Tree tribes and lower bounds for switching lemmas

We show tight upper and lower bounds for switching lemmas obtained by the action of random p-restrictions on boolean functions that can be expressed as decision trees in which every vertex is at a distance of at most t from some leaf, also called t-clipped decision trees. More specifically, we show the following: 1. If a boolean function f can be expressed as a t-clipped decision tree, then under the action of a random p-restriction ρ, the probability that the smallest depth decision tree for f |ρ has depth greater than d is upper bounded by (4p2t)d. 2. For every t, there exists a function gt that can be expressed as a t-clipped decision tree, such that under the action of a random p-restriction ρ, the probability that the smallest depth decision tree for gt|ρ has depth greater than d is lower bounded by (c0 p2t)d, for 0 ≤ p ≤ cp2−t and 0 ≤ d ≤ cd log n 2t log t , where c0, cp, cd are universal constants. ∗California Institute of Technology. 1 ar X iv :1 70 3. 00 04 3v 1 [ cs .C C ] 2 8 Fe b 20 17