In this position paper, we argue that understanding the relation between structure in the data distribution and structure in trained models is central to AI alignment. First, we discuss how two neural networks can have equivalent performance on the training set but compute their outputs in essentially different ways and thus generalise differently. For this reason, standard testing and evaluation are insufficient for obtaining assurances of safety for widely deployed generally intelligent systems. We argue that to progress beyond evaluation to a robust mathematical science of AI alignment, we need to develop statistical foundations for an understanding of the relation between structure in the data distribution, internal structure in models, and how these structures underlie generalisation.

Modern deep neural networks display striking examples of rich internal computational structure. Uncovering principles governing the development of such structure is a priority for the science of deep learning. In this paper, we explore the transient ridge phenomenon: when transformers are trained on in-context linear regression tasks with intermediate task diversity, they initially behave like ridge regression before specializing to the tasks in their training distribution. This transition from a general solution to a specialized solution is revealed by joint trajectory principal component analysis. Further, we draw on the theory of Bayesian internal model selection to suggest a general explanation for the phenomena of transient structure in transformers, based on an evolving tradeoff between loss and complexity. We empirically validate this explanation by measuring the model complexity of our transformers as defined by the local learning coefficient.

Mechanistic interpretability aims to understand the computational mechanisms underlying neural networks' capabilities in order to accomplish concrete scientific and engineering goals. Progress in this field thus promises to provide greater assurance over AI system behavior and shed light on exciting scientific questions about the nature of intelligence. Despite recent progress toward these goals, there are many open problems in the field that require solutions before many scientific and practical benefits can be realized: Our methods require both conceptual and practical improvements to reveal deeper insights; we must figure out how best to apply our methods in pursuit of specific goals; and the field must grapple with socio-technical challenges that influence and are influenced by our work. This forward-facing review discusses the current frontier of mechanistic interpretability and the open problems that the field may benefit from prioritizing. This review collects the perspectives of its various authors and represents a synthesis of their views by Apollo Research on behalf of Schmidt Sciences. The perspectives presented here do not necessarily reflect the views of any individual author or the institutions with which they are affiliated.

Structure in the data distribution has long been recognized as central to the development of internal structure in artificial and biological neural networks (Rumelhart et al., 1986; Olshausen & Field, 1996; Rogers & McClelland, 2004). Recent observations have renewed interest in this topic: language models progress through distinct stages of development during training, acquiring increasingly sophisticated linguistic and reasoning abilities in ways that seem to reflect the structure of the data distribution (Olsson et al., 2022; Chen et al., 2024; Belrose et al., 2024; Tigges et al., 2024; Edelman et al., 2024; Hoogland et al., 2024). A deeper understanding of how structure in the data determines internal structure in trained models requires tools that provide information about which components of a model are being shaped in response to what structure in the data distribution. Our foundation for the study of such questions begins with the local learning coefficient (LLC; Lau et al. 2023) from singular learning theory (SLT; Watanabe 2009), which is a measure of model complexity. In this paper, we introduce the refined local learning coefficient (rLLC), which measures the complexity of a component of the model with respect to an arbitrary data distribution. We focus mainly on the rLLCs of individual attention heads and demonstrate the utility of these metrics in studying the progressive differentiation and specialization of heads. The diversity of attention heads at the end of training has been established in recent years through mechanistic interpretability, which has provided numerous examples of attention heads that appear to have specialized functions, including previous-token heads (Voita et al., 2019; Clark et al., 2019) and induction heads (Olsson et al., 2022) among other kinds (Wang et al., 2023; Gould et al., 2024).

We show that in-context learning emerges in transformers in discrete developmental stages, when they are trained on either language modeling or linear regression tasks. We introduce two methods for detecting the milestones that separate these stages, by probing the geometry of the population loss in both parameter space and function space. We study the stages revealed by these new methods using a range of behavioral and structural metrics to establish their validity.