Policy Gradient Methods

We consider the policy evaluation and control in a finite horizon reinforcement learning (RL) setting under an expectile-based objective. First, we derive the mean-squared error (MSE) and concentration bounds for the classic estimator of expectiles based on independent and identically distributed (i.i.d.) samples. To the best of our knowledge, expectiles have not been analyzed in the non-asymptotic regime and the bounds we derive may be of independent interest. Next, we analyze a Monte Carlo type estimator of expectile of the Markov chain underlying a given policy. We derive upper bounds that hold in expectation as well as with high probability for this estimator. Further, we show the order-optimality of our estimator by deriving a lower bound on expectile-based policy evaluation. For the problem of control, we adopt a policy gradient approach and derive a policy gradient theorem for expectiles. Using this result, we propose a gradient estimator with a $O\left(1/m\right)$ mean-squared error bounds, where $m$ is the number of trajectories. Further, under standard assumptions for policy gradient-type algorithms, we establish smoothness of the expectile-sensitive objective, in turn leading to stationary convergence rate bounds for the overall risk-sensitive policy gradient algorithm that we propose. Finally, we conduct numerical experiments to show the utility of expectiles on popular RL benchmarks.

Reinforcement Learning (RL) for Large Language Models (LLMs) often suffers from training collapse in long-horizon tasks due to exploding gradient variance. To mitigate this, a baseline is commonly introduced for advantage computation; however, traditional value models remain difficult to optimize, and standard group-based baselines overlook sequence heterogeneity. Although classic optimal baseline theory can achieve global variance reduction, it neglects token heterogeneity and requires prohibitive gradient-based computation. In this work, we derive the Optimal Token Baseline (OTB) from first principles, proving that gradient updates should be weighted inversely to their cumulative gradient norm. To ensure efficiency, we propose the Logit-Gradient Proxy that approximates the gradient norm using only forward-pass probabilities. Our method achieves training stability and matches the performance of large group sizes ($N=32$) with only $N=4$, reducing token consumption by over 65% across single-turn and tool-integrated reasoning tasks.

Reinforcement learning has been widely applied to diffusion and flow models for visual tasks such as text-to-image generation. However, these tasks remain challenging because diffusion models have intractable likelihoods, which creates a barrier for directly applying popular policy-gradient type methods. Existing approaches primarily focus on crafting new objectives built on already heavily engineered LLM objectives, using ad hoc estimators for likelihood, without a thorough investigation into how such estimation affects overall algorithmic performance. In this work, we provide a systematic analysis of the RL design space by disentangling three factors: i) policy-gradient objectives, ii) likelihood estimators, and iii) rollout sampling schemes. We show that adopting an evidence lower bound (ELBO) based model likelihood estimator, computed only from the final generated sample, is the dominant factor enabling effective, efficient, and stable RL optimization, outweighing the impact of the specific policy-gradient loss functional. We validate our findings across multiple reward benchmarks using SD 3.5 Medium, and observe consistent trends across all tasks. Our method improves the GenEval score from 0.24 to 0.95 in 90 GPU hours, which is 4.6 times more efficient than FlowGRPO and $2\times$ more efficient than the SOTA method without reward hacking.

Stabilizing a dynamical system is a fundamental problem that serves as a cornerstone for many complex tasks in the field of control systems. The problem becomes challenging when the system model is unknown. Among the Reinforcement Learning (RL) algorithms that have been successfully applied to solve problems pertaining to unknown linear dynamical systems, the policy gradient (PG) method stands out due to its ease of implementation and can solve the problem in a model-free manner. However, most of the existing works on PG methods for unknown linear dynamical systems assume full-state feedback. In this paper, we take a step towards model-free learning for partially observed linear dynamical systems with output feedback and focus on the fundamental stabilization problem of the system. We propose an algorithmic framework that stretches the boundary of PG methods to the problem without global convergence guarantees. We show that by leveraging zeroth-order PG update based on system trajectories and its convergence to stationary points, the proposed algorithms return a stabilizing output feedback policy for discrete-time linear dynamical systems. We also explicitly characterize the sample complexity of our algorithm and verify the effectiveness of the algorithm using numerical examples.

Policy gradient methods for Large Language Models optimize a policy $π_θ$ via a surrogate objective computed from samples of a rollout policy $π_{\text{roll}}$. However, modern LLM-RL pipelines suffer from unavoidable implementation divergences -- backend discrepancies, Mixture-of-Experts routing discontinuities, and distributed training staleness -- causing off-policy mismatch ($π_{\text{roll}} \neq π_θ$) and approximation errors between the surrogate and the true objective. We demonstrate that classical trust region bounds on this error scale as $O(T^2)$ with sequence length $T$, rendering them vacuous for long-horizon tasks. To address this, we derive a family of bounds -- both KL-based and TV-based -- including a Pinsker-Marginal bound ($O(T^{3/2})$), a Mixed bound ($O(T)$), and an Adaptive bound that strictly generalizes the Pinsker-Marginal bound via per-position importance-ratio decomposition. Taking the minimum over all bounds yields the tightest known guarantee across all divergence regimes. Crucially, all bounds depend on the maximum token-level divergence $D_{\mathrm{KL}}^{\mathrm{tok,max}}$ (or $D_{\mathrm{TV}}^{\mathrm{tok,max}}$), a sequence-level quantity that cannot be controlled by token-independent methods like PPO clipping. We propose Trust Region Masking (TRM), which masks entire sequences violating the trust region, enabling the first non-vacuous monotonic improvement guarantees for long-horizon LLM-RL.

Negative learning has never been more important than it is today, particularly in large language model (LLM) fine-tuning, where it underpins machine unlearning, preference- and reward-based optimization, and exploratory data generation. Yet its functional role remains a subject of ongoing debate, with existing work variously characterizing it as forgetting, regularization, or target shaping. However, analytical support focuses primarily on the single-update scale, while empirical results at the trajectory scale are typically obtained under negative-only training and may not extend to composite objectives in which it interacts with other components. To develop a deeper understanding, we introduce a gradient-based endpoint probing framework that can examine negative updates separately throughout training. Our analysis reveals that negative learning plays different roles across algorithms and training phases. For example, in off-policy DPO, it acts more like a regularizer early in training and shows evidence of target shaping only toward out-of-distribution endpoints and only in later phases. In on-policy PPO and GRPO, by contrast, negative updates offset endpoint-aligned positive learning without stable target shaping, consistent with a role in sustaining exploration. Guided by these findings, we develop a family of stage-aware methods that modulate the strength of negative updates throughout training, improving fine-tuning performance across tasks.

In-context reinforcement learning (ICRL) has emerged as an effective paradigm for test time adaptation to unseen tasks without parameter updates. However, existing ICRL methods can exhibit brittle and unstable adaptions, and the mechanisms underlying such adaptions remain poorly understood. We provide a policy-gradient view of ICRL and argue that relying on trajectory-level return feedback can lead to high-variance updates when test time rollout is limited. Motivated by the actor--critic principle, we propose Context Value Informed ICRL (CV-ICRL), which equips the in-context policy with an explicit context value: the expected discounted return conditioned on the current state and accumulated interaction context. CV-ICRL trains a value head and writes its prediction back into the context as a value token, enabling TD-style return targets for lower-variance policy updates. Experiments on the Dark Room, Minigrid, and Procgen testbeds show that CV-ICRL substantially improves the stability of test time adaptation and achieves higher returns across tasks and environments. Code and data are available at https://github.com/Bluixe/CV-ICRL .

We consider solutions to the linear quadratic Gaussian (LQG) regulator problem via policy gradient (PG) methods. Although PG methods have demonstrated strong theoretical guarantees in solving the linear quadratic regulator (LQR) problem, despite its nonconvex landscape, their theoretical understanding in the LQG setting remains limited. Notably, the LQG problem lacks gradient dominance in the classical parameterization, i.e., with a dynamic controller, which hinders global convergence guarantees. In this work, we study a PG method for the LQG problem by adopting an alternative parameterization of the set of stabilizing controllers and employing a lifting argument. We refer to this parameterization as a history representation of the control input as it is parameterized by past input and output data from the previous p time-steps. Under this parameterization, the infinite-history LQG optimization reduces to an equivalent static state-feedback LQR problem, allowing us to recover gradient dominance and smoothness. For finite history length p, we establish model-based and model-free convergence guarantees up to a geometrically decaying history-truncation error.

Policy gradient (PG) methods have played an essential role in the empirical successes of reinforcement learning. In order to handle large state-action spaces, PG methods are typically used with function approximation. In this setting, the approximation error in modeling problem-dependent quantities is a key notion for characterizing the global convergence of PG methods. We study Softmax PG with linear function approximation (referred to as $\texttt{Lin-SPG}$) and demonstrate that the approximation error is irrelevant to the algorithm's global convergence even in the bandit setting. Consequently, we rethink the effect of approximation error in the standard stochastic multi-armed bandit problem. We first identify the conditions on the policy feature representation that can guarantee the asymptotic global convergence of $\texttt{Lin-SPG}$. Under these feature conditions, we further prove that $T$ iterations of $\texttt{Lin-SPG}$ with a problem-specific learning rate result in an $O(1/T)$ convergence to the optimal policy. Moreover, we prove that $\texttt{Lin-SPG}$ with an arbitrary constant learning rate can ensure asymptotic convergence to the optimal policy.

Post-training plays a crucial role in refining and aligning large language models to meet specific tasks and human preferences. While recent advancements in post-training techniques, such as Group Relative Policy Optimization (GRPO), leverage increased sampling with relative reward scoring to achieve superior performance, these methods often suffer from training instability that limits their practical adoption. As a next step, we present Group Variance Policy Optimization (GVPO). GVPO incorporates the analytical solution to KL-constrained reward maximization directly into its gradient weights, ensuring alignment with the optimal policy. The method provides intuitive physical interpretations: its gradient mirrors the mean squared error between the central distance of implicit rewards and that of actual rewards. GVPO offers two key advantages: (1) it guarantees a unique optimal solution, exactly the KL-constrained reward maximization objective, (2) it supports flexible sampling distributions that avoids on-policy and importance sampling limitations. By unifying theoretical guarantees with practical adaptability, GVPO establishes a new paradigm for reliable and versatile LLM post-training.

Policy gradient methods are among the most successful approaches for solving challenging reinforcement learning problems. Despite their empirical success, many state-of-the-art policy gradient algorithms for discounted problems estimate update directions using a sampling distribution that differs from the discounted state distribution required by the policy gradient theorem. In this work, we analyze the impact of this distribution mismatch on policy gradient methods. Specifically, {for tabular direct and softmax parameterizations, we establish global convergence of objective values under biased updates}. {For tabular softmax policies, we further derive the exact leading-order coefficients of the asymptotic objective errors under biased and normalized exact updates. Their ratio provides an explicit criterion for determining when distribution mismatch accelerates or slows asymptotic convergence.} We then extend the analysis to more general policy parameterizations by deriving explicit bounds on both the state distribution mismatch and the induced normalized policy-gradient mismatch in episodic and continuing MDPs. These mismatches vanish at least linearly as the discount factor approaches one. Under an additional smoothness assumption, we further show that the limit inferior of the squared norm of the normalized exact policy-gradient direction along the biased iterates is bounded by $O((1-γ)^2)$ in both settings. Our results provide theoretical insights into the robustness of policy gradient methods to distribution mismatch and shed light on the gap between their theoretical formulations and practical implementations.

In many control problems, actions consume a resource budget, such as a wear allowance, and the probability of exhausting it within a given horizon must stay below a tolerance. A simple, widely used approach is to augment the state with the remaining budget, subtract an interior barrier from the utility and penalize exhaustion, so that policy-gradient methods can be applied unchanged. We clip the combined one-step reward from below at a level m to keep the returns of exploratory policies finite. The clip level is a price of failure risk, since the clipped value decomposes exactly into the utility before failure, the cost of the interior margin and m times a geometric failure risk. Consequently, an approximately optimal policy maximizes utility net of margin cost among policies with no greater risk. Its risk exceeds that of any comparison policy by at most order 1/m and can be certified from its value or from validation data. Our main result proves exponential convergence of the population policy-gradient flow for every parameter regularizer that is convex along generalized geodesics and sufficiently strong relative to the price. This extends the mean-field analysis from relative entropy to weight decay and to finite networks. Relative entropy adds parameter noise and a pull toward a prior, whereas weight decay provides only the pull. Action regularization bounds changes in failure risk by the relative entropy to a reference policy. The bounds are evaluated on an exact maintenance benchmark, a fuel-tank controller, a solvable flow and a benchmark-suite task.

On-policy reinforcement learning methods, like Trust Region Policy Optimization (TRPO) and Proximal Policy Optimization (PPO), often demand extensive data per update, leading to sample inefficiency. This paper introduces Reflective Policy Optimization (RPO), a novel on-policy extension that amalgamates past and future state-action information for policy optimization. This approach empowers the agent for introspection, allowing modifications to its actions within the current state. Theoretical analysis confirms that policy performance is monotonically improved and contracts the solution space, consequently expediting the convergence procedure. Empirical results demonstrate RPO's feasibility and efficacy in two reinforcement learning benchmarks, culminating in superior sample efficiency. The source code of this work is available at https://github.com/Edgargan/RPO.

Risk-sensitive reinforcement learning (RL) is crucial for maintaining reliable performance in high-stakes applications. While traditional RL methods aim to learn a point estimate of the random cumulative cost, distributional RL seeks to estimate the entire distribution of it, leading to a unified framework for handling different risk measures. However, developing policy gradient methods for risk-sensitive distributional RL is inherently more complex as it often involves finding the gradient of a probability measure. This paper introduces a new distributional policy gradient framework for risk-sensitive RL, where we derive an analytical gradient of the probability measure of the cumulative cost. For practical implementation, we further design a categorical distributional policy gradient algorithm (CDPG) that approximates arbitrary distributions using a categorical family supported on fixed points. Using Conditional Value-at-Risk (CVaR) as the objective, we prove that the proposed CDPG converges to stationary points and establish its iteration complexities under inexact policy evaluation. Through experiments in a stochastic Cliffwalk environment, we demonstrate the effectiveness of the proposed algorithm and highlight the benefits of incorporating risk sensitivity into distributional RL.

The policy gradient theorem describes the gradient of the expected discounted return with respect to an agent's policy parameters. However, many policy gradient methods omit the outer discount factor, yielding an update that need not equal the gradient of the discounted objective. Is the resulting update the gradient of another objective? This has been an open question for several years, and this lack of theoretical clarity has led to an abundance of misstatements in the literature. We answer this question for the resulting episode-summed, on-policy update by proving that it need not equal the gradient of any function. Further, we construct a counterexample in which exact expected updates drive the policy toward a deterministic limit that is globally pessimal with respect to both the discounted and undiscounted objectives, while the parameter diverges. We motivate this work by surveying the literature and showing that there remains a widespread misunderstanding regarding discounted policy gradient methods, with errors present even in highly-cited papers published at top conferences.