Sparse High-Dimensional Regression: Exact Scalable Algorithms and Phase Transitions
This work addresses the scalability and optimality challenges in sparse regression for high-dimensional data, offering a superior alternative to heuristic methods, though it is incremental in improving existing algorithms.
The authors tackled the sparse high-dimensional regression problem by developing a novel binary convex reformulation and cutting plane method, achieving provable optimality for sample sizes and regressors in the 100,000s, which is two orders of magnitude better than the state of the art, and observed new phase transition phenomena where the problem becomes easier with more samples.
We present a novel binary convex reformulation of the sparse regression problem that constitutes a new duality perspective. We devise a new cutting plane method and provide evidence that it can solve to provable optimality the sparse regression problem for sample sizes n and number of regressors p in the 100,000s, that is two orders of magnitude better than the current state of the art, in seconds. The ability to solve the problem for very high dimensions allows us to observe new phase transition phenomena. Contrary to traditional complexity theory which suggests that the difficulty of a problem increases with problem size, the sparse regression problem has the property that as the number of samples $n$ increases the problem becomes easier in that the solution recovers 100% of the true signal, and our approach solves the problem extremely fast (in fact faster than Lasso), while for small number of samples n, our approach takes a larger amount of time to solve the problem, but importantly the optimal solution provides a statistically more relevant regressor. We argue that our exact sparse regression approach presents a superior alternative over heuristic methods available at present.