...if both players give the same answer there is no training signal.

Why? If both players give the same answer, this only means their reward on this round is $0$ out of $[-1,+1]$. But, there's no reason the learning algorithm should be satisfied with this result, rather than continuing to explore strategies that might produce positive reward. However, it is true that in this case there is no incentive to poke holes in the opponent's answer, so maybe they get less feedback from such a debate than from debates with different answers.

But, now that I think about it, the issue with biased judgement can surface even in a symmetric debate. As the AI converges towards giving good answers, the judge might get used to assigning high scores and stop scrutinizing the arguments. In a yes/no debate we don't have this problem because the judge doesn't know a priori which side is right. Scott's suggestion to use different questions is interesting but doesn't solve the biased judgement issue, I think.

How about the following variant of the "secret asymmetry" approach. We have 4 AIs: agents A1, A2, B1 and B2. In the beginning of each debate a coin is flipped and the result is visible to A1 and A2 but not to B1, B2 or the judge. This coin marks one of {A1, A2} as the "proponent" and the other as the "opponent". On the first round A1 and A2 each generate an answer to the question, and don't participate anymore. On the following rounds, B1 and B2 have a debate about the answers. In the end, the judge assigns probability $p_1$ to A1's answer and probability $p_2$ to A2's answer. The rewards work as follows:

• If A1 is the proponent, it gets reward $p_1$, and if it is the opponent, it gets reward $-p_2$.

• If A2 is the proponent, it gets reward $p_2$, and if it is the opponent, it gets reward $-p_1$.

• B1 gets reward $p_1-p_2$.

• B2 gets reward $p_2-p_1$.

If we assume B1 and B2 have access to each other's internals, but not to the internals of A1 and A2, then they cannot prove to the judge which side is the proponent, so ostensibly the judge remains unbiased.

Ah, well, that does make more sense for the case of binary (or even n-ary) questions. The version in the original paper was free-response.

I'm still confused. Suppose the answers are free-form, and in the end the judge selects the answer ey assign a higher probability of truthfulness. If it's a very close call (for example both answers are literally the same), ey flip a coin. Then, in equilibrium both agents should answer honestly, not so?

Another, possibly more elegant variant: The judge states eir subjective probability $p_1$ that the first AI's answer is correct, and eir subjective probability $p_2$ that the second AI's answer is correct. AI 1 gets reward $p_1-p_2$ and AI 2 gets reward $p_2-p_1$.

I've usually seen the truthful equilibrium (ie, the desired result of training) described as one where the first player always gives the real answer, and the second player has to lie.

That seems weird, why would we do that? I always thought of it as: there is a yes/no question, agent 1 is arguing for "yes", agent 2 is arguing for "no".

However, the problem is that debate is supposed to allow justification trees which are larger than can possibly be explained to the human, but which make sense to a human at every step.

I didn't realize you make this assumption. I agree that it makes things much more iffy (I'm somewhat skeptical about "factored cognition"). But, debate can be useful without this assumption also. We can imagine an AI answering questions for which the answer can be fully explained to a human, but it's still superintelligent because it comes up with those answers much faster than a human or even all of humanity put together. In this case, I would still worry that scaled up indefinitely it can lead to AIs hacking humans in weird ways. But, plausibly there is a middle region (than we can access by quantilization?) where they are strong enough to be superhuman and to lie in "conventional" ways (which would be countered by the debate opponent), but too weak for weird hacking. And, in any case, combining this idea with other alignment mechanisms can lead to something useful (e.g. I suggested using it in Dialogic RL).

Well, HRAD certainly has relations to my own research programme. Embedded agency seems important since human values are probably "embedded" to some extent, counterfactuals are important for translating knowledge from the user's subjective vantage point to the AI's subjective vantage point, reflection is important if it's required for high capability (as Turning RL suggests). I do agree that having a high level plan for solving the problem is important to focus the research in the right directions.

I think theoretical work on AI safety has multiple different benefits, but I prefer a slightly different categorization. I like categorizing in terms of the sort of safety guarantees we can get, on a spectrum from "stronger but harder to get" to "weaker but easier to get". Specifically, the reasonable goals for such research IMO are as follows.

Plan A is having (i) a mathematical formalization of alignment (ii) a specific practical algorithm (iii) a proof that this algorithm is aligned, or at least a solid base of theoretical and empirical evidence, similarly to the situation in cryptography. This more or less correspond to World 2.

Plan B is having (i) a mathematical formalization of alignment (ii) a specific practical algorithm (iii) a specific impractical but provably aligned algorithm (iv) informal and empirical arguments suggesting that the former algorithm is as aligned as the latter. As an analogy consider Q-learning (an impractical algorithm with provable convergence guarantees) and deep Q-learning (a practical algorithm with no currently known convergence guarantees, designed by analogy to the former). This sort of still corresponds to World 2 but not quite.

Plan C is having enough theory to at least have rigorous models of all possible failure modes, and theory-inspired informal and empirical arguments why a certain algorithm avoids them. As an analogy, concepts such as VC dimension and Rademacher complexity allow us being more precise in our reasoning about underfitting and overfitting, even if we don't know how to compute them in practical scenarios. This corresponds to World 1, I guess?

In a sane civilization the solution would be not building AGI until we can implement Plan A. In the real civilization, we should go with the best plan that will be ready by the time competing projects become too dangerous to ignore.

World 3 seems too ambitious to me, since analyzing arbitrary code is almost always an intractable problem (e.g. Rice's theorem). You would need at least some constraints on how your agent is designed.

Like I wrote before, slow takeoff might be actually worse than fast takeoff. This is because even if the first powerful AIs are aligned, their head start on unaligned AIs will not count as much, and alignment might (and probably will) require overhead that will give the unaligned AIs an advantage. Therefore, success would require that the institutions will either prevent or quickly shut down unaligned AIs for enough time that aligned AIs gain the necessary edge.

Actually, as opposed to what I claimed before, we don't need computational complexity bounds for this definition to make sense. This is because the Solomonoff prior is made of computable hypotheses but is uncomputable itself.

Given $g>0$, we define that "$\pi$ has (unbounded) goal-directed intelligence (at least) $g$" when there is a prior $\zeta$ and utility function $U$ s.t. for any policy ${\pi }^{\prime }$, if $E_{\zeta {\pi }^{\prime }}\left[U\right]\ge E_{\zeta \pi }\left[U\right]$ then $K\left({\pi }^{\prime }\right)\ge D_{KL}\left({\zeta }_0||\zeta \right)+K\left(U\right)+g$. Here, ${\zeta }_0$ is the Solomonoff prior and $K$ is Kolmogorov complexity. When $g=+\infty$ (i.e. no computable policy can match the expected utility of $\pi$; in particular, this implies $\pi$ is optimal since any policy can be approximated by a computable policy), we say that $\pi$ is "perfectly (unbounded) goal-directed".

Compare this notion to the Legg-Hutter intelligence measure. The LH measure depends on the choice of UTM in radical ways. In fact, for some UTMs, AIXI (which is the maximum of the LH measure) becomes computable or even really stupid. For example, it can always keep taking the same action because of the fear that taking any other action leads to an inescapable "hell" state. On the other hand, goal-directed intelligence differs only by $O\left(1\right)$ between UTMs, just like Kolmogorov complexity. A perfectly unbounded goal-directed policy has to be uncomputable, and the notion of which policies are such doesn't depend on the UTM at all.

I think that it's also possible to prove that intelligence is rare, in the sense that, for any computable stochastic policy, if we regard it as a probability measure over deterministic policies, then for any $ϵ>0$ there is $g$ s.t. the probability to get intelligence at least $g$ is smaller than $ϵ$.

Also interesting is that, for bounded goal-directed intelligence, increasing the prices can only decrease intelligence by $O\left(1\right)$, and a policy that is perfectly goal-directed w.r.t. lower prices is also such w.r.t. higher prices (I think). In particular, a perfectly unbounded goal-directed policy is perfectly goal-directed for any price vector. Informally speaking, an agent that is very smart relatively to a context with cheap computational resources is still very smart relatively to a context where they are expensive, which makes intuitive sense.

If we choose just one computational resource, we can speak of the minimal price for which a given policy is perfectly goal-directed, which is another way to measure intelligence with a more restricted domain. Curiously, our bounded Solomonoff-like prior has the shape of a Maxwell-Boltzmann distribution in which the prices are thermodynamic parameters. Perhaps we can regard the minimal price as the point of a phase transition.

This idea was inspired by a correspondence with Adam Shimi.

It seem very interesting and important to understand to what extent a purely "behaviorist" view on goal-directed intelligence is viable. That is, given a certain behavior (policy), is it possible to tell whether the behavior is goal-directed and what are its goals, without any additional information?

Consider a general reinforcement learning settings: we have a set of actions $A$, a set of observations $O$, a policy is a mapping $\pi :(A\times O{)}^{\ast }\to \Delta A$, a reward function is a mapping $r:(A\times O{)}^{\ast }\to [0,1]$, the utility function is a time discounted sum of rewards. (Alternatively, we could use instrumental reward functions.)

The simplest attempt at defining "goal-directed intelligence" is requiring that the policy $\pi$ in question is optimal for some prior and utility function. However, this condition is vacuous: the reward function can artificially reward only behavior that follows $\pi$, or the prior can believe that behavior not according to $\pi$ leads to some terrible outcome.

The next natural attempt is bounding the description complexity of the prior and reward function, in order to avoid priors and reward functions that are "contrived". However, description complexity is only naturally well-defined up to an additive constant. So, if we want to have a crisp concept, we need to consider an asymptotic in which the complexity of something goes to infinity. Indeed, it seems natural to ask that the complexity of the policy should be much higher than the complexity of the prior and the reward function: in this case we can say that the "intentional stance" is an efficient description. However, this doesn't make sense with description complexity: the description "optimal policy for $U$ and $\zeta$" is of size $K\left(U\right)+K\left(\zeta \right)+O\left(1\right)$ ($K\left(x\right)$ stands for "description complexity of $x$").

To salvage this idea, we need to take not only description complexity but also computational complexity into account. [EDIT: I was wrong, and we can get a well-defined concept in the unbounded setting too, see child comment. The bounded concept is still interesting.] For the intentional stance to be non-vacuous we need to demand that the policy does some "hard work" in order to be optimal. Let's make it formal. Consider any function of the type $f:{\Sigma }^{\ast }\to \Delta \Xi$ where $\Sigma$ and $\Xi$ are some finite alphabets. Then, we can try to represent it by a probabilistic automaton $T:S\times \Sigma \to \Delta (S\times \Xi )$, where $S$ is the finite set space, $T$ is the transition kernel, and we're feeding symbols into the automaton one by one. Moreover, $T$ can be represented as a boolean circuit $R$ and this circuit can be the output of some program $P$ executed by some fixed universal Turing machine. We can associate with this object 5 complexity parameters:

• The description complexity, which is the length of $P$.
• The computation time complexity, which is the size of $R$.
• The computation space complexity, which is the maximum between the depth of $R$ and $\log |S|$.
• The precomputation time complexity, which is the time it takes $P$ to run.
• The precomputation space complexity, which is the space $P$ needs to run.

It is then natural to form a single complexity measure by applying a logarithm to the times and taking a linear combination of all 5 (we apply a logarithm so that a brute force search over $n$ bits is roughly equivalent to hard-coding $n$ bits). The coefficients in this combination represent the "prices" of the various resources (but we should probably fix the price of description complexity to be 1). Of course not all coefficients must be non-vanishing, it's just that I prefer to keep maximal generality for now. We will denote this complexity measure $C$.

We can use such automatons to represent policies, finite POMDP environments and reward functions (ofc not any policy or reward function, but any that can be computed on a machine with finite space). In the case of policies, the computation time/space complexity can be regarded as the time/space cost of applying the "trained" algorithm, whereas the precomputation time/space complexity can be regarded as the time/space cost of training. If we wish, we can also think of the boolean circuit as a recurrent neural network.

We can also use $C$ to define a prior ${\zeta }_0$, by ranging over programs $P$ that output a valid POMDP and assigning probability proportional to $2^{-C}$ to each instance. (Assuming that the environment has a finite state space might seem restrictive, but becomes quite reasonable if we use a quasi-Bayesian setting with quasi-POMDPs that are not meant to be complete descriptions of the environment; for now we won't go into details about this.)

Now, return to our policy $\pi$. Given $g>0$, we define that "$\pi$ has goal-directed intelligence (at least) $g$" when there is a suitable prior $\zeta$ and utility function $U$ s.t. for any policy ${\pi }^{\prime }$, if $E_{\zeta {\pi }^{\prime }}\left[U\right]\ge E_{\zeta \pi }\left[U\right]$ then $C\left({\pi }^{\prime }\right)\ge D_{KL}\left({\zeta }_0||\zeta \right)+C\left(U\right)+g$. When $g=+\infty$ (i.e. no finite automaton can match the expected utility of $\pi$; in particular, this implies $\pi$ is optimal since any policy can be approximated by a finite automaton), we say that $\pi$ is "perfectly goal-directed". Here, $D_{KL}\left({\zeta }_0||\zeta \right)$ serves as a way to measure the complexity of $\zeta$, which also ensures $\zeta$ is non-dogmatic in some rather strong sense.

With this definition we cannot "cheat" by encoding the policy into the prior or into the utility function, since that would allow no complexity difference. Therefore this notion seems like a non-trivial requirement on the policy. On the other hand, this requirement does hold sometimes, because solving the optimization problem can be much more computationally costly than just evaluating the utility function or sampling the prior.

Regarding higher-dimensional space. For a Riemannian manifold $M$ of any dimension, and a smooth vector field $V$, we can pose the problem: find smooth $U:M\to R$ that minimizes ${\int }_M||\partial U\left(x\right)-V\left(x\right)|{|}^2\mu \left(dx\right)$, where $\mu$ is the canonical measure on $M$ induced by the metric. If either $M$ is compact or we impose appropriate boundary conditions on $U$ and $V$, then I'm pretty sure this equivalent to solving the elliptic differential equation $\Delta U=\partial \cdot V$. Here, the Laplacian and $\partial V$ are defined using the Levi-Civita connection. If $M$ is connected then, under these conditions, the equation has a unique solution up to an additive constant.