An Analysis of Self-supervised Pre-training with Dependent Samples

Self-supervised learning matches supervised accuracy from a fraction of the labels, but the labeled-sample efficiency behind this has lacked a theoretical explanation. We provide one. Data augmentation induces a similarity graph on the unlabeled data, so downstream learning on that graph is graph-Laplacian-regularized learning. We prove a fast transductive rate, $O(1/n_L)$ in the number of labels, in place of the supervised $O(1/\sqrt{n_L})$, by carrying the leave-one-out stability apparatus of Johnson and Zhang (JMLR 2007) over to the augmentation graph, and without the unrealistic assumptions of limit-based analyses (exact kernel, generalizing features). The bound makes augmentation quality explicit: the expected error is at most $C/n_L + R_{\mathrm{DA}}(y)$, where the data-augmentation alignment error $R_{\mathrm{DA}}(y)$ is proportional to the graph-cut mass of augmentations that cross a label boundary, so good augmentations let few labels suffice. The analysis uses a streamlined loss that drops the projector, negative-sample, and orthogonality overhead of standard objectives yet still recovers the top-$K$ ideal features in the infinite-data limit, the augmentation-kernel eigenspace studied by Zhai et al. The bound gives a mechanistic account of the accuracy-versus-label-count curve through augmentation quality, verified in a controlled model where the constants are known.

Self-supervised pre-training, where large corpora of unlabeled data are used to learn representations for downstream fine-tuning, has become a cornerstone of modern machine learning. While a growing body of theoretical work has begun to analyze this paradigm, existing bounds leave open the question of how sharp the current rates are, and whether they accurately capture the complex interaction between pre-training and fine-tuning. In this paper, we address this gap by developing an asymptotic theory of pre-training via two-stage M-estimation. A key challenge is that the pre-training estimator is often identifiable only up to a group symmetry, a feature common in representation learning that requires careful treatment. We address this issue using tools from Riemannian geometry to study the intrinsic parameters of the pre-training representation, which we link with the downstream predictor through a notion of orbit-invariance, precisely characterizing the limiting distribution of the downstream test risk. We apply our main result to several case studies, including spectral pre-training, factor models, and Gaussian mixture models, and obtain substantial improvements in problem-specific factors over prior art when applicable.

Self-supervision is a powerful technique for learning visual representations from unlabeled data. Existing techniques primarily adopt a two-stage approach for self-supervised learning (SSL): a pretraining stage on unlabeled data followed by a finetuning stage on labeled data. While this pipeline has demonstrated extreme effectiveness, the interaction between self-supervised and supervised learning objectives remains insufficiently understood. In this work, we systematically investigate whether jointly optimizing the self-supervised and supervised objectives during training provides a better alternative. We compare two training paradigms: (1) the aforementioned pretraining followed by finetuning (PFT) and (2) joint training (JT), where self-supervised and supervised losses are optimized simultaneously in the same network. Across eight representative SSL methods and diverse computer vision tasks on natural, medical, crisis response, and remote sensing data, we evaluate performance under varying percentages of labeled data. Our results reveal that the relative effectiveness of PFT and JT depends strongly on the task at hand, the availability of labeled data, and the complexity of the domain. We find that JT consistently improves data and training efficiency while being robust in low-label settings, while PFT is more reliable in more specialized domains. We further analyze representation quality, robustness, and cross-domain generalization, providing new insights into how self-supervised and supervised objectives interact during optimization. We establish a comprehensive empirical benchmark for hybrid SSL-based semi-supervised learning and offer practical guidance for selecting appropriate training strategies across diverse vision applications.