How do you estimate the entropy of this distribution?

If you want to force the computation to end up with no garbage, then this significantly restricts the range of functions you compute, since the sampling process was invertible. In particular, they need to be functions such that you can compute the agent’s entire computation history only given the output, which feels like a restriction that may be dodging the hard cases of the informed oversight problem.

If you allow the agent to end up with garbage, then you can calculate the probability of that particular garbage+outputs, but the entropy of that distribution will be unboundedly larger than the entropy of the distribution over outputs.

The approach I have in mind is (roughly) to let the agent output some number of bits of garbage, but penalize for the number of bits of garbage (so generating additional uniformly random garbage doesn’t make a difference to the score). I think this can be done using autoencoders (use layer n+1 to compress layer n into a small number of bits of garbage). It’s not clear whether this approach is practical for complex agents, though.

In the OWF example, the garbage is necessarily low-entropy though (at least k bits short on entropy, where k is the size of advice needed to invert the OWF). Right?

How do you estimate the entropy of this distribution?

If you want to force the computation to end up with no garbage, then this significantly restricts the range of functions you compute, since the sampling process was invertible. In particular, they need to be functions such that you can compute the agent’s entire computation history only given the output, which feels like a restriction that may be dodging the hard cases of the informed oversight problem.

If you allow the agent to end up with garbage, then you can calculate the probability of that particular garbage+outputs, but the entropy of that distribution will be unboundedly larger than the entropy of the distribution over outputs.

The approach I have in mind is (roughly) to let the agent output some number of bits of garbage, but penalize for the number of bits of garbage (so generating additional uniformly random garbage doesn’t make a difference to the score). I think this can be done using autoencoders (use layer n+1 to compress layer n into a small number of bits of garbage). It’s not clear whether this approach is practical for complex agents, though.

In the OWF example, the garbage is necessarily low-entropy though (at least k bits short on entropy, where k is the size of advice needed to invert the OWF). Right?

Yes, that seems right. So this won’t work for that example.