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A Statistical Theory of Contrastive Learning via Approximate Sufficient Statistics
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Develops a statistical theory of SimCLR-style contrastive learning by generalizing approximate sufficient statistics to f-divergences and bounding downstream regression/classification risk by encoder sufficiency plus augmentation error.
Problem
Theoretical understanding of why data-augmentation-based contrastive learning (SimCLR) yields representations that transfer to downstream tasks, and how sufficiency and augmentation error control downstream risk.
Approach
Extends approximate sufficient statistics (previously defined for CLIP with KL divergence) to three equivalent forms and general f-divergences, showing SimCLR/f-contrastive loss minimization yields approximately sufficient encoders whose downstream risk is bounded by sufficiency plus augmentation-induced error.
Key results
- Linear regression on KL- or chi-square-contrastive representations gives much lower excess risk than direct linear regression at small downstream sample sizes (m=150, 500).
- Direct linear regression's excess risk converges to zero as m grows, while contrastive-representation excess risk converges to a nonzero constant, consistent with the theory's sufficiency and augmentation error terms.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use KL (InfoNCE)-pretrained representations instead of raw features | Downstream linear regression · Synthetic linear regression (m = 150, 500) | excess risk | Helped |
| use χ²-pretrained representations instead of raw features | Downstream linear regression · Synthetic linear regression (m = 150, 500) | excess risk | Helped |
| use χ² contrastive learning instead of KL (InfoNCE) contrastive learning | Downstream linear regression · Synthetic linear regression | excess risk | No clear effect |
| increase downstream sample size | Direct LR and contrastively pretrained representations · Synthetic linear regression | excess risk | Mixed |
Surprising findings
- Representation-based excess risk plateaus at a nonzero constant as downstream samples grow, so pretrained encoders can be outperformed asymptotically by direct regression.
Limitations
- General f-divergence case (beyond chi-square) left for future work because the optimal S_x lacks closed form in the finite-sample estimator.
- Experiments are small synthetic settings with a two-layer network; no real-data SimCLR evaluation.
- Results concern ERM encoders and do not address minimal sufficiency or training dynamics.
Experimental setup
Datasets: synthetic Gaussian data with random augmentation
Benchmarks: synthetic linear regression excess risk
Metrics: excess risk
Baselines: InfoNCE (KL) contrastive loss Direct linear regression on raw features
Builds on: SimCLR InfoNCE CLIP sufficiency analysis (approximate sufficient statistics) InfoMax principle
Closest papers by meaning
- Contrastive Learning with Data Misalignment: Feature Purity, Training Dynamics and Theoretical Generalization Guarantees (NeurIPS 2025 · similarity 0.89)
- Chaos is a Ladder: A New Theoretical Understanding of Contrastive Learning via Augmentation Overlap (ICLR 2022 · similarity 0.89)
- Variational Supervised Contrastive Learning (NeurIPS 2025 · similarity 0.88)
- $\mathbb{X}$-Sample Contrastive Loss: Improving Contrastive Learning with Sample Similarity Graphs (ICLR 2025 · similarity 0.88)
- On the Similarities of Embeddings in Contrastive Learning (ICML 2025 · similarity 0.88)
- An Augmentation-Aware Theory for Self-Supervised Contrastive Learning (ICML 2025 · similarity 0.88)
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