, offering an efficient solution to the exploration-exploitation dilemma in RL. EPO is evaluated on the Atari Pong and Breakout benchmarks. Experimental results show that EPO improves both policy quality and sample efficiency compared to standard PG and EC methods, making it effective for tasks that require both exploration and local optimization.
We prove that, for finite-arm bandits with linear function approximation, the global convergence of policy gradient (PG) methods depends on inter-related properties between the policy update and the representation. textcolor{blue}{First}, we establish a few key observations that frame the study: \textbf{(i)} Global convergence can be achieved under linear function approximation without policy or reward realizability, both for the standard Softmax PG and natural policy gradient (NPG). \textbf{(ii)} Approximation error is not a key quantity for characterizing global convergence in either algorithm. \textbf{(iii)} The conditions on the representation that imply global convergence are different between these two algorithms. Overall, these observations call into question approximation error as an appropriate quantity for characterizing the global convergence of PG methods under linear function approximation. \textcolor{blue}{Second}, motivated by these observations, we establish new general results: \textbf{(i)} NPG with linear function approximation achieves global convergence \emph{if and only if} the projection of the reward onto the representable space preserves the optimal action's rank, a quantity that is not strongly related to approximation error. \textbf{(ii)} The global convergence of Softmax PG occurs if the representation satisfies a non-domination condition and can preserve the ranking of rewards, which goes well beyond policy or reward realizability. We provide experimental results to support these theoretical findings.
We propose a decentralized reinforcement learning solution for multi-agent shepherding of non-cohesive targets using policy-gradient methods. Our architecture integrates target-selection with target-driving through Proximal Policy Optimization, overcoming discrete-action constraints of previous Deep Q-Network approaches and enabling smoother agent trajectories. This model-free framework effectively solves the shepherding problem without prior dynamics knowledge. Experiments demonstrate our method's effectiveness and scalability with increased target numbers and limited sensing capabilities.
Policy gradient methods are one of the most successful methods for solving challenging reinforcement learning problems. However, despite their empirical successes, many SOTA policy gradient algorithms for discounted problems deviate from the theoretical policy gradient theorem due to the existence of a distribution mismatch. In this work, we analyze the impact of this mismatch on the policy gradient methods. Specifically, we first show that in the case of tabular parameterizations, the methods under the mismatch remain globally optimal. Then, we extend this analysis to more general parameterizations by leveraging the theory of biased stochastic gradient descent. Our findings offer new insights into the robustness of policy gradient methods as well as the gap between theoretical foundations and practical implementations.
Reinforcement Learning and Imitation Learning have achieved widespread success in many domains but remain constrained during real-world deployment. One of the main issues is the additional requirements that were not considered during training. To address this challenge, policy customization has been introduced, aiming to adapt a prior policy while preserving its inherent properties and meeting new task-specific requirements. A principled approach to policy customization is Residual Q-Learning (RQL), which formulates the problem as a Markov Decision Process (MDP) and derives a family of value-based learning algorithms. However, RQL has not yet been applied to policy gradient methods, which restricts its applicability, especially in tasks where policy gradient has already proven more effective. In this work, we first derive a concise form of Soft Policy Gradient as a preliminary. Building on this, we introduce Residual Policy Gradient (RPG), which extends RQL to policy gradient methods, allowing policy customization in gradient-based RL settings. With the view of RPG, we rethink the KL-regularized objective widely used in RL fine-tuning. We show that under certain assumptions, KL-regularized objective leads to a maximum-entropy policy that balances the inherent properties and task-specific requirements on a reward-level. Our experiments in MuJoCo demonstrate the effectiveness of Soft Policy Gradient and Residual Policy Gradient.
Diffusion models have revolutionized generative modeling in continuous domains like image, audio, and video synthesis. However, their iterative sampling process leads to slow generation and inefficient training, challenges that are further exacerbated when incorporating Reinforcement Learning from Human Feedback (RLHF) due to sparse rewards and long time horizons. Consistency models address these issues by enabling single-step or efficient multi-step generation, significantly reducing computational costs. In this work, we propose a direct reward optimization framework for applying RLHF to consistency models, incorporating distributional regularization to enhance training stability and prevent reward hacking. We investigate various $f$-divergences as regularization strategies, striking a balance between reward maximization and model consistency. Unlike policy gradient methods, our approach leverages first-order gradients, making it more efficient and less sensitive to hyperparameter tuning. Empirical results show that our method achieves competitive or superior performance compared to policy gradient based RLHF methods, across various automatic metrics and human evaluation. Additionally, our analysis demonstrates the impact of different regularization techniques in improving model generalization and preventing overfitting.
Policy gradient (PG) methods are the backbone of many reinforcement learning algorithms due to their good performance in policy optimization problems. As a gradient-based approach, PG methods typically rely on knowledge of the system dynamics. However, this information is not always available, and in such cases, trajectory data can be utilized to approximate first-order information. When the data are noisy, gradient estimates become inaccurate and a formal investigation that encompasses uncertainty estimation and the analysis of its propagation through the algorithm is currently missing. To address this, our work focuses on the Linear Quadratic Regulator (LQR) problem for systems subject to additive stochastic noise. After briefly summarizing the state of the art for cases with a known model, we focus on scenarios where the system dynamics are unknown, and approximate gradient information is obtained using zeroth-order optimization techniques. We analyze the theoretical properties by computing the error in the estimated gradient and examining how this error affects the convergence of PG algorithms. Additionally, we provide global convergence guarantees for various versions of PG methods, including those employing adaptive step sizes and variance reduction techniques, which help increase the convergence rate and reduce sample complexity. One contribution of this work is the study of the robustness of model-free PG methods, aiming to identify their limitations in the presence of noise and propose improvements to enhance their applicability. Numerical simulations show that these theoretical analyses provide valuable guidance in tuning the algorithm parameters, thereby making these methods more reliable in practically relevant scenarios.
Recently, enhancing the numerical and logical reasoning capability of Large Language Models (LLMs) has emerged as a research hotspot. Existing methods face several limitations: inference-phase techniques (e.g., Chain of Thoughts) rely on prompt selection and the pretrained knowledge; sentence-level Supervised Fine-Tuning (SFT) and Direct Preference Optimization (DPO) struggle with step-wise mathematical correctness and depend on stronger models distillation or human annotations; while Reinforcement Learning (RL) approaches incur high GPU memory costs and unstable training. To address these, we propose \textbf{S}elf-training framework integrating \textbf{P}rocess \textbf{P}reference learning using \textbf{D}ynamic value margin (SPPD). SPPD leverages a process-based Markov Decision Process (MDP) and Bellman optimality equation to derive \textbf{dynamic value margin} on step-level preference optimization, which employs tree-based self-sampling on model responses \textbf{without any distillation} from other models. Furthermore, we theoretically prove that SPPD is \textbf{equivalent to on-policy policy gradient methods} under reward constraints. Experiments on 7B-scale models demonstrate superior performance across in-domain and out-domain mathematical benchmarks. We open-source our code at \href{https://anonymous.4open.science/r/SSDPO-D-DCDD}{https://anonymous.4open.science/r/SPPD-DCDD}.