Efficient Logistic Regression with Mixture of Sigmoids

• URL: http://arxiv.org/abs/2604.02920v1
• Date: Fri, 03 Apr 2026 09:36:34 GMT
• Title: Efficient Logistic Regression with Mixture of Sigmoids
• Abstract summary: We show that the near-optimal worst-case regret bound $O(dlog(Bn))$ for EW can be achieved with total worst-case computational complexity $O(B3 n5)$.<n>Our results show that EW can be both computationally tractable and geometrically adaptive in online classification.
• Score: 15.14559143371469
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: This paper studies the Exponential Weights (EW) algorithm with an isotropic Gaussian prior for online logistic regression. We show that the near-optimal worst-case regret bound $O(d\log(Bn))$ for EW, established by Kakade and Ng (2005) against the best linear predictor of norm at most $B$, can be achieved with total worst-case computational complexity $O(B^3 n^5)$. This substantially improves on the $O(B^{18}n^{37})$ complexity of prior work achieving the same guarantee (Foster et al., 2018). Beyond efficiency, we analyze the large-$B$ regime under linear separability: after rescaling by $B$, the EW posterior converges as $B\to\infty$ to a standard Gaussian truncated to the version cone. Accordingly, the predictor converges to a solid-angle vote over separating directions and, on every fixed-margin slice of this cone, the mode of the corresponding truncated Gaussian is aligned with the hard-margin SVM direction. Using this geometry, we derive non-asymptotic regret bounds showing that once $B$ exceeds a margin-dependent threshold, the regret becomes independent of $B$ and grows only logarithmically with the inverse margin. Overall, our results show that EW can be both computationally tractable and geometrically adaptive in online classification.

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