I got annoyed at yet another primer that does not put information theory as foundational, so I decided to write my own.

The Little Pamphlet of AI Foundations: An Unintuitive Philosophical Primer

Losses

Intelligence is the ability to explain phenomena. One is more intelligent where they can better explain phenomena.

A phenomenon is any correspondence between data, and an explanation is an encoding of that correspondence. There is an infinite amount of data so an intelligence must be a program that generates new explanations when it sees new data.

A program is a string of bits run by an interpreter. Since strings of bits are data, programs may be generated by other programs, and a program's interpreter may itself be another program or intelligence. This recursion eventually bottoms out in physical ware. Human and artificial intelligences differ mainly in their ware, though human intelligences also have an impredicative recursion property that artificial intelligences are only expected to gain in 2027.

At a low level of intelligence, a calculator takes in numbers and operations through electrical impulses, moves electrons through a series of transistors, which creates new outputs to be displayed on the screen.

At a higher level, a neural network training algorithm takes in examples and outputs weights, usually through preconditioned gradient descent.

At an even higher level, a language model takes in words and outputs possible continuations. To convert this to the correct continuation, an additional error correction code must be provided for every word it outputs. This unexplained gap in the language model, or 'loss', dominates in the limit of a large number of words. To consistently provide short explanations, a model must have a small loss.

At the highest level, an explanation of an environment must encode all data in that environment, including any intelligences. An intelligence encompasses a larger share of its environment if it is harder to encode. Under a fixed loss budget, the hardest intelligence to encode provides explanations weighted exponentially by their length. This is the origin of the common mantras that a short explanation is exponentially more likely to be the correct one, colloquially known as Occam's razor, Solomonoff's lightsaber, and Kolmogorov's complexity.

The important takeaway is that a loss represents the average size of an error correction code, and a smaller loss is better.

Architecture

An intelligence is a program that generates encodings of data correspondences. Choosing an architecture with better data topologies enables it to do so more efficiently.

A data topology consists of data and possible connections between the data, with computational costs determined by the architecture. As computational resources grow superexponentially, the goal of architecting is generally to lower the computational complexity of creating connections, encapsulated colloquially as 'The Bitter Lesson'. To give a few examples:

1. The limited liability corporation lowered the personal risk of starting a business from the size of the business to a fixed cost, making economic connections cheaper to form.

2. Residual networks directly form connections between neuron data and output gradients, bypassing the exponentially weakening connections that previously made deep learning impossible.

3. Flow transport models improve upon continuous normalizing flows by giving target distribution data along the entire generation process.

4. Quadratic attention connects every input datum to every input datum, instead of relying on fringe connections from the nonlinearity in activation functions. Linear attention further lowers this cost by adaptively selecting data to go into a memory bank, and only forming connections with the memory bank.

5. A gradient descent preconditioner modifies the weight flow density, so distributions of weights are squeezed closer together or spread further apart. In the extreme, this no longer looks like gradient descent and looks closer to an intelligence directly writing weight updates based on new examples.

Learning Theory

A perfect intelligence provides explanations weighted exponentially by their length. A Bayesian neural network is a network generated by a perfect intelligence as an explanation for a dataset, and its distribution of weights—its 'partition function'—will be concentrated in regions with exponentially low loss. Analogous to physics, much can be gleaned from the partition function.

In adversarial contexts such as generative adversarial networks or reinforcement learning, examples rotate through a cyclic group of six elements, being positive, neutral, or negative examples depending on what the other intelligences have learned. This creates an anyonic partition function for each learnable feature and the familiar sextessential steep upwards loss curve before a plateau. Similarly, an alternating cycle of learning and forgetting gives the fermionic partition function and a loss curve colloquially referred to as grokking. Most learning is just learning and looks bosonic in nature: losses quickly go down, then settle into an exponential curve.

In the limit of a large number of training examples, the network's loss dominates the size of the explanation, but it still takes a logarithmically increasing number of bits to describe the path the weights take along the weight flow. To be precise, the explanation length grows as

  nL(w) + λ log n − (m − 1) log log n

where n is the number of training examples, L is the network's loss, λ the real log canonical threshold, and m its multiplicity. It is possible to subsume the logarithmic terms into the loss by adding normal noise to the weights after each training iteration with variance dηT/λn, where η is the learning rate, T the temperature, and d the model dimension. In the nonsingular case λ = d/2.

The Little Book of Generative AI Foundations: An Intuitive Mathematical Primer