On the Asymptotics of Self-Supervised Pre-training: Two-Stage M-Estimation and Representation Symmetry

Learning to generalise from limited data is a fundamental challenge for both artificial and biological systems. A common strategy is to extract reusable structure from abundant unlabelled data, enabling efficient adaptation to new tasks from limited labelled data. This two-stage paradigm is now standard in modern training pipelines, where pretraining is followed by fine-tuning or linear probing. We provide an analytical model of this process: structure extraction is formalized as principal component analysis on unlabelled data, and downstream learning as linear regression on a separate labelled dataset. In the high-dimensional regime, we derive exact expressions for training and generalisation error showcasing their dependence on representation dimensionality, unlabelled and labelled sample sizes, and task alignment. Our results show that pretrained representations strongly influence downstream generalisation, and we characterize the optimal representation size as a function of task parameters: with abundant pretraining data but scarce downstream data, maximally compressed representations are optimal, whereas with limited pretraining data, higher-dimensional representations generalise better. Furthermore, we establish an exact trade-off between pretraining and supervision, quantifying how much unlabelled data is required to replace a single labelled sample. Beyond our idealised model, we observe similar phenomenology in autoencoders and pretrained LLMs. Altogether, we highlight that optimising representation size is critical, giving conditions for when compression during pretraining improves generalisation.

A self-supervised encoder is trained once, frozen, and reused through lightweight probes on tasks nobody named at training time; the practitioner's question is when the off-the-shelf features are good enough and when they need fixing. Canonical correlation analysis, HGR maximal correlation, and the population optimum of the spectral contrastive loss all return the top-$k$ singular subspace of a cross-view dependence operator, justified by isotropy: if the task prior has no directional preference, that subspace is universally optimal. We show isotropy is the wrong hypothesis. The prior enters the transfer risk only through the task covariance $Λ=\mathbb{E}[ΔΔ^\top]$, and only through its compression onto the operator's leading singular directions; what matters is not whether $Λ$ is isotropic but whether its preferred directions are ordered consistently with the operator's spectrum. We prove matching two-sided rates---worst-case regret is exactly $1-1/κ(Λ)$, refines to $1-A_k$ for an alignment coefficient $A_k$, localizes to the top-$2k$ subspace, becomes second order under a spectral gap, and is improvable by no task-agnostic representation---and show why alignment is generic: incoherent preferences cancel in high dimension, and $T$ diverse tasks force $α=\widetilde O(\sqrt{d_x/T})$, a quantitative account of why task diversity, not symmetry, makes self-supervised features transfer. The governing statistics cost $O(kd_x^2)$, and when they signal misalignment a one-line reweighting of the positive-pair term provably restores exact optimality. The result is a diagnostic that answers the practitioner's question from a small labelled budget and refuses when the task bank cannot support the width requested; on controlled data it takes a regret of $0.86$ down to $0.003$, and on a CIFAR-100 encoder it correctly predicts that no correction is needed.

Self-supervised learning (SSL) has emerged as a powerful paradigm for learning meaningful representations from unlabeled data. However, the standard protocol for evaluating these representations, linear probing, is computationally expensive, sensitive to hyperparameters, and provides limited insight into the geometric structure of the representation space. In this work, motivated by connections between neural network generalization and intrinsic dimension (ID) we propose IdEst, a method for estimating the ID of SSL representations via the Minimum Spanning Tree dimension estimator ($\mathrm{dim}_\mathrm{MST}$). Across diverse datasets, architectures, and SSL pretraining objectives, we show that IdEst strongly correlates with downstream linear probe performances. Furthermore, we demonstrate that IdEst enables efficient hyperparameter selection, significantly reducing the computational cost compared to supervised alternatives. Our results highlight intrinsic dimensionality as a principled geometric proxy for assessing SSL representations, complementing standard supervised probing protocols.