Related papers: Adaptive Online Learning with Varying Norms

Adaptive Online Learning with Varying Norms

URL: http://arxiv.org/abs/2002.03963v1
Date: Mon, 10 Feb 2020 17:22:08 GMT
Title: Adaptive Online Learning with Varying Norms
Authors: Ashok Cutkosky
Abstract summary: We provide an online convex optimization algorithm that outputs points $w_t$ in some domain $W$. We apply this result to obtain new "full-matrix"-style regret bounds.
Score: 45.11667443216861
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
Abstract: 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.

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