Acceleration of Descent-based Optimization Algorithms via Carath\'eodory's Theorem

We propose a new technique to accelerate algorithms based on Gradient Descent using Carath\'eodory's Theorem. In the case of the standard Gradient Descent algorithm, we analyse the theoretical convergence of the approach under convexity assumptions and empirically display its ameliorations. As a core contribution, we then present an application of the acceleration technique to Block Coordinate Descent methods. Experimental comparisons on least squares regression with a LASSO regularisation term show remarkably improved performance on LASSO than the ADAM and SAG algorithms.