Recap and context

• Last lecture: we introduced Online Mirror Descent (OMD) as a general framework that encapsulates the OGD and MW algorithms as special cases
• This lecture: our goal is to give a general analysis of OMD, from which we will derive the OGD and MW regret bounds directly

Recap: Online Mirror Descent (OMD)

Recall the general OMD template:

<aside> ⚙ Meta-Algorithm: Online Mirror Descent (OMD)

Parameters: strictly convex regularization $R$ over $W \subseteq \R^d$, stepsize $\eta>0$

• Initialize $w_1 \in W$

• For $t=1,2,\ldots,T$:

  1. play $w_t \in W$, receive loss $f_t$
  2. compute $g_t \in \partial f_t(w_t)$
  3. update:

  $$ \newcommand{\E}{\mathbb E} \begin{align*} \nabla R(w_{t+1}') &= \nabla R(w_t) - \eta g_t \tag{step} \\ w_{t+1} \qquad\;&= \argmin_{w \in W} \; D_R(w,w_{t+1}') \tag{projection} \end{align*} $$

</aside>

Recall:

<aside> 💡 Definition: Bregman divergence

The Bregman divergence $D_f$ associated with a convex and differentiable $f: S \to \R$ is:

$$ \begin{align*} \forall x,y \in S:\quad D_f(y,x) = f(y)-f(x)-\nabla f(x) \cdot (y-x) \end{align*} $$

</aside>

Recap: Strong convexity, generalized

We will analyze OMD for regularizers $R$ that are strongly convex with respect to a norm $\|\cdot\|$ :