Much research remains to be done within this space of reinforcement learning methods. For example, even for the tabular case no control method using multistep backups has been proved to converge to an optimal policy. Among planning methods, basic ideas such as trajectory sampling and focusing sample backups are almost completely unexplored. On closer inspection, parts of the space will undoubtedly turn out to have far greater complexity and greater internal structure than is now apparent. There are also other dimensions along which reinforcement learning can be extended, we have not yet mentioned, that lead to a much larger space of methods. Here we identify some of these dimensions and note some of the open questions and frontiers that have been left out of the preceding chapters.

One of the most important extensions of reinforcement learning beyond what we have treated in this book is to eliminate the requirement that the state representation have the Markov property. There are a number of interesting approaches to the non-Markov case. Most strive to construct from the given state signal and its past values a new signal that is Markov, or more nearly Markov. For example, one approach is based on the theory of partially observable MDPs (POMDPs). POMDPs are finite MDPs in which the state is not observable, but another "sensation" signal stochastically related to the state is observable. The theory of POMDPs has been extensively studied for the case of complete knowledge of the dynamics of the POMDP. In this case, Bayesian methods can be used to compute at each time step the probability of the environment's being in each state of the underlying MDP. This probability distribution can then be used as a new state signal for the original problem. The downside for the Bayesian POMDP approach is its computational expense and its strong reliance on complete environment models. Some of the recent work pursuing this approach is by Littman, Cassandra, and Kaelbling (1995), Parr and Russell (1995), and Chrisman (1992). If we are not willing to assume a complete model of a POMDP's dynamics, then existing theory seems to offer little guidance. Nevertheless, one can still attempt to construct a Markov state signal from the sequence of sensations. Various statistical and ad hoc methods along these lines have been explored (e.g., McCallum, 1992, 1993, 1995; Lin and Mitchell, 1992; Chapman and Kaelbling, 1991; Moore, 1994; Rivest and Schapire, 1987; Colombetti and Dorigo, 1994; Whitehead and Ballard, 1991; Hochreiter and Schmidhuber, 1997).

All of the above methods involve constructing an improved state representation from the non-Markov one provided by the environment. Another approach is to leave the state representation unchanged and use methods that are not too adversely affected by its being non-Markov (e.g., Singh, Jaakkola, and Jordan, 1994, 1995; Jaakkola, Singh and Jordan, 1995). In fact, most function approximation methods can be viewed in this way. For example, state aggregation methods for function approximation are in effect equivalent to a non-Markov representation in which all members of a set of states are mapped into a common sensation. There are other parallels between the issues of function approximation and non-Markov representations. In both cases the overall problem divides into two parts: constructing an improved representation, and making do with the current representation. In both cases the "making do" part is relatively well understood, whereas the constructive part is unclear and wide open. At this point we can only guess as to whether or not these parallels point to any common solution methods for the two problems.

Another important direction for extending reinforcement learning beyond what we have covered in this book is to incorporate ideas of modularity and hierarchy. Introductory reinforcement learning is about learning value functions and one-step models of the dynamics of the environment. But much of what people learn does not seem to fall exactly into either of these categories. For example, consider what we know about tying our shoes, making a phone call, or traveling to London. Having learned how to do such things, we are then able to choose among them and plan as if they were primitive actions. What we have learned in order to do this are not conventional value functions or one-step models. We are able to plan and learn at a variety of levels and flexibly interrelate them. Much of our learning appears not to be about learning values directly, but about preparing us to quickly estimate values later in response to new situations or new information. Considerable reinforcement learning research has been directed at capturing such abilities (e.g., Watkins, 1989; Dayan and Hinton, 1993; Singh, 1992a, 1992b; Ring, 1994, Kaelbling, 1993b; Sutton, 1995).

Researchers have also explored ways of using the structure of particular tasks to advantage. For example, many problems have state representations that are naturally lists of variables, like the readings of multiple sensors or actions that are lists of component actions. The independence or near independence of some variables from others can sometimes be exploited to obtain more efficient special forms of reinforcement learning algorithms. It is sometimes even possible to decompose a problem into several independent subproblems that can be solved by separate learning agents. A reinforcement learning problem can usually be structured in many different ways, some reflecting natural aspects of the problem, such as the existence of physical sensors, and others being the result of explicit attempts to decompose the problem into simpler subproblems. Possibilities for exploiting structure in reinforcement learning and related planning problems have been studied by many researchers (e.g., Boutilier, Dearden, and Goldszmidt, 1995; Dean and Lin, 1995). There are also related studies of multiagent or distributed reinforcement learning (e.g., Littman, 1994; Markey, 1994; Crites and Barto, 1996; Tan, 1993).

Finally, we want to emphasize that reinforcement learning is meant to be a
*general* approach to learning from interaction. It is general
enough not to require special-purpose teachers and domain knowledge, but also
general enough to utilize such things if they are available. For example, it
is often possible to accelerate reinforcement learning by giving advice or
hints to the agent (Clouse and Utgoff, 1992; Maclin and Shavlik, 1994) or by
demonstrating instructive behavioral trajectories
(Lin, 1992). Another way to make learning easier, related to "shaping" in
psychology, is to give the learning
agent a series of relatively easy problems building up to the harder
problem of ultimate interest (e.g., Selfridge, Sutton, and Barto, 1985).
These methods, and others not yet developed, have the potential to give the
machine-learning terms *training* and *teaching* new meanings that are
closer to their meanings for animal and human learning.