Frank-Wolfe with Subsampling Oracle
This work addresses efficiency in optimization for machine learning by reducing iteration costs, though it is incremental as it builds on existing Frank-Wolfe methods.
The paper tackles the computational cost of Frank-Wolfe algorithms by proposing randomized variants that use subsampling to reduce the linear minimization step, achieving sublinear and linear convergence rates similar to deterministic versions, with computational gains demonstrated on regression problems involving l1 and latent group lasso penalties.
We analyze two novel randomized variants of the Frank-Wolfe (FW) or conditional gradient algorithm. While classical FW algorithms require solving a linear minimization problem over the domain at each iteration, the proposed method only requires to solve a linear minimization problem over a small \emph{subset} of the original domain. The first algorithm that we propose is a randomized variant of the original FW algorithm and achieves a $\mathcal{O}(1/t)$ sublinear convergence rate as in the deterministic counterpart. The second algorithm is a randomized variant of the Away-step FW algorithm, and again as its deterministic counterpart, reaches linear (i.e., exponential) convergence rate making it the first provably convergent randomized variant of Away-step FW. In both cases, while subsampling reduces the convergence rate by a constant factor, the linear minimization step can be a fraction of the cost of that of the deterministic versions, especially when the data is streamed. We illustrate computational gains of the algorithms on regression problems, involving both $\ell_1$ and latent group lasso penalties.