#### Authors

Eyal Even-dar, Sham M. Kakade, Yishay Mansour

#### Abstract

We consider an MDP setting in which the reward function is allowed to change during each time step of play (possibly in an adversarial manner), yet the dynamics remain fixed. Similar to the experts setting, we address the question of how well can an agent do when compared to the reward achieved under the best stationary policy over time. We provide efficient algorithms, which have regret bounds with no dependence on the size of state space. Instead, these bounds depend only on a certain horizon time of the process and logarithmically on the number of actions. We also show that in the case that the dynamics change over time, the problem becomes computationally hard.

1 Introduction

There is an inherent tension between the objectives in an expert setting and those in a re- inforcement learning setting. In the experts problem, during every round a learner chooses one of n decision making experts and incurs the loss of the chosen expert. The setting is typically an adversarial one, where Nature provides the examples to a learner. The stan- dard objective here is a myopic, backwards looking one -- in retrospect, we desire that our performance is not much worse than had we chosen any single expert on the sequence of examples provided by Nature. In contrast, a reinforcement learning setting typically makes the much stronger assumption of a fixed environment, typically a Markov decision pro- cess (MDP), and the forward looking objective is to maximize some measure of the future reward with respect to this fixed environment.

The motivation of this work is to understand how to efficiently incorporate the benefits of existing experts algorithms into a more adversarial reinforcement learning setting, where certain aspects of the environment could change over time. A naive way to implement an experts algorithm is to simply associate an expert with each fixed policy. The running time of such algorithms is polynomial in the number of experts and the regret (the difference from the optimal reward) is logarithmic in the number of experts. For our setting the num- ber of policies is huge, namely #actions#states, which renders the naive experts approach computationally infeasible.

Furthermore, straightforward applications of standard regret algorithms produce regret bounds which are logarithmic in the number of policies, so they have linear dependence

   This work was supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778, by a grant from the Israel Science Foundation and an IBM faculty award. This publication only reflects the authors' views.


on the number of states. We might hope for a more effective regret bound which has no dependence on the size of state space (which is typically large).

The setting we consider is one in which the dynamics of the environment are known to the learner, but the reward function can change over time. We assume that after each time step the learner has complete knowledge of the previous reward functions (over the entire environment), but does not know the future reward functions.

As a motivating example one can consider taking a long road-trip over some period of time T . The dynamics, namely the roads, are fixed, but the road conditions may change frequently. By listening to the radio, one can get (effectively) instant updates of the road and traffic conditions. Here, the task is to minimize the cost during the period of time T . Note that at each time step we select one road segment, suffer a certain delay, and need to plan ahead with respect to our current position.

This example is similar to an adversarial shortest path problem considered in Kalai and Vempala [2003]. In fact Kalai and Vempala [2003], address the computational difficulty of handling a large number of experts under certain linear assumptions on the reward func- tions. However, their algorithm is not directly applicable to our setting, due to the fact that in our setting, decisions must be made with respect to the current state of the agent (and the reward could be changing frequently), while in their setting the decisions are only made with respect to a single state.

McMahan et al. [2003] also considered a similar setting -- they also assume that the reward function is chosen by an adversary and that the dynamics are fixed. However, they assume that the cost functions come from a finite set (but are not observable) and the goal is to find a min-max solution for the related stochastic game.

In this work, we provide efficient ways to incorporate existing best experts algorithms into the MDP setting. Furthermore, our loss bounds (compared to the best constant policy) have no dependence on the number of states and depend only on on a certain horizon time of the environment and log(#actions). There are two sensible extensions of our setting. The first is where we allow Nature to change the dynamics of the environment over time. Here, we show that it becomes NP-Hard to develop a low regret algorithm even for oblivious adversary. The second extension is to consider one in which the agent only observes the rewards for the states it actually visits (a generalization of the multi-arm bandits problem). We leave this interesting direction for future work.