When asked about what it means for a system to be goal-directed, one common answer draws on some version of Dennett’s intentional stance: a goal-directed system is a system such that modeling it as having a goal provides accurate and efficient predictions about its behavior. I agree up to that point. But then, some people follow up by saying that the prediction is that the system will accomplish its goal. For example, it makes sense to model AlphaGo as goal-directed towards winning at Go, because it will eventually win. And taking the intentional stance allows me to predict that.

But what if I make AlphaGo play against AlphaZero, which is strictly better at Go? Then AlphaGo will consistently lose. Does it mean that it’s no longer goal-directed towards winning?

What feels wrong to me is the implicit link drawn between goal-directedness and competence. A bad Go player will usually lose, but it doesn’t seem any less goal-directed to me than a stronger one that consistently wins.

Competence is thus not the whole story. It might be useful to compute goal-directedness; reaching some lower-bound of competency might even be a necessary condition for goal-directedness (play badly enough and it becomes debatable whether you're even trying to win). But when forcing together the two, I feel like something important is lost.

To solve this problem, I propose a new metric of goal-directedness, focus: how much is the system trying to accomplish a certain goal. Focus is not the whole story about being goal-directed, but I think computing the focus of a system for some goal (details in the next paragraph) gives useful information about its goal-directedness.

Given a system (as a function from states or histories to actions) and a goal (as a set of states), here are the steps to compute the focus of towards .

- I define a reward function over states valued 1 at states in and 0 at all other states.
- Then I define be the set of all policies that can be generated by Reinforcement Learning (RL) on . I’ll go into details about below, but the most important part here is that it isn't limited to optimal policies; I also consider policies of RL with “few resources”. Basically all policies at intermediary steps of the RL training are in .
- Lastly, I pick a distance between policies. If the two policies are deterministic, a Hamming distance will do. If they are stochastic, maybe some vector distance based on the Kullback-Leibler divergence.
- Then, the focus of towards is inversely proportional to the distance between and .

The intuition here is that any policy that result from training on this reward function is aiming maximally towards the goal, by definition. And by taking the appropriate distance, we can measure how far our system is from such a fully focused policy. The distance captures the proportion of actions taken by the policy that fits with aiming towards the specific goal.

Of course, there are many points that need further thought:

- What does “all policies given by using RL” mean in this case? The easy answer is all policies resulting from taking any RL method and any initial conditions, and training for any amount of resources on the reward function of the goal. But not only is this really, really uncomputable, I’m not sure it’s well defined enough what are “all methods of RL”?). Ideally, I would want to limit the study to one specific RL algorithm (SARSA for example) and then the set of generated policies would be well-defined. But I’m not sure if I’m losing any policy by doing so.
- Even when fixing some RL algorithm, it is completely unfeasible to consider all initial conditions and amounts of resources. Yet this is the obvious way to compute the set of maximally-focused policies. Here I hope for either a dense subset (or a good approximation) of this set of policies, or even an analytical characterization if ones exists.
- The ghost of competence strikes back here, because I cannot really consider any amount of resources; if I did, then every policy would be maximally-focused for the goal, as it would be generated by taking the policy as an initial condition and using no resources at all. My intuition for dealing with this is that there should be a meaningful lower bound on the amount of resources the RL algorithm has to use before the resulting policy is indeed maximally-focused. Maybe enough resource for all state values or state-action pairs value to have been updated at least once?

Finally, assuming we are able to compute the focus of any system for any goal, how to interpret the results we get? Focus is not divided between goals like probability: for example, the full goal consisting of all possible states always has maximal focus, as all policies are optimal for the corresponding reward; but other goals might also have the same focus. This entails that finding the most representative goal is not only about focus, but also about the triviality of the goal.

My far less clean intuition here is that the “triviality” of the goal should weight its focus. That is, the goal consisting of all possible states is trivial, whereas the one consisting of exactly one state is not trivial at all. Thus even if the former has stronger focus than the latter, it has to be really, really stronger to compensate its triviality. Or said another way, a non-trivial goal with a small but not negligible focus exhibits goal-directedness than a trivial goal with enormous focus.

Even with all those uncertainties, I still believe focus is a step in the right direction. It trims down competence to the part that seems the most relevant to goal-directedness. That being said, I am very interested in any weakness of the idea, or any competing intuition.

*Thanks to Jérémy Perret for feedback on the writing, and to Joe Collman, Michele Campolo and Sabrina Tang for feedback on the idea.*

Thanks for writing up this post! It's really similar in spirit to some research I've been working on with others, which you can find on the ArXiv here: https://arxiv.org/abs/2006.07532 We also model bounded goal-directed agents by assuming that the agent is running some algorithm given bounded compute, but our approach differs in the following ways:

The abstract for the paper is below -- hope it's useful to read, and I'd be curious to hear your thoughts:

Sorry for the delay in answering.

Your paper looks great! It seems to tackle in a clean and formal way what I was vaguely pointing at. We're currently reading a lot of papers and blog posts to prepare for an in-depth literature review about goal-directedness, and I added your paper to the list. I'll try to come back here and comment after I read it.

It seems like there's an implicit model here which is potentially a lot more interesting than the definition of focus itself. Conceptually, the idea is that we define directedness-toward-goal-X by looking at the set of attractors of RL run with X as a goal. Setting aside the whole question of a metric on the space of policies, what can we say about the set of attractors of RL algorithms?

For instance, things like dutch book theorems seem like they should apply to the attractors of some (but not all) RL algorithms and goals. What class of algorithms/goals do they apply to? When they do apply, and can we say anything about what world-models and utility functions the attractors display? What exact conditions on the RL algorithm/goal make them not apply?

I'd imagine that there's other general properties of the attractors of broad classes of RL algorithms/goals as well.

Yes, that's definitely a question I asked myself. All the discussions about minimal amount of resources and choice of RL policies boil down to defining the attractors such that they're neither trivial nor too restrictive. I'd be very interested by any work in this direction.

Planned summary for the Alignment Newsletter:

Planned opinion:

Thanks for the summary and opinion!

On the summary, I would say that the following sentence

is written as if goal-directedness is a binary condition, just a more lenient one. I think your next sentence clarifies this a bit, but it might be worth it to mention at this point that the notion of goal-directedness considered here is more like a spectrum/scale.

For your opinion, I agree that this specific formalization is not meant to capture all of goal-directedness, just one aspect that I find important. (It's also an element of answer to your question on the difference between "being good" and "trying hard").

That being said, one point I disagree with in your opinion is about generalization. I'm relatively sure that focus doesn't capture all of the generalization part of goal-directedness; but if we include model-based RL in the process, we might have some generalization. The trick is that the set of policies considered is the one generated by all possible RL methods on the reward. This is arguably very vague and hard to construct, which is why I mentioned the hope of reducing it to the policies generated by one or a few RL methods.

In light off this, I interpret your comment as pointing that limiting ourselves to SARSA makes us lose a lot, and thus is not a good idea. By the way, do you have a reference on that? That would be very useful, thanks.

Lastly, I find your example about my condition on the resources spot on. Even as I wrote it, I didn't notice that requiring every state to be updated means that the policy "has seen it all" in some sense. This indeed limits the use of focus. That being said, your "eat sweet things" behavior might still have very good focus towards this goal, if your "wrong" exploratory behavior happens rarely enough.

When you encounter a particular state, you only update the Q-value of that state in the table, and don't do anything to the Q-values of any other state. Therefore, seeing one state will make no difference to your policy on any other state, i.e. no generalization.

You need to use function approximators of some sort to see generalization to new states. (This doesn't have to be a neural net -- you could approximate the Q-function as a linear function over handcoded features, and this would also give you some generalization to new states.)

Yeah, I was ignoring the "all possible RL methods" because it was vague (and I also expect it not to work for any specific formalization, e.g. you'd have to rule out RL methods that say "if the goal is G, then output <specific policy>, otherwise do regular RL", which seems non-trivial). If you use only a few RL methods, then I think I would stick with my claim about generalization:

I could add the sentence "Alternatively, if you try to use "all possible" RL algorithms, I expect that there will be many pathological RL algorithms that effectively make any policy goal-directed", if you wanted me to, but I think the version with a small set of RL algorithms seems better to me and I'd rather keep the focus on that.

I updated the summary to:

Okay, thanks for the explanation!

Yes, trying to ensure that not all policies are generated is indeed the main issue here. It also underlies the resource condition. This makes me think that maybe using RL is not the appropriate way. That being said, I still think an approach exists for computing focus instead of competence. I just don't know it yet.

I agree that keeping the focus (!) on the more realistic case makes more sense here.

I think it might actually be very difficult to come up with a distance metric between policies that corresponds even reasonably well to behavioral similarity. I imagine that flipping the sign on a single crucial parameter in a neural net could completely change its behavior, or at least break it sufficiently that it goes from highly goal oriented behavior to random/chaotic behavior.

By analogy, imagine trying to come up with a distance metric between python source files in a way that captures behavioral similarity. Very subtle changes to source code can completely alter behavior, while drastic refactorings can leave behavior unchanged.

Ultimately we'd like to be able to handle cases where we're using network architectures that permit arbitrary Turing machines to emerge as policies, in which case determining behavioral similarity by comparing source code is equivalent to the halting problem.

Sorry for the delay in answering.

In this post, I assume that a policy is a description of its behavior (like a function from state to action or distribution over action), and thus the distances mentioned indeed capture behavioral similarity. That being said, you're right that a similar concept of distance between the internal structure of the policies would prove difficult, eventually butting against uncomputability.