Contrastive and Non-Contrastive Self-Supervised Learning Recover Global and Local Spectral Embedding Methods (2205.11508v3)

Abstract: Self-Supervised Learning (SSL) surmises that inputs and pairwise positive relationships are enough to learn meaningful representations. Although SSL has recently reached a milestone: outperforming supervised methods in many modalities\dots the theoretical foundations are limited, method-specific, and fail to provide principled design guidelines to practitioners. In this paper, we propose a unifying framework under the helm of spectral manifold learning to address those limitations. Through the course of this study, we will rigorously demonstrate that VICReg, SimCLR, BarlowTwins et al. correspond to eponymous spectral methods such as Laplacian Eigenmaps, Multidimensional Scaling et al. This unification will then allow us to obtain (i) the closed-form optimal representation for each method, (ii) the closed-form optimal network parameters in the linear regime for each method, (iii) the impact of the pairwise relations used during training on each of those quantities and on downstream task performances, and most importantly, (iv) the first theoretical bridge between contrastive and non-contrastive methods towards global and local spectral embedding methods respectively, hinting at the benefits and limitations of each. For example, (i) if the pairwise relation is aligned with the downstream task, any SSL method can be employed successfully and will recover the supervised method, but in the low data regime, VICReg's invariance hyper-parameter should be high; (ii) if the pairwise relation is misaligned with the downstream task, VICReg with small invariance hyper-parameter should be preferred over SimCLR or BarlowTwins.

• The paper unifies contrastive and non-contrastive SSL methods by mapping techniques like SimCLR, BarlowTwins, and VICReg to global and local spectral embedding methods.
• It derives analytical solutions that align learned representation singular vectors with those of target data to ensure optimal downstream task performance.
• The paper offers practical guidelines on handling invariance parameters, recommending VICReg in scenarios with unclear positive pair relations.

Overview of Self-Supervised Learning: Spectral Embeddings and Optimal Representations

The paper entitled "Contrastive and Non-Contrastive Self-Supervised Learning Recover Global and Local Spectral Embedding Methods" offers a comprehensive theoretical framework that unifies a variety of existing self-supervised learning (SSL) techniques through the lens of spectral manifold learning. The authors, Randall Balestriero and Yann LeCun from Meta AI Research, aim to address the theoretical limitations in SSL by demonstrating the equivalence between popular SSL methods and known spectral embedding techniques. Three primary SSL methods are analyzed: VICReg, SimCLR, and BarlowTwins. The paper establishes a connection between these methods and spectral embedding approaches, extending and generalizing previous insights into the optimization and structure of SSL representations.

Main Contributions

• Unified Framework: The paper provides a unification of SSL methods within the context of spectral manifold learning. Through this framework, the authors identify both contrastive and non-contrastive SSL techniques with specific spectral methods, such as Laplacian Eigenmaps and Multidimensional Scaling.
• Analytical Solutions: The paper offers analytical solutions for the optimal representations produced by SSL methods and delineates the spectral properties shared across these methods. It demonstrates that these methods uniformly aim to align the left singular vectors of the learned representation with the target data's singular vectors.
• Optimal Task Alignment: It is shown that all SSL methods studied can yield optimal representations for specific downstream tasks, provided that the positive pair relations align with the task's singular vectors. This alignment ensures task solvability with zero loss, emphasizing the critical role of the similarity matrix in SSL.
• Spectral Embedding Relations: VICReg is aligned with local spectral methods, like Laplacian Eigenmaps, while SimCLR and its variants correlate with global spectral methods, such as ISOMAP. BarlowTwins are linked to Canonical Correlation Analysis (CCA) and Kernel CCA.

Important Findings and Implications

• Parameter Optimization: The paper provides closed-form solutions for the weights and representations learned by SSL methods in both non-linear and linear regimes. It offers insights into the optimization process of SSL, revealing the shared properties in learned representations.
• Practical Recommendations: Given that SSL representations can collapse in rank, the paper discusses trade-offs between full-rank and collapsible representations depending on the available pairwise relational data and its alignment with target tasks. VICReg methods, because of their ability to manage invariance parameters, are suggested as preferable when positive pair relations are unknown.
• Spectral Method Analogy: Through the exploration of SSL methods as adaptations of global and local spectral embedding techniques, the paper provides a novel perspective on designing SSL architectures by leveraging insights from spectral methods.

Future Directions

The authors speculate on the potential of cross-pollinating techniques and results between SSL frameworks and spectral methods, aiming for practical improvements in complex datasets where spectral methods traditionally fall short. Furthermore, the application of insights from the paper could guide the design and selection of more effective SSL strategies tailored to specific data geometries and downstream tasks.

In summary, this comprehensive theoretical treatment of SSL through spectral learning theory bridges several existing gaps, providing robust guidelines and insights for optimizing SSL techniques and understanding their capabilities relative to various data tasks. Such unification efforts are crucial in advancing our theoretical understanding and practical applications within the expansive domain of machine learning.