We study non-parametric bandits in which arms are evaluated using Cumulative Prospect Theory (CPT) rather than expected reward. We consider continuous action spaces with arm-dependent noise and propose Non-Parametric CPT Bandits ($\mathtt{NPCB}$), an optimistic algorithm that decouples mean estimation from CPT-based decision-making. We prove CPT regret of order $\widetilde{\mathcal O}\bigl(\sqrt{T\,\dim_E(\mathcal F,T^{-2})\,\log N(\mathcal F,T^{-2},\|\cdot\|_\infty)}\bigr)$, where $\dim_E$ is the Eluder dimension of the function class $\mathcal F$, $T$ is the time horizon, and $N$ is its covering number in the sup norm. This matches the order-wise regret of classical non-parametric bandits, showing that CPT risk sensitivity incurs no additional order-wise regret cost. We extend $\mathtt{NPCB}$ to unknown noise CDFs, retaining the known-CDF mean-learning term and adding a CDF-learning cost governed by a distributional Eluder dimension. We also study Kernel-CPT bandits and obtain information-gain-based regret bounds for a broad range of kernels, matching the corresponding risk-neutral kernel rates up to logarithmic factors.
Let $P$ be a finite point set in $\mathbb{R}^d$. A weak $\varepsilon$-net for $P$ with respect to convex sets is a finite set $X\subset\mathbb{R}^d$ intersecting every convex set containing at least $\varepsilon|P|$ points of $P$. Determining their optimal worst-case size was described by Alon, Kalai, Matousek, and Meshulam [Adv. Appl. Math., 2002] as "one of the truly important open problems in combinatorial geometry." We give elementary constructions which establish the first polynomially superlinear lower bounds in fixed dimension. Already in $\mathbb{R}^3$, we construct point sets for which every weak $\varepsilon$-net has size $Ω(\varepsilon^{-3/2})$. More generally, for every fixed $d\ge3$, we obtain a lower bound of $Ω_d(\varepsilon^{-γ_d})$, where $γ_d=\sqrt{d/2}+O(1)$. Consequently, no upper bound of the form $O_d(\varepsilon^{-C})$ can hold with an absolute constant $C$ which is independent of the dimension. Despite the recent progress on upper bounds, the strongest previous general lower bound remained $Ω_d(\varepsilon^{-1}\log^{d-1}(1/\varepsilon))$, due to Bukh, Matousek, and Nivasch [Israel J. Math., 2011]. To obtain our results, we apply a quadratic lift to grid-based configurations with many point-line incidences. The bounds also yield polynomial separations between fractional and integral piercing numbers of convex families, and lower bounds for Hadwiger-Debrunner numbers.