We study reinforcement learning (RL) with linear function approximation. For episodic time-inhomogeneous linear Markov decision processes (linear MDPs) whose transition dynamic can be parameterized as a linear function of a given feature mapping, we propose the first computationally efficient algorithm that achieves the nearly minimax optimal regret $\tilde O(d\sqrt{H^3K})$, where $d$ is the dimension of the feature mapping, $H$ is the planning horizon, and $K$ is the number of episodes. Our algorithm is based on a weighted linear regression scheme with a carefully designed weight, which depends on a new variance estimator that (1) directly estimates the variance of the \emph{optimal} value function, (2) monotonically decreases with respect to the number of episodes to ensure a better estimation accuracy, and (3) uses a rare-switching policy to update the value function estimator to control the complexity of the estimated value function class. Our work provides a complete answer to optimal RL with linear MDPs, and the developed algorithm and theoretical tools may be of independent interest.
We study federated contextual linear bandits, where $M$ agents cooperate with each other to solve a global contextual linear bandit problem with the help of a central server. We consider the asynchronous setting, where all agents work independently and the communication between one agent and the server will not trigger other agents' communication. We propose a simple algorithm named \texttt{FedLinUCB} based on the principle of optimism. We prove that the regret of \texttt{FedLinUCB} is bounded by $\tilde{O}(d\sqrt{\sum_{m=1}^M T_m})$ and the communication complexity is $\tilde{O}(dM^2)$, where $d$ is the dimension of the contextual vector and $T_m$ is the total number of interactions with the environment by $m$-th agent. To the best of our knowledge, this is the first provably efficient algorithm that allows fully asynchronous communication for federated contextual linear bandits, while achieving the same regret guarantee as in the single-agent setting.
We study the linear contextual bandit problem in the presence of adversarial corruption, where the reward at each round is corrupted by an adversary, and the corruption level (i.e., the sum of corruption magnitudes over the horizon) is $C\geq 0$. The best-known algorithms in this setting are limited in that they either are computationally inefficient or require a strong assumption on the corruption, or their regret is at least $C$ times worse than the regret without corruption. In this paper, to overcome these limitations, we propose a new algorithm based on the principle of optimism in the face of uncertainty. At the core of our algorithm is a weighted ridge regression where the weight of each chosen action depends on its confidence up to some threshold. We show that for both known $C$ and unknown $C$ cases, our algorithm with proper choice of hyperparameter achieves a regret that nearly matches the lower bounds. Thus, our algorithm is nearly optimal up to logarithmic factors for both cases. Notably, our algorithm achieves the near-optimal regret for both corrupted and uncorrupted cases ($C=0$) simultaneously.
We consider learning a stochastic bandit model, where the reward function belongs to a general class of uniformly bounded functions, and the additive noise can be heteroscedastic. Our model captures contextual linear bandits and generalized linear bandits as special cases. While previous works (Kirschner and Krause, 2018; Zhou et al., 2021) based on weighted ridge regression can deal with linear bandits with heteroscedastic noise, they are not directly applicable to our general model due to the curse of nonlinearity. In order to tackle this problem, we propose a multi-level learning framework for the general bandit model. The core idea of our framework is to partition the observed data into different levels according to the variance of their respective reward and perform online learning at each level collaboratively. Under our framework, we first design an algorithm that constructs the variance-aware confidence set based on empirical risk minimization and prove a variance-dependent regret bound. For generalized linear bandits, we further propose an algorithm based on follow-the-regularized-leader (FTRL) subroutine and online-to-confidence-set conversion, which can achieve a tighter variance-dependent regret under certain conditions.
We study the stochastic shortest path (SSP) problem in reinforcement learning with linear function approximation, where the transition kernel is represented as a linear mixture of unknown models. We call this class of SSP problems the linear mixture SSP. We propose a novel algorithm for learning the linear mixture SSP, which can attain a $\tilde O(d B_{\star}^{1.5}\sqrt{K/c_{\min}})$ regret. Here $K$ is the number of episodes, $d$ is the dimension of the feature mapping in the mixture model, $B_{\star}$ bounds the expected cumulative cost of the optimal policy, and $c_{\min}>0$ is the lower bound of the cost function. Our algorithm also applies to the case when $c_{\min} = 0$, where a $\tilde O(K^{2/3})$ regret is guaranteed. To the best of our knowledge, this is the first algorithm with a sublinear regret guarantee for learning linear mixture SSP. In complement to the regret upper bounds, we also prove a lower bound of $\Omega(d B_{\star} \sqrt{K})$, which nearly matches our upper bound.
Reinforcement learning (RL) algorithms can be used to provide personalized services, which rely on users' private and sensitive data. To protect the users' privacy, privacy-preserving RL algorithms are in demand. In this paper, we study RL with linear function approximation and local differential privacy (LDP) guarantees. We propose a novel $(\varepsilon, \delta)$-LDP algorithm for learning a class of Markov decision processes (MDPs) dubbed linear mixture MDPs, and obtains an $\tilde{\mathcal{O}}( d^{5/4}H^{7/4}T^{3/4}\left(\log(1/\delta)\right)^{1/4}\sqrt{1/\varepsilon})$ regret, where $d$ is the dimension of feature mapping, $H$ is the length of the planning horizon, and $T$ is the number of interactions with the environment. We also prove a lower bound $\Omega(dH\sqrt{T}/\left(e^{\varepsilon}(e^{\varepsilon}-1)\right))$ for learning linear mixture MDPs under $\varepsilon$-LDP constraint. Experiments on synthetic datasets verify the effectiveness of our algorithm. To the best of our knowledge, this is the first provable privacy-preserving RL algorithm with linear function approximation.
The success of deep reinforcement learning (DRL) is due to the power of learning a representation that is suitable for the underlying exploration and exploitation task. However, existing provable reinforcement learning algorithms with linear function approximation often assume the feature representation is known and fixed. In order to understand how representation learning can improve the efficiency of RL, we study representation learning for a class of low-rank Markov Decision Processes (MDPs) where the transition kernel can be represented in a bilinear form. We propose a provably efficient algorithm called ReLEX that can simultaneously learn the representation and perform exploration. We show that ReLEX always performs no worse than a state-of-the-art algorithm without representation learning, and will be strictly better in terms of sample efficiency if the function class of representations enjoys a certain mild "coverage'' property over the whole state-action space.
We study reinforcement learning (RL) with linear function approximation. Existing algorithms for this problem only have high-probability regret and/or Probably Approximately Correct (PAC) sample complexity guarantees, which cannot guarantee the convergence to the optimal policy. In this paper, in order to overcome the limitation of existing algorithms, we propose a new algorithm called FLUTE, which enjoys uniform-PAC convergence to the optimal policy with high probability. The uniform-PAC guarantee is the strongest possible guarantee for reinforcement learning in the literature, which can directly imply both PAC and high probability regret bounds, making our algorithm superior to all existing algorithms with linear function approximation. At the core of our algorithm is a novel minimax value function estimator and a multi-level partition scheme to select the training samples from historical observations. Both of these techniques are new and of independent interest.
We study the reinforcement learning for finite-horizon episodic Markov decision processes with adversarial reward and full information feedback, where the unknown transition probability function is a linear function of a given feature mapping. We propose an optimistic policy optimization algorithm with Bernstein bonus and show that it can achieve $\tilde{O}(dH\sqrt{T})$ regret, where $H$ is the length of the episode, $T$ is the number of interaction with the MDP and $d$ is the dimension of the feature mapping. Furthermore, we also prove a matching lower bound of $\tilde{\Omega}(dH\sqrt{T})$ up to logarithmic factors. To the best of our knowledge, this is the first computationally efficient, nearly minimax optimal algorithm for adversarial Markov decision processes with linear function approximation.
Reinforcement learning (RL) with linear function approximation has received increasing attention recently. However, existing work has focused on obtaining $\sqrt{T}$-type regret bound, where $T$ is the number of steps. In this paper, we show that logarithmic regret is attainable under two recently proposed linear MDP assumptions provided that there exists a positive sub-optimality gap for the optimal action-value function. In specific, under the linear MDP assumption (Jin et al. 2019), the LSVI-UCB algorithm can achieve $\tilde{O}(d^{3}H^5/\text{gap}_{\text{min}}\cdot \log(T))$ regret; and under the linear mixture model assumption (Ayoub et al. 2020), the UCRL-VTR algorithm can achieve $\tilde{O}(d^{2}H^5/\text{gap}_{\text{min}}\cdot \log^3(T))$ regret, where $d$ is the dimension of feature mapping, $H$ is the length of episode, and $\text{gap}_{\text{min}}$ is the minimum of sub-optimality gap. To the best of our knowledge, these are the first logarithmic regret bounds for RL with linear function approximation.