We study online learning with bandit feedback (i.e. learner has access to only zeroth-order oracle) where cost\/reward functions $\\f_t$ admit a “pseudo-1d” structure, i.e. $\\f_t(\\w) = \\loss_t(\\pred_t(\\w))$ where the output of $\\pred_t$ is one-dimensional. At each round, the learner observes context $\\x_t$, plays prediction $\\pred_t(\\w_t; \\x_t)$ (e.g. $\\pred_t(\\cdot)=\\langle \\x_t, \\cdot\\rangle$) for some $\\w_t \\in \\mathbb{R}^d$ and observes loss $\\loss_t(\\pred_t(\\w_t))$ where $\\loss_t$ is a convex Lipschitz-continuous function. The goal is to minimize the standard regret metric. This pseudo-1d bandit convex optimization problem (\\SBCO) arises frequently in domains such as online decision-making or parameter-tuning in large systems. For this problem, we first show a lower bound of $\\min(\\sqrt{dT}, T^{3\/4})$ for the regret of any algorithm, where $T$ is the number of rounds. We propose a new algorithm \\sbcalg that combines randomized online gradient descent with a kernelized exponential weights method to exploit the pseudo-1d structure effectively, guaranteeing the {\\em optimal} regret bound mentioned above, up to additional logarithmic factors. In contrast, applying state-of-the-art online convex optimization methods leads to $\\tilde{O}\\left(\\min\\left(d^{9.5}\\sqrt{T},\\sqrt{d}T^{3\/4}\\right)\\right)$ regret, that is significantly suboptimal in $d$.<\/p>\n","protected":false},"excerpt":{"rendered":"
We study online learning with bandit feedback (i.e. learner has access to only zeroth-order oracle) where cost\/reward functions $\\f_t$ admit a “pseudo-1d” structure, i.e. $\\f_t(\\w) = \\loss_t(\\pred_t(\\w))$ where the output of $\\pred_t$ is one-dimensional. At each round, the learner observes context $\\x_t$, plays prediction $\\pred_t(\\w_t; \\x_t)$ (e.g. $\\pred_t(\\cdot)=\\langle \\x_t, \\cdot\\rangle$) for some $\\w_t \\in \\mathbb{R}^d$ 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