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 ( η n ) n ≥ 0 {\displaystyle (\eta _{n})_{n\geq 0}} applied to a differentiable function F {\displaystyle F} , one starts with a guess x 0 {\displaystyle \mathbf {x} _{0}} for a local minimum of F , {\displaystyle F,} and considers the sequence x 0 , x 1 , x 2 , … {\displaystyle \mathbf {x} _{0},\mathbf {x} _{1},\mathbf {x} _{2},\ldots } such that
x n + 1 = x n − η n ∇ F ( x n ) , n ≥ 0. {\displaystyle \mathbf {x} _{n+1}=\mathbf {x} _{n}-\eta _{n}\nabla F(\mathbf {x} _{n}),\ n\geq 0.}
This can be reformulated by noting that
x n + 1 = arg min x ( F ( x n ) + ∇ F ( x n ) T ( x − x n ) + 1 2 η n ‖ x − x n ‖ 2 ) {\displaystyle \mathbf {x} _{n+1}=\arg \min _{\mathbf {x} }\left(F(\mathbf {x} _{n})+\nabla F(\mathbf {x} _{n})^{T}(\mathbf {x} -\mathbf {x} _{n})+{\frac {1}{2\eta _{n}}}\|\mathbf {x} -\mathbf {x} _{n}\|^{2}\right)}
In other words, x n + 1 {\displaystyle \mathbf {x} _{n+1}} minimizes the first-order approximation to F {\displaystyle F} at x n {\displaystyle \mathbf {x} _{n}} with added proximity term ‖ x − x n ‖ 2 {\displaystyle \|\mathbf {x} -\mathbf {x} _{n}\|^{2}} . 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 f {\displaystyle f} to optimize over a convex set K ⊂ R n {\displaystyle K\subset \mathbb {R} ^{n}} , and given some norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} on R n {\displaystyle \mathbb {R} ^{n}} . We are also given differentiable convex function h : R n → R {\displaystyle h\colon \mathbb {R} ^{n}\to \mathbb {R} } , α {\displaystyle \alpha } -strongly convex with respect to the given norm. This is called the distance-generating function, and its gradient ∇ h : R n → R n {\displaystyle \nabla h\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}} is known as the mirror map. Starting from initial x 0 ∈ K {\displaystyle x_{0}\in K} , in each iteration of Mirror Descent:
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