Reflective oracles accurately answer questions about what arbitrary halting probabilistic oracle machines output. It is possible to make a variant of a reflective oracle that accurately answers questions about what sufficiently short-running Turing machines with access to the same oracle output.

We introduce a variant of the concept of a “quantilizer” for the setting of choosing a policy for a finite Markov decision process (MDP), where the generic unknown cost is replaced by an unknown penalty term in the reward function. This is essentially a generalization of quantilization in repeated games with a cost independence assumption. We show that the “quantilal” policy shares some properties with the ordinary optimal policy, namely that (i) it can always be chosen to be Markov (ii) it can be chosen to be stationary when time discount is geometric (iii) the “quantilum” value of an MDP with geometric time discount is a continuous piecewise rational function of the parameters, and it converges when the discount parameter \(\lambda\) approaches 1. Finally, we demonstrate a polynomial-time algorithm for computing the quantilal policy, showing that quantilization is not qualitatively harder than ordinary optimization.

Reflective oracles can be approximated by computing Nash equilibria. But is there some procedure that produces a Pareto-optimal equilibrium in a game, aka, a point produced by a Cooperative oracle? It turns out there is. There are some interesting philosophical aspects to it, which will be typed up in the next post.

The result is not original to me, it’s been floating around MIRI for a while. I think Scott, Sam, and Abram worked on it, but there might have been others. All I did was formalize it a bit, and generalize from the 2-player 2-move case to the n-player n-move case. With the formalism here, it’s a bit hard to intuitively understand what’s going on, so I’ll indicate where to visualize an appropriate 3-dimensional object.

Crossposted at Lesserwrong.

**tl;dr**: if an agent has a biased learning process, it may choose actions that are worse (with certainty) for every possible reward function it could be learning.

I’d like to credit Daniel Demski for helpful discussion.

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.

Edit: This article has major flaws. See my comment below.

This idea was informed by discussions with Abram Demski, Scott Garrabrant, and the MIRIchi discussion group.

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.