Fast K-Means Clustering with Anderson Acceleration

We propose a novel method to accelerate Lloyd's algorithm for K-Means clustering. Unlike previous acceleration approaches that reduce computational cost per iterations or improve initialization, our approach is focused on reducing the number of iterations required for convergence. This is achieved by treating the assignment step and the update step of Lloyd's algorithm as a fixed-point iteration, and applying Anderson acceleration, a well-established technique for accelerating fixed-point solvers. Classical Anderson acceleration utilizes m previous iterates to find an accelerated iterate, and its performance on K-Means clustering can be sensitive to choice of m and the distribution of samples. We propose a new strategy to dynamically adjust the value of m, which achieves robust and consistent speedups across different problem instances. Our method complements existing acceleration techniques, and can be combined with them to achieve state-of-the-art performance. We perform extensive experiments to evaluate the performance of the proposed method, where it outperforms other algorithms in 106 out of 120 test cases, and the mean decrease ratio of computational time is more than 33%

Globally Convergent Type-I Anderson Acceleration for Non-Smooth Fixed-Point Iterations

We consider the application of the type-I Anderson acceleration to solving general non-smooth fixed-point problems. By interleaving with safe-guarding steps, and employing a Powell-type regularization and a re-start checking for strong linear independence of the updates, we propose the first globally convergent variant of Anderson acceleration assuming only that the fixed-point iteration is non-expansive. We show by extensive numerical experiments that many first order algorithms can be improved, especially in their terminal convergence, with the proposed algorithm. Our proposed method of acceleration is being implemented in SCS 2.0, one of the default solvers used in the convex optimization parser-solver CVXPY 1.0.Comment: 47 page