# Mathematical Measures of Optimization Power

In explorations of AI risk, it is helpful to formalize concepts. One particularly important concept is intelligence. How can we formalize it, or better yet, measure it? “Intelligence” is often considered mysterious or is anthropomorphized. One way to [taboo](/lw/nu/taboo_your_words/) “intelligence” is to talk instead about optimization processes. An optimization process (OP, also optimization power) selects some futures from a space of possible futures. It does so according to some criterion; that is, it optimizes for _something_. Eliezer Yudkowsky [spends](/lw/tx/optimization/) [a few](/lw/v7/expected_creative_surprises/) [of the](/lw/v8/belief_in_intelligence/) [sequence](/lw/v9/aiming_at_the_target/) [posts](/lw/va/measuring_optimization_power/) [discussing](/lw/vb/efficient_crossdomain_optimization/) [the nature](/lw/vc/economic_definition_of_intelligence/) and importance of this concept for understanding AI risk. In them, he informally describes a way to measure the power of an OP. We consider mathematical formalizations of this measure.

Here's EY's [original description](/lw/tx/optimization/) of his measure of OP.

Put a measure on the state space - if it's discrete, you can just count. Then collect all the states which are equal to or greater than the observed outcome, in that optimization process's implicit or explicit preference ordering. Sum or integrate over the total size of all such states. Divide by the total volume of the state space. This gives you the power of the optimization process measured in terms of the improbabilities that it can produce - that is, improbability of a random selection producing an equally good result, relative to a measure and a preference ordering.

If you prefer, you can take the reciprocal of this improbability (1/1000 becomes 1000) and then take the logarithm base 2. This gives you the power of the optimization process in bits.

Let's say that at time ![](http://www.codecogs.com/png.latex?t=0) we have a formalism to specify all possible world states ![](http://www.codecogs.com/png.latex?w \in W) at some future time ![](http://www.codecogs.com/png.latex?t=1). Perhaps it is a list of particle locations and velocities, or perhaps it is a list of all possible universal wave functions. Or maybe we're working in a limited domain, and it's a list of all possible next-move chess boards. Let's also assume that we have a [well-justified prior](/lw/dhg/an_intuitive_explanation_of_solomonoff_induction/) ![](http://www.codecogs.com/png.latex?p(w)) over these states being the next ones to occur in the absence of an OP (more on that later).

We order ![](http://www.codecogs.com/png.latex?W) according to the OP's preferences. For the moment, we actually don't care about the density, or “measure” of our ordering. Now we have a probability distribution over ![](http://www.codecogs.com/png.latex?W). The integral from ![](http://www.codecogs.com/png.latex?w=a) to ![](http://www.codecogs.com/png.latex?w=b) over this represents the probability that the worldstate at ![](http://www.codecogs.com/png.latex?t=1) will be better than ![](http://www.codecogs.com/png.latex?a), and worse than ![](http://www.codecogs.com/png.latex?b). When time continues, and the OP acts to bring about some worldstate ![](http://www.codecogs.com/png.latex?\omega), we can calculate the probability of an equal or better outcome occurring;

![](http://www.codecogs.com/png.latex?\int_\omega^\infty p(w)dw)

This is a simple generalization of what EY describes above. Here are some things I am confused about.

Finding a specification for all possible worldstates is hard, but it's been done before. There are many ways to reasonably represent this. What I can't figure out is how to specify possible worldstates “in the absence of an OP”. This phrase hides tons of complexity. How can we formally construct this counterfactual? Is the matter that composes the OP no longer present? Is it present but “not acting”? What constitutes a null action? Are we considering the expected worldstate distribution as if the OP never existed? If the OP is some kind of black-box AI agent, it's easier to imagine this. But if the OP is evolution, or a forest fire, it's harder to imagine. Furthermore, is the specification [dualist](http://en.wikipedia.org/wiki/Cartesian_dualism), or is the agent part of the worldstates? If it's dualist, this is a fundamental falseness which can have lots of bad implications. If the agent is part of the worldstates, how do we represent them “in absence of an OP”?

But for the rest of this article, let's pretend we have such a specification. There's also a loss from ignoring the [cardinal utility](http://en.wikipedia.org/wiki/Cardinal_utility) of the worldstates. Let's say you have the two distributions of utility over sets ![](http://www.codecogs.com/png.latex?W), representing two different OPs. In both, the OP choose a ![](http://www.codecogs.com/png.latex?w) with the same utility ![](http://www.codecogs.com/png.latex?u). The distributions are the same on the left side of ![](http://www.codecogs.com/png.latex?u), and the second distribution has a longer tail on the right. It seems like the OP in distribution 1 was more impressive; the second OP missed all the available higher utility. We could make the expected utility of the second distribution arbitrarily high, while maintaining the same fraction of probability mass above the achieved worldstate. Conversely, we could instead extend the left tail of the second distribution, and say that the second OP was more impressive because it managed to avoid all the bad worlds.

Perhaps it is more natural to consider two distributions; the distribution of utility over entire world _futures_ assuming the OP isn't present, versus the distribution after the OP takes its action. So instead of selecting a single possibility with certainty, the probabilities have just shifted. 

How should we reduce this distribution shift to a single number which we call OP? Any shift of probability mass upwards in utility should increase the measure of OP, and vice versa. I think also that an increase in the [expected](http://en.wikipedia.org/wiki/Expected_value) [utility](http://en.wikipedia.org/wiki/Expected_utility) (EU) of these distributions should be measured as a positive OP, and vice versa. EU seems like the critical metric to use. Let's generalize a little further, and say that instead of measuring OP between two points in time, we let the time difference go to zero, and measure instantaneous OP. Therefore we're interested in some equation which has the same sign as

![](http://www.codecogs.com/png.latex?\frac{dEU}{dt} = EU').

Besides that, I'm not exactly sure which specific equation should equal OP. I seem to have two contradicting desires;

1a) The sign of ![](http://www.codecogs.com/png.latex?EU') should be the sign of the OP.

  b) Negative ![](http://www.codecogs.com/png.latex?EU) and ![](http://www.codecogs.com/png.latex?EU') should be possible.

2) Constant positive OP should imply exponentially increasing ![](http://www.codecogs.com/png.latex?EU).

Criterion 1) feels pretty obvious. Criterion 2) feels like a recognition of what is “natural” for OPs; to improve upon themselves, so that they can get better and better returns. The simplest differential equation that represents positive feedback yields exponentials, and is used across many domains because of its universal nature.

![](http://www.codecogs.com/png.latex?y' = cy)

This intuition certainly isn't anthropocentric, but it might be this-universe biased. I'd be interested in seeing if it is natural in other computable environments.

If we just use ![](http://www.codecogs.com/png.latex?OP = EU'), then criterion 2) is not satisfied. If we use ![](http://www.codecogs.com/png.latex?OP = \log EU'), then decreases in EU are not defined, and constant EU is negative infinite OP, violating 1). [If we use](http://en.wikipedia.org/wiki/Logarithmic_derivative) ![](http://www.codecogs.com/png.latex?OP = EU'/EU), then 2) is satisfied, but negative and decreasing EU give positive OP, violating 1a). If we use ![](http://www.codecogs.com/png.latex?OP = EU''/EU'), then 2) is still satisfied, but ![](http://www.codecogs.com/png.latex?EU = at + b) gives ![](http://www.codecogs.com/png.latex?OP = 0), violating 1a). Perhaps the only consistent equation would be ![](http://www.codecogs.com/png.latex?OP = EU'/|EU|). But seriously, who uses absolute values? I can't recall a fundamental equation that relied on them. They feel totally ad hoc. Plus, there's this weird singularity at ![](http://www.codecogs.com/png.latex?EU = 0). What's up with that?

Classically, utility is invariant up to [positive affine transformations](http://en.wikipedia.org/wiki/Cardinal_utility). Criterion 1) respects this because the derivative removes the additive constant, but 2) doesn't. It is still scale invariant, but it has an intrinsic zero. This made me consider the nature of “zero utility”. At least for humans, there is an intuitive sign to utility. We wouldn't say that stubbing your toe is 1,000,000 utils, and getting a car is 1,002,000 utils. It seems to me, especially after reading [Omohundro's “Basic AI Drives”](http://selfawaresystems.com/2007/11/30/paper-on-the-basic-ai-drives/), that there is in some sense an intrinsic zero utility for all OPs.

All OPs need certain initial conditions to even exist. After that, they need resources. AIs need computer hardware and energy. Evolution needed certain chemicals and energy. Having _no_ resources makes it impossible, in general, to do anything. If you have literally zero resources, you are not a "thing" which "does". So that is a type of intrinsic zero utility. Then what would having negative utility mean? It would mean the OP anti-exists. It's making it even less likely for it to be able to _start_ working toward its utility function. What would exponentially decreasing utility mean? It would mean that it is a constant OP for the _negative_ of the utility function that we are considering. So, it doesn't really have negative optimization power; if that's the result of our calculation, we should negate the utility function, and say it has positive OP. And that singularity at ![](http://www.codecogs.com/png.latex?EU = 0)? When you go from the positive side, getting closer and closer to 0 is _really bad_, because you're destroying the last bits of your resources; your last chance of doing any optimization. And going from negative utility to positive is infinite impressive, because you bootstrapped from optimizing away from your goal to optimizing toward your goal.

So perhaps we should drop the part of 1b) that says negative EU can exist. Certainly world-states can exist that are terrible for a given utility function, but if an OP with that utility function exists, then the expected utility of the future is positive.

If this is true, then it seems there is more to the concept of utility than the [von Neumann-Morgenstern](http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem) axioms.

How do people feel about criterion 2), and my proposal that ![](http://www.codecogs.com/png.latex?OP = EU'/|EU|) ?