Learning Exponential Families in High-Dimensions: Strong Convexity and Sparsity

The versatility of exponential families, along with their attendant convexity properties, make them a popular and effective statistical model. A central issue is learning these models in high-dimensions, such as when there is some sparsity pattern of the optimal parameter. This work characterizes a certain strong convexity property of general exponential families, which allow their generalization ability to be quantified. In particular, we show how this property can be used to analyze generic exponential families under L_1 regularization.

💡 Research Summary

The paper addresses two intertwined challenges that arise when fitting exponential‑family models in high‑dimensional settings: the lack of a strong curvature guarantee for the log‑likelihood and the difficulty of exploiting sparsity in the true parameter vector. The authors first revisit the canonical form of an exponential family, where the log‑likelihood for a single observation is L(θ;x)=⟨θ,T(x)⟩−A(θ). The Hessian of the empirical loss is the empirical Fisher information matrix Î_n(θ)=∇²A(θ), and its population counterpart I(θ)=E