1.An Untrollable Mathematician
post by Abram Demski 59 days ago | Alex Appel, Sam Eisenstat, Vadim Kosoy, Jack Gallagher, Jessica Taylor, Paul Christiano, Scott Garrabrant and Vladimir Slepnev like this | 1 comment

Follow-up to All Mathematicians are Trollable.

It is relatively easy to see that no computable Bayesian prior on logic can converge to a single coherent probability distribution as we update it on logical statements. Furthermore, the non-convergence behavior is about as bad as could be: someone selecting the ordering of provable statements to update on can drive the Bayesian’s beliefs arbitrarily up or down, arbitrarily many times, despite only saying true things. I called this wild non-convergence behavior “trollability”. Previously, I showed that if the Bayesian updates on the provabilily of a sentence rather than updating on the sentence itself, it is still trollable. I left open the question of whether some other side information could save us. Sam Eisenstat has closed this question, providing a simple logical prior and a way of doing a Bayesian update on it which (1) cannot be trolled, and (2) converges to a coherent distribution.

2.More precise regret bound for DRL
post by Vadim Kosoy 91 days ago | Alex Appel likes this | discuss

We derive a regret bound for DRL reflecting dependence on:

• Number of hypotheses

• Mixing time of MDP hypotheses

• The probability with which the advisor takes optimal actions

That is, the regret bound we get is fully explicit up to a multiplicative constant (which can also be made explicit). Currently we focus on plain (as opposed to catastrophe) and uniform (finite number of hypotheses, uniform prior) DRL, although this result can and should be extended to the catastrophe and/or non-uniform settings.

3.Policy Selection Solves Most Problems
post by Abram Demski 116 days ago | Alex Appel and Vladimir Slepnev like this | 4 comments

It seems like logically updateless reasoning is what we would want in order to solve many decision-theory problems. I show that several of the problems which seem to require updateless reasoning can instead be solved by selecting a policy with a logical inductor that’s run a small amount of time. The policy specifies how to make use of knowledge from a logical inductor which is run longer. This addresses the difficulties which seem to block logically updateless decision theory in a fairly direct manner. On the other hand, it doesn’t seem to hold much promise for the kind of insights which we would want from a real solution.

4.Reflective oracles as a solution to the converse Lawvere problem
post by Sam Eisenstat 128 days ago | Alex Mennen, Alex Appel, Vadim Kosoy, Abram Demski, Jessica Taylor, Scott Garrabrant and Vladimir Slepnev like this | discuss

1 Introduction

Before the work of Turing, one could justifiably be skeptical of the idea of a universal computable function. After all, there is no computable function $$f\colon\mathbb{N}\times\mathbb{N}\to\mathbb{N}$$ such that for all computable $$g\colon\mathbb{N}\to\mathbb{N}$$ there is some index $$i_{g}$$ such that $$f\left(i_{g},n\right)=g\left(n\right)$$ for all $$n$$. If there were, we could pick $$g\left(n\right)=f\left(n,n\right)+1$$, and then $g\left(i_{g}\right)=f\left(i_{g},i_{g}\right)+1=g\left(i_{g}\right)+1,$ a contradiction. Of course, universal Turing machines don’t run into this obstacle; as Gödel put it, “By a kind of miracle it is not necessary to distinguish orders, and the diagonal procedure does not lead outside the defined notion.” [1]

The miracle of Turing machines is that there is a partial computable function $$f\colon\mathbb{N}\times\mathbb{N}\to\mathbb{N}\cup\left\{ \bot\right\}$$ such that for all partial computable $$g\colon\mathbb{N}\to\mathbb{N}\cup\left\{ \bot\right\}$$ there is an index $$i$$ such that $$f\left(i,n\right)=g\left(n\right)$$ for all $$n$$. Here, we look at a different “miracle”, that of reflective oracles [2,3]. As we will see in Theorem 1, given a reflective oracle $$O$$, there is a (stochastic) $$O$$-computable function $$f\colon\mathbb{N}\times\mathbb{N}\to\left\{ 0,1\right\}$$ such that for any (stochastic) $$O$$-computable function $$g\colon\mathbb{N}\to\left\{ 0,1\right\}$$, there is some index $$i$$ such that $$f\left(i,n\right)$$ and $$g\left(n\right)$$ have the same distribution for all $$n$$. This existence theorem seems to skirt even closer to the contradiction mentioned above.

We use this idea to answer “in spirit” the converse Lawvere problem posed in [4]. These methods also generalize to prove a similar analogue of the ubiquitous converse Lawvere problem from [5]. The original questions, stated in terms of topology, remain open, but I find that the model proposed here, using computability, is equally satisfying from the point of view of studying reflective agents. Those references can be consulted for more motivation on these problems from the perspective of reflective agency.

Section 3 proves the main lemma, and proves the converse Lawvere theorem for reflective oracles. In section 4, we use that to give a (circular) proof of Brouwer’s fixed point theorem, as mentioned in [4]. In section 5, we prove the ubiquitous converse Lawvere theorem for reflective oracles.

5.Comparing LICDT and LIEDT
post by Abram Demski 153 days ago | Alex Appel likes this | discuss

Attempted versions of CDT and EDT can be constructed using logical inductors, called LICDT and LIEDT. It is shown, however, that LICDT fails XOR Blackmail, and LIEDT fails Newcomb. One interpretation of this is that LICDT and LIEDT do not implement CDT and EDT very well. I argue that they are indeed forms of CDT and EDT, but stray from expectations because they also implement the ratifiability condition I discussed previously. Continuing the line of thinking from that post, I discuss conditions in which LICDT=LIEDT, and try to draw out broader implications for decision theory.

6.Delegative Inverse Reinforcement Learning
post by Vadim Kosoy 265 days ago | Alex Appel likes this | 11 comments

We introduce a reinforcement-like learning setting we call Delegative Inverse Reinforcement Learning (DIRL). In DIRL, the agent can, at any point of time, delegate the choice of action to an “advisor”. The agent knows neither the environment nor the reward function, whereas the advisor knows both. Thus, DIRL can be regarded as a special case of CIRL. A similar setting was studied in Clouse 1997, but as far as we can tell, the relevant literature offers few theoretical results and virtually all researchers focus on the MDP case (please correct me if I’m wrong). On the other hand, we consider general environments (not necessarily MDP or even POMDP) and prove a natural performance guarantee.

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This is somewhat related to
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This uses logical inductors
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Nice writeup. Is one-boxing
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Hi Alex! The definition of
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A summary that might be
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