1. Ryan ’Donnell Carnegie Mellon University Institute for Advanced Study O

2. def: For f : {−1,1} n → ℝ, “δ-noisy-influence of i on f” is “Invariance Principles” usually work under the assumption of “low noisy-influences”:

3. Theorem: [O–Servedio–Tan–Wan] ∀, δ, γ > 0, ∀ f : {−1,1} n → ℝ with E[f 2 ] = 1, ∃ a decision tree of depth ≤ s.t. all but a γ fraction of subfunctions f L satisfy

4. Proof: “Energy increment” argument. Build tree inductively. Energy = ∈ [0,1]. Recall Stab 1−δ [g] = If a leaf L has replacing it with a query to x j increases the leaf’s contribution to the energy by ≥ δ. If doable for a γ fraction of leaves, make the replacements and increase energy by ≥ δγ.

5. Applications:

7. “Majority Is Stablest Theorem” [MOO’05]: If Inf i [f] = o(1) ∀ i, Stab ρ [f] ≤ Stab ρ [Majority] + o(1). One application: Can weaken hypothesis to