Let $d$ be the lowest $L_1$ distance to which a $k$-symbol distribution $p$ can be estimated from $m$ batches of $n$ samples each, when up to $\beta m$ batches may be adversarial. For $\beta<1/2$, Qiao and Valiant (2017) showed that $d=\Omega(\beta/\sqrt{n})$ and requires $m=\Omega(k/\beta^2)$ batches. For $\beta<1/900$, they provided a $d$ and $m$ order-optimal algorithm that runs in time exponential in $k$. For $\beta<0.5$, we propose an algorithm with comparably optimal $d$ and $m$, but run-time polynomial in $k$ and all other parameters.

The title of "The Bad Batch," Ana Lily Amirpour's arid and feverish new movie, refers to the assorted undesirables who have been exiled by the U.S. government to a vast and barely habitable stretch of Texas wasteland. Under a merciless sun, a sullen new arrival named Arlen (the British actress Suki Waterhouse) is promptly captured by a gang of iron-pumping cannibals who tie her up, drug her and divest her of an arm and a leg. Arlen escapes, barely, and finds her way to a makeshift town of losers and drifters, noodle carts and shipping containers laughably known as Comfort. Ruled over by a self-styled messiah/drug dealer/harem-keeper known as the Dream (Keanu Reeves), Comfort is a slight improvement on Arlen's previous situation. But it's still no country for old men or young women -- or, for that matter, a little girl named Honey (Jayda Fink) and her bunny rabbit, both token symbols of innocence in this dust-choked dystopia.