is the first in a series of essays that aim to derive a regret bound for DRL that depends on finer attributes of the prior than just the number of hypotheses. Specifically, we consider the entropy of the prior and a certain learning-theoretic dimension parameter. As a "by product", we derive a new regret bound for ordinary RL without resets and without traps. In this chapter, we open the series by demonstrating the latter, under the significant simplifying assumption that the MDPs are deterministic.
The regret bound we previously derived for DRL grows as a power law with the number of hypotheses. In contrast, the RL literature is usually concerned with considering all transition kernels on a fixed state space satisfying some simple assumption (e.g. a bound on the diameter or bias span). In particular, the number of hypotheses is uncountable. While the former seems too restrictive (although it's relatively simple to generalize it to countable hypothesis classes), the latter seems too coarse. Indeed, we expect a universally intelligent agent to detect patterns in the data, i.e. follow some kind of simplicity prior rather than a uniform distribution over transition kernels.
The underlying technique of our proof was lower bounding the information gain in a single episode of posterior sampling by the expected regret incurred during this episode. Although we have not previously stated it in this form, the resulting regret bound depends on the entropy of the prior (we considered a uniform prior instead). This idea (unbeknownst to us) appeared earlier in Russo and Van Roy. Moreover, later Russo and Van Roy used to it formulate a generalization of posterior sampling they call "information-directed sampling" the can produce far better regret bounds in certain scenarios. However, to the best of our knowledge, this technique was not previously used to analyze reinforcement learning (as opposed to bandits). Therefore, it seems natural to derive such an "entropic" regret bound for ordinary RL, before extending it to DRL.
Now, Osband and Van Roy derived a regret bound for priors supported on some space of transition kernels as a function of its Kolmogorov dimension and "eluder" dimension (the latter introduced previously by Russo and Van Roy). They also consider a continuous state space. This is a finer approach than considering nearly arbitrary transition kernels on a fixed state space, but it still doesn't distinguish between different priors with the same support. Our new results involve a parameter similar to eluder dimension, but instead of Kolmogorov dimension we use entropy (in the following chapters we will see that Kolmogorov dimension is, in some sense, an upper bound on entropy). As opposed to Osband and Van Roy, we currently limit ourselves to finite state spaces, but on the other hand we consider no resets (at the price of a "no traps" assumption).
In this chapter we derive the entropic regret bound for deterministic MDPs. (In the following we will call them deterministic decision processes (DDPs) since they have little to do with Markov.) This latter restriction significantly simplifies the analysis. In the following chapters, we will extend it to stochastic MDPs, however the resulting regret bound will be somewhat weaker.
We start by introducing a new learning-theoretic concept of dimension. It is similar to eluder dimension, but is adapted to the discrete deterministic setting and also somewhat stronger (i.e. smaller: more environment classes are low dimensional w.r.t. this concept than w.r.t. eluder dimension).
Consider sets , and non-empty. Given and , we say that is -dependent of when for any s.t. for any it holds , we have . Otherwise, we say that is -independent of .
The prediction dimension of (denoted ) is the supremum of the set of for which there is a sequence s.t. for every , is -independent of .
Fix non-empty finite sets (the set of states) and (the set of actions). Denote , where the second factor is regarded as the space of reward values. Given , and s.t. , we regard as the (deterministic) transition kernel and as the reward function associated with "DDP hypothesis" . This allows us to speak of the dimension of a DDP hypothesis class (i.e. some ). We now give some examples for dimensions of particular function/hypothesis classes.
Given any , and , we have .
Given any , and , we have . In particular, given as above, .
We now consider deterministic Markov decision processes that are cellular automata. Consider finite sets (the set of cells), (the set of neighbor types) and a mapping (which cell is the neighbor of which cell). For example, might be a subset of a group acting on . More specifically, can be acting on itself, corresponding to a -dimensional toroidal cellular automaton of size .
Consider another set (the set of cell states) and suppose that . Given any , and , define by . Given any , define by . Given any , define by . Define by
That is, is the set of transition kernels and reward functions that are local in the sense defined by . Then, .
In Proposition 3, it might seem like, although the rules of the automaton are local, the influence of the agent is necessarily global, because the dependence on the action appears in all cells. However, this is not really a restriction: the state of the cells can encode a particular location for the agent and the rules might be such that the agent's influence is local around this location. More unrealistic is the full observability. Dealing with partially observable cellular automata is outside the scope of this essay.
The central lemma in the proof of the regret bound for RL is a regret bound in its own right, in the setting of (deterministic) contextual bandits. Since this lemma might be of independent interest, we state it already here.
Let (contexts), (arms) and (outcomes) be non-empty sets. Fix a function (the reward function). For any (a fixed sequence of contexts), (outcome rule) and (policy), we define to be the resulting distribution over outcome histories. Given , we define (the utility function) by
Consider a countable non-empty set of hypotheses and some (the prior). For each , define by
Let and suppose that is countable [this assumption is to simplify the proof and is not really necessary]. Then, there exists s.t. for any and
Note that the expression on the left hand side is the Bayesian regret. On the right hand side, stands for the Shannon entropy of . In particular we have
Also, it's not hard to see that and therefore
Finally, the policy we actually consider in the proof is Thompson sampling.
Now we proceed to studying reinforcement learning. First, we state a regret bound for RL with resets. Now stands for the set of states and for the set of actions. We fix a sequence of initial states . For any (environment), (policy), and we define to be the resulting distribution over histories, assuming that the state is reset to and the reward to every time period of length . In particular, we have
Given we define
Consider a countable non-empty set of hypotheses and some . Let . Then, for any and , there exists s.t. for any
Note that is actually a probability measure concentrated on a single history, sincee is deterministic: we didn't make it explicit only to avoid introducing new notation.
Finally, we give the regret bound without resets. For any and , we define to be the resulting distribution over histories, given initial state and no resets.
Consider a countable non-empty set of hypotheses and some . Let . Assume that for any and , [ was defined here in "Definition 1"; so was the value function used below] (i.e. there are no traps). For any we define by
Then, for any s.t. there exists s.t.
Note that decreases with so this factor doesn't make the qualitative dependence on any worse.
Both Theorem 1 and Theorem 2 have anytime variants in which the policy doesn't depend on at the price of a slightly (within a constant factor) worse regret bound, but for the sake of brevity we don't state them (our ultimate aim is DRL which is not anytime anyway). In Theorem 2 we didn't specify the constant, so it is actually true verbatim without the dependence in (but we still leave the dependence to simplify the proof a little). It is also possible to spell out the assumption on in Theorem 2.
Consider sets , and non-empty. A sequence is said to be -independent, when for every , is -independent of .
Consider sets , . Given and , suppose are s.t.:
- For any ,
Then, are said to shatter .
Given sequences and , is said to shatter when for any , shatters .
Proof of Proposition 1
Consider an -independent sequence. We will now construct s.t. , which is sufficient.
By the definition of -independence, there is a sequence that shatters . We have and therefore either or . Without loss of generality, assume that for each , . It follows that for any and , and therefore .
If then there is nothing to prove since is non-empty, hence we can assume . For any , and therefore . Also, and therefore . We now take , completing the proof.
Proof of Proposition 2
Consider an -independent sequence and that shatters it. For any and , we have but , implying that . It follows that .
Proof of Proposition 3
Consider an -independent sequence and that shatters it. Define by
Since shatters , we have .
Consider any . Obviously, there is s.t.
Denote . We have . On the other hand, for any , the shattering implies and in particular . Therefore, . We conclude that .
Now consider any . Define by
(These are dot products in .) On the other hand, for any , the shattering implies and therefore . Therefore, is not in the linear span of and hence the set is linearly independent. We conclude that .
Putting everything together, we get .
Consider a countable set , a set , a non-empty countable set , some , some and some . Denote
Then, for any , we can choose and s.t. , and for any and , .
Proof of Proposition A.1
We define by
The fact that follows from the definition of .
Denote and enumerate as . For each , define recursively by
Set . Define by
Denote . For any , we denote by the -th number in in increasing order. Denote and . By the definition of , for each we can choose s.t. . Moreover, it also follows from the definition that for every and , (using only the fact that and ). Therefore, shatters and hence .
By the definition of , for any , . On the other hand, for any , and in particular . We conclude that
Consider a countable set , a set , a non-empty countable set , some and some . Consider s.t.
[ stands for the mutual information between and the joint random variables .]
Proof of Proposition A.2
By the chain rule for mutual information
The first term on the right hand side obviously vanishes, and the second term can be expressed in terms of KL-divergence. For any , Define by . We get
For any and , by definition of . If , then
It follows that, in this case, . We conclude
Consider a countable set , a set , a non-empty countable set , some and some . Let be s.t.
Proof of Proposition A.3
Denote , and . By Proposition A.2, there is s.t.
By Proposition A.1 (setting ), there are and s.t. , and for any and , . Define . We have .
Denote . For brevity, we will also use the notation . We get, using the bound on
Using the properties of and we get
The expression inside the expected values in the first term on the right hand side is negative, by the property of . We conclude
Proof of Lemma 1
For each we choose some s.t.
We now take to be Thompson sampling. More formally, we construct a probability space with random variables (the true environment) and (the hypothesis sampled on round ). We define (the action taken on round ) and (the observation made on round ) by
We also define by
is thus the set of hypotheses consistent with the previous outcomes . The random variables have to satisfy (the distribution of the true environment is the prior) and
That is, the distribution of conditional on and for is given by the prior updated by the previous outcomes. is then defined s.t. for any , and :
Now we need to prove the regret bound. We denote the Bayesian regret by :
The construction of implies that
Define (the belief state of the agent on round ) and (the distribution over actions on round ) by
Using expectation conditional on we can rewrite the equation for as
It follows that
We now apply Proposition A.3 and get
Given any , we will use the notation .
Fix and consider a non-empty set of DDP hypotheses . For each we define recursively by
(That is, is the history resulting from DDP interacting with action sequence starting from initial state .) Define
Then, for any , .
Here, the subscript has the same meaning as in Lemma 1.
Proof of Proposition A.4
Fix . Let be an -independent sequence. We will now construct a sequence
s.t shatters . The latter implies that is an -independent sequence, establishing the claim.
For brevity, we will use the notation (here, and ). For each define by
Note that the set above is indeed non-empty because is -independent.
Given , we will use the notational convention .