Lecture 2: Multiplicative Weights and Mirror Descent

In the last lecture, we considered thematrix scaling problem: Given non-negativematrices X, T ∈ ’n×n + , our goal was to find non-negative diagonal matrices D1 ,D2 so that D1XD2 had the same row and column sums as the target matrix T. In other words, we sought to weight the rows and columns of X by positive numbers in order to achieve this. We used entropy optimization to prove the following theorem.