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Improved Confidence Bounds for the Linear Logistic Model and Applications to Bandits
Kwang-Sung Jun · Lalit Jain · Blake Mason · Houssam Nassif

Wed Jul 21 07:25 AM -- 07:30 AM (PDT) @
We propose improved fixed-design confidence bounds for the linear logistic model. Our bounds significantly improve upon the state-of-the-art bound by Li et al. (2017) via recent developments of the self-concordant analysis of the logistic loss (Faury et al., 2020). Specifically, our confidence bound avoids a direct dependence on $1/\kappa$, where $\kappa$ is the minimal variance over all arms' reward distributions. In general, $1/\kappa$ scales exponentially with the norm of the unknown linear parameter $\theta^*$. Instead of relying on this worst case quantity, our confidence bound for the reward of any given arm depends directly on the variance of that arm's reward distribution. We present two applications of our novel bounds to pure exploration and regret minimization logistic bandits improving upon state-of-the-art performance guarantees. For pure exploration we also provide a lower bound highlighting a dependence on $1/\kappa$ for a family of instances.

Author Information

Kwang-Sung Jun (University of Arizona)
Lalit Jain (University of Washington)
Blake Mason (University of Wisconsin, Madison)
Houssam Nassif (amazon)

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