This paper studies efficient online algorithms for multinomial logistic bandits (MLogB), where the feedback distribution over $K+1$ outcomes follows a multinomial logistic model of $d$-dimensional action vectors. A representative UCB-type algorithm, OFUL-MLogB, achieves a regret bound of $\tilde{\mathcal{O}}(Kd\sqrt{T})$, but still requires $\mathcal{O}(K^3d^3)$ time and $\mathcal{O}(K^2d^2)$ space per round due to parameter estimation and optimistic reward construction, which is prohibitive in high-dimensional settings. To address this limitation, we propose EOFD-MLogB, which integrates frequent directions matrix sketching into OFUL-MLogB. By maintaining a low-rank SVD sketch of the accumulated Hessian, constrained online Newton updates in parameter estimation and $Kd \times K$ spectral-norm computations in the reward bonus are reduced to one-dimensional root-finding tasks and $K \times K$ eigenvalue computations, respectively. This yields dominant per-round time complexity $\mathcal{O}(Kd(m+K)^2)$ and space complexity $\mathcal{O}(Kd(m+K))$, where $m \ll d$ is the sketch size. We further prove a regret bound of $\tilde{\mathcal{O}}(Δ_T(Kd\lnΔ_T+m)\sqrt{T})$, where the sketching error factor $Δ_T$ is controlled by the $m$-truncated spectral tail of the Hessian. Thus, when the Hessian is approximately low-rank, the regret is close to that of OFUL-MLogB. Experiments validate the computational efficiency and competitive performance.

翻译：本文研究多项逻辑斯蒂强盗的高效在线算法；在该问题中，$K+1$个结果的反馈分布服从$d$维动作向量的多项逻辑斯蒂模型。一类代表性的UCB型算法OFUL-MLogB可获得$\tilde{\mathcal{O}}(Kd\sqrt{T})$的遗憾界，但由于参数估计和乐观奖励构造，每轮仍需$\mathcal{O}(K^3d^3)$时间和$\mathcal{O}(K^2d^2)$空间，在高维场景下难以承受。为解决此局限，我们提出EOFd-MLogB，它将频繁方向矩阵素描技术融入OFUL-MLogB。通过维护累积Hessian矩阵的低秩SVD素描，参数估计中的约束在线牛顿更新和奖励奖励中的$Kd \times K$谱范数计算分别被简化为单变量求根问题与$K \times K$特征值计算。由此，主要每轮时间复杂度降至$\mathcal{O}(Kd(m+K)^2)$，空间复杂度为$\mathcal{O}(Kd(m+K))$，其中$m \ll d$为素描大小。我们进一步证明了$\tilde{\mathcal{O}}(Δ_T(Kd\lnΔ_T+m)\sqrt{T})$的遗憾界，其中素描误差因子$Δ_T$由Hessian矩阵的$m$截断谱尾控制。因此，当Hessian近似低秩时，其遗憾接近OFUL-MLogB。实验验证了其计算效率与竞争性表现。