  Given any increasing sequence of norms $\|\cdot\|_0,\dots,\|\cdot\|_{T-1}$,
we provide an online convex optimization algorithm that outputs points $w_t$ in
some domain $W$ in response to convex losses $\ell_t:W\to \mathbb{R}$ that
guarantees regret $R_T(u)=\sum_{t=1}^T \ell_t(w_t)-\ell_t(u)\le \tilde
O\left(\|u\|_{T-1}\sqrt{\sum_{t=1}^T \|g_t\|_{t-1,\star}^2}\right)$ where $g_t$
is a subgradient of $\ell_t$ at $w_t$. Our method does not require tuning to
the value of $u$ and allows for arbitrary convex $W$. We apply this result to
obtain new "full-matrix"-style regret bounds. Along the way, we provide a new
examination of the full-matrix AdaGrad algorithm, suggesting a better learning
rate value that improves significantly upon prior analysis. We use our new
techniques to tune AdaGrad on-the-fly, realizing our improved bound in a
concrete algorithm.
