We present the first efficient averaging sampler that achieves asymptotically optimal randomness complexity and near-optimal sample complexity for natural parameter choices. Specifically, for any constant $\alpha > 0$, for $\delta > 2^{-\mathrm{poly}(1 / \varepsilon)}$, it uses $m + O(\log (1 / \delta))$ random bits to output $t = O(\log(1 ... more >>>