We consider a general-sum N-player linear-quadratic game with stochastic dynamics over a finite horizon and prove the global convergence of the natural policy gradient method to the Nash equilibrium. In order to prove the convergence of the method, we require a certain amount of noise in the system. We give a condition, essentially a lower bound on the covariance of the noise in terms of the model parameters, in order to guarantee convergence. We illustrate our results with numerical experiments to show that even in situations where the policy gradient method may not converge in the deterministic setting, the addition of noise leads to convergence.
Learning continuous control in high-dimensional sparse reward settings, such as robotic manipulation, is a challenging problem due to the number of samples often required to obtain accurate optimal value and policy estimates. While many deep reinforcement learning methods have aimed at improving sample efficiency through replay or improved exploration techniques, state of the art actor-critic and policy gradient methods still suffer from the hard exploration problem in sparse reward settings. Motivated by recent successes of value-based methods for approximating state-action values, like RBF-DQN, we explore the potential of value-based reinforcement learning for learning continuous robotic manipulation tasks in multi-task sparse reward settings. On robotic manipulation tasks, we empirically show RBF-DQN converges faster than current state of the art algorithms such as TD3, SAC, and PPO. We also perform ablation studies with RBF-DQN and have shown that some enhancement techniques for vanilla Deep Q learning such as Hindsight Experience Replay (HER) and Prioritized Experience Replay (PER) can also be applied to RBF-DQN. Our experimental analysis suggests that value-based approaches may be more sensitive to data augmentation and replay buffer sample techniques than policy-gradient methods, and that the benefits of these methods for robot manipulation are heavily dependent on the transition dynamics of generated subgoal states.
Policy gradient methods are appealing in deep reinforcement learning but suffer from high variance of gradient estimate. To reduce the variance, the state value function is applied commonly. However, the effect of the state value function becomes limited in stochastic dynamic environments, where the unexpected state dynamics and rewards will increase the variance. In this paper, we propose to replace the state value function with a novel hindsight value function, which leverages the information from the future to reduce the variance of the gradient estimate for stochastic dynamic environments. Particularly, to obtain an ideally unbiased gradient estimate, we propose an information-theoretic approach, which optimizes the embeddings of the future to be independent of previous actions. In our experiments, we apply the proposed hindsight value function in stochastic dynamic environments, including discrete-action environments and continuous-action environments. Compared with the standard state value function, the proposed hindsight value function consistently reduces the variance, stabilizes the training, and improves the eventual policy.
In recent years, reinforcement learning (RL) has gained increasing attention in control engineering. Especially, policy gradient methods are widely used. In this work, we improve the tracking performance of proximal policy optimization (PPO) for arbitrary reference signals by incorporating information about future reference values. Two variants of extending the argument of the actor and the critic taking future reference values into account are presented. In the first variant, global future reference values are added to the argument. For the second variant, a novel kind of residual space with future reference values applicable to model-free reinforcement learning is introduced. Our approach is evaluated against a PI controller on a simple drive train model. We expect our method to generalize to arbitrary references better than previous approaches, pointing towards the applicability of RL to control real systems.
Approximate Policy Iteration (API) algorithms alternate between (approximate) policy evaluation and (approximate) greedification. Many different approaches have been explored for approximate policy evaluation, but less is understood about approximate greedification and what choices guarantee policy improvement. In this work, we investigate approximate greedification when reducing the KL divergence between the parameterized policy and the Boltzmann distribution over action values. In particular, we investigate the difference between the forward and reverse KL divergences, with varying degrees of entropy regularization. We show that the reverse KL has stronger policy improvement guarantees, but that reducing the forward KL can result in a worse policy. We also demonstrate, however, that a large enough reduction of the forward KL can induce improvement under additional assumptions. Empirically, we show on simple continuous-action environments that the forward KL can induce more exploration, but at the cost of a more suboptimal policy. No significant differences were observed in the discrete-action setting or on a suite of benchmark problems. Throughout, we highlight that many policy gradient methods can be seen as an instance of API, with either the forward or reverse KL for the policy update, and discuss next steps for understanding and improving our policy optimization algorithms.
Data augmentation is a commonly used approach to improving the generalization of deep learning models. Recent works show that learned data augmentation policies can achieve better generalization than hand-crafted ones. However, most of these works use unified augmentation policies for all samples in a dataset, which is observed not necessarily beneficial for all labels in multi-label classification tasks, i.e., some policies may have negative impacts on some labels while benefitting the others. To tackle this problem, we propose a novel Label-Based AutoAugmentation (LB-Aug) method for multi-label scenarios, where augmentation policies are generated with respect to labels by an augmentation-policy network. The policies are learned via reinforcement learning using policy gradient methods, providing a mapping from instance labels to their optimal augmentation policies. Numerical experiments show that our LB-Aug outperforms previous state-of-the-art augmentation methods by large margins in multiple benchmarks on image and video classification.
We propose policy gradient algorithms which learn risk-sensitive policies in a reinforcement learning (RL) framework. Our proposed algorithms maximize the distortion risk measure (DRM) of the cumulative reward in an episodic Markov decision process in on-policy and off-policy RL settings, respectively. We derive a variant of the policy gradient theorem that caters to the DRM objective, and integrate it with a likelihood ratio-based gradient estimation scheme. We derive non-asymptotic bounds that establish the convergence of our proposed algorithms to an approximate stationary point of the DRM objective.
Having access to an exploring restart distribution (the so-called wide coverage assumption) is critical with policy gradient methods. This is due to the fact that, while the objective function is insensitive to updates in unlikely states, the agent may still need improvements in those states in order to reach a nearly optimal payoff. For this reason, wide coverage is used in some form when analyzing theoretical properties of practical policy gradient methods. However, this assumption can be unfeasible in certain environments, for instance when learning is online, or when restarts are possible only from a fixed initial state. In these cases, classical policy gradient algorithms can have very poor convergence properties and sample efficiency. In this paper, we develop Curious Explorer, a novel and simple iterative state space exploration strategy that can be used with any starting distribution $\rho$. Curious Explorer starts from $\rho$, then using intrinsic rewards assigned to the set of poorly visited states produces a sequence of policies, each one more exploratory than the previous one in an informed way, and finally outputs a restart model $\mu$ based on the state visitation distribution of the exploratory policies. Curious Explorer is provable, in the sense that we provide theoretical upper bounds on how often an optimal policy visits poorly visited states. These bounds can be used to prove PAC convergence and sample efficiency results when a PAC optimizer is plugged in Curious Explorer. This allows to achieve global convergence and sample efficiency results without any coverage assumption for REINFORCE, and potentially for any other policy gradient method ensuring PAC convergence with wide coverage. Finally, we plug (the output of) Curious Explorer into REINFORCE and TRPO, and show empirically that it can improve performance in MDPs with challenging exploration.
Many transfer problems require re-using previously optimal decisions for solving new tasks, which suggests the need for learning algorithms that can modify the mechanisms for choosing certain actions independently of those for choosing others. However, there is currently no formalism nor theory for how to achieve this kind of modular credit assignment. To answer this question, we define modular credit assignment as a constraint on minimizing the algorithmic mutual information among feedback signals for different decisions. We introduce what we call the modularity criterion for testing whether a learning algorithm satisfies this constraint by performing causal analysis on the algorithm itself. We generalize the recently proposed societal decision-making framework as a more granular formalism than the Markov decision process to prove that for decision sequences that do not contain cycles, certain single-step temporal difference action-value methods meet this criterion while all policy-gradient methods do not. Empirical evidence suggests that such action-value methods are more sample efficient than policy-gradient methods on transfer problems that require only sparse changes to a sequence of previously optimal decisions.
Neural models and symbolic algorithms have recently been combined for tasks requiring both perception and reasoning. Neural models ground perceptual input into a conceptual vocabulary, on which a classical reasoning algorithm is applied to generate output. A key limitation is that such neural-to-symbolic models can only be trained end-to-end for tasks where the output space is symbolic. In this paper, we study neural-symbolic-neural models for reasoning tasks that require a conversion from an image input (e.g., a partially filled sudoku) to an image output (e.g., the image of the completed sudoku). While designing such a three-step hybrid architecture may be straightforward, the key technical challenge is end-to-end training -- how to backpropagate without intermediate supervision through the symbolic component. We propose NSNnet, an architecture that combines an image reconstruction loss with a novel output encoder to generate a supervisory signal, develops update algorithms that leverage policy gradient methods for supervision, and optimizes loss using a novel subsampling heuristic. We experiment on problem settings where symbolic algorithms are easily specified: a visual maze solving task and a visual Sudoku solver where the supervision is in image form. Experiments show high accuracy with significantly less data compared to purely neural approaches.
Reinforcement Learning in large action spaces is a challenging problem. Cooperative multi-agent reinforcement learning (MARL) exacerbates matters by imposing various constraints on communication and observability. In this work, we consider the fundamental hurdle affecting both value-based and policy-gradient approaches: an exponential blowup of the action space with the number of agents. For value-based methods, it poses challenges in accurately representing the optimal value function. For policy gradient methods, it makes training the critic difficult and exacerbates the problem of the lagging critic. We show that from a learning theory perspective, both problems can be addressed by accurately representing the associated action-value function with a low-complexity hypothesis class. This requires accurately modelling the agent interactions in a sample efficient way. To this end, we propose a novel tensorised formulation of the Bellman equation. This gives rise to our method Tesseract, which views the Q-function as a tensor whose modes correspond to the action spaces of different agents. Algorithms derived from Tesseract decompose the Q-tensor across agents and utilise low-rank tensor approximations to model agent interactions relevant to the task. We provide PAC analysis for Tesseract-based algorithms and highlight their relevance to the class of rich observation MDPs. Empirical results in different domains confirm Tesseract's gains in sample efficiency predicted by the theory.
This paper provides the details of implementing two important policy gradient methods to solve the inverted pendulum problem. These are namely the Deep Deterministic Policy Gradient (DDPG) and the Proximal Policy Optimization (PPO) algorithm. The problem is solved by using an actor-critic model where an actor-network is used to learn the policy function and a critic network is to evaluate the actor-network by learning to estimate the Q function. Apart from briefly explaining the mathematics behind these two algorithms, the details of python implementation are provided which helps in demystifying the underlying complexity of the algorithm. In the process, the readers will be introduced to OpenAI/Gym, Tensorflow 2.x and Keras utilities used for implementing the above concepts.
Markov Decision Processes are classically solved using Value Iteration and Policy Iteration algorithms. Recent interest in Reinforcement Learning has motivated the study of methods inspired by optimization, such as gradient ascent. Among these, a popular algorithm is the Natural Policy Gradient, which is a mirror descent variant for MDPs. This algorithm forms the basis of several popular Reinforcement Learning algorithms such as Natural actor-critic, TRPO, PPO, etc, and so is being studied with growing interest. It has been shown that Natural Policy Gradient with constant step size converges with a sublinear rate of O(1/k) to the global optimal. In this paper, we present improved finite time convergence bounds, and show that this algorithm has geometric (also known as linear) asymptotic convergence rate. We further improve this convergence result by introducing a variant of Natural Policy Gradient with adaptive step sizes. Finally, we compare different variants of policy gradient methods experimentally.
Policy gradient methods are an attractive approach to multi-agent reinforcement learning problems due to their convergence properties and robustness in partially observable scenarios. However, there is a significant performance gap between state-of-the-art policy gradient and value-based methods on the popular StarCraft Multi-Agent Challenge (SMAC) benchmark. In this paper, we introduce semi-on-policy (SOP) training as an effective and computationally efficient way to address the sample inefficiency of on-policy policy gradient methods. We enhance two state-of-the-art policy gradient algorithms with SOP training, demonstrating significant performance improvements. Furthermore, we show that our methods perform as well or better than state-of-the-art value-based methods on a variety of SMAC tasks.
Despite the increasing popularity of policy gradient methods, they are yet to be widely utilized in sample-scarce applications, such as robotics. The sample efficiency could be improved by making best usage of available information. As a key component in reinforcement learning, the reward function is usually devised carefully to guide the agent. Hence, the reward function is usually known, allowing access to not only scalar reward signals but also reward gradients. To benefit from reward gradients, previous works require the knowledge of environment dynamics, which are hard to obtain. In this work, we develop the \textit{Reward Policy Gradient} estimator, a novel approach that integrates reward gradients without learning a model. Bypassing the model dynamics allows our estimator to achieve a better bias-variance trade-off, which results in a higher sample efficiency, as shown in the empirical analysis. Our method also boosts the performance of Proximal Policy Optimization on different MuJoCo control tasks.
Policy gradient methods can solve complex tasks but often fail when the dimensionality of the action-space or objective multiplicity grow very large. This occurs, in part, because the variance on score-based gradient estimators scales quadratically. In this paper, we address this problem through a factor baseline which exploits independence structure encoded in a novel action-target influence network. Factored policy gradients (FPGs), which follow, provide a common framework for analysing key state-of-the-art algorithms, are shown to generalise traditional policy gradients, and yield a principled way of incorporating prior knowledge of a problem domain's generative processes. We provide an analysis of the proposed estimator and identify the conditions under which variance is reduced. The algorithmic aspects of FPGs are discussed, including optimal policy factorisation, as characterised by minimum biclique coverings, and the implications for the bias-variance trade-off of incorrectly specifying the network. Finally, we demonstrate the performance advantages of our algorithm on large-scale bandit and traffic intersection problems, providing a novel contribution to the latter in the form of a spatial approximation.
The softmax policy gradient (PG) method, which performs gradient ascent under softmax policy parameterization, is arguably one of the de facto implementations of policy optimization in modern reinforcement learning. For $\gamma$-discounted infinite-horizon tabular Markov decision processes (MDPs), remarkable progress has recently been achieved towards establishing global convergence of softmax PG methods in finding a near-optimal policy. However, prior results fall short of delineating clear dependencies of convergence rates on salient parameters such as the cardinality of the state space $\mathcal{S}$ and the effective horizon $\frac{1}{1-\gamma}$, both of which could be excessively large. In this paper, we deliver a pessimistic message regarding the iteration complexity of softmax PG methods, despite assuming access to exact gradient computation. Specifically, we demonstrate that the softmax PG method with stepsize $\eta$ can take \[ \frac{1}{\eta} |\mathcal{S}|^{2^{\Omega\big(\frac{1}{1-\gamma}\big)}} ~\text{iterations} \] to converge, even in the presence of a benign policy initialization and an initial state distribution amenable to exploration (so that the distribution mismatch coefficient is not exceedingly large). This is accomplished by characterizing the algorithmic dynamics over a carefully-constructed MDP containing only three actions. Our exponential lower bound hints at the necessity of carefully adjusting update rules or enforcing proper regularization in accelerating PG methods.
This paper presents a multi-agent reinforcement learning algorithm to represent strategic bidding behavior in freight transport markets. Using this algorithm, we investigate whether feasible market equilibriums arise without any central control or communication between agents. Studying behavior in such environments may serve as a stepping stone towards self-organizing logistics systems like the Physical Internet. We model an agent-based environment in which a shipper and a carrier actively learn bidding strategies using policy gradient methods, posing bid- and ask prices at the individual container level. Both agents aim to learn the best response given the expected behavior of the opposing agent. A neutral broker allocates jobs based on bid-ask spreads. Our game-theoretical analysis and numerical experiments focus on behavioral insights. To evaluate system performance, we measure adherence to Nash equilibria, fairness of reward division and utilization of transport capacity. We observe good performance both in predictable, deterministic settings (~95% adherence to Nash equilibria) and highly stochastic environments (~85% adherence). Risk-seeking behavior may increase an agent's reward share, as long as the strategies are not overly aggressive. The results suggest a potential for full automation and decentralization of freight transport markets.
Policy-based methods with function approximation are widely used for solving two-player zero-sum games with large state and/or action spaces. However, it remains elusive how to obtain optimization and statistical guarantees for such algorithms. We present a new policy optimization algorithm with function approximation and prove that under standard regularity conditions on the Markov game and the function approximation class, our algorithm finds a near-optimal policy within a polynomial number of samples and iterations. To our knowledge, this is the first provably efficient policy optimization algorithm with function approximation that solves two-player zero-sum Markov games.
We obtain global, non-asymptotic convergence guarantees for independent learning algorithms in competitive reinforcement learning settings with two agents (i.e., zero-sum stochastic games). We consider an episodic setting where in each episode, each player independently selects a policy and observes only their own actions and rewards, along with the state. We show that if both players run policy gradient methods in tandem, their policies will converge to a min-max equilibrium of the game, as long as their learning rates follow a two-timescale rule (which is necessary). To the best of our knowledge, this constitutes the first finite-sample convergence result for independent policy gradient methods in competitive RL; prior work has largely focused on centralized, coordinated procedures for equilibrium computation.