# Ω 1

Crossposted from the AI Alignment Forum. May contain more technical jargon than usual.

Appendix A contains the proofs and Appendix B contains propositions proved before.

# Notation

The notation means is Markov kernel from to . When is a finite set, this is the same as a measurable function , and we use these notations interchangeably. Given and (corr. ), we will use the notation (corr ) to stand for (corr. ). Given is a finite set, , and , the notation means .

Given a measurable space, , , , finite sets and , random variables (measurable mappings), the mutual information between the joint distribution of the and the joint distribution of the will be denoted

We will parameterize our geometric time discount by , thus all functions that were previously defined to depend on are now considered functions of .

# Results

We start by explaining the relation between the formalism of general environments we used before and the formalism of finite MDPs.

## Definition 1

A finite Markov Decision Process (MDP) is a tuple

Here, is a finite set (the set of states), is a finite set (the set of actions), is the transition kernel and is the reward function.

A stationary policy for is any . The space of stationary policies is denoted . Given , we define by

We define and by

Here, is just the -th power of in the sense of Markov kernel composition.

As well known, and are rational functions in for , therefore in this limit we have the Taylor expansions

Given any , we define recursively by

All MDPs will be assumed to be finite, so we drop the adjective "finite" from now on.

## Definition 2

Let be an interface. An -universe is said to be an -realization of MDP with state function when and for any , and

Now now define the relevant notion of a "good advisor."

## Definition 3

Let be a universe and . A policy is said to be -sane for when there are , s.t. is an -realization of with state function and for any

i.

ii.

We can now formulate the regret bound.

## Theorem 1

Fix an interface and . Consider for some (note that doesn't depend on ). Assume that for each , is an -sane policy for . Then, there is an -metapolicy s.t. for any

## Corollary 1

Fix an interface and . Consider . Assume that for each , is a -sane policy for . Define . Then, is learnable.

Now, we deal with corrupt states.

## Definition 4

Let be a universe and . A policy is said to be locally -sane for when there are , and (the set of uncorrupt states) s.t. is an -realization of with state function , and for any , if then

i. If and then .

ii.

iii.

iv.

Of course, this requirement is still unrealistic for humans in the real world. In particular, it makes the formalism unsuitable for modeling the use of AI for catastrophe mitigation (which is ultimately what we are interested in!) since it assumes the advisor is already capable of avoiding any catastrophe. In following work, we plan to relax the assumptions further.

## Corollary 2

Fix an interface and . Consider for some . Assume that for each , is locally -sane for . For each , let be the corresponding set of uncorrupt states. Assume further that for any and , if and , then . Then, there is an -policy s.t. for any

## Corollary 3

Assume the same conditions as in Corollary 2, except that may be countable infinite. Then, is learnable.

# Appendix A

First, we prove an information theoretic bound that shows that for Thompson sampling, the expected information gain is bounded below by a function of the loss.

## Proposition A.1

Consider a probability space , , a finite set and random variables , and . Assume that and . Then

## Proof of Proposition A.1

We have

Using Pinsker's inequality, we get

Denote . We get

Now, we describe a "delegation routine" that can transform any "proto-policy" that recommends some set of actions from into an actual -policy s.t (i) with high probability, on each round, either a "safe" recommended action is taken, or all recommended actions are "unsafe" or delegation is performed and (ii) the expected number of delegations is small. For technical reasons, we also need to the modified routines which behave the same way as except for some low probability cases.

## Proposition A.2

Fix an interface , , , . Consider some . Then, there exist and with the following properties. Given , we denote its projection to . Thus, . Given an -environment, , and , we can define as follows

We require that for every , and as above, the following conditions hold

i.

ii.

iii. For all , if then

iv. For all , if then

In order to prove Proposition A.2, we need another mutual information bound.

## Proposition A.3

Consider , a finite set, , , , and . Suppose that for every

Then

## Proof of Proposition A.3

We have

Define by

Let be s.t. and either or every has . For every , denote

If then either or there is s.t. and . This implies

We have

It follows that

## Proof of Proposition A.2

We define , , , , , and recursively. For each , , , and , we require

Denote . Proposition A.3 implies that for any

This gives us condition i.

Condition ii follows because the only difference is in the equation for vs. . That is, may "discard" the "correct" element of but this happens with probability at most per discard and there are at most discards.

Conditions iii and iv are obvious from the definition.

## Definition A.1

Consider and a universe that is an -realization of with state function . A policy is called -optimal for when for any

is called Blackwell optimal for when it is -optimal for any . Obviously, a stationary Blackwell optimal policy always exists. It is a standard (up to straightforward adaption to our formalism) result in MDP theory that any Blackwell optimal policy has maximal expected utility for any sufficiently close to 1.

The following proposition relates optimality in terms of expected utility to expected truncated utility, where truncated utility is defined by only summing rewards within a time duration .

## Proposition A.4

Fix an MDP . Then, for any , , universe that is an -realization of with state function , a -optimal policy for and a Blackwell optimal policy for , we have

## Proof of Proposition A.4

Let be a policy s.t. for any

By Proposition B.1

Using the Blackwell optimality of and 1-optimality of , we get that for

Denote , . Using again the Blackwell optimality of

Since both and are in particular 0-optimal, we have

It follows

The following shows that for any policy that doesn't make "irreversible errors," regret can be approximated by "episodic regret" for sufficiently large episode duration.

## Proposition A.5

Consider a universe that is an -realization of with state function . Suppose that is a Blackwell optimal policy for and is a -optimal policy for . For any , let be a policy s.t. for any

Then, for any and

## Proof of Proposition A.5

By Proposition B.1, for any

becomes Blackwell optimal after , therefore for ,