Gödel Machine vs AIXI etc.

Unlike Gödel machines,
Hutter's recent *AIXI model* [15]
generally needs *unlimited* computational resources per input update.
It combines Solomonoff's universal prediction
scheme [47,48] with an *expectimax*
computation. In discrete cycle
,
action results
in perception and reward , both
sampled from the unknown
(reactive) environmental probability distribution . AIXI defines
a mixture distribution as a weighted sum of distributions
, where is any class of distributions that includes the
true environment . For example, may
be a sum of all computable
distributions [47,48], where the sum of
the weights does not exceed 1. In cycle , AIXI
selects as next action the first in an action sequence maximizing
-predicted reward up to some given horizon.
Recent work [17] demonstrated AIXI's optimal
use of observations as follows. The Bayes-optimal policy based on
the mixture is self-optimizing in the sense that its average
utility value converges asymptotically for all
to the
optimal value achieved by the (infeasible) Bayes-optimal policy
which knows in advance. The necessary condition that
admits self-optimizing policies is also sufficient.
Furthermore, is Pareto-optimal
in the sense that there is no other policy yielding higher or equal
value in *all* environments
and a strictly higher
value in at least one [17].

While AIXI clarifies certain
theoretical limits of machine learning, it
is computationally intractable, especially when
includes all computable
distributions. This drawback
motivated work on the time-bounded, asymptotically optimal
AIXI system [15]
and the related HSEARCH [16], both
already discussed in the introduction. Both methods could be used
to seed the Gödel machine with an initial policy.
Unlike AIXI and HSEARCH, however, the Gödel machine
is not only *asymptotically* optimal but optimal relative to
any given
formal self-improvement criterion (which does not have to ignore
constant slowdowns just because they are constant).
It is defined this way.
Unlike AIXI it does not have to make the unrealistic assumption
that all events of the entire life are memorizable (which is
actually a minor issue considering the other costs of AIXI).
Moreover, the Gödel machine neither requires
AIXI's assumption of an enumerable environmental
distribution nor AIXI's particular utility function--we may
plug in other types of utility functions as well.

While both HSEARCH and AIXI feature hardwired `meta-algorithms' that
carefully allocate time to various subroutines in an unmodifiable
way, the Gödel machine is *self-referential*
and does not have any unchangeable software.
It can replace the proof searcher
itself by a faster, perhaps more focused or more
domain-specific proof searcher in an online fashion,
once the initial axioms together with
experiences collected so far yield a proof
there is one.
It is precisely these *self-referential* aspects
of the Gödel machine that relieve us of much of the burden of careful
algorithm design--they make the Gödel machine both
conceptually simpler *and* more
general than HSEARCH.

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