White-Box Transformers via Sparse Rate Reduction (2306.01129v1)

Abstract: In this paper, we contend that the objective of representation learning is to compress and transform the distribution of the data, say sets of tokens, towards a mixture of low-dimensional Gaussian distributions supported on incoherent subspaces. The quality of the final representation can be measured by a unified objective function called sparse rate reduction. From this perspective, popular deep networks such as transformers can be naturally viewed as realizing iterative schemes to optimize this objective incrementally. Particularly, we show that the standard transformer block can be derived from alternating optimization on complementary parts of this objective: the multi-head self-attention operator can be viewed as a gradient descent step to compress the token sets by minimizing their lossy coding rate, and the subsequent multi-layer perceptron can be viewed as attempting to sparsify the representation of the tokens. This leads to a family of white-box transformer-like deep network architectures which are mathematically fully interpretable. Despite their simplicity, experiments show that these networks indeed learn to optimize the designed objective: they compress and sparsify representations of large-scale real-world vision datasets such as ImageNet, and achieve performance very close to thoroughly engineered transformers such as ViT. Code is at \url{https://github.com/Ma-Lab-Berkeley/CRATE}.

• The paper shows that transformer architectures can be derived by optimizing a unified sparse rate reduction objective that integrates compression and sparsification.
• It introduces an unrolled optimization approach where multi-head self-attention acts as a gradient descent step on coding rate functions, enhanced by an ISTA-inspired sparsification.
• Experimental validations reveal that these white-box transformers achieve competitive performance on benchmark tasks while offering improved interpretability.

Overview of "White-Box Transformers via Sparse Rate Reduction"

The paper "White-Box Transformers via Sparse Rate Reduction" presents a novel framework that mathematically interprets transformer architectures through the lens of sparse rate reduction. This approach conceptualizes deep learning as an iterative optimization process aiming for compressive and sparse data representation. The research delineates how well-known transformer architectures, particularly those utilizing multi-head self-attention, can be derived from optimizing a unified objective function: sparse rate reduction. This involves compressing input representations to facilitate efficient downstream processing.

Technical Contribution

The authors propose a novel objective function termed sparse rate reduction, combining two key principles: maximal coding rate reduction (MCR2) and sparsification. The framework suggests that the ultimate goal of learning representations is to transform the data distribution into a mixture of low-dimensional Gaussian distributions on incoherent subspaces, a characterization deemed amenable to compact representation.

Unrolled Optimization

To achieve this transformation, the paper outlines an unrolled optimization approach that can be integrated into deep network architectures. Such an approach leads to layers performing incremental optimization steps:

1. Compression: This is realized using a derivative calculation from a coding rate function that approximates the input set against learned subspaces. The multi-head self-attention mechanism is interpreted as a gradient descent step minimizing the coding rate of these subspaces. The introduction of a Subspace Self-Attention (SSA) operator and its multi-head version (MSSA) formalizes this compression perspective.
2. Sparsification: Following compression, the representation is further sparsified. Here, the authors employ a proximate version of the Iterative Shrinkage-Thresholding Algorithm (ISTA) to encourage sparsity within a learned dictionary. This addresses the sparsification term in the sparse rate reduction objective, ensuring a parsimonious final representation.

Experimental Validation

The paper reports that, despite their simplicity, networks instantiated from the proposed framework achieve performance comparable to established models like ViT on vision datasets such as ImageNet. The authors confirm these results by evaluating intermediate representations for their compressive qualities and sparsity, observing that these metrics generally improve with depth, aligning with the theoretical predictions.

Implications and Future Directions

The principal contribution of this research is the formal interpretability it provides to transformer-like architectures. The insights offer a white-box perspective, contrasting the typical black-box paradigms of deep learning. The implication is twofold:

• Theoretical Implications: By framing learning as an iterative optimization problem, the research paves the way for more principled architecture design. It unifies diverse deep learning approaches under a common mathematical framework, potentially extending its applicability across modalities.
• Practical Implications: The outcomes suggest practical avenues for developing interpretable, efficient transformer architectures. The insights into layer function could fuel the design of architectures that mirror human intuition and domain understanding more closely.

Importantly, the paper hints at the vast unexplored potential within this framework for tuning hyperparameters and altering architecture constituents to enhance performance across tasks. Future developments may focus on refining the models and exploring more complex settings that could further exploit the power of sparse rate reduction optimization in representation learning.

Conclusion

In sum, this paper offers a rigorous theoretical framework for understanding and designing transformer architectures. By rooting itself in sparse rate reduction, it provides a comprehensive, interpretable methodology that maintains competitive performance on real-world tasks. The integration of compression and sparsification within a mathematically principled context stands as a notable stride in the quest for transparent and effective deep learning architectures.