Versions of AIXI can be arbitrarily stupid

18


Many people (including me) had the impression that AIXI was ideally smart. Sure, it was uncomputable, and there might be "up to finite constant" issues (as with anything involving Kolmogorov complexity), but it was, informally at least, "the best intelligent agent out there". This was reinforced by Pareto-optimality results, namely that there was no computable policy that performed at least as well as AIXI in all environments, and strictly better in at least one.

However, Jan Leike and Marcus Hutter have proved that AIXI can be, in some sense, arbitrarily bad. The problem is that AIXI is not fully specified, because the universal prior is not fully specified. It depends on a choice of a initial computing language (or, equivalently, of an initial Turing machine).

For the universal prior, this will only affect it up to a constant (though this constant could be arbitrarily large). However, for the agent AIXI, it could force it into continually bad behaviour that never ends.

For illustration, imagine that there are two possible environments:

  1. The first one is Hell, which will give ε reward if the AIXI outputs "0", but, the first time it outputs "1", the environment will give no reward for ever and ever after that.
  2. The second is Heaven, which gives ε reward for outputting "0" and 1 reward for outputting "1", and is otherwise memoryless.

Now simply choose a language/Turing machine such that the ratio P(Hell)/P(Heaven) is higher than the ratio 1/ε. In that case, for any discount rate, the AIXI will always output "0", and thus will never learn whether its in Hell or not (because its too risky to do so). It will observe the environment giving reward ε after receiving "0", behaviour which is compatible with both Heaven and Hell. Thus keeping P(Hell)/P(Heaven) constant, and ensuring the AIXI never does anything else.

In fact, it's worse than this. If you use the prior to measure intelligence, then an AIXI that follows one prior can be arbitrarily stupid with respect to another.