Structured sparsity-inducing norms through submodular functions (1008.4220v3)

Abstract: Sparse methods for supervised learning aim at finding good linear predictors from as few variables as possible, i.e., with small cardinality of their supports. This combinatorial selection problem is often turned into a convex optimization problem by replacing the cardinality function by its convex envelope (tightest convex lower bound), in this case the L1-norm. In this paper, we investigate more general set-functions than the cardinality, that may incorporate prior knowledge or structural constraints which are common in many applications: namely, we show that for nondecreasing submodular set-functions, the corresponding convex envelope can be obtained from its \lova extension, a common tool in submodular analysis. This defines a family of polyhedral norms, for which we provide generic algorithmic tools (subgradients and proximal operators) and theoretical results (conditions for support recovery or high-dimensional inference). By selecting specific submodular functions, we can give a new interpretation to known norms, such as those based on rank-statistics or grouped norms with potentially overlapping groups; we also define new norms, in particular ones that can be used as non-factorial priors for supervised learning.

• The paper introduces a convex framework that transforms non-decreasing submodular functions into structured sparsity-inducing norms using the Lovász extension.
• The paper outlines efficient algorithmic tools, including subgradient and proximal operators, to enable robust high-dimensional inference and support recovery.
• The paper interprets classical norms and proposes new ones, broadening applications in supervised learning and non-factorial priors for improved model interpretability.

Structured Sparsity-inducing Norms Through Submodular Functions

The paper "Structured sparsity-inducing norms through submodular functions," authored by Francis Bach, explores the integration of structured sparsity into convex optimization frameworks using submodular functions. The motivation stems from the desire to impose parsimony in models, not merely by reducing cardinality, but by incorporating prior knowledge and structural constraints that are prevalent in many practical applications.

Key Contributions

This paper makes several substantial contributions to the field of sparsity-inducing norms:

• Convex Envelopes via Lovász Extension: The paper generalizes the process of obtaining the convex envelope of set-functions associated with supervised learning problems. Specifically, it shows that nondecreasing submodular functions can be transformed into a family of structured norms through the Lovász extension, laying a theoretical foundation for utilizing submodular functions in defining sparse models.
• Algorithmic and Theoretical Framework: The authors provide generic algorithmic tools, including subgradient and proximal operators, essential for the efficient implementation of the proposed norms. Additionally, theoretical results are presented that extend classical high-dimensional inference techniques, focusing on support recovery conditions and estimation consistency.
• New Norm Interpretations: By selecting specific submodular functions, the paper interprets known norms, like those based on rank-statistics or overlapping group norms, and introduces new norms. These new norms serve as alternatives for non-factorial priors in supervised learning, demonstrating their versatility and broader applicability.

Methodological Insights

The paper's methodology revolves around the notion of submodular set-functions, which are critical due to their extensive utility in combinatorial optimization. The norms defined in this work diverge from traditional cardinality minimization and instead encapsulate sophisticated dependencies by leveraging the properties of submodular functions. The paper explores:

• Polyhedral Norms: Through the Lovász extension, the norms are inherently polyhedral, allowing for a richer, structured form of sparsity. The unit ball of these norms comes with a robust geometric interpretation, characterized by potentially exponential faces and vertices.
• Mathematical Properties: Detailed analysis is conducted on the properties of these norms, including decomposition, dual norms, and characterizations of extreme points, which are crucial for understanding the behavior of the norms under different conditions.

Implications and Future Directions

The implications of incorporating such structured sparsity are multifaceted:

• Enhanced Model Interpretability: By embedding meaningful structures into sparsity constraints, models can be better aligned with the intrinsic properties of the data, offering greater interpretability in fields like bioinformatics and image processing.
• Flexibility and Expandability: The framework is adaptable to various structures inherent in different data types, making it a valuable tool across domains where structured sparsity is desired.
• Potential in Non-factorial Priors: The introduction of non-factorial priors through these norms opens up new avenues in Bayesian analysis, particularly in supervised learning contexts where traditional priors may fall short.

Conclusion

The paper sets a precedent for broadening the scope of sparsity-inducing norms through submodular analysis, which could significantly influence future research in sparse learning paradigms. The robust theoretical and algorithmic foundation laid out invites further exploration into more complex applications and extension of the framework, potentially advancing the capabilities of machine learning models in handling structured data. This work paves the way for the development of more sophisticated, context-aware sparsity methods that capitalize on submodular function properties.