Improved Regret Bounds for Linear Adversarial MDPs via Linear Optimization

Reinforcement learning (RL) describes the interaction between a learning agent and an unknown environment, where the agent aims to maximize the cumulative reward through trial and error Sutton and Barto (2018). It has achieved great success in many real applications, such as games (Mnih et al., 2013; Silver et al., 2016), robotics (Kober et al., 2013; Lillicrap et al., 2015), autonomous driving (Kiran et al., 2021) and recommendation systems (Afsar et al., 2022; Lin et al., 2021). The interaction in RL is commonly portrayed by Markov decision processes (MDP). Most of the works study the stochastic setting, where the reward is sampled from a fixed distribution (Azar et al., 2017; Jin et al., 2018; Simchowitz and Jamieson, 2019; Yang et al., 2021). RL in real applications is in general more challenging than the stochastic setting, as the environment could be non-stationary and the reward function could be adaptive towards the agent’s policy. For example, a scheduling algorithm will be deployed to self-interested parties, and recommendation algorithms will face strategic users.

To design robust algorithms that work under non-stationary environments, a line of works focuses on the adversarial setting, where the reward function could be arbitrarily chosen by an adversary (Yu et al., 2009; Rosenberg and Mansour, 2019; Jin et al., 2020a; Chen et al., 2021; Luo et al., 2021a). Many works in adversarial MDPs optimize the policy by learning the value function with a tabular representation. In this case, both their computation complexity and their regret bounds depend on the state space and action space sizes. In real applications however, the state and action spaces could be exponentially large or even infinite, such as in the game of Go and in robotics. Such cost of computation and the performance are then inadequate.

To cope with the curse of dimensionality, function approximation methods are widely deployed to approximate the value functions with learnable structures. Great empirical success has proved its efficacy in a wide range of areas. Despite this, theoretical understandings of MDP with general function approximation are yet to be available. As an essential step towards understanding function approximation, linear MDP has been an important setting and has received significant attention from the community. It presumes that the transition and reward functions in MDP follow a linear structure with respect to a known feature (Jin et al., 2020b; He et al., 2021; Hu et al., 2022). The stochastic setting in linear MDP has been well studied and near-optimal results are available (Jin et al., 2020b; Hu et al., 2022). The adversarial setting in linear MDP is much more challenging since the underlying linear parameters of the loss function and transition kernel are especially hard to estimate in a varying environment.

The research on linear adversarial MDPs remains open. Early works have proposed algorithms when the transition function is known. (Neu and Olkhovskaya, 2021) Several recent works explore the problem without a known transition function and derive policy optimization algorithms with the state-of-the-art regret of $\tilde{\mathcal{O}}\left(K^{6/7}\right)$ (Luo et al., 2021a; Dai et al., 2023; Sherman et al., 2023). While the optimal regret in tabular MDPs is of order $\tilde{\mathcal{O}}\left({K}^{1/2}\right)$ (Jin et al., 2020a), the regret upper bounds available for linear adversarial MDPs seem to admit a large room for improvement.

In this paper, we investigate linear adversarial MDPs with unknown transition. We propose a new view of the problem and design an algorithm based on our view. The idea is to reduce the MDP setting to a linear optimization problem by subtly setting the feature maps of the bandit arms of linear optimization. In this way, we operate on a set of policies and optimize the probability distribution of which policy to execute. By carefully balancing the suboptimality in policy execution, the suboptimality in policy construction, and the suboptimality in feature visitation estimation, we deduce new analyses of the problem. Improved regret bounds are obtained both when we have and do not have a simulator. In particular, we conclude the first $\tilde{\mathcal{O}}\left(K^{4/5}\right)$ regret bound for linear adversarial MDPs without a simulator.

Let $d$ be the feature dimension and $H$ be the length of each episode. Details of our contributions are as follows.

• •

  With an exploratory assumption (Assumption 1), we obtain an $\tilde{\mathcal{O}}(d^{7/5}H^{12/5}K^{4/5})$ regret upper bound for linear adversarial MDP. As compared in Table 1, this is the first regret bound that achieves $\tilde{\mathcal{O}}(K^{4/5})$ order when a simulator of the transition is not provided. We also want to note that our exploratory assumption that only ensures the MDP is learnable is much weaker than previous works (Neu and Olkhovskaya, 2021; Luo et al., 2021a). Under a weaker exploratory assumption, our result achieves a significant improvement over the $\tilde{\mathcal{O}}(K^{6/7})$ regret in Luo et al. (2021a) and also removes the dependence on $\lambda$ which is the minimum eigenvalue of the exploratory policy’s covariance that can be small.

• •

  In a simpler setting where the agent has access to a simulator, our regret can be further improved to $\tilde{\mathcal{O}}(\sqrt{d^{2}H^{5}K})$. This result also removes the dependence on $\lambda$ in previous works (Neu and Olkhovskaya, 2021; Luo et al., 2021a). Compared with Luo et al. (2021a), our required simulator is also weaker: we only need to have access to the trajectory when given any policy $\pi$; while Luo et al. (2021a) requires the next state $s^{\prime}$ when given any state-action pair $(s,a)$.

• •

  Technically, we provide a new tool for linear MDP problems by exploiting the linear features of the MDP and transforming it into a linear optimization problem. This tool could be of independent interest and might be useful in other problems that possess a linear structure.

In this work, we study the episodic adversarial Markov Decision Processes (MDP) denoted by
$\mathcal{M}(\mathcal{S},\mathcal{A},H,\left\{P_{h}\right\}_{h=1}^{H},\left\{\ell_{k}\right\}_{k=1}^{K})$ where $\mathcal{S}$ is the state space, $\mathcal{A}$ is the action space, $H$ is the horizon of each episode, $P_{h}:\mathcal{S}\times\mathcal{A}\times\mathcal{S}\to[0,1]$ is the transition kernel of step $h$ with $P_{h}(s^{\prime}\mid s,a)$ representing the transition probability from $s$ to $s^{\prime}$ by taking action $a$ at step $h$, and $\ell_{k}$ is the loss function at episode $k$. We denote $\pi_{k}:\left\{\pi_{k,h}\right\}_{h=1}^{H}$ as the learner’s policy at each episode $k$, where $\pi_{k,h}$ is a mapping from each state to a distribution over the action space. Let $\pi_{k,h}(a\mid s)$ represent the selecting probability of action $a$ at state $s$ by following policy $\pi_{k}$ at step $h$.

The learner interacts with the MDP $\mathcal{M}$ for $K$ episodes. At each episode $k=1,2,\ldots,K$, the environment (adversary) first chooses the loss function $\ell_{k}:=\left\{\ell_{k,h}\right\}_{h=1}^{H}$ which may be probably based on the history information before episode $k$. The learner simultaneously decides its policy $\pi_{k}$. For each step $h=1,2,\ldots,H$, the learner observes the current state $s_{k,h}$, taking action $a_{k,h}$ based on $\pi_{k,h}$, and observe the loss $\ell_{k,h}(s_{k,h},a_{k,h})$. The environment will transit to the next state $s_{k,h+1}$ at the end of the step based on the transition kernel $P_{h}(\cdot\mid s_{k,h},a_{k,h})$.

The performance of a policy $\pi$ over episode $k$ can be evaluated by its value function, which is the expected cumulative loss,

where the expectation is taken from the randomness in the transition and the policy $\pi$. Denote $\pi^{*}\in\operatorname*{argmin}_{\pi}\sum_{k=1}^{K}V^{\pi}_{k}$ as the optimal policy that suffers the least expected loss over $K$ episodes. The objective of the learner is to minimize the cumulative regret,

which is defined as the cumulative difference between the value of the taken policies and that of the optimal policy $\pi^{*}$.

Linear adversarial MDP denotes an MDP where both the transition kernel and the loss function are linearly depending on a feature mapping. We give a formal definition as follows.

Given a policy $\pi$, its feature visitation vector at step $h$ is the expected feature mapping this policy encounters at step $h$: $\phi_{\pi,h}=\mathbb{E}_{\pi}\left[\phi\left(s_{h},a_{h}\right)\right]$. With this definition, the expected loss that the policy $\pi$ receives at step $h$ of episode $k$ can be written as

and the value of policy $\pi$ can be expressed as

For simplicity, we also define $\phi_{\pi,h}\left(s\right)=\mathbb{E}_{a\sim\pi_{h}\left(\cdot|s\right)}\left[\phi(s,a)\right]$ to represent the expected feature visitation of state $s$ at step $h$ by following $\pi$.

To ensure that the linear MDP is learnable, we make the following exploratory assumption, which analogs to those assumptions made in previous works that study the function approximation setting (Neu and Olkhovskaya, 2021; Luo et al., 2021a, b; Hao et al., 2021; Agarwal et al., 2021). For any policy $\pi$, define $\mathbf{\Lambda}_{\pi,h}:=\mathbb{E}_{\pi}\left[\phi(s_{h},a_{h})\phi(s_{h},a_{h})^{\top}\right]$ as the expected covariance of $\pi$ at step $h$. Let $\lambda_{\min,h}^{*}=\sup_{\pi}\lambda_{\min}\left(\mathbf{\Lambda_{\pi,h}}\right)$, where $\lambda_{\min}(\cdot)$ denotes the smallest eigenvalue of a matrix, and $\lambda_{\min}^{*}=\min_{h}\lambda_{\min,h}^{*}$. To ensure that the linear MDP is learnable, we assume that there exists a policy that generates full rank covariance matrices.

When the assumption is reduced to the tabular setting, where $\phi(s,a)$ is a basis vector in $\mathbb{R}^{\mathcal{S}\times\mathcal{A}}$, this assumption becomes $\mu_{\min}:=\max_{\pi}\min_{s,a}\mu^{\pi}(s,a)>0$, where $\mu^{\pi}(s,a)$ is the probability of visiting the state-action pair $(s,a)$ under the trajectory induced by $\pi$. It simply means that there exists a policy with positive visitation probability for all state-action pairs, which is standard (Li et al., 2020). In the linear setting, it guarantees that all the directions in $\mathbb{R}^{d}$ is able to be visited by some policy.

We point out that this assumption is weaker than the exploratory assumptions used in previous works (Neu and Olkhovskaya, 2021; Luo et al., 2021a, b), in which they assume such exploratory policy $\pi_{0}$, satisfying $\lambda_{\min}\left(\mathbf{\Lambda}_{\pi_{0},h}\right)\geq\lambda_{0}>0$ for all $h\in[H]$, is given as input to the algorithm. Since finding such an exploratory policy is extremely difficult, our assumption, which only requires the transition of the MDP itself to satisfy this constraint, is more preferred.

In this section, we introduce the proposed algorithm (Algorithm 1). The algorithm takes a finite policy class $\Pi$ and the feature visitation estimators $\left\{\hat{\phi}_{\pi,h}:\pi\in\Pi,\,h\in[H]\right\}$ as input and selects a policy $\pi_{k}\in\Pi$ in each episode $k\in[K]$. The acquisition of $\Pi$ and $\left\{\hat{\phi}_{\pi,h}:\pi\in\Pi,\,h\in[H]\right\}$ will be introduced in Section 4.1 and Section 4.2, respectively.

Recall that the loss value $\ell_{k,h}(s,a)$ is an inner product between the feature $\phi(s,a)$ and the loss vector $\theta_{k,h}$. According to this structure, we investigate ridge linear regression to estimate the unknown loss vector. To be specific, in each episode after executing policy $\pi_{k}$, the observed loss value can be used to estimate the loss vector $\hat{\theta}_{k,h}$ and the value function $\hat{V}_{k}^{\pi}$ can be estimated for each policy $\pi$ (line 9). We then adopt an optimistic strategy toward the value of policies and an optimistic estimation of the policy $\pi$’s value (line 10). Based on the optimistic value, the exploitation probability $w(\pi)$ of a policy $\pi$ follows the EXP3-type update rule (line 11). To better explore each dimension of the linear space, the final selecting probability is defined as the weighted combination of the exploitation probability and an exploration probability $g(\pi)$, where the weight $\gamma$ is an input parameter (line 7). Here the exploration probability $g_{h}:=\left\{g_{h}(\pi)\right\}_{\pi\in\Pi}$ is derived by computing the G-optimal design problem to minimize the uncertainty of all policies, i.e.,

where $\mathbf{V}_{h}(p)=\sum_{\pi\in\Pi}p(\pi)\phi_{\pi,h}\phi_{\pi,h}^{\top}$.

If the input $\hat{\phi}_{\pi,h}$ is the true feature visitation ${\phi}_{\pi,h}$, we can ensure that the regret of the algorithm, compared with the optimal policy in $\Pi$, is upper bounded. Now it suffices to bound the additional regret caused by the sub-optimality of the best policy in $\Pi$, and the bias of the feature visitation estimators, which will be discussed in the following sections.

This section provides the regret guarantees for the proposed algorithm as well as the proof sketch.

We consider Algorithm 1 with the policy set constructed in Section 4.1 and the feature of policies estimated in Section 4.2 as input. Suppose we run Algorithm 1 for $K$ rounds, the regret compared with any fixed policy $\pi\in\Pi$ in these $K$ rounds can be bounded as below.

Recall that when the policy set is constructed as section 4.1, the difference between the global regret defined in Equation (1) and the regret compared with the policy in $\Pi$ is just a constant. So the global regret can also be bounded as above Lemma 3.

Similar to previous works on linear adversarial MDP (Luo et al., 2021a, b; Sherman et al., 2023; Dai et al., 2023) that discuss the cases of whether a transition simulator is available, we define the simulator that may help in the following assumption. Note that this simulator is weaker than Luo et al. (2021a, b); Dai et al. (2023) because their simulator could generate a next state given any state-action pair.

If the learning agent has access to such a simulator described in Assumption 2, then the feature estimation process in Section 4.2 can be regarded as regret-free and the final regret is just as shown in Lemma 3. Otherwise, there is an additional $\tilde{\mathcal{O}}(d^{4}H^{3}/\epsilon^{2})$ regret term. Balancing the choice of $\gamma$ and $\epsilon$ yields the following regret upper bound.

The proof of the main results is deferred to Appendix A.

In this paper, we investigate linear adversarial MDP with bandit feedback. We propose a new view of linear MDP, where optimizing policies in linear MDP can be regarded as a linear optimization problem. Based on this new insight we propose an algorithm by constructing a set of policies and deploying a probability distribution over the policies to execute. With an exploratory assumption, our algorithm yields the first $\tilde{\mathcal{O}}(K^{4/5})$ regret without access to a simulator. Compared to the results in Luo et al. (2021a), our algorithm enjoys a weaker assumption, a better regret bound with respect to both $K$ and $\lambda$, and a weaker simulator with fewer queries if it uses one.

Our view contributes a new approach to linear MDP, which could be of independent interest. We demonstrated how our algorithm under this view is generalized to infinite action spaces. Future implications of this technique could involve solving other adversarial settings, such as when the loss function is corrupted up to a budget, and solving robust linear MDP where the transition kernel could change over episodes.