by jhanschoo | 2025-10-25 11:37:23+00:00
I'm going to go into my own perspective of it; it is not reflective of what it discusses.

The linked multimedia article gives a narrative of intelligent systems, but Hutter and AIXI give a (noncomputable) definition of an ideal intelligent agent. The book situates the definitions in a reinforcement learning setting, but the core idea is succinctly expressed in a supervised learning setting.

The idea is this: given a dataset with yes/no labels (and no repeats in the features), and a commonsense encoding of turing machines as a binary string, the ideal map from input to probability distribution model is defined by

1. taking all turing machines that decide the input space and agree with the labels of the training set, and

2. the inference algorithm is that on new input, the output is exactly the distribution by counting all such machines that assent vs. reject the input, with their mass being weighted by the reciprocal of 2 to the power of the length, then the weighted counts normalized. This is of course a noncomputable algorithm.

The intuition is that if a simply-patterned function from input to output exists in the training set, then there is a simply/shortly described turing machine that captures that function, and so that machine's opinion on the new input is given a lot of weight. But there exist plausible more complex patterns, and we also consider them.

What I like about this abstract definition is that it is not in reference to "human intelligence" or "animal intelligence" or some other anthropic or biological notion. Rather, you can use these ideas anytime you isolate a notion of agent from an environment/data, and want to evaluate how the agent interacts/predicts intelligently against novel input from the environment/data, under the limited input that it has. It is a precise formalization of inductive thinking / Occam's razor.

Another thing I like about this is that it gives theoretical justification for the double-descent phenomenon. It is a (noncomputable) algorithm to give the best predictor, but it is defined in reference to the largest hypothesis space (all turing machines that decide on the input space). It suggests that whereas prior ML methods got better results with architectures that are carefully designed to make bad predictors unrepresentable, it is also not idle, if you have a lot of computational resources, to have an architecture that defines an expressive hypothesis space, and instead softly prioritizing simpler hypotheses through the learning algorithms (e.g. an approximation of which is regularization). This allows your model to learn complex patterns defined by the data that you did not anticipate, if that evidence in the data justifies it, whereas a small, biased hypothesis space would not be able to represent such a pattern if not anticipated but significant.

Note that under this definition, you might want to talk about a situation where the observations are noisy but you want to learn the trend of it without the noise. You can adapt the definition to be over noisy input by for example accompanying each input with distinct sequence numbers or random salts, then consider the marginal distribution for numbers/salts not in the training set (there are some technical issues of convergence, but the general approach is feasible), and this models the noise distribution as well.