A brief note on naming: Solomonoff exhibited an uncomputable algorithm that does idealized induction, which we call Solomonoff induction. Garrabrant exhibited a computable algorithm that does logical induction, which we have named Garrabrant induction.
This seems misleading. Solomonoff induction has computable versions obtained by imposing a complexity bound on the programs. Garrabrant induction has uncomputable versions obtained by removing the complexity bound from the traders. The important difference between Solomonoff and Garrabrant is not computable v.s uncomputable. Also I feel that it would be appropriate to mention defensive forecasting as a historical precursor of Garrabrant induction.
Sorry if this is a stupid question but wouldn’t “LI with no complexity bound on the traders” be trivial? Like, there’s a noncomputable trader (brute force proof search + halting oracle) that can just look at any statement and immediately declare whether it’s provably false, provably true, or neither. So wouldn’t the prices collapse to their asymptotic value after a single step and then nothing else ever happens?
First, “no complexity bounds on the trader” doesn’t mean we allow uncomputable traders, we just don’t limit their time or other resources (exactly like in Solomonoff induction). Second, even having a trader that knows everything doesn’t mean all the prices collapse in a single step. It does mean that the prices will converge to knowing everything with time. GI guarantees no budget-limited trader will make an infinite profit, it doesn’t guarantee no trader will make a profit at all (indeed guaranteeing the later is impossible).
This seems misleading. Solomonoff induction has computable versions obtained by imposing a complexity bound on the programs. Garrabrant induction has uncomputable versions obtained by removing the complexity bound from the traders. The important difference between Solomonoff and Garrabrant is not computable v.s uncomputable. Also I feel that it would be appropriate to mention defensive forecasting as a historical precursor of Garrabrant induction.
Sorry if this is a stupid question but wouldn’t “LI with no complexity bound on the traders” be trivial? Like, there’s a noncomputable trader (brute force proof search + halting oracle) that can just look at any statement and immediately declare whether it’s provably false, provably true, or neither. So wouldn’t the prices collapse to their asymptotic value after a single step and then nothing else ever happens?
First, “no complexity bounds on the trader” doesn’t mean we allow uncomputable traders, we just don’t limit their time or other resources (exactly like in Solomonoff induction). Second, even having a trader that knows everything doesn’t mean all the prices collapse in a single step. It does mean that the prices will converge to knowing everything with time. GI guarantees no budget-limited trader will make an infinite profit, it doesn’t guarantee no trader will make a profit at all (indeed guaranteeing the later is impossible).