## Understanding Deep Contrastive Learning via Coordinate-wise Optimization

### Yuandong Tian

##### Hall J #803

Keywords: [ contrastive learning ] [ deep linear network ] [ principal component analysis ] [ landscape analysis ] [ loss design ] [ Representation Learning ] [ Self-supervised learning ]

[ Abstract ]
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Tue 29 Nov 9 a.m. PST — 11 a.m. PST

Abstract: We show that Contrastive Learning (CL) under a broad family of loss functions (including InfoNCE) has a unified formulation of coordinate-wise optimization on the network parameter $\vtheta$ and pairwise importance $\alpha$, where the \emph{max player} $\vtheta$ learns representation for contrastiveness, and the \emph{min player} $\alpha$ puts more weights on pairs of distinct samples that share similar representations. The resulting formulation, called \boldmethod{}, unifies not only various existing contrastive losses, which differ by how sample-pair importance $\alpha$ is constructed, but also is able to extrapolate to give novel contrastive losses beyond popular ones, opening a new avenue of contrastive loss design. These novel losses yield comparable (or better) performance on CIFAR10, STL-10 and CIFAR-100 than classic InfoNCE. Furthermore, we also analyze the max player in detail: we prove that with fixed $\alpha$, max player is equivalent to Principal Component Analysis (PCA) for deep linear network, and almost all local minima are global and rank-1, recovering optimal PCA solutions. Finally, we extend our analysis on max player to 2-layer ReLU networks, showing that its fixed points can have higher ranks. Codes are available in https://github.com/facebookresearch/luckmatters/tree/main/ssl/real-dataset.

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