Understanding Selection Theorems

28th May 2022

5johnswentworth

5philh

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This is probably the best-written post I've seen yet from the Distillation Contest. Well done.

There are some technical points which aren't quite right. Most notably:

In terms of the type signature of A's goal component, the fact that it took the

current world stateas input proved to be a problem. Agent B's goals were more stable since they were not dependent on the current state of the room...

B's type signature is still (probably) a function of room-state, e.g. it has a utility function mapping room-state to a real number. A's type signature might also be a function of room-state, e.g. it might map each room-state to an action or a preferred next state. But A's type signature does not map room-state to a real number, which it then acts to increase.

Anyway, your examples are great, and even in places where I don't quite agree with the wording of technical statements the examples are still basically correct and convey the right intuition. Great job!

The quote specifically mentioned the type signature of A's goal component. The idea I took from it was that B is doing something like

```
utility = compare(world_state, goal)
```

and A is doing something like

```
utility = compare(world_state, goal(world_state))
```

where the second thing can be unstable in ways that the first can't.

...But maybe talking about the type signature of components is cheating? We probably don't want to assume that all relevant agents will have something we can consider a "goal component", though conceivably a selection theorem might let us do that in some situations.

Thank you! I'd be glad to include this and any other corrections in an edit once contest results are released. Are there any other errors which catch your eye?

This post is a distillation of a corpus of ideas on Selection Theorems byjohnswentworth, mainlythis post. It was made for EA UC Berkeley'sDistillation Contest.## Introduction

Selection Theorems are tools for answering the question, "What will be the likely features of agents we might encounter in an environment?" Roughly speaking, if an environment selects for agents according to some metric, then Selection Theorems try to give us some guarantees or heuristics about the classes of agents that will be highly fit in the environment.

This selective pressure need not be biological. If an AI researcher scraps some agent designs and keeps others, or a set of market traders can thrive or go bankrupt based on their trading performance, then these might also be situations where Selection Theorems can apply.

The value of strong Selection Theorems is quite alluring: if we can make specific claims about the types of agents that will arise in particular environments, it might become easier to know what aligned systems look like and how to make them. They may also be instrumental in resolving our general confusion about the nature of agency.

The goals of this post will be to provide intuition and examples for the notion of agent

type signatures, define and describe Selection Theorems through an example, and discuss what future research might look like. I do not assume any technical background from readers, other than a baseline familiarity with ideas like agency and expected utility.## A Toy Scenario

Suppose I am trying to build a robotic agent which mops and polishes floors. I have two prototype mopping agents, A and B, with different internal architectures:

Both agents are relatively safe and competent at this job, but there are a few key differences:

There is a sense in which Agent A is less effective and Agent B is more effective with respect to my desires. Even though A and B may have broadly similar views of what makes a clean room, Agent B seems better suited to complete the task in the real world. As I continue prototyping these agents, preferring only the most robust cleaning robots, the end result will be agents structurally less like A and more like B.

## Agent Type Signatures

The above observations are a good segue to what John calls the

type signatureof an agent. In computer science, type signatures refer to the input and output data structures of functions. For example, a function with a type signature R→R takes real numbers as input and outputs a real number. The functions "add 1" and "multiply by 2" both share this type signature, despite being different functions. Type signatures are interested in capturing the general structure of a function rather than its specific behavior.Insofar as agents might just be special types of functions, thinking about their type signature makes sense. In particular, John sees an agent's type signature as collectively referring to the

subcomponentsof agents, thedata structuresor mathematical objects that represent these components, the high-levelinteractionsbetween subcomponents, as well as theembeddedrelationbetween components and what they are actually supposed to represent in the environment.As an example for part of an agent's type signature, many agents have "goal" components that rank world states (which are themselves components represented by their own data structures). One can then represent goals through the data structure of utility functions, or alternatively as committees of utility functions, or something entirely different.

Type signatures can make or break the effectiveness of an agent. Let us again consider the mopping agents, and in particular the data structures representing their key components:

current world stateas input proved to be a problem. Agent B's goals were more stable since they were not dependent on the current state of the room, reflecting a more robust underlying data structure with a consistent preference ordering.—its momentary visual view of rooms. Although vision of things and things themselves are tightly related in most situations, Agent A was still vulnerable to tricks which prevented it from doing what I wanted. The value of Agent B's more successful world model came from its ability to distinguish its perception from its model of the environment, and leverage perception-independent memories of its surroundings. Agent B internally represented its environment on a lower level, and was thus able to recognize and account for incomplete or unreliable perceptual inputs.Another aspect of agent type signatures is the manner in which component outputs directly interact with the environment. Consider a third cleaning robot, Agent C, whose type signature has a decision-making component that outputs natural language directives (like "put the mop in the water bucket" and " walk forward three steps") rather than motor controls. Agent C would be wholly useless to me: it would just stand unmoving, with a mop lying beside it, repeating "pick up mop," "pick up mop."

We see from these considerations that a well-chosen type signature plays a large part in the success of an agent. Of course, the value of type signatures is completely environment-dependent. For example, although Agent A was ineffective because it represented world states as pixel arrays, an agent playing Pac-Man would do well with such a world state data structure, since a picture of a Pac-Man screen tells an agent everything they need to know to play well.

## Selection Theorems and an Example

In any environment that exerts a selection pressure on agents, we might expect high-fit or optimal agents (with respect to the selection criteria) to exhibit particular sorts of behaviors or type signatures. Selection Theorems are any result that shows what these optima looks like for a particular class of environments. According to John:

John gives many examples of Selection Theorems. I will focus only on coherence theorems in this post; they strike me as the archetypal example.

Much has already been said on LessWrong about coherence theorems, and so I will not go into specifics or proofs. Instead, my goal here is to explicitly reframe them in the language of Selection Theorems. To do so, we are forced to ask, "What exactly do the coherence theorems assume about agent type signatures, and what do they derive?" Let us list these assumptions in brief for a slightly simplified formulation of a coherence theorem:

world states.Actionsfor agents are represented as bets, where a bet might cost resources upfront but might pay out resources based on the resulting world state.Once all of these assumptions are satisfied, it is mathematically guaranteed that the optimal decision-making mechanism for an agent this environment will be equivalent to an

expected utility maximizer, with respect to the resource that is selected for! In other words, agents which do not have decision-making components of this form are strictly outcompeted by alternate agents which do—this is the mechanism of the selection.## Next Steps for Selection Theorems

What are some key takeaways from coherence theorems? For one thing, the stringency of the assumptions means that, in their current form, coherence theorems

do notdirectly apply to agents in the real world. Many people make the mistake of invoking them without making sure assumptions apply. Of course, coherence theorems certainly have useful implications, insofar as the real world has high-level equivalents to resources, and insofar as many real-world actions can be modeled as bets. But as John elaborates in this comment, they are flawed. Among other things:derivethe notion of a resource within the Selection Theorem itself. In other words, the notion of resources should arise without assumption from the proof, in the same way that the notion of probability arises without prior assumption from current coherence theorems.notthe optimal, or at least not the only optimal, type signature that will be selected for. See John's Why Subagents? post for details.These concerns are paralleled in many other existing Selection Theorems: their starting assumptions about the environment and agency often too narrow and hard to apply. What we ultimately want is a family of Selection Theorems which make as

few assumptionsas possible about the environment or agent type signature, while deducingas muchas possible about the optimal type signature and behavior of agents. Further, in John's words:In these respects, coherence theorems (and our current scattering of existing Selection Theorems in general) are woefully incomplete.

Aside from merely improving existing theorems, new Selection Theorems might venture to explain and sharpen broad classes of fuzzy observations about existing agents, answering questions like "Why don't plants develop central nervous systems (brains)? What assumptions about plants and their environments imply brains are being selected against, if at all?" John gives a detailed overview of what the Selection Theorems research program might investigate and what sorts of results it might eventually deduce. I suggest that the interested reader continue there.

Overall, I see Selection Theorems as a promising approach to chipping away at our confusion behind agency. Future research may involve an empirical component, but the conceptual nature of this research program suggests that new paradigms for understanding agency may be a couple good insights away.