A very nice reference for this post is [this blog post].
Mirror Descent is a first-order algorithm to solve a convex optimization problem 
In this post we will learn several equivalent formulations of this algorithm and discuss how they are related to each other.
Let 
be a distance-generating function (d.-g.f.), 1-strongly convex on the domain 
and differentiable on its interior, and consider the Bregman divergence on 
Projection formulation
Here 
is any subgradient of the objective function, and 
is the stepsize parameter. This formulation, used extensively in [Bubeck 2015], permits to draw analogy with the Projected Subgradient Descent for which 
and 
Nevertheless, this formulation is a bit delusive since it is not clear how to compute the point 
itself, and under what conditions such exists. Actually the existence of is guaranteed as soon as is strongly convex and proper on 
Supposing it is indeed the case, let us obtain another formulation by some elementary convex analysis. Consider the Legendre conjugate of 
with respect to 
Since 
is the pointwise maximum of convex functions, we have
where 
is the corresponding maximizer, unique due to the strong convexity of 
Combining that observation with the previous formulation,
we obtain
Mirror formulation
![\displaystyle x^+ = \nabla\omega_X^*[\nabla\omega(x) - t \nabla f(x)]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+x%5E%2B+%3D+%5Cnabla%5Comega_X%5E%2A%5B%5Cnabla%5Comega%28x%29+-+t+%5Cnabla+f%28x%29%5D&bg=ffffff&fg=333333&s=0 "\displaystyle x^+ = \nabla\omega_X^*[\nabla\omega(x) - t \nabla f(x)]")
The interesting point is that actually this formulation is more general: it does not require (in contrast with the first one) any assumptions besides strong convexity. Moreover, it is easy to see from the first-order optimality conditions that the map 
is given by the following relation:
where 
is the indicator function, 
is the normal cone, and the multi-map 
is invertible (injective and surjective). The mirror formulation reads: first go ‘beyond the mirror’ by the map 
on 
then take a step beyond the mirror, and finally go back by the inverse map.
Yet another formulation is obtained from the last line of the chaining argument above.
Proximal point formulation
In other words, the algorithm is trying to minimize the local linear model of the function while staying close to the previous search point. Finally, elementary algebra gives the following
Regularization formulation
It is most often used to actually implement the algorithm.
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About Dmitry Ostrovsky
PhD student with interests in statistics, optimization, and machine learning.
