Properly Learning Decision Trees in almost Polynomial Time

We give an n O (log log ) -time membership query algorithm for properly and agnostically learning decision trees under the uniform distribution over { ± 1} . Even in realizable setting, previous fastest runtime was , a consequence of classic Ehrenfeucht Haussler. Our shares similarities with practical heuristics trees, which we augment additional ideas to circumvent known lower bounds against these heuristics. To analyze our algorithm, prove new structural result that strengthens theorem O’Donnell, Saks, Schramm, Servedio. While OSSS says every tree has influential variable, show how can be “pruned” so variable resulting is influential.