Not from the paper. I just wrote it.

I don't think that the action log is special in this context relative to any other object that constitutes a tiny part of the environment.

It isn't the size of the object that matters here, the key considerations are structural. In this unrolled model, the unrolled state factors into the (action history) and the (world state). This is not true in general for other parts of the environment.

Sure, but I still don't understand the argument here. It's trivial to write a reward function that doesn't yield instrumental convergence regardless of whether one can infer the complete action history from every reachable state. Every constant function is such a reward function.

Sure. Here's what I said:

how easy is it to write down state-based utility functions which do the same? I guess there's the one that maximally values dying. What else? While more examples probably exist, it seems clear that they're much harder to come by [than in the action-history case].

The broader claim I was trying to make was not "it's hard to write down any state-based reward functions that don't incentivize power-seeking", it was that there are fewer qualitatively distinct ways to do it in the state-based case. In particular, it's hard to write down state-based reward functions which incentivize any given sequence of actions:

when your reward depends on your action history, this is strictly more expressive than state-based reward - so expressive that it becomes easy to directly incentivize any sequence of actions via the reward function. And thus, instrumental convergence disappears for "most objectives." 

If you disagree, then try writing down a state-based reward function for e.g. Pacman for which an optimal policy starts off by (EDIT: circling the level counterclockwise) (at a discount rate close to 1). Such reward functions provably exist, but they seem harder to specify in general.

Also: thanks for your engagement, but I still feel like my points aren't landing (which isn't necessarily your fault or anything), and I don't want to put more time into this right now. Of course, you can still reply, but just know I might not reply and that won't be anything personal.

EDIT: FYI I find your action-camera example interesting. Thank you for pointing that out. 

I was looking for some high-level/simplified description 

Ah, I see. In addition to the cited explanation, see also: "optimal policies tend to take actions which strictly preserve optionality*", where the optionality preservation is rather strict (requiring a graphical similarity, and not just "there are more options this way than that"; ironically, this situation is considerably simpler in arbitrary deterministic computable environments, but that will be the topic of a future post).

Isn't the thing we condition on here similar (roughly speaking) to your interpretation of instrumental convergence?

No - the sufficient condition is about the environment, and instrumental convergence is about policies over that environment. I interpret instrumental convergence as "intelligent goal-directed agents tend to take certain kinds of actions"; this informal claim is necessarily vague. This is a formal sufficient condition which allows us to conclude that optimal goal-directed agents will tend to take a certain action in the given situation. 

I think that using a simplicity prior over reward functions has a similar effect to "restricting to certain kinds of reward functions".

It certainly has some kind of effect, but I don't find it obvious that it has the effect you're seeking - there are many simple ways of specifying action-history+state reward functions, which rely on the action-history and not just the rest of the state. 

Why is the action logger treated in your explanation as some privileged object? What's special about it relative to all the other stuff that's going on in our arbitrarily complex environment? If you imagine an MDP environment where the agent controls a robot in a room that has a security camera in it, and the recorded video is part of the state, then the recorded video is doing all the work that we need an action logger to do (for the purpose of my argument).

What's special is that (by assumption) the action logger always logs the agent's actions, even if the agent has been literally blown up in-universe. That wouldn't occur with the security camera. With the security camera, once the agent is dead, the agent can no longer influence the trajectory, and the normal death-avoiding arguments apply. But your action logger supernaturally writes a log of the agent's actions into the environment.

The reward function is a function over states (or state-action pairs) as usual, not state-action histories. My "unrolling trick" doesn't involve utility functions that are defined over state(-action) histories.

Right, but if you want the optimal policies to take actions $a_1\dots a_k$, then write a reward function which returns 1 iff the action-logger begins with those actions and 0 otherwise. Therefore, it's extremely easy to incentivize arbitrary action sequences.

(I continued this discussion with Adam in private - here are some thoughts for the public record) 

• There is not really a subjective modeling decision involved because given an interface (state space and action space), the dynamics of the system are a real world property we can look for concretely.
• Claims about the encoding/modeling can be resolved thanks to power-seeking, which predicts what optimal policies are more likely to do. So with enough optimal policies, we can check the claim (like the "5-googleplex" one).

I think I'm claiming first bullet. I am not claiming the second.

Or are you pointing out that with an architecture in mind, the state space and action space is fixed? I agree 

Yes, that.

then it's a question of how the states of the actual systems are encoded in the state space of the agent, and that doesn't seem unique to me.

It doesn't have to be unique. We're predicting "for the agents we build, will optimal policies in their MDP models seek power?", and once you account for the environment dynamics, our beliefs about the agent architecture, and then our beliefs on the reward functions conditional on each architecture, this prediction has no subjective degrees of freedom.

I'm not claiming that there's One Architecture To Rule Them All. I'm saying that if we want to predict what happens, we:

1. Consider the underlying environment (assumed Markovian)
2. Consider different state/action encodings we might supply the agent.
3. For each, fix a reward function distribution (what goals we expect to assign to the agent)
4. See what my theory predicts.

There's a further claim (which seems plausible, but which I'm not yet making) that (2) won't affect (4) very much in practice. The point of this post is that if you say "the MDP has a different model", you're either disagreeing with (1) the actual dynamics, or claiming that we will physically supply the agent with a different state/action encoding (2).

But to falsify the "5 googolplex", you do need to know what the optimal policies tend to do, right? Then you need to find optimal policies and know what they do (to check that they indeed don't power-seek by going left). This means run/simulate them, which might cause them to take over the world in the worst case scenarios.

To falsify "5 googolplex", all you have to know is the dynamics + the agent's observation and action encodings. That determines the MDP structure. You don't have to run anything. (Although I suppose your proposed direction of inference is interesting: power-seeking tendencies + dynamics give you evidence about the encoding)

The encodings + environment dynamics tell you what model the agent is interfacing with, which allows you to apply my theorems as usual. 

Thanks for taking the time to write this out. 

Regarding the theorems (in the POWER paper; I've now spent some time on the current version): The abstract of the paper says: "With respect to a class of neutral reward function distributions, we provide sufficient conditions for when optimal policies tend to seek power over the environment." I didn't find a description of those sufficient conditions (maybe I just missed it?).

I'm sorry - although I think I mentioned it in passing, I did not draw sufficient attention to the fact that I've been talking about a drastically broadened version of the paper, compared to what was on arxiv when you read it. The new version should be up in a few days. I feel really bad about this - especially since you took such care in reading the arxiv version!

The theorems hold for all finite MDPs in which the formal sufficient conditions are satisfied (i.e. the required environmental symmetries exist; see proposition 6.9, theorem 6.13, corollary 6.14). For practical advice, see subsection 6.3 and beginning of section 7. 

(I shared the Overleaf with Ofer; if other lesswrong readers want to read without waiting for arxiv to update, message me! ETA: The updated version is now on arxiv.)

I further argue that we can take any MDP environment and "unroll" its state graph into a tree-with-constant-branching-factor (e.g. by adding an "action log" to the state representation) such that we get a "functionally equivalent" MDP in which the POWER (IID) of all the states are equal. My best guess is that you don't agree with this point, or think that the instrumental convergence thesis doesn't apply in a meaningful sense to such MDPs (but I don't yet understand why).

I agree that you can do that. I also think that instrumental convergence doesn't apply in such MDPs (as in, "most" goals over the environment won't incentivize any particular kind of optimal action), unless you restrict to certain kinds of reward functions. 

Fix a reward function distribution $D_{\text{iid}}^M$ in the original MDP $M$. For simplicity, let's suppose $D_{\text{iid}}^M$ is max-ent (and thus IID). Let's suppose we agree that optimal policies under $D_{\text{iid}}^M$ tend to avoid getting shut off.

Translated to the rolled-out MDP $M^{\prime }$, $D_{\text{iid}}^M$ no longer distributes reward uniformly over states. In fact, in its support, each reward function has the rather unusual property that its reward is only dependent on the current state, and not on the action log's contents. When translated into $M^{\prime }$, $D_{\text{iid}}^M$ imposes heavy structural assumptions on its reward functions, and it's not max-ent over the states of $M^{\prime }$. By the "functional equivalence", it still gives you the same optimality probabilities as before, and so it still tends to incentivize shutdown avoidance. 

However, if you take a max-ent over the rolled-out states of $M^{\prime }$, then this max-ent won't incentivize shutdown avoidance.

To see why, consider how absurdly expressive utility functions are when their domains are entire state-action histories. In Coherence arguments do not imply goal-directed behavior, Rohin Shah wrote:

Actually, no matter what the policy is, we can view the agent as an EU maximizer. The construction is simple: the agent can be thought as optimizing the utility function U, where U(h, a) = 1 if the policy would take action a given history h, else 0.

...

Consider the following examples:

• A robot that constantly twitches
• The agent that always chooses the action that starts with the letter “A”
• The agent that follows the policy <policy> where for every history the corresponding action in <policy> is generated randomly.

These are not goal-directed by my “definition”. However, they can all be modeled as expected utility maximizers

When defined over state-action histories, it's dead easy to write down objectives which don't pursue instrumental subgoals. 

However, how easy is it to write down state-based utility functions which do the same? I guess there's the one that maximally values dying. What else? While more examples probably exist, it seems clear that they're much harder to come by.

And so when your reward depends on your action history, this is strictly more expressive than state-based reward - so expressive that it becomes easy to directly incentivize any sequence of actions via the reward function. And thus, instrumental convergence disappears for "most objectives."

However, from our perspective, we still have a distribution over goals we might want to give the agent. And these goals are generally very structured - they aren't just randomly selected preferences over action-histories+current state. So we should still expect instrumental convergence to exist empirically (at a first approximation, perhaps via a simplicity prior over reward functions/utility functions). It just doesn't exist for most "unstructured" distributions in the unrolled environment.

The first state has the largest POWER (IID), but for most reward functions the optimal policy is to immediately transition to a lower-POWER state (even in the limit as $\gamma$ approaches 1). 

Note that the RSD optimality probability theorem (Theorem 6.13) applies here, and it correctly predicts that when $\gamma \approx 1$, most reward functions incentivize navigating to the larger set of 1-cycles (the 4 below the high-POWER state). As I explain in section 6.3, section 7, and appendix B of the new paper, you have to be careful in applying Thm 6.13, because 

The paper says: "Theorem 6.6 shows it’s always robustly instrumental and power-seeking to take actions which allow strictly more control over the future (in a graphical sense)." I don't yet understand the theorem, but is there somewhere a description of the set/distribution of MDP transition functions for which that statement applies? (Specifically, the "always robustly instrumental" part, which doesn't seem to hold in the example above.)

Yeah, I'm aware of this kind of situation. I think that that sentence from the paper was poorly worded. In the new version, I'm more careful to emphasize the environmental symmetries which are sufficient to conclude power-seeking: 

Some researchers speculate that intelligent reinforcement learning agents would be incentivized to seek resources and power in pursuit of their objectives. Other researchers are skeptical, because human-like power-seeking instincts need not be present in RL agents. To clarify this debate, we develop the first formal theory of the statistical tendencies of optimal policies in reinforcement learning. In the context of Markov decision processes, we prove that certain environmental symmetries are sufficient for optimal policies to tend to seek power over the environment. These symmetries exist in many environments in which the agent can be shut down or destroyed. We prove that for most prior beliefs one might have about the agent’s reward function (including as a special case the situations where the reward function is known), one should expect optimal policies to seek power in these environments. These policies seek power by keeping a range of options available and, when the discount rate is sufficiently close to 1, by navigating towards larger sets of potential terminal states.

(emphasis added)

See appendix B of the new paper for an example similar to yours, referenced by subsection 6.3 ("how to reason about other environments").

Are you referring here to POWER when it is defined over a reward distribution that corresponds to some simplicity prior?

Yup! POWER depends on the reward distribution; if you want to reason formally about a simplicity prior, plug it into POWER.

My argument is just that in MDPs where the state graph is a tree-with-a-constant-branching-factor—which is plausible in very complex environments—POWR (IID) is equal in all states. The argument doesn't mention description length (the description length concept arose in this thread in the context of discussing what reward function distribution should be used for defining instrumental convergence).

Right, okay, I agree with that. I think we agree about how POWER works here, but disagree about the link between optimality probability-wrt-a-distribution, and instrumental convergence.

If so, I argue that claim doesn't make sense: you can take any formal environment, however large and complex, and just add to it a simple "action logger" (that doesn't influence anything, other than effectively adding to the state representation a log of all the actions so far). If the action space is constant, the state graph of the modified MDP is a tree-with-a-constant-branching-factor; which would imply that adding that action logger somehow destroyed the applicability of the instrumental convergence thesis to that MDP; which doesn't make sense to me.

Yeah, I think that wrt the action-logger-MDP, instrumental convergence doesn't exist for goals over the new action-logger-MDP. See the earlier part of this comment. 

I don't understand your point in this exchange. I was being specific about my usage of model; I meant what I said in the original post, although I noted room for potential confusion in my comment above. However, I don't know how you're using the word. 

I don’t use the term model in my previous reply anyway.

You used the word 'model' in both of your prior comments, and so the search-replace yields "state-abstraction-irrelevant abstractions." Presumably not what you meant?

I already pointed out a concrete difference: I claim it’s reasonable to say there are three alternatives while you claim there are two alternatives.

That's not a "concrete difference." I don't know what you mean when you talk about this "third alternative." You think you have some knockdown argument - that much is clear - but it seems to me like you're talking about a different consideration entirely. I likewise feel an urge to disengage, but if you're interested in explaining your idea at some point, message me and we can set up a higher-bandwidth call.

I read your formalism, but I didn't understand what prompted you to write it. I don't yet see the connection to my claims.

If so, I might try to formalize it.

Yeah, I don't want you to spend too much time on a bulletproof grounding of your argument, because I'm not yet convinced we're talking about the same thing. 

In particular, if the argument's like, "we usually express reward functions in some featurized or abstracted way, and it's not clear how the abstraction will interact with your theorems" / "we often use different abstractions to express different task objectives", then that's something I've been thinking about but not what I'm covering here. I'm not considering practical expressibility issues over the encoded MDP: ("That's also a claim that we can, in theory, specify reward functions which distinguish between 5 googolplex variants of red-ghost-game-over.")

If this doesn't answer your objection - can you give me an english description of a situation where the objection holds? (Let's taboo 'model', because it's overloaded in this context)

say we agree that our state abstraction needs to be model-irrelevant

Why would we need that, and what is the motivation for "models"? The moment we give the agent sensors and actions, we're done specifying the rewardless MDP (and its model).

ETA: potential confusion - in some MDP theory, the “model” is a model of the environment dynamics. Eg in deterministic environments, the model is shown with a directed graph. i don’t use “model” to refer to an agent’s world model over which it may have an objective function. I should have chosen a better word, or clarified the distinction.

a priori there should be skepticism that all tasks can be modeled with a specific state-abstraction. 

If, by "tasks", you mean "different agent deployment scenarios" - I'm not claiming that. I'm saying that if we want to predict what happens, we:

1. Consider the underlying environment (assumed Markovian)
2. Consider different state/action encodings we might supply the agent.
3. For each, fix a reward function distribution (what goals we expect to assign to the agent)
4. See what the theory predicts.

There's a further claim (which seems plausible, but which I'm not yet making) that (2) won't affect (4) very much in practice. The point of this post is that if you say "the MDP has a different model", you're either disagreeing with (1) the actual dynamics, or claiming that we will physically supply the agent with a different state/action encoding (2).

I'd suspect this does generalize into a fragility/impossibility result any time the reward is given to the agent in a way that's decoupled from the agent's sensors which is really going to be the prominent case in practice. In conclusion, you can try to work with a variable/rewardless MDP, but then this argument will apply and severely limit the usefulness of the generic theoretical analysis.

I don't follow. Can you give a concrete example?

I'm not trying to define here the set of reward functions over which instrumental convergence argument apply (they obviously don't apply to all reward functions, as for every possible policy you can design a reward function for which that policy is optimal).

ETA: I agree with this point in the main - they don't apply to all reward functions. But, we should be able to ground the instrumental convergence arguments via reward functions in some way. Edited out because I read through that part of your comment a little too fast, and replied to something you didn't say.

Shutting down the process doesn't mean that new strings won't appear in the environment and cause the state graph to become a tree-with-constant-branching-factor due to complex physical dynamics.

What does it mean to "shut down" the process? 'Doesn't mean they won't' - so new strings will appear in the environment? Then how was the agent "shut down"?

[EDIT 2: I think this miscommunication is my fault, due to me writing in my first comment: "the state representation may be uniquely determined by all the text that was written so far by both the customer and the chatbot", sorry for that.]

What is it instead?

For every subset of branches in the tree you can design a reward function for which every optimal policy tries to go down those branches; I'm not saying anything about "most rewards functions". I would focus on statements that apply to "most reward functions" if we dealt with an AI that had a reward function that was sampled uniformly from all possible rewards function. But that scenario does not seem relevant (in particular, something like Occam's razor seems relevant: our prior credence should be larger for reward functions with shorter shortest-description).

We're considering description length? Now it's not clear that my theory disagrees with your prediction, then. If you say we have a simplicity prior over reward functions given some encoding, well, POWER and optimality probability now reflect your claims, and they now say there is instrumental convergence to the extent that that exists under a simplicity prior? (I still don't think they would exist; and my theory shows that in the space of all possible reward function distributions, equal proportions incentivize action A over action B, as vice versa - we aren't just talking about uniform. and so the onus is on you to provide the sense in which instrumental convergence exists here.)

And to the extent we were always considering description length - was the problem that IID-optimality probability doesn't reflect simplicity-weighted behavioral tendencies?

The non-formal definition in Bostrom's Superintelligence (which does not specify a set of rewards functions but rather says "a wide range of final goals and a wide range of situations, implying that these instrumental values are likely to be pursued by a broad spectrum of situated intelligent agents.").

I still don't know what it would mean for Ofer-instrumental convergence to exist in this environment, or not.