Efficient Multinomial Logistic Bandit Algorithm Using Frequent Directions Matrix Sketching
Researchers have proposed EOFD-MLogB, a new algorithm for multinomial logistic bandits that integrates frequent directions matrix sketching to dramatically reduce per-round time and space complexity. The existing state-of-the-art UCB-type algorithm, OFUL-MLogB, achieves strong regret bounds but requires cubic time and quadratic space per round, making it impractical in high-dimensional settings. The new approach reduces these costs by exploiting low-rank structure in the accumulated Hessian, making the method scalable while preserving near-optimal regret guarantees.
Multinomial logistic bandits (MLogB) model sequential decision-making problems where feedback over multiple outcomes follows a multinomial logistic distribution, with applications in areas such as recommendation systems and clinical trials. The leading prior algorithm, OFUL-MLogB, achieves a regret bound of Õ(Kd√T) but incurs O(K³d³) time and O(K²d²) space per round, which becomes prohibitive as the action dimension d or number of outcomes K grows. The proposed EOFD-MLogB algorithm addresses this by maintaining a low-rank SVD sketch of the accumulated Hessian via the frequent directions technique, reducing expensive matrix operations to one-dimensional root-finding and small K×K eigenvalue computations. The resulting dominant per-round complexity is O(Kd(m+K)²) in time and O(Kd(m+K)) in space, where the sketch size m is much smaller than d. The regret bound for EOFD-MLogB is Õ(Δ_T(Kd ln Δ_T + m)√T), where the sketching error factor Δ_T depends on the truncated spectral tail of the Hessian, meaning that when the Hessian is approximately low-rank, regret closely matches that of the exact method. Experiments reported by the authors validate both the computational efficiency and competitive predictive performance of the new algorithm.
What's missing
The paper does not detail the specific datasets or benchmark tasks used in experiments, nor does it provide empirical comparisons against alternative sketching or approximation methods beyond OFUL-MLogB. The work is a preprint and has not yet undergone peer review.
What different sources said
- arXiv cs.LGCenter
Efficient Multinomial Logistic Bandit via Frequent Directions
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