A lasso for hierarchical interactions (1205.5050v3)

Abstract: We add a set of convex constraints to the lasso to produce sparse interaction models that honor the hierarchy restriction that an interaction only be included in a model if one or both variables are marginally important. We give a precise characterization of the effect of this hierarchy constraint, prove that hierarchy holds with probability one and derive an unbiased estimate for the degrees of freedom of our estimator. A bound on this estimate reveals the amount of fitting "saved" by the hierarchy constraint. We distinguish between parameter sparsity - the number of nonzero coefficients - and practical sparsity - the number of raw variables one must measure to make a new prediction. Hierarchy focuses on the latter, which is more closely tied to important data collection concerns such as cost, time and effort. We develop an algorithm, available in the R package hierNet, and perform an empirical study of our method.

• The paper introduces a lasso variant with convex hierarchical constraints that include an interaction term only if its main effects are present.
• It develops an efficient algorithm using generalized gradient descent and ADMM, implemented in the hierNet R package.
• The study provides an unbiased degrees of freedom estimate and demonstrates improved predictive performance under hierarchical constraints.

Overview of "A Lasso for Hierarchical Interactions"

The paper "A Lasso for Hierarchical Interactions" by Jacob Bien, Jonathan Taylor, and Robert Tibshirani, introduces a regularization method that extends the lasso to accommodate hierarchical interaction terms in regression models. This approach is implemented through a set of convex constraints, leading to sparse models where an interaction term is included only if its corresponding main effects are present, ensuring either strong or weak hierarchy.

Key Contributions

1. Hierarchical Constraints: The authors propose a method that incorporates convex constraints into the lasso objective function. This effectively enforces a hierarchy where interaction terms depend on the presence of main effects. The primary forms discussed are strong hierarchy (both main effects present if the interaction term is) and weak hierarchy (at least one main effect present).
2. Estimation and Algorithm: The paper develops an efficient algorithm, provided as the hierNet package in R, to fit these hierarchical models. This uses a combination of generalized gradient descent and an Alternating Direction Method of Multipliers (ADMM) approach for optimization, particularly addressing the computational challenges posed by strong hierarchy constraints.
3. Degrees of Freedom: An unbiased estimate for the degrees of freedom of the proposed estimator is provided. The authors also offer a bound on this estimate, highlighting how the hierarchical constraints affect the model's complexity and the "fitting" resources saved.
4. Theoretical Guarantees: The hierarchy constraint is shown to hold with probability one, given standard assumptions about the data, ensuring that the model respects the assumed hierarchical structure.

Numerical Results and Implications

The empirical comparisons show that models incorporating hierarchical constraints, particularly in the presence of truly hierarchical interaction terms, achieve better performance in prediction tasks. However, even in situations where the true model does not adhere to the hierarchical structure, the proposed method, especially under weak hierarchy constraints, remains competitive by effectively managing the trade-off between identifying main effects and interaction terms.

Practical and Theoretical Implications

The introduction of hierarchy into interaction modeling is of significant practical concern, particularly in fields like genomics and epidemiology, where interactions are critical but require substantial data collection effort. The focus on practical sparsity aligns well with scenarios where measurement costs are high.

Theoretically, this work contributes to the field of structured sparsity by providing a clear linkage between the hierarchy in interaction terms and regularization approaches. It expands on the flexibility of the lasso by accommodating complex, structured sparsity patterns without relinquishing the interpretability and computational advantages of convex optimization.

Future Directions

The paper opens avenues for extending hierarchical modeling to more complex hierarchical structures and broader classes of models, such as generalized additive models or models with nonlinear interactions. Additionally, exploring the impact of these methods in large-scale data environments and integrating the approach with other advanced machine learning techniques could yield substantial benefits.

In conclusion, this paper advances the state-of-the-art in regression modeling by effectively integrating hierarchical structure into the field of sparse models, offering both theoretical rigor and practical utility. The hierNet package makes the authors' contributions accessible, facilitating further research and application in domains where interaction hierarchies are pivotal.