Efficient Euclidean Projections onto the Intersection of Norm Balls
This work addresses a computational bottleneck in projection-based methods for learning tasks with sparse-inducing norm constraints, offering incremental improvements in efficiency for machine learning applications.
The paper tackles the problem of efficiently computing Euclidean projections onto the intersection of ℓ₁ and ℓ₁,q norm balls, proposing a method that reduces the projection to finding the root of a piecewise smooth monotonic function using a bisection algorithm. The result shows time complexities of O(n+g log g) for q=2 and O(n log n) for q=∞, with empirical improvements in running time and memory usage over classical methods.
Using sparse-inducing norms to learn robust models has received increasing attention from many fields for its attractive properties. Projection-based methods have been widely applied to learning tasks constrained by such norms. As a key building block of these methods, an efficient operator for Euclidean projection onto the intersection of $\ell_1$ and $\ell_{1,q}$ norm balls $(q=2\text{or}\infty)$ is proposed in this paper. We prove that the projection can be reduced to finding the root of an auxiliary function which is piecewise smooth and monotonic. Hence, a bisection algorithm is sufficient to solve the problem. We show that the time complexity of our solution is $O(n+g\log g)$ for $q=2$ and $O(n\log n)$ for $q=\infty$, where $n$ is the dimensionality of the vector to be projected and $g$ is the number of disjoint groups; we confirm this complexity by experimentation. Empirical study reveals that our method achieves significantly better performance than classical methods in terms of running time and memory usage. We further show that embedded with our efficient projection operator, projection-based algorithms can solve regression problems with composite norm constraints more efficiently than other methods and give superior accuracy.