Approximate Carath\'eodory bounds via Discrepancy Theory

The approximate Carath\'eodory problem in general form is as follows: Given two symmetric convex bodies $P,Q \subseteq \mathbb{R}^m$, a parameter $k \in \mathbb{N}$ and $\mathbf{z} \in \textrm{conv}(X)$ with $X \subseteq P$, find $\mathbf{v}_1,\ldots,\mathbf{v}_k \in X$ so that $\|\mathbf{z} - \frac{1}{k}\sum_{i=1}^k \mathbf{v}_i\|_Q$ is minimized. Maurey showed that if both $P$ and $Q$ coincide with the $\| \cdot \|_p$-ball, then an error of $O(\sqrt{p/k})$ is possible. We prove a reduction to the vector balancing constant from discrepancy theory which for most cases can provide tight bounds for general $P$ and $Q$. For the case where $P$ and $Q$ are both $\| \cdot \|_p$-balls we prove an upper bound of $\sqrt{ \frac{\min\{ p, \log (\frac{2m}{k}) \}}{k}}$. Interestingly, this bound cannot be obtained taking independent random samples; instead we use the Lovett-Meka random walk. We also prove an extension to the more general case where $P$ and $Q$ are $\|\cdot \|_p$ and $\| \cdot \|_q$-balls with $2 \leq p \leq q \leq \infty$.