# Tight PAC-Bayesian Risk Certificates for Contrastive Learning

arXiv: https://arxiv.org/abs/2412.03486 · HTML: https://mlsanity.com/paper/2412.03486/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 58 references that are in the map.
- First posted on arXiv 4 Dec 2024.

## Builds on (most cited first, 30 of 40)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [Exploring Simple Siamese Representation Learning](https://mlsanity.com/paper/2011.10566/) (2020)
- [SimCSE: Simple Contrastive Learning of Sentence Embeddings](https://mlsanity.com/paper/2104.08821/) (2021)
- [Unsupervised Feature Learning via Non-Parametric Instance-level Discrimination](https://mlsanity.com/paper/1805.01978/) (2018)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [VICReg: Variance-Invariance-Covariance Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2105.04906/) (2021)
- [Self-Supervised Learning from Images with a Joint-Embedding Predictive Architecture](https://mlsanity.com/paper/2301.08243/) (2023)
- [Contrastive Learning of Medical Visual Representations from Paired Images and Text](https://mlsanity.com/paper/2010.00747/) (2020)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [A Theoretical Analysis of Contrastive Unsupervised Representation Learning](https://mlsanity.com/paper/1902.09229/) (2019)
- [GCC: Graph Contrastive Coding for Graph Neural Network Pre-Training](https://mlsanity.com/paper/2006.09963/) (2020)
- [Understanding Dimensional Collapse in Contrastive Self-supervised Learning](https://mlsanity.com/paper/2110.09348/) (2021)
- [Keystroke Dynamics as Part of Lifelogging](https://mlsanity.com/paper/2010.10144/) (2020)
- [Provable Guarantees for Self-Supervised Deep Learning with Spectral Contrastive Loss](https://mlsanity.com/paper/2106.04156/) (2021)
- [Theoretical Analysis of Self-Training with Deep Networks on Unlabeled Data](https://mlsanity.com/paper/2010.03622/) (2020)
- [Contrastive learning, multi-view redundancy, and linear models](https://mlsanity.com/paper/2008.10150/) (2020)
- [Non-Vacuous Generalization Bounds at the ImageNet Scale: A PAC-Bayesian Compression Approach](https://mlsanity.com/paper/1804.05862/) (2018)
- [Tighter risk certificates for neural networks](https://mlsanity.com/paper/2007.12911/) (2020)
- [Chaos is a Ladder: A New Theoretical Understanding of Contrastive Learning via Augmentation Overlap](https://mlsanity.com/paper/2203.13457/) (2022)
- [Towards the Generalization of Contrastive Self-Supervised Learning](https://mlsanity.com/paper/2111.00743/) (2021)
- [To Compress or Not to Compress- Self-Supervised Learning and Information Theory: A Review](https://mlsanity.com/paper/2304.09355/) (2023)
- [XSimGCL: Towards Extremely Simple Graph Contrastive Learning for Recommendation](https://mlsanity.com/paper/2209.02544/) (2022)
- [Understanding Negative Samples in Instance Discriminative Self-supervised Representation Learning](https://mlsanity.com/paper/2102.06866/) (2021)
- [Simpler PAC-Bayesian Bounds for Hostile Data](https://mlsanity.com/paper/1610.07193/) (2016)
- [Beyond Separability: Analyzing the Linear Transferability of Contrastive Representations to Related Subpopulations](https://mlsanity.com/paper/2204.02683/) (2022)
- [A Theoretical Study of Inductive Biases in Contrastive Learning](https://mlsanity.com/paper/2211.14699/) (2022)

## Built on by (most cited first, 6 of 6)

- [A Statistical Theory of Contrastive Learning via Approximate Sufficient Statistics](https://mlsanity.com/paper/2503.17538/) (2025)
- [Global Minimizers of Sigmoid Contrastive Loss](https://mlsanity.com/paper/2509.18552/) (2025)
- [Theoretical Foundations of Representation Learning using Unlabeled Data: Statistics and Optimization](https://mlsanity.com/paper/2509.18997/) (2025)
- [Expressivity In Multimodal Contrastive Learning](https://mlsanity.com/paper/2608.17203/) (2026)
- [An Analysis of Self-supervised Pre-training with Dependent Samples](https://mlsanity.com/paper/2609.05031/) (2026)
- [Certification Frontiers for Gaussian LoRA: Independent Priors, Posterior Risk, and Prediction-Preserving Balancing](https://mlsanity.com/paper/2609.32271/) (2026)

## Abstract (arXiv)

Contrastive representation learning is a modern paradigm for learning representations of unlabeled data via augmentations -- precisely, contrastive models learn to embed semantically similar pairs of samples (positive pairs) closer than independently drawn samples (negative samples). In spite of its empirical success and widespread use in foundation models, statistical theory for contrastive learning remains less explored. Recent works have developed generalization error bounds for contrastive losses, but the resulting risk certificates are either vacuous (certificates based on Rademacher complexity or f-divergence) or require strong assumptions about samples that are unreasonable in practice. The present paper develops non-vacuous PAC-Bayesian risk certificates for contrastive representation learning, considering the practical considerations of the popular SimCLR framework. Notably, we take into account that SimCLR reuses positive pairs of augmented data as negative samples for other data, thereby inducing strong dependence and making classical PAC or PAC-Bayesian bounds inapplicable. We further refine existing bounds on the downstream classification loss by incorporating SimCLR-specific factors, including data augmentation and temperature scaling, and derive risk certificates for the contrastive zero-one risk. The resulting bounds for contrastive loss and downstream prediction are much tighter than those of previous risk certificates, as demonstrated by experiments on CIFAR-10.