A quadratic mirror map gives a projected gradient step

Let $d\ge 1$, let $K\subseteq {\mathbb{R}}^d$ be nonempty, bounded, closed, and convex with nonempty interior, choose $x_0\in {\mathbb{R}}^d$, and set $R(x)=\frac{1}{2}\mathrm{\parallel }x-x_0{\mathrm{\parallel }}_2^2$ on $U={\mathbb{R}}^d$. Initialize at a minimizer of $R$ on $K$. At any round with current decision $x_t\in K$ and gradient $g_t$, the agile mirror step with parameter $\eta >0$ has $y_{t+1}=x_t-\eta g_t$ and

$x_{t+1}=\arg {\min }_{x\in K}{B}_R(x\mathrm{\parallel }{y}_{t+1})=\arg {\min }_{x\in K}\frac{1}{2}\mathrm{\parallel }x-(x_t-\eta g_t){\mathrm{\parallel }}_2^2\mathrm{.}$

Thus its one-step update map is exactly the projected gradient update with the same step size. If $\eta$ is set to the current projected-gradient step size, the two displayed one-step maps agree.

Agile online mirror descent: defines the agile dual step and Bregman projection.

Compactness makes the Bregman projection well-defined: establishes existence and uniqueness of the minimizer over $K$.

Mirror maps, Bregman divergence, and local Hessian norms: defines the quadratic regularizer's Bregman divergence and Hessian norms.

Projected online gradient descent: defines the projected gradient update with the current step size.

The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$: defines the Euclidean norm and inner product used in the quadratic.

A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points: supplies the segment condition for the decision set.

The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement: closedness is in the Euclidean metric topology.

Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space: boundedness is with respect to the Euclidean metric.