One thing which is clear on a little reflection is that there exists a single master learning problem capable of encoding essentially all learning problems. This problem is of course a very general sort of reinforcement learning where the world interacts with an agent as:

- The world announces an observation
*x*. - The agent makes a choice
*a*. - The world announces a reward
*r*.

The goal here is to maximize the sum of the rewards over the time of the agent. No particular structure relating *x* to *a* or *a* to *r* is implied by this setting so we do not know effective general algorithms for the agent. It’s very easy to prove lower bounds showing that an agent cannot hope to succeed here—just consider the case where actions are unrelated to rewards. Nevertheless, there is a real sense in which essentially all forms of life are agents operating in this setting, somehow succeeding. The gap between these observations drives research—How can we find tractable specializations of the master problem general enough to provide an effective solution in real problems?

The process of specializing is a tricky business, as you want to simultaneously achieve tractable analysis, sufficient generality to be useful, and yet capture a new aspect of the master problem not otherwise addressed. Consider: How is it even possible to choose a setting where analysis is tractable before you even try to analyze it? What follows is my mental map of different specializations.

### Online Learning

The online learning setting is perhaps the most satisfying specialization more general than standard batch learning at present, because it turns out to additionally provide tractable algorithms for many batch learning settings.

Standard online learning models specialize in two ways: You assume that the choice of action in step 2 does not influence future observations and rewards, and you assume additional information is available in step 3, a retrospectively available reward for each action. The algorithm for an agent in this setting typically has a given name—gradient descent, weighted majority, Winnow, etc…

The general algorithm here is a more refined version of follow-the-leader than in batch learning, with online update rules. An awesome discovery about this setting is that it’s possible to compete with a set of predictors even when the world is totally adversarial, substantially strengthening our understanding of what learning is and where it might be useful. For this adversarial setting, the algorithm alters into a form of follow-the-perturbed leader, where the learning algorithm randomizes it’s action amongst the set of plausible alternatives in order to defeat an adversary.

The standard form of argument in this setting is a potential argument, where at each step you show that if the learning algorithm performs badly, there is some finite budget from which an adversary deducts it’s ability. The form of the final theorem is that you compete with the accumulated reward of a set any one-step policies *h:X – > A*, with a dependence log(#policies) or weaker in regret, a measure of failure to compete.

A good basic paper to read here is:

Nick Littlestone and Manfred Warmuth, The Weighted Majority Algorithm, which shows the basic information-theoretic claim clearly. Vovk‘s page on aggregating algorithms is also relevant, although somewhat harder to read.

Provably computationally tractable special cases all have linear structure, either on rewards or policies. Good results are often observed empirically by applying backpropagation for nonlinear architectures, with the danger of local minima understood.

### Bandit Analysis

In the bandit setting, step 1 is omitted, and the difficulty of the problem is weakened by assuming that action in step (2) don’t alter future rewards. The goal is generally to compete with all constant arm strategies.

Analysis in this basic setting started very specialized with Gittin’s Indicies and gradually generalized over time to include IID and fully adversarial settings, with EXP3 a canonical algorithm. If there are *k* strategies available, the standard theorem states that you can compete with the set of all constant strategies up to regret *k*. The most impressive theoretical discovery in this setting is that the dependence on *T*, the number of timesteps, is not substantially worse than supervised learning despite the need to explore.

Given the dependence on *k* all of these algorithms are computationally tractable.

However, the setting is flawed, because the set of constant strategies is inevitably too weak in practice—it’s an example of optimal decision making given that you ignore almost all information. Adding back the observation in step 1 allows competing with a large set of policies, while the regret grows only as log(#policies) or weaker. Canonical algorithms here are EXP4 (computationally intractable, but information theoretically near-optimal), Epoch-Greedy (computationally tractable given an oracle optimizer), and the Offset Tree providing a reduction to supervised binary classification.

### MDP analysis

A substantial fraction of reinforcement learning has specialized on the Markov Decision Process setting, where the observation *x* is a state *s*, which is a sufficient statistic for predicting all future observations. Compared to the previous settings, dealing with time dependence is explicitly required, but learning typically exists in only primitive forms.

The first work here was in the 1950’s where the actual MDP was assumed known and the problem was simply computing a good policy, typically via dynamic programming style solutions. More recently, principally in the 1990’s, the setting where the MDP was not assumed known was analyzed. A very substantial theoretical advancement was the *E ^{3}* algorithm which requires only

*O(S*experience to learn a near-optimal policy where the world is an MDP with

^{2}A)*S*state and

*A*actions per state. A further improvement on this is Delayed Q-Learning, where only

*O(SA)*experience is required. There are many variants on the model-based approach and not much for the model-free approach. Lihong Li‘s thesis probably has the best detailed discussion at present.

There are some unsatisfactory elements of the analysis here. First, I’ve suppressed the dependence on the definition of “approximate” and the typical time horizon, for which the dependence is often bad and the optimality is unclear. The second is the dependence on *S*, which is intuitively unremovable, with this observation formalized in the lower bound Sham and I worked on (section 8.6 of Sham’s thesis). Empirically, these and related algorithms are often finicky, because in practice the observation isn’t a sufficient statistic and the number of states isn’t small, so approximating things as such is often troublesome.

A very different variant of this setting is given by Control theory, which I know less about than I should. The canonical setting for control theory is with a known MDP having linear transition dynamics. More exciting are the system identification problems where the system must be first identified. I don’t know any good relatively assumption free results for this setting.

### Oracle Advice Shortcuts

Techniques here specialize the setting to situations in which some form of oracle advice is available when a policy is being learned. A good example of this is an oracle which provides samples from the distribution of observations visited by a good policy. Using this oracle, conservative policy iteration is guaranteed to perform well, so long as a base learning algorithm can predict well. This algorithm was refined and improved a bit by PSDP, which works via dynamic programming, improving guarantees to work with regret rather than errors.

An alternative form of oracle is provide by access to a good policy at training time. In this setting, Searn has similar provable guarantees with a similar analysis.

The oracle based algorithms appear to work well anywhere these oracles are available.

### Uncontrolled Delay

In the uncontrolled delay setting, step (2) is removed, and typically steps (1) and (3) are collapsed into one observation, where the goal becomes state tracking. Most of the algorithms for state tracking are heavily model dependent, implying good success within particular domains. Examples include Kalman filters, hidden markov models, and particle filters which typical operate according to an explicit probabilistic model of world dynamics.

Relatively little is known for a nonparametric version of this problem. One observation is that the process of predicting adjacent observations well forms states as a byproduct when the observations are sufficiently rich as detailed here.

A basic question is: What’s missing from the above? A good answer is worth a career.