Practical Acceleration of the Condat-Vũ Algorithm

The Condat-Vũ algorithm is a widely used primal-dual method for optimizing composite objectives of three functions. Several algorithms for optimizing composite objectives of two functions are special cases of Condat-Vũ, including proximal gradient descent (PGD). It is well-known that PGD exhibits suboptimal performance, and a simple adjustment to PGD can accelerate its convergence rate from 𝒪(1/T) to 𝒪(1/T^2) on convex objectives, and this accelerated rate is optimal. In this work, we show that a simple adjustment to the Condat-Vũ algorithm allows it to recover accelerated PGD (APGD) as a special case, instead of PGD. We prove that this accelerated Condat–Vũ algorithm achieves optimal convergence rates and significantly outperforms the traditional Condat-Vũ algorithm in regimes where the Condat–Vũ algorithm approximates the dynamics of PGD. We demonstrate the effectiveness of our approach in various applications in machine learning and computational imaging.