I like the idea of adding in additional constraints.

I think it is complicated a lot by the boundedness of the agent—finding the actual optimum will naturally be intractable, so the question is what it looks like to imperfectly optimize this objective. I’m not sure it does what we want (in the toy model from my post, I think that you may just end up with O(zxx) for a random z).

But I think the most fundamental difficulty is approximating the entropy during training.

One approach is to train a second model to estimate the density of d, and then use the density estimates to compute the payoff −log(d(y)). But this actually won’t work here, for precisely the same reason that providing oversight is so hard. Our model for estimating the density of dalso can’t tell the difference between the good actions and the bad actions, even if d is exclusively outputting bad actions, and so can’t actually penalize d for outputting a bad action (unless it also penalizes d for outputting a good action).

As far as I can tell, most approaches to training d run into this problem in one form or another.

finding the actual optimum will naturally be intractable, so the question is what it looks like to imperfectly optimize this objective

Actually even getting the actual optimum doesn’t seem to really address the problem.

You are then OK if the unintended solutions are rare. But it’s easy to imagine cases where they are dense (e.g. where good plans are highly constrained, but plans with a subtle bug in one step have lots of degrees of freedom in the other steps).

So at a minimum it seems like this will only address some cases, or would require some additional analysis.

This seems correct. While an approach like this could isolate one particular distribution over solutions to the problem as the optimal one, you also have to engineer the problem such that this distribution over solutions is desirable.

Suppose d is outputting a binary string of length n. Then it starts by outputting a probability. The first bit gets sampled using this probability. The agent is told this bit, and then it outputs the probability for the next bit. Then the second bit gets sampled using this probability. And so on, until n bits are output. Now it’s easy to estimate the entropy of d.

Perhaps outputting the distribution over strings bit-by-bit like this is difficult. But there are other ways of representing distributions that allows sampling from them and estimating their entropy. Consider the following algebraic data type, meant to represent distributions:

data Distr a where
Unit :: Distr ()
Flip :: Double -> Distr Bool
InvFmap :: (a <-> b) -> Distr a -> Distr b
Samp :: (a -> Distr b) -> Distr a -> Distr (a, b)

Unit is a delta distribution on (); Flip flips a coin with a certain probability; InvFmap composes a distribution with an invertible function (which could be represented as a pair of functions f and g; to run the invertible function on an input x, take f(x) and make sure that g(f(x))=x or else throw an error); Samp is similar to the monadic bind >>= but keeps the original sample of a around; a sample from Samp f a is (x, y) where x∼a and y∼f(x). The intention is to have everything be fully reversible, so there is no way to throw away information.

(note that it’s also possible to have Unsamp :: (a -> Distr b) -> Distr (a, b) -> Distr a, intended as an inverse of Samp; I can give more details if you’re interested).

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?

Additional note: I think you can implement the approach I sketched (which I’m calling “reversible probabilistic programming”) using autoencoders. Represent each layer of the autoencoder by a single-layer neural net f : a -> Distr b and approximate inverse g : b -> Distr a. Given a distribution for the previous layer x : Distr a, get the distribution for the next layer by taking Unsamp g (InvFmap (\(x, y) -> (y, x)) (Samp f x)) :: Distr b. Compose a lot of these together to get a multi-layer generative model. This seems simple enough that there’s probably a simple more direct way to estimate the entropy of a generative model represented by an autoencoder.

(actually, I think this might not work, because the inverses won’t be very accurate, but maybe something like this works?)

I like the idea of adding in additional constraints.

I think it is complicated a lot by the boundedness of the agent—finding the actual optimum will naturally be intractable, so the question is what it looks like to imperfectly optimize this objective. I’m not sure it does what we want (in the toy model from my post, I think that you may just end up with O(zxx) for a random z).

But I think the most fundamental difficulty is approximating the entropy during training.

One approach is to train a second model to estimate the density of d, and then use the density estimates to compute the payoff −log(d(y)). But this actually won’t work here, for precisely the same reason that providing oversight is so hard. Our model for estimating the density of d

alsocan’t tell the difference between the good actions and the bad actions, even if d is exclusively outputting bad actions, and so can’t actually penalize d for outputting a bad action (unless it also penalizes d for outputting a good action).As far as I can tell, most approaches to training d run into this problem in one form or another.

Actually even getting the actual optimum doesn’t seem to really address the problem.

You are then OK if the unintended solutions are rare. But it’s easy to imagine cases where they are dense (e.g. where good plans are highly constrained, but plans with a subtle bug in one step have lots of degrees of freedom in the other steps).

So at a minimum it seems like this will only address some cases, or would require some additional analysis.

This seems correct. While an approach like this could isolate one particular distribution over solutions to the problem as the optimal one, you also have to engineer the problem such that this distribution over solutions is desirable.

I propose training d using something like this:

Suppose d is outputting a binary string of length n. Then it starts by outputting a probability. The first bit gets sampled using this probability. The agent is told this bit, and then it outputs the probability for the next bit. Then the second bit gets sampled using this probability. And so on, until n bits are output. Now it’s easy to estimate the entropy of d.

Perhaps outputting the distribution over strings bit-by-bit like this is difficult. But there are other ways of representing distributions that allows sampling from them and estimating their entropy. Consider the following algebraic data type, meant to represent distributions:

`Unit`

is a delta distribution on ();`Flip`

flips a coin with a certain probability;`InvFmap`

composes a distribution with an invertible function (which could be represented as a pair of functions f and g; to run the invertible function on an input x, take f(x) and make sure that g(f(x))=x or else throw an error);`Samp`

is similar to the monadic bind`>>=`

but keeps the original sample of a around; a sample from`Samp f a`

is`(x, y)`

where x∼a and y∼f(x). The intention is to have everything be fully reversible, so there is no way to throw away information.(note that it’s also possible to have

`Unsamp :: (a -> Distr b) -> Distr (a, b) -> Distr a`

, intended as an inverse of`Samp`

; I can give more details if you’re interested).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.

Additional note: I think you can implement the approach I sketched (which I’m calling “reversible probabilistic programming”) using autoencoders. Represent each layer of the autoencoder by a single-layer neural net

`f : a -> Distr b`

and approximate inverse`g : b -> Distr a`

. Given a distribution for the previous layer`x : Distr a`

, get the distribution for the next layer by taking`Unsamp g (InvFmap (\(x, y) -> (y, x)) (Samp f x)) :: Distr b`

. Compose a lot of these together to get a multi-layer generative model. This seems simple enough that there’s probably a simple more direct way to estimate the entropy of a generative model represented by an autoencoder.(actually, I think this might not work, because the inverses won’t be very accurate, but maybe something like this works?)