Efficient Optimal Transport Algorithm by Accelerated Gradient descent
This work addresses efficiency and accuracy bottlenecks in optimal transport for machine learning applications, representing an incremental improvement over existing methods like Sinkhorn.
The paper tackles the challenge of computing discrete optimal transport plans efficiently and accurately for large-scale problems by proposing a novel algorithm based on Nesterov's smoothing technique, achieving a computational complexity of O(n^(5/2) sqrt(log n)/ε) and demonstrating faster convergence and better accuracy than the Sinkhorn algorithm in experiments.
Optimal transport (OT) plays an essential role in various areas like machine learning and deep learning. However, computing discrete optimal transport plan for large scale problems with adequate accuracy and efficiency is still highly challenging. Recently, methods based on the Sinkhorn algorithm add an entropy regularizer to the prime problem and get a trade off between efficiency and accuracy. In this paper, we propose a novel algorithm to further improve the efficiency and accuracy based on Nesterov's smoothing technique. Basically, the non-smooth c-transform of the Kantorovich potential is approximated by the smooth Log-Sum-Exp function, which finally smooths the original non-smooth Kantorovich dual functional (energy). The smooth Kantorovich functional can be optimized by the fast proximal gradient algorithm (FISTA) efficiently. Theoretically, the computational complexity of the proposed method is given by $O(n^{\frac{5}{2}} \sqrt{\log n} /ε)$, which is lower than that of the Sinkhorn algorithm. Empirically, compared with the Sinkhorn algorithm, our experimental results demonstrate that the proposed method achieves faster convergence and better accuracy with the same parameter.