Title: Primal Optimism in Online Optimization
Primary area: optimization
Keywords: online convex optimization; online learning; convex optimization; stochastic optimization; stochastic convex optimization

Abstract: We consider the classic online convex optimization problem in which an algorithm outputs vectors $z_t$ in response to vectors $g_t$. Our algorithm seeks to improve the regret when it has access to a sequence of "hint" vectors $v_t$ that estimate the location of the final optimal parameter value $u$. Specifically, we provide an online linear optimization algorithm that guarantees regret $R_T(u)=\sum_{t=1}^T \langle g_t, z_t -u\rangle \le \sqrt{\sum_{t=1}^T \|g_t\|^2\|v_t-u\|^2}$ for any comparison point $u$ and any sequence of vectors $v_t$, so long as $v_t$ is available before we commit to $z_{t+1}$.