We give a treatment of advisor corruption in DIRL, more elegant and general than our previous formalism.
The following definition replaces the original Definition 5.
Consider a meta-universe and . A metapolicy is called -rational for (as opposed to before, we assume is an -metapolicy rather than an -metapolicy; this is purely for notational convenience and it is straightforward to generalize the definition) when there exists s.t.
i. For any and , there is s.t. .
iii. For any and
In condition ii, is understood to mean . Conditions i+ii can be seen as the definition of given . A notable special case of condition iii is when for any
As a simple example, we can have a set of corrupt states in which the behavior of the advisor becomes arbitrary, but for each there is s.t. and (i.e., to corrupt the advisor one has to take an action that the advisor would never take). As opposed to before, this formalism can also account for partial corruption, e.g. if for each and , we have (like in strict -rationality) whereas for , we only have for some constant , then to ensure -rationality, it is sufficient that for each :
Consider a countable family of -meta-universes and s.t. . Let be a family of -metapolicies s.t. for every , is -rational for . Define . Then, is learnable.
Proof of Theorem
We don't spell out the proof in detail, but only the modifications with respect to the original.
As in the proof of the original theorem, we can assume without loss of generality that is finite. Define the same way as in Lemma A, but with redefined as
Similarly, define the same way as in the proof of Lemma A, but with redefined as
As in the proof of Lemma A, we have
Using condition iii in the Definition, we conclude that for some function with
We can now repeat the same arguments as in the proof of Lemma A to get
The desired result follows.