Actor-critic algorithms are widely used in reinforcement learning, but are challenging to mathematically analyze due to the online arrival of non-i.i.d. data samples. The distribution of the data samples dynamically changes as the model is updated, introducing a complex feedback loop between the data distribution and the reinforcement learning algorithm. We prove that, under a time rescaling, the online actor-critic algorithm with tabular parametrization converges to an ordinary differential equations (ODEs) as the number of updates becomes large. The proof first establishes the geometric ergodicity of the data samples under a fixed actor policy. Then, using a Poisson equation, we prove that the fluctuations of the data samples around a dynamic probability measure, which is a function of the evolving actor model, vanish as the number of updates become large. Once the ODE limit has been derived, we study its convergence properties using a two time-scale analysis which asymptotically de-couples the critic ODE from the actor ODE. The convergence of the critic to the solution of the Bellman equation and the actor to the optimal policy are proven. In addition, a convergence rate to this global minimum is also established. Our convergence analysis holds under specific choices for the learning rates and exploration rates in the actor-critic algorithm, which could provide guidance for the implementation of actor-critic algorithms in practice.
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Scaling reinforcement learning (RL) to recommender systems (RS) is promising since maximizing the expected cumulative rewards for RL agents meets the objective of RS, i.e., improving customers' long-term satisfaction. A key approach to this goal is offline RL, which aims to learn policies from logged data. However, the high-dimensional action space and the non-stationary dynamics in commercial RS intensify distributional shift issues, making it challenging to apply offline RL methods to RS. To alleviate the action distribution shift problem in extracting RL policy from static trajectories, we propose Value Penalized Q-learning (VPQ), an uncertainty-based offline RL algorithm. It penalizes the unstable Q-values in the regression target by uncertainty-aware weights, without the need to estimate the behavior policy, suitable for RS with a large number of items. We derive the penalty weights from the variances across an ensemble of Q-functions. To alleviate distributional shift issues at test time, we further introduce the critic framework to integrate the proposed method with classic RS models. Extensive experiments conducted on two real-world datasets show that the proposed method could serve as a gain plugin for existing RS models.
Nonlinear trajectory optimization algorithms have been developed to handle optimal control problems with nonlinear dynamics and nonconvex constraints in trajectory planning. The performance and computational efficiency of many trajectory optimization methods are sensitive to the initial guess, i.e., the trajectory guess needed by the recursive trajectory optimization algorithm. Motivated by this observation, we tackle the initialization problem for trajectory optimization via policy optimization. To optimize a policy, we propose a guided policy search method that has two key components: i) Trajectory update; ii) Policy update. The trajectory update involves offline solutions of a large number of trajectory optimization problems from different initial states via Sequential Convex Programming (SCP). Here we take a single SCP step to generate the trajectory iterate for each problem. In conjunction with these iterates, we also generate additional trajectories around each iterate via a feedback control law. Then all these trajectories are used by a stochastic gradient descent algorithm to update the neural network policy, i.e., the policy update step. As a result, the trained policy makes it possible to generate trajectory candidates that are close to the optimality and feasibility and that provide excellent initial guesses for the trajectory optimization methods. We validate the proposed method via a real-world 6-degree-of-freedom powered descent guidance problem for a reusable rocket.
We study infinite horizon discounted Mean Field Control (MFC) problems with common noise through the lens of Mean Field Markov Decision Processes (MFMDP). We allow the agents to use actions that are randomized not only at the individual level but also at the level of the population. This common randomization allows us to establish connections between both closed-loop and open-loop policies for MFC and Markov policies for the MFMDP. In particular, we show that there exists an optimal closed-loop policy for the original MFC. Building on this framework and the notion of state-action value function, we then propose reinforcement learning (RL) methods for such problems, by adapting existing tabular and deep RL methods to the mean-field setting. The main difficulty is the treatment of the population state, which is an input of the policy and the value function. We provide convergence guarantees for tabular algorithms based on discretizations of the simplex. Neural network based algorithms are more suitable for continuous spaces and allow us to avoid discretizing the mean field state space. Numerical examples are provided.
We study the model-based reward-free reinforcement learning with linear function approximation for episodic Markov decision processes (MDPs). In this setting, the agent works in two phases. In the exploration phase, the agent interacts with the environment and collects samples without the reward. In the planning phase, the agent is given a specific reward function and uses samples collected from the exploration phase to learn a good policy. We propose a new provably efficient algorithm, called UCRL-RFE under the Linear Mixture MDP assumption, where the transition probability kernel of the MDP can be parameterized by a linear function over certain feature mappings defined on the triplet of state, action, and next state. We show that to obtain an $\epsilon$-optimal policy for arbitrary reward function, UCRL-RFE needs to sample at most $\tilde O(H^5d^2\epsilon^{-2})$ episodes during the exploration phase. Here, $H$ is the length of the episode, $d$ is the dimension of the feature mapping. We also propose a variant of UCRL-RFE using Bernstein-type bonus and show that it needs to sample at most $\tilde O(H^4d(H + d)\epsilon^{-2})$ to achieve an $\epsilon$-optimal policy. By constructing a special class of linear Mixture MDPs, we also prove that for any reward-free algorithm, it needs to sample at least $\tilde \Omega(H^2d\epsilon^{-2})$ episodes to obtain an $\epsilon$-optimal policy. Our upper bound matches the lower bound in terms of the dependence on $\epsilon$ and the dependence on $d$ if $H \ge d$.
This paper proves the global convergence of a triangularized orthogonalization-free method (TriOFM). TriOFM, in general, applies a triangularization idea to the gradient of an objective function and removes the rotation invariance in minimizers. More precisely, in this paper, the TriOFM works as an eigensolver for sizeable sparse matrices and obtains eigenvectors without any orthogonalization step. Due to the triangularization, the iteration is a discrete-time flow in a non-conservative vector field. The global convergence relies on the stable manifold theorem, whereas the convergence to stationary points is proved in detail in this paper. We provide two proofs inspired by the noisy power method and the noisy optimization method, respectively.
The Q-learning algorithm is known to be affected by the maximization bias, i.e. the systematic overestimation of action values, an important issue that has recently received renewed attention. Double Q-learning has been proposed as an efficient algorithm to mitigate this bias. However, this comes at the price of an underestimation of action values, in addition to increased memory requirements and a slower convergence. In this paper, we introduce a new way to address the maximization bias in the form of a "self-correcting algorithm" for approximating the maximum of an expected value. Our method balances the overestimation of the single estimator used in conventional Q-learning and the underestimation of the double estimator used in Double Q-learning. Applying this strategy to Q-learning results in Self-correcting Q-learning. We show theoretically that this new algorithm enjoys the same convergence guarantees as Q-learning while being more accurate. Empirically, it performs better than Double Q-learning in domains with rewards of high variance, and it even attains faster convergence than Q-learning in domains with rewards of zero or low variance. These advantages transfer to a Deep Q Network implementation that we call Self-correcting DQN and which outperforms regular DQN and Double DQN on several tasks in the Atari 2600 domain.
Deep reinforcement learning suggests the promise of fully automated learning of robotic control policies that directly map sensory inputs to low-level actions. However, applying deep reinforcement learning methods on real-world robots is exceptionally difficult, due both to the sample complexity and, just as importantly, the sensitivity of such methods to hyperparameters. While hyperparameter tuning can be performed in parallel in simulated domains, it is usually impractical to tune hyperparameters directly on real-world robotic platforms, especially legged platforms like quadrupedal robots that can be damaged through extensive trial-and-error learning. In this paper, we develop a stable variant of the soft actor-critic deep reinforcement learning algorithm that requires minimal hyperparameter tuning, while also requiring only a modest number of trials to learn multilayer neural network policies. This algorithm is based on the framework of maximum entropy reinforcement learning, and automatically trades off exploration against exploitation by dynamically and automatically tuning a temperature parameter that determines the stochasticity of the policy. We show that this method achieves state-of-the-art performance on four standard benchmark environments. We then demonstrate that it can be used to learn quadrupedal locomotion gaits on a real-world Minitaur robot, learning to walk from scratch directly in the real world in two hours of training.
Recent successes of value-based multi-agent deep reinforcement learning employ optimism in value function by carefully controlling learning rate(Omidshafiei et al., 2017) or reducing update prob-ability (Palmer et al., 2018). We introduce a de-centralized quantile estimator: Responsible Implicit Quantile Network (RIQN), while robust to teammate-environment interactions, able to reduce the amount of imposed optimism. Upon benchmarking against related Hysteretic-DQN(HDQN) and Lenient-DQN (LDQN), we findRIQN agents more stable, sample efficient and more likely to converge to the optimal policy.
This work considers the problem of provably optimal reinforcement learning for episodic finite horizon MDPs, i.e. how an agent learns to maximize his/her long term reward in an uncertain environment. The main contribution is in providing a novel algorithm --- Variance-reduced Upper Confidence Q-learning (vUCQ) --- which enjoys a regret bound of $\widetilde{O}(\sqrt{HSAT} + H^5SA)$, where the $T$ is the number of time steps the agent acts in the MDP, $S$ is the number of states, $A$ is the number of actions, and $H$ is the (episodic) horizon time. This is the first regret bound that is both sub-linear in the model size and asymptotically optimal. The algorithm is sub-linear in that the time to achieve $\epsilon$-average regret for any constant $\epsilon$ is $O(SA)$, which is a number of samples that is far less than that required to learn any non-trivial estimate of the transition model (the transition model is specified by $O(S^2A)$ parameters). The importance of sub-linear algorithms is largely the motivation for algorithms such as $Q$-learning and other "model free" approaches. vUCQ algorithm also enjoys minimax optimal regret in the long run, matching the $\Omega(\sqrt{HSAT})$ lower bound. Variance-reduced Upper Confidence Q-learning (vUCQ) is a successive refinement method in which the algorithm reduces the variance in $Q$-value estimates and couples this estimation scheme with an upper confidence based algorithm. Technically, the coupling of both of these techniques is what leads to the algorithm enjoying both the sub-linear regret property and the asymptotically optimal regret.
Policy gradient methods are widely used in reinforcement learning algorithms to search for better policies in the parameterized policy space. They do gradient search in the policy space and are known to converge very slowly. Nesterov developed an accelerated gradient search algorithm for convex optimization problems. This has been recently extended for non-convex and also stochastic optimization. We use Nesterov's acceleration for policy gradient search in the well-known actor-critic algorithm and show the convergence using ODE method. We tested this algorithm on a scheduling problem. Here an incoming job is scheduled into one of the four queues based on the queue lengths. We see from experimental results that algorithm using Nesterov's acceleration has significantly better performance compared to algorithm which do not use acceleration. To the best of our knowledge this is the first time Nesterov's acceleration has been used with actor-critic algorithm.
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