by Vadim Kosoy 262 days ago | link | parent After thinking some more, maybe the following is natural way towards formalizing the optimism condition. Let $$H$$ be the space of hypotheses and $$\xi_0 \in \Delta H$$ be the “unbiased” universal prior. Given any $$\zeta \in \Delta H$$, we denote $$\hat{\zeta} = E_{\mu \sim \zeta}[\mu]$$, i.e. the environment resulting from mixing the environments in the belief state $$\zeta$$. Given an environment $$\mu$$, let $$\pi^\mu$$ be the Bayes-optimal policy for $$\mu$$ and $$\pi^\mu_\theta$$ the perturbed Bayes-optimal policy for $$\mu$$, where $$\theta$$ is a perturbation parameter. Here, “perturbed” probably means something like softmax expected utility, but more thought is needed. Then, the “optimistic” prior $$\xi$$ is defined as a solution to the following fixed point equation: $\xi(\mu) = Z^{-1} \xi_0(\mu) \exp(\beta(E_{\mu\bowtie\pi^{\hat{\xi}}_\theta}[U]-E_{\mu\bowtie\pi^\mu}[U]))$ Here, $$Z$$ is a normalization constant and $$\beta$$ is an additional parameter. This equation defines something like a softmax Nash equilibrium in a cooperative game of two players where, one player chooses $$\mu$$ (so that $$\xi$$ is eir mixed strategy), another player chooses $$\pi$$ and the utility is minus regret (alternatively, we might want to choose only Pareto efficient Nash equilibria). The parameter $$\beta$$ controls optimism regarding the ability to learn the environment, whereas the parameter $$\theta$$ represents optimism regarding the presence of slack: ability to learn despite making some errors or random exploration (how to choose these parameters is another question). Possibly, the idea of exploring the environment “layer by layer” can be recovered from combining this with hierarchy assumptions.

 by Jessica Taylor 261 days ago | link This seems like a hack. The equilibrium policy is going to assume that the environment is good to it in general in a magical fashion, rather than assuming the environment is good to it in the specific ways we should expect given our own knowledge of how the environment works. It’s kind of like assuming “things magically end up lower than you expected on priors” instead of having a theory of gravity. I think there is something like a theory of gravity here. The things I would note about our universe that make it possible to avoid a lot of traps include: Physical laws are symmetric across spacetime. Physical laws are spacially local. The predictable effects of a local action are typically local; most effects “dissipate” after a while (e.g. into heat). The butterfly effect is evidence for this rather than against this, since it means many effects are unpredictable and so can be modeled thermodynamically. When small changes have big and predictable effects (e.g. in a computer), there is often agentic optimization power towards the creation and maintenance of this system of effects, and in these cases it is possible for at least some agents to understand important things about how the system works. Some “partially-dissipated” effects are statistical in nature. For example, an earthquake hitting an area has many immediate effects, but over the long term the important effects are things like “this much local productive activity was disrupted”, “this much local human health was lost”, etc. You have the genes that you do because evolution, which is similar to a reinforcement learning algorithm, believed that these genes would cause you to survive and reproduce. If we construct AI systems, we will give them code (including a prior) that we expect to cause them to do something useful for us. In general, the agency of an agent’s creator should affect the agent’s beliefs. If there are many copies of an agent, and successful agents are able to repurpose the resources of unsuccessful ones, then different copies can try different strategies; some will fail but the successful ones can then repurpose their resources. (Evolution can be seen as a special case of this) Some phenemona have a “fractal” nature, where a small thing behaves similar to a big thing. For example, there are a lot of similarities between the dynamics of a nation and the dynamics of a city. Thus small things can be used as models of big things. If your interests are aligned with those of agents in your local vicinity, then they will mostly try to help you. (This applies to parents making their children’s environment safe) I don’t have an elegant theory yet but these observations seem like a reasonable starting point for forming one. reply
 by Vadim Kosoy 260 days ago | link I think that we should expect evolution to give us a prior that is a good lossy compression of actual physics (where “actual physics” means, those patterns the universe has that can be described within our computational complexity bounds). Meaning that, on the one hand it should be low description complexity (otherwise it will be hard for evolution to find it), and on the other hand it should be assign high probability to the true environment (in other words, the KL divergence of the true environment from the prior should be small). And also it should be approximately learnable, otherwise it won’t go from assigning high probability to actually performing well. The principles you outlined seem reasonable overall. Note that the locality/dissipation/multiagent assumptions amount to a special case of “the environment is effectively reversible (from the perspective of the human species as a whole) as long as you don’t apply too much optimization power” (“optimization power” probably translates to divergence from some baseline policy plus maybe computational complexity considerations). Now, as you noted before, actual macroscopic physics is not reversible, but it might still be effectively reversible if you have a reliable long-term source of negentropy (like the sun). Maybe we can also slightly relax them by allowing irreversible changes as long as they are localized and the available space is sufficiently big. “If we construct AI systems, we will give them code (including a prior) that we expect to cause them to do something useful for us. In general, the agency of an agent’s creator should affect the agent’s beliefs” is essentially what DRL does: allows transferring our knowledge to the AI without hard-coding it by hand. “When small changes have big and predictable effects (e.g. in a computer), there is often agentic optimization power towards the creation and maintenance of this system of effects, and in these cases it is possible for at least some agents to understand important things about how the system works” seems like it would allow us to go beyond effective reversibility, but I’m not sure how to formalize it or whether it’s a justified assumption. One way towards formalizing it is, the prior is s.t. studying the initial state approximate communication class allows determining the entire environment, but this seems to point at a very broad class of approximately learnable priors w/o specifying a criterion how to choose among them. Another principle that we can try to use is, the ubiquity of analytic functions. Analytic functions have the property that, knowing the function in a bounded domain allows extrapolating it everywhere. This is different from allowing arbitrary computable functions which may have “if” clauses, so that studying the function in a bounded domain is never enough to be sure about its behavior outside it. In particular, this line of inquiry seems relatively easy to formalize using continuous MDPs (although we run into the problem that finding the optimal policy is infeasible, in general). Also, it might have something to do with the effectiveness of neural networks (although the popular ReLU response function is not analytic). reply

### NEW DISCUSSION POSTS

[Note: This comment is three
 by Ryan Carey on A brief note on factoring out certain variables | 0 likes

There should be a chat icon
 by Alex Mennen on Meta: IAFF vs LessWrong | 0 likes

Apparently "You must be
 by Jessica Taylor on Meta: IAFF vs LessWrong | 1 like

There is a replacement for
 by Alex Mennen on Meta: IAFF vs LessWrong | 1 like

Regarding the physical
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

I think that we should expect
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

I think I understand your
 by Jessica Taylor on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

This seems like a hack. The
 by Jessica Taylor on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

After thinking some more,
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

Yes, I think that we're
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

My intuition is that it must
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

To first approximation, a
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

Actually, I *am* including
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

Yeah, when I went back and
 by Alex Appel on Optimal and Causal Counterfactual Worlds | 0 likes

> Well, we could give up on
 by Jessica Taylor on The Learning-Theoretic AI Alignment Research Agend... | 0 likes