Transformers represent belief state geometry in their residual stream

What computational structure are we building into large language models when we train them on next-token prediction? Here, we present evidence that this structure is given by the meta-dynamics of belief updating over hidden states of the data-generating process. Leveraging the theory of optimal prediction, we anticipate and then find that belief states are linearly represented in the residual stream of transformers, even in cases where the predicted belief state geometry has highly nontrivial fractal structure. We investigate cases where the belief state geometry is represented in the final residual stream or distributed across the residual streams of multiple layers, providing a framework to explain these observations. Furthermore we demonstrate that the inferred belief states contain information about the entire future, beyond the local next-token prediction that the transformers are explicitly trained on. Our work provides a framework connecting the structure of training data to the computational structure and representations that transformers use to carry out their behavior.

Geometry and Dynamics of LayerNorm

A technical note aiming to offer deeper intuition for the LayerNorm function common in deep neural networks. LayerNorm is defined relative to a distinguished 'neural' basis, but it does more than just normalize the corresponding vector elements. Rather, it implements a composition -- of linear projection, nonlinear scaling, and then affine transformation -- on input activation vectors. We develop both a new mathematical expression and geometric intuition, to make the net effect more transparent. We emphasize that, when LayerNorm acts on an N-dimensional vector space, all outcomes of LayerNorm lie within the intersection of an (N-1)-dimensional hyperplane and the interior of an N-dimensional hyperellipsoid. This intersection is the interior of an (N-1)-dimensional hyperellipsoid, and typical inputs are mapped near its surface. We find the direction and length of the principal axes of this (N-1)-dimensional hyperellipsoid via the eigen-decomposition of a simply constructed matrix.

Ultimate limit on learning non-Markovian behavior: Fisher information rate and excess information

We address the fundamental limits of learning unknown parameters of any stochastic process from time-series data, and discover exact closed-form expressions for how optimal inference scales with observation length. Given a parametrized class of candidate models, the Fisher information of observed sequence probabilities lower-bounds the variance in model estimation from finite data. As sequence-length increases, the minimal variance scales as the square inverse of the length -- with constant coefficient given by the information rate. We discover a simple closed-form expression for this information rate, even in the case of infinite Markov order. We furthermore obtain the exact analytic lower bound on model variance from the observation-induced metadynamic among belief states. We discover ephemeral, exponential, and more general modes of convergence to the asymptotic information rate. Surprisingly, this myopic information rate converges to the asymptotic Fisher information rate with exactly the same relaxation timescales that appear in the myopic entropy rate as it converges to the Shannon entropy rate for the process. We illustrate these results with a sequence of examples that highlight qualitatively distinct features of stochastic processes that shape optimal learning.

Complexity-calibrated Benchmarks for Machine Learning Reveal When Next-Generation Reservoir Computer Predictions Succeed and Mislead

Recurrent neural networks are used to forecast time series in finance, climate, language, and from many other domains. Reservoir computers are a particularly easily trainable form of recurrent neural network. Recently, a "next-generation" reservoir computer was introduced in which the memory trace involves only a finite number of previous symbols. We explore the inherent limitations of finite-past memory traces in this intriguing proposal. A lower bound from Fano's inequality shows that, on highly non-Markovian processes generated by large probabilistic state machines, next-generation reservoir computers with reasonably long memory traces have an error probability that is at least ~ 60% higher than the minimal attainable error probability in predicting the next observation. More generally, it appears that popular recurrent neural networks fall far short of optimally predicting such complex processes. These results highlight the need for a new generation of optimized recurrent neural network architectures. Alongside this finding, we present concentration-of-measure results for randomly-generated but complex processes. One conclusion is that large probabilistic state machines -- specifically, large $\epsilon$-machines -- are key to generating challenging and structurally-unbiased stimuli for ground-truthing recurrent neural network architectures.