Mirror descent
In mathematics, mirror descent is an iterative optimization algorithm for finding a local minimum of a differentiable function.
It generalizes algorithms such as gradient descent and multiplicative weights.
History
Mirror descent was originally proposed by Nemirovski and Yudin in 1983.Motivation
In gradient descent with the sequence of learning rates applied to a differentiable function, one starts with a guess for a local minimum of and considers the sequence such thatThis can be reformulated by noting that
In other words, minimizes the first-order approximation to at with added proximity term.
This squared Euclidean distance term is a particular example of a Bregman distance. Using other Bregman distances will yield other algorithms such as Hedge which may be more suited to optimization over particular geometries.
Formulation
We are given convex function to optimize over a convex set, and given some norm on.We are also given differentiable convex function, -strongly convex with respect to the given norm. This is called the distance-generating function, and its gradient is known as the mirror map.
Starting from initial, in each iteration of Mirror Descent:
- Map to the dual space:
- Update in the dual space using a gradient step:
- Map back to the primal space:
- Project back to the feasible region :, where is the Bregman divergence.
Connections with other methods and extensions
Mirror descent in the online optimization setting is known as Online Mirror Descent.