[hodgehog11]

Fine, that's fair. I believe the statement that you made is countered by my claim, which is:

Theorem. For any tolerance epsilon > 0, there exists a transformer neural network of sufficient size that follows, up to the factor epsilon, the policy that most optimally achieves arbitrary goals in arbitrary stochastic environments.

Proof (sketch). For any stochastic environment with a given goal, there exists a model that maximizes expected return under this goal (not necessarily unique, but it exists). From Solomonoff's convergence theorem (Theorem 3.19 in [1]), Bayes-optimal predictors under the universal Kolmogorov prior converge with increasing context to this model. Consequently, there exists an agent (called the AIXI agent) that is Pareto-optimal for arbitrary goals (Theorem 5.23 in [1]). This agent is a sequence-to-sequence map with some mild regularity, and satisfies the conditions of Theorem 3 in [2]. From this universal approximation theorem (itself proven in Appendices B and C in [2]), there exists a transformer neural network of a sufficient size that replicates the AIXI agent up to the factor epsilon.

This is effectively the argument made in [3], although I'm not fond of their presentation. Now, practitioners still cry foul because existence doesn't guarantee a procedure to find this particular architecture (this is the constructive bit). This is where the neural scaling law comes in. The trick is to work with a linearization of the network, called the neural tangent kernel; it's existence is guaranteed from Theorem 7.2 of [4]. The NTK predictors are also universal and are a subset of the random feature models treated in [5], which derives the neural scaling laws for these models. Extrapolating these laws out as per [6] for specific tasks shows that the "floor" is always below human error rates, but this is still empirical because it works with the ill-defined definition of superintelligence that is "better than humans in all contexts".

[1] Hutter, M. (2005). Universal artificial intelligence: Sequential decisions based on algorithmic probability. Springer Science & Business Media.

[2] https://arxiv.org/abs/1912.10077

[3] https://openreview.net/pdf?id=Vib3KtwoWs

[4] https://arxiv.org/abs/2006.14548

[5] https://arxiv.org/abs/2210.16859

[6] https://arxiv.org/abs/2001.08361

[measurablefunc]

How do you reconcile that w/ the fact that optimal probabilistic planning¹ is actually undecidable?

¹https://www.sciencedirect.com/science/article/pii/S000437020...

[hodgehog11]

Good question. It's because we don't need to be completely optimal in practice, only epsilon close to it. Optimality is undecidable, but epsilon close is not, and that's what the claim says that NNs can provide.