Intelligent Agent Foundations Forumsign up / log in
A failed attempt at Updatelessness using Universal Inductors
discussion post by Scott Garrabrant 408 days ago | Jessica Taylor and Patrick LaVictoire like this | 1 comment

Here, I present a failed attempt to build an updateless decision theory out of universal inductors. It fails because it is is mistaking updatelessness about which logical theory it is in for true logical updatelessness about computations. I will use Tsvi’s notation.

Fix a UGI, \((\mathbb{P}_n)\). Fix a sequence of utility functions \((U_n:2^\omega\rightarrow \mathbb{R})\), which assigns a utility to all propositionally consistent worlds (represented by infinite bit strings). We assume that \(U_n(W)\) is computable function of \(n\) and the first \(k(n)\) bits for some computable function \(k\). In the simplest example, \(U_n\) is just equal to a single bit in the string.

We define a sequence of agents \((A_n)\) which output a single bit \(1\) or \(0\). These agents will be broken into two pieces, a deductive process, which outputs a bunch of logical facts, and a decision process, which chooses a policy in the form of a function from the possible outputs of the deductive process to \(\{0,1\}\).

Let \(P_n\) denote the set of policies that the decision process can output. There is a computable partition of worlds into sets of worlds where each policy is output, \(S_n:2^\omega\rightarrow P_n\). For each \(p\in P_n\), we can compute the expectation, \(\mathbb{E}(U_n(W)|S_n(W)=p)\), where W is sampled according to \(\mathbb{P}_n\). The decision process outputs the policy \(p\) which maximizes \(\mathbb{E}(U_n(W)|S_n(W)=p)\), and the agent \(A_n\) outputs the result of applying that policy to the output of the deductive process.

There are actually many things wrong with the above proposal, and many similar proposals that fail in similar or different ways. However, I want to focus on the one problem that proposals like this have in common:

Universal Induction is a model for uncertainty about what theory/model you are in; it is not a model for uncertainty about the output of computations.

It is easiest to see why this is a problem using the counterfactual mugging problem. We would like to use a universal inductor to be uncertain about a digit of \(\pi\), and thus reason about the world in which it went another way. The problem is that a powerful universal inductor has the digits of \(\pi\) in its probabilities, even if it does not know that it is in PA. This is because the Kolomorogov complexity of a infinite string of the digits of \(\pi\) is very low, while the Kolomorogov complexity of a string that looks like \(\pi\) for a very long time, and then changes is high. We do not have to direct our UGI at PA for it to have good beliefs about late bits in a string that starts out looking like \(\pi\).

I will use the phrase “logical updatelessness” to refer updatelessness about computations. I think updatelessness about the logical system is mostly a distraction from the more important concept of logical updatelessness. (Similarly, I believe that early work in logical uncertainty about distributions over complete theories was mostly a distraction from the later work that focused on uncertainty about computations.)



by Paul Christiano 406 days ago | link

From my perspective, the point of reasoning about complete theories isn’t that we actually care about them, it’s that “what does this halting oracle output?” might be a useful analogy for “what does this long-running computation output?” I still think it is/was a useful analogy, though the time eventually came to move on to smaller and better things.

reply



NEW LINKS

NEW POSTS

NEW DISCUSSION POSTS

RECENT COMMENTS

This is exactly the sort of
by Stuart Armstrong on Being legible to other agents by committing to usi... | 0 likes

When considering an embedder
by Jack Gallagher on Where does ADT Go Wrong? | 0 likes

The differences between this
by Abram Demski on Policy Selection Solves Most Problems | 0 likes

Looking "at the very
by Abram Demski on Policy Selection Solves Most Problems | 0 likes

Without reading closely, this
by Paul Christiano on Policy Selection Solves Most Problems | 1 like

>policy selection converges
by Stuart Armstrong on Policy Selection Solves Most Problems | 0 likes

Indeed there is some kind of
by Vadim Kosoy on Catastrophe Mitigation Using DRL | 0 likes

Very nice. I wonder whether
by Vadim Kosoy on Hyperreal Brouwer | 0 likes

Freezing the reward seems
by Vadim Kosoy on Resolving human inconsistency in a simple model | 0 likes

Unfortunately, it's not just
by Vadim Kosoy on Catastrophe Mitigation Using DRL | 0 likes

>We can solve the problem in
by Wei Dai on The Happy Dance Problem | 1 like

Maybe it's just my browser,
by Gordon Worley III on Catastrophe Mitigation Using DRL | 2 likes

At present, I think the main
by Abram Demski on Looking for Recommendations RE UDT vs. bounded com... | 0 likes

In the first round I'm
by Paul Christiano on Funding opportunity for AI alignment research | 0 likes

Fine with it being shared
by Paul Christiano on Funding opportunity for AI alignment research | 0 likes

RSS

Privacy & Terms