A unifying mutual information view of metric learning: cross-entropy vs. pairwise losses (2003.08983v3)

Abstract: Recently, substantial research efforts in Deep Metric Learning (DML) focused on designing complex pairwise-distance losses, which require convoluted schemes to ease optimization, such as sample mining or pair weighting. The standard cross-entropy loss for classification has been largely overlooked in DML. On the surface, the cross-entropy may seem unrelated and irrelevant to metric learning as it does not explicitly involve pairwise distances. However, we provide a theoretical analysis that links the cross-entropy to several well-known and recent pairwise losses. Our connections are drawn from two different perspectives: one based on an explicit optimization insight; the other on discriminative and generative views of the mutual information between the labels and the learned features. First, we explicitly demonstrate that the cross-entropy is an upper bound on a new pairwise loss, which has a structure similar to various pairwise losses: it minimizes intra-class distances while maximizing inter-class distances. As a result, minimizing the cross-entropy can be seen as an approximate bound-optimization (or Majorize-Minimize) algorithm for minimizing this pairwise loss. Second, we show that, more generally, minimizing the cross-entropy is actually equivalent to maximizing the mutual information, to which we connect several well-known pairwise losses. Furthermore, we show that various standard pairwise losses can be explicitly related to one another via bound relationships. Our findings indicate that the cross-entropy represents a proxy for maximizing the mutual information -- as pairwise losses do -- without the need for convoluted sample-mining heuristics. Our experiments over four standard DML benchmarks strongly support our findings. We obtain state-of-the-art results, outperforming recent and complex DML methods.

• The paper demonstrates that minimizing cross-entropy acts as a majorize-minimize strategy for pairwise loss, enhancing intra-class compactness and inter-class separation.
• It reveals that cross-entropy implicitly maximizes mutual information between embeddings and labels, balancing discriminative and generative objectives in DML.
• Experiments on benchmark datasets show that cross-entropy achieves state-of-the-art metric learning performance while simplifying training protocols.

A Mutual Information Perspective on Metric Learning: Cross-Entropy vs. Pairwise Losses

The research paper by Malik Boudiaf et al. provides a rigorous analysis of the relationship between the standard cross-entropy loss and various pairwise losses used in Deep Metric Learning (DML). The authors challenge the conventional segregation of classification and metric learning paradigms by demonstrating that cross-entropy, typically overlooked in the metric learning domain, implicitly aligns with the fundamental goals of DML when viewed through the lens of mutual information.

Theoretical Insights and Connections

The authors propose two primary perspectives to illustrate the connections between cross-entropy and pairwise losses. The first perspective is rooted in optimization theory. The authors reveal that minimizing cross-entropy can be interpreted as an upper bound on a pairwise loss with an inherent structure conducive to maximizing intra-class compactness and inter-class separation. This insight is significant as it portrays cross-entropy as a majorize-minimize algorithm for an underlying pairwise loss, thus aligning it with the objectives of DML.

The second perspective leverages information theory, notably mutual information (MI), to underline the equivalence of cross-entropy minimization and mutual information maximization. By maximizing the MI between the embeddings and their labels, cross-entropy inherently seeks both discriminative and generative balance—a key pursuit in metric learning frameworks. The paper bridges the gap between DML's feature shaping losses and classification losses by showing they both optimize the MI, albeit from different stances.

Experimental Validation

Empirical results on benchmark DML datasets such as CUB200, Cars-196, Stanford Online Product, and In-Shop demonstrate that cross-entropy can yield state-of-the-art performance without the intricate sample mining and weighting schemes commonly employed by pairwise losses. This contrast highlights the potential efficiency and simplicity of leveraging cross-entropy in situations traditionally reserved for pairwise approaches.

Implications and Future Directions

The paper has profound implications for the development and application of DML algorithms. By establishing a clear theoretical foundation that supports the use of cross-entropy in metric learning tasks, the authors suggest that practitioners might be able to simplify model design and training procedures without sacrificing performance. This could lead to more robust metric learning models with broader applicability in areas like image retrieval, face verification, and zero-shot learning, where sample relationships are crucial yet complex.

Moreover, these insights open avenues for future exploration of other classification losses under the mutual information framework, potentially uncovering new paradigms that further unify disparate areas of machine learning. There is also the opportunity to extend this approach to unsupervised and semi-supervised learning scenarios where metrics are not explicitly provided.

Conclusion

The paper by Boudiaf et al. presents a nuanced and theoretically-rich perspective on the intersections between cross-entropy loss and pairwise losses. By integrating mutual information theory with practical optimization insights, the authors reshape our understanding of metric learning tasks, advocating for a reevaluation of traditional methodologies. This work not only uncovers the latent capabilities of cross-entropy in DML but also sets the stage for future research that could extend these principles across different machine learning domains and applications.