Complexity Analysis of the Lasso Regularization Path
This addresses computational bottlenecks for statisticians and machine learning practitioners using Lasso regularization, providing both theoretical limits and practical approximations.
The paper analyzes the computational complexity of computing the exact Lasso regularization path, proving it has exponential worst-case complexity in the number of variables, but shows an approximate path with O(1/√ε) segments can achieve ε-optimality.
The regularization path of the Lasso can be shown to be piecewise linear, making it possible to "follow" and explicitly compute the entire path. We analyze in this paper this popular strategy, and prove that its worst case complexity is exponential in the number of variables. We then oppose this pessimistic result to an (optimistic) approximate analysis: We show that an approximate path with at most O(1/sqrt(epsilon)) linear segments can always be obtained, where every point on the path is guaranteed to be optimal up to a relative epsilon-duality gap. We complete our theoretical analysis with a practical algorithm to compute these approximate paths.