Sinkhorn Algorithm as a Special Case of Stochastic Mirror Descent
This provides a theoretical insight for researchers in optimization and optimal transport, but it is incremental as it reframes an existing algorithm.
The paper shows that the Sinkhorn algorithm is a special case of stochastic mirror descent, enabling new methods for optimal transport and an extension beyond two constraints.
We present a new perspective on the celebrated Sinkhorn algorithm by showing that is a special case of incremental/stochastic mirror descent. In order to see this, one should simply plug Kullback-Leibler divergence in both mirror map and the objective function. Since the problem has unbounded domain, the objective function is neither smooth nor it has bounded gradients. However, one can still approach the problem using the notion of relative smoothness, obtaining that the stochastic objective is 1-relative smooth. The discovered equivalence allows us to propose 1) new methods for optimal transport, 2) an extension of Sinkhorn algorithm beyond two constraints.