Designing a safe policy for uncertain environments is crucial in real-world control systems. However, this challenge remains inadequately addressed within the Markov decision process (MDP) framework. This paper presents the first algorithm guaranteed to identify a near-optimal policy in a robust constrained MDP (RCMDP), where an optimal policy minimizes cumulative cost while satisfying constraints in the worst-case scenario across a set of environments. We first prove that the conventional policy gradient approach to the Lagrangian max-min formulation can become trapped in suboptimal solutions. This occurs when its inner minimization encounters a sum of conflicting gradients from the objective and constraint functions. To address this, we leverage the epigraph form of the RCMDP problem, which resolves the conflict by selecting a single gradient from either the objective or the constraints. Building on the epigraph form, we propose a bisection search algorithm with a policy gradient subroutine and prove that it identifies an $\varepsilon$-optimal policy in an RCMDP with $\tilde{\mathcal{O}}(\varepsilon^{-4})$ robust policy evaluations.
Recently, reinforcement learning (RL) has proved a promising alternative for conventional local heuristics in score-based approaches to learning directed acyclic causal graphs (DAGs) from observational data. However, the intricate acyclicity constraint still challenges the efficient exploration of the vast space of DAGs in existing methods. In this study, we introduce ALIAS (reinforced dAg Learning wIthout Acyclicity conStraints), a novel approach to causal discovery powered by the RL machinery. Our method features an efficient policy for generating DAGs in just a single step with an optimal quadratic complexity, fueled by a novel parametrization of DAGs that directly translates a continuous space to the space of all DAGs, bypassing the need for explicitly enforcing acyclicity constraints. This approach enables us to navigate the search space more effectively by utilizing policy gradient methods and established scoring functions. In addition, we provide compelling empirical evidence for the strong performance of ALIAS in comparison with state-of-the-arts in causal discovery over increasingly difficult experiment conditions on both synthetic and real datasets.
We study the problem of computing deterministic optimal policies for constrained Markov decision processes (MDPs) with continuous state and action spaces, which are widely encountered in constrained dynamical systems. Designing deterministic policy gradient methods in continuous state and action spaces is particularly challenging due to the lack of enumerable state-action pairs and the adoption of deterministic policies, hindering the application of existing policy gradient methods. To this end, we develop a deterministic policy gradient primal-dual method to find an optimal deterministic policy with non-asymptotic convergence. Specifically, we leverage regularization of the Lagrangian of the constrained MDP to propose a deterministic policy gradient primal-dual (D-PGPD) algorithm that updates the deterministic policy via a quadratic-regularized gradient ascent step and the dual variable via a quadratic-regularized gradient descent step. We prove that the primal-dual iterates of D-PGPD converge at a sub-linear rate to an optimal regularized primal-dual pair. We instantiate D-PGPD with function approximation and prove that the primal-dual iterates of D-PGPD converge at a sub-linear rate to an optimal regularized primal-dual pair, up to a function approximation error. Furthermore, we demonstrate the effectiveness of our method in two continuous control problems: robot navigation and fluid control. This appears to be the first work that proposes a deterministic policy search method for continuous-space constrained MDPs.
Recent advances in Behavior Cloning (BC) have made it easy to teach robots new tasks. However, we find that the ease of teaching comes at the cost of unreliable performance that saturates with increasing data for tasks requiring precision. The performance saturation can be attributed to two critical factors: (a) distribution shift resulting from the use of offline data and (b) the lack of closed-loop corrective control caused by action chucking (predicting a set of future actions executed open-loop) critical for BC performance. Our key insight is that by predicting action chunks, BC policies function more like trajectory "planners" than closed-loop controllers necessary for reliable execution. To address these challenges, we devise a simple yet effective method, ResiP (Residual for Precise Manipulation), that overcomes the reliability problem while retaining BC's ease of teaching and long-horizon capabilities. ResiP augments a frozen, chunked BC model with a fully closed-loop residual policy trained with reinforcement learning (RL) that addresses distribution shifts and introduces closed-loop corrections over open-loop execution of action chunks predicted by the BC trajectory planner. Videos, code, and data: https://residual-assembly.github.io.
This paper introduces PG-Rainbow, a novel algorithm that incorporates a distributional reinforcement learning framework with a policy gradient algorithm. Existing policy gradient methods are sample inefficient and rely on the mean of returns when calculating the state-action value function, neglecting the distributional nature of returns in reinforcement learning tasks. To address this issue, we use an Implicit Quantile Network that provides the quantile information of the distribution of rewards to the critic network of the Proximal Policy Optimization algorithm. We show empirical results that through the integration of reward distribution information into the policy network, the policy agent acquires enhanced capabilities to comprehensively evaluate the consequences of potential actions in a given state, facilitating more sophisticated and informed decision-making processes. We evaluate the performance of the proposed algorithm in the Atari-2600 game suite, simulated via the Arcade Learning Environment (ALE).
While contemporary reinforcement learning research and applications have embraced policy gradient methods as the panacea of solving learning problems, value-based methods can still be useful in many domains as long as we can wrangle with how to exploit them in a sample efficient way. In this paper, we explore the chaotic nature of DQNs in reinforcement learning, while understanding how the information that they retain when trained can be repurposed for adapting a model to different tasks. We start by designing a simple experiment in which we are able to observe the Q-values for each state and action in an environment. Then we train in eight different ways to explore how these training algorithms affect the way that accurate Q-values are learned (or not learned). We tested the adaptability of each trained model when retrained to accomplish a slightly modified task. We then scaled our setup to test the larger problem of an autonomous vehicle at an unprotected intersection. We observed that the model is able to adapt to new tasks quicker when the base model's Q-value estimates are closer to the true Q-values. The results provide some insights and guidelines into what algorithms are useful for sample efficient task adaptation.
Deterministic policies are often preferred over stochastic ones when implemented on physical systems. They can prevent erratic and harmful behaviors while being easier to implement and interpret. However, in practice, exploration is largely performed by stochastic policies. First-order Bayesian Optimization (BO) methods offer a principled way of performing exploration using deterministic policies. This is done through a learned probabilistic model of the objective function and its gradient. Nonetheless, such approaches treat policy search as a black-box problem, and thus, neglect the reinforcement learning nature of the problem. In this work, we leverage the performance difference lemma to introduce a novel mean function for the probabilistic model. This results in augmenting BO methods with the action-value function. Hence, we call our method Augmented Bayesian Search~(ABS). Interestingly, this new mean function enhances the posterior gradient with the deterministic policy gradient, effectively bridging the gap between BO and policy gradient methods. The resulting algorithm combines the convenience of the direct policy search with the scalability of reinforcement learning. We validate ABS on high-dimensional locomotion problems and demonstrate competitive performance compared to existing direct policy search schemes.
Recently proposed reward-conditioned policies (RCPs) offer an appealing alternative in reinforcement learning. Compared with policy gradient methods, policy learning in RCPs is simpler since it is based on supervised learning, and unlike value-based methods, it does not require optimization in the action space to take actions. However, for multi-armed bandit (MAB) problems, we find that RCPs are slower to converge and have inferior expected rewards at convergence, compared with classic methods such as the upper confidence bound and Thompson sampling. In this work, we show that the performance of RCPs can be enhanced by constructing policies through the marginalization of rewards using normalized weight functions, whose sum or integral equal $1$, although the function values may be negative. We refer to this technique as generalized marginalization, whose advantage is that negative weights for policies conditioned on low rewards can make the resulting policies more distinct from them. Strategies to perform generalized marginalization in MAB with discrete action spaces are studied. Through simulations, we demonstrate that the proposed technique improves RCPs and makes them competitive with classic methods, showing superior performance on challenging MABs with large action spaces and sparse reward signals.
Digital Twins have gained attention in various industries for simulation, monitoring, and decision-making, relying on ever-improving machine learning models. However, agricultural Digital Twin implementations are limited compared to other industries. Meanwhile, machine learning, particularly reinforcement learning, has shown potential in agricultural applications like optimizing decision-making, task automation, and resource management. A key aspect of Digital Twins is representing physical assets or systems in a virtual environment, which aligns well with reinforcement learning's need for environment representations to learn the best policy for a task. Reinforcement learning in agriculture can thus enable various Digital Twin applications in agricultural domains. This review aims to categorize existing research employing reinforcement learning in agricultural settings by application domains like robotics, greenhouse management, irrigation systems, and crop management, identifying potential future areas for reinforcement learning-based Digital Twins. It also categorizes the reinforcement learning techniques used, including tabular methods, Deep Q-Networks (DQN), Policy Gradient methods, and Actor-Critic algorithms, to overview currently employed models. The review seeks to provide insights into the state-of-the-art in integrating Digital Twins and reinforcement learning in agriculture, identifying gaps and opportunities for future research, and exploring synergies to tackle agricultural challenges and optimize farming, paving the way for more efficient and sustainable farming methodologies.
We study the error introduced by entropy regularization in infinite-horizon, discrete, discounted Markov decision processes. We show that this error decreases exponentially in the inverse regularization strength both in a weighted KL-divergence and in value with a problem-specific exponent. This is in contrast to previously known estimates, of the order $O(\tau)$, where $\tau$ is the regularization strength. We provide a lower bound matching our upper bound up to a polynomial term, thereby characterizing the exponential convergence rate for entropy regularization. Our proof relies on the observation that the solutions of entropy-regularized Markov decision processes solve a gradient flow of the unregularized reward with respect to a Riemannian metric common in natural policy gradient methods. This correspondence allows us to identify the limit of this gradient flow as the generalized maximum entropy optimal policy, thereby characterizing the implicit bias of this gradient flow, which corresponds to a time-continuous version of the natural policy gradient method. We use our improved error estimates to show that for entropy-regularized natural policy gradient methods, the overall error decays exponentially in the square root of the number of iterations, improving over existing sublinear guarantees. Finally, we extend our analysis to settings beyond the entropy. In particular, we characterize the implicit bias regarding general convex potentials and their resulting generalized natural policy gradients.
Neural Network based approximations of the Value function make up the core of leading Policy Based methods such as Trust Regional Policy Optimization (TRPO) and Proximal Policy Optimization (PPO). While this adds significant value when dealing with very complex environments, we note that in sufficiently low State and action space environments, a computationally expensive Neural Network architecture offers marginal improvement over simpler Value approximation methods. We present an implementation of Natural Actor Critic algorithms with actor updates through Natural Policy Gradient methods. This paper proposes that Natural Policy Gradient (NPG) methods with Linear Function Approximation as a paradigm for value approximation may surpass the performance and speed of Neural Network based models such as TRPO and PPO within these environments. Over Reinforcement Learning benchmarks Cart Pole and Acrobot, we observe that our algorithm trains much faster than complex neural network architectures, and obtains an equivalent or greater result. This allows us to recommend the use of NPG methods with Linear Function Approximation over TRPO and PPO for both traditional and sparse reward low dimensional problems.
Entropy regularization has been widely used in policy optimization algorithms to enhance exploration and the robustness of the optimal control; however it also introduces an additional regularization bias. This work quantifies the impact of entropy regularization on the convergence of policy gradient methods for stochastic exit time control problems. We analyze a continuous-time policy mirror descent dynamics, which updates the policy based on the gradient of an entropy-regularized value function and adjusts the strength of entropy regularization as the algorithm progresses. We prove that with a fixed entropy level, the mirror descent dynamics converges exponentially to the optimal solution of the regularized problem. We further show that when the entropy level decays at suitable polynomial rates, the annealed flow converges to the solution of the unregularized problem at a rate of $\mathcal O(1/S)$ for discrete action spaces and, under suitable conditions, at a rate of $\mathcal O(1/\sqrt{S})$ for general action spaces, with $S$ being the gradient flow running time. The technical challenge lies in analyzing the gradient flow in the infinite-dimensional space of Markov kernels for nonconvex objectives. This paper explains how entropy regularization improves policy optimization, even with the true gradient, from the perspective of convergence rate.
Estimating a policy that maps states to actions is a central problem in reinforcement learning. Traditionally, policies are inferred from the so called value functions (VFs), but exact VF computation suffers from the curse of dimensionality. Policy gradient (PG) methods bypass this by learning directly a parametric stochastic policy. Typically, the parameters of the policy are estimated using neural networks (NNs) tuned via stochastic gradient descent. However, finding adequate NN architectures can be challenging, and convergence issues are common as well. In this paper, we put forth low-rank matrix-based models to estimate efficiently the parameters of PG algorithms. We collect the parameters of the stochastic policy into a matrix, and then, we leverage matrix-completion techniques to promote (enforce) low rank. We demonstrate via numerical studies how low-rank matrix-based policy models reduce the computational and sample complexities relative to NN models, while achieving a similar aggregated reward.
Policy gradient methods have enabled deep reinforcement learning (RL) to approach challenging continuous control problems, even when the underlying systems involve highly nonlinear dynamics that generate complex non-smooth optimization landscapes. We develop a rigorous framework for understanding how policy gradient methods mollify non-smooth optimization landscapes to enable effective policy search, as well as the downside of it: while making the objective function smoother and easier to optimize, the stochastic objective deviates further from the original problem. We demonstrate the equivalence between policy gradient methods and solving backward heat equations. Following the ill-posedness of backward heat equations from PDE theory, we present a fundamental challenge to the use of policy gradient under stochasticity. Moreover, we make the connection between this limitation and the uncertainty principle in harmonic analysis to understand the effects of exploration with stochastic policies in RL. We also provide experimental results to illustrate both the positive and negative aspects of mollification effects in practice.
Stochastic gradient methods are among the most important algorithms in training machine learning problems. While classical assumptions such as strong convexity allow a simple analysis they are rarely satisfied in applications. In recent years, global and local gradient domination properties have shown to be a more realistic replacement of strong convexity. They were proved to hold in diverse settings such as (simple) policy gradient methods in reinforcement learning and training of deep neural networks with analytic activation functions. We prove almost sure convergence rates $f(X_n)-f^*\in o\big( n^{-\frac{1}{4\beta-1}+\epsilon}\big)$ of the last iterate for stochastic gradient descent (with and without momentum) under global and local $\beta$-gradient domination assumptions. The almost sure rates get arbitrarily close to recent rates in expectation. Finally, we demonstrate how to apply our results to the training task in both supervised and reinforcement learning.
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 (DRL) seeks to estimate the entire distribution of it, which leads to a unified framework for handling different risk measures. However, developing policy gradient methods for risk-sensitive DRL is inherently more complex as it involves finding the gradient of a probability measure. This paper introduces a new policy gradient method for risk-sensitive DRL with general coherent risk measures, where we provide an analytical form of the probability measure's gradient for any distribution. For practical use, we design a categorical distributional policy gradient algorithm (CDPG) that approximates any distribution by a categorical family supported on some fixed points. We further provide a finite-support optimality guarantee and a finite-iteration convergence guarantee under inexact policy evaluation and gradient estimation. Through experiments on stochastic Cliffwalk and CartPole environments, we illustrate the benefits of considering a risk-sensitive setting in DRL.
Multi-agent reinforcement learning (MARL) has exploded in popularity in recent years. While numerous approaches have been developed, they can be broadly categorized into three main types: centralized training and execution (CTE), centralized training for decentralized execution (CTDE), and decentralized training and execution (DTE). CTE methods assume centralization during training and execution (e.g., with fast, free, and perfect communication) and have the most information during execution. CTDE methods are the most common, as they leverage centralized information during training while enabling decentralized execution -- using only information available to that agent during execution. Decentralized training and execution methods make the fewest assumptions and are often simple to implement. This text is an introduction to cooperative MARL -- MARL in which all agents share a single, joint reward. It is meant to explain the setting, basic concepts, and common methods for the CTE, CTDE, and DTE settings. It does not cover all work in cooperative MARL as the area is quite extensive. I have included work that I believe is important for understanding the main concepts in the area and apologize to those that I have omitted. Topics include simple applications of single-agent methods to CTE as well as some more scalable methods that exploit the multi-agent structure, independent Q-learning and policy gradient methods and their extensions, as well as value function factorization methods including the well-known VDN, QMIX, and QPLEX approaches, and centralized critic methods including MADDPG, COMA, and MAPPO. I also discuss common misconceptions, the relationship between different approaches, and some open questions.
We study a Federated Reinforcement Learning (FedRL) problem with constraint heterogeneity. In our setting, we aim to solve a reinforcement learning problem with multiple constraints while $N$ training agents are located in $N$ different environments with limited access to the constraint signals and they are expected to collaboratively learn a policy satisfying all constraint signals. Such learning problems are prevalent in scenarios of Large Language Model (LLM) fine-tuning and healthcare applications. To solve the problem, we propose federated primal-dual policy optimization methods based on traditional policy gradient methods. Specifically, we introduce $N$ local Lagrange functions for agents to perform local policy updates, and these agents are then scheduled to periodically communicate on their local policies. Taking natural policy gradient (NPG) and proximal policy optimization (PPO) as policy optimization methods, we mainly focus on two instances of our algorithms, ie, {FedNPG} and {FedPPO}. We show that FedNPG achieves global convergence with an $\tilde{O}(1/\sqrt{T})$ rate, and FedPPO efficiently solves complicated learning tasks with the use of deep neural networks.
Policy-based methods have achieved remarkable success in solving challenging reinforcement learning problems. Among these methods, off-policy policy gradient methods are particularly important due to that they can benefit from off-policy data. However, these methods suffer from the high variance of the off-policy policy gradient (OPPG) estimator, which results in poor sample efficiency during training. In this paper, we propose an off-policy policy gradient method with the optimal action-dependent baseline (Off-OAB) to mitigate this variance issue. Specifically, this baseline maintains the OPPG estimator's unbiasedness while theoretically minimizing its variance. To enhance practical computational efficiency, we design an approximated version of this optimal baseline. Utilizing this approximation, our method (Off-OAB) aims to decrease the OPPG estimator's variance during policy optimization. We evaluate the proposed Off-OAB method on six representative tasks from OpenAI Gym and MuJoCo, where it demonstrably surpasses state-of-the-art methods on the majority of these tasks.
The paper studies information-theoretic opacity, an information-flow privacy property, in a setting involving two agents: A planning agent who controls a stochastic system and an observer who partially observes the system states. The goal of the observer is to infer some secret, represented by a random variable, from its partial observations, while the goal of the planning agent is to make the secret maximally opaque to the observer while achieving a satisfactory total return. Modeling the stochastic system using a Markov decision process, two classes of opacity properties are considered -- Last-state opacity is to ensure that the observer is uncertain if the last state is in a specific set and initial-state opacity is to ensure that the observer is unsure of the realization of the initial state. As the measure of opacity, we employ the Shannon conditional entropy capturing the information about the secret revealed by the observable. Then, we develop primal-dual policy gradient methods for opacity-enforcement planning subject to constraints on total returns. We propose novel algorithms to compute the policy gradient of entropy for each observation, leveraging message passing within the hidden Markov models. This gradient computation enables us to have stable and fast convergence. We demonstrate our solution of opacity-enforcement control through a grid world example.