We study a \emph{max-risk} objective for active learning in $d$-armed bandits: a learner adaptively allocates a budget of $T$ samples across $d$ groups to minimize the worst-case per-group uncertainty index $\max_{k\in[d]}\sigma_k^2/n_k$. We develop a local minimax framework and prove the first general lower bound for this objective, valid for any finite-variance hypothesis class $\mathcal H$. The bound separates difficulty into three orthogonal factors: a \emph{budget} term, a \emph{heteroscedasticity} index measuring how unevenly the uncertainty is spread across arms, and a model-dependent curvature functional, the \emph{Variance Local Curvature} ($\mathrm{VLC}$), which captures how much information a local change of variance creates inside $\mathcal H$. For smooth classes, the $\mathrm{VLC}$ is a reparametrization of a variance–Fisher information, with closed-form values for common families. Benchmarking against the strongest available upper bound shows near-optimality up to logarithmic factors in broad regimes, and pinpoints a systematic gap in highly heterogeneous instances. Our proof introduces two key ingredients: a loss-induced $\ell_1$ geometry on the decision space, and a representation-based instance generator that reduces hard-instance construction to an explicit random matrix calculation.

We consider a fundamental problem where there are multiple groups whose data distributions are unknown, and an analyst would like to learn the mean of each group. We consider an active learning framework to sequentially collect $T$ samples with bandit, each period observing a sample from a chosen group. After observing a sample, the analyst may update their estimate of the mean and variance of that group and choose the next group accordingly. The objective is to dynamically collect samples to minimize the $p$-norm of the vector of variances of our mean estimators after $T$ rounds. We propose an algorithm, Variance-UCB, that selects groups according to a an upper bound on the variance estimate adjusted to the $p$-norm chosen. We show that the regret of Variance-UCB is $O(T^{-2})$ for finite $p$, and prove that no algorithm can do better. When $p$ is infinite, we recover the $O(T^{-1.5})$ obtained in \cite{activelearning, carpentier2011upper} and provide a new lower bound showing that no algorithm can do better.

In this paper, we study a dynamic assortment optimization problem under bandit feedback, where a seller with a fixed initial inventory of N substitutable products faces a sequence of i.i.d. customer arrivals (with an unknown distribution) over a time horizon of T periods, and needs to decide in each period on an assortment of products to offer to the customer to maximize the total expected revenue. Such a problem arises in many applications including online retail and recommendations. The seller has initially no (or only limited) information about the customer's preferences and needs to learn them through repeated interaction with the i.i.d. customers. Specifically, in each period, the seller offers an assortment to the customer; the customer makes a choice from the assortment according to the unknown preferences or choice model, and the seller only observes the eventual choice from the given assortment and needs to update the estimate and future actions under this bandit feedback. Therefore, this problem exemplifies the classical trade-off between exploitation and exploration: the seller needs to simultaneously gain information about the customer's preferences and offer revenue-maximizing assortments, while respecting the resource constraints.