Local Correction of Juntas

A Boolean function f over n variables is said to be q-locally correctable if, given a black-box access to a function g which is “close” to an isomorphism f_sigma of f, we can compute f_sigma(x) for any x in Z_2^n with good probability using q queries to g. We observe that any k-junta, that is, any function which depends only on k of its input variables, is O(2^k)-locally correctable. Moreover, we show that there are examples where this is essentially best possible, and locally correcting some k-juntas requires a number of queries which is exponential in k. These examples, however, are far from being typical, and indeed we prove that for almost every k-junta, O(k log k) queries suffice.

💡 Research Summary

The paper studies the problem of locally correcting Boolean functions, focusing on the class of k‑juntas—functions that depend on only k out of n input variables. A function f is said to be q‑locally correctable (for a given error parameter ε) if, given black‑box access to a function g that is ε‑close to some isomorphic copy fσ of f (i.e., f after a permutation σ of its variables), one can recover the value fσ(x) for any specified input x with probability at least 2/3 while making at most q queries to g.

The authors first observe that any Boolean function of degree at most k is O(2^k)‑locally correctable, a result that follows from known techniques for testing low‑degree polynomials. Since every k‑junta can be expressed as a degree‑k polynomial, this yields an upper bound of O(2^k) queries for all k‑juntas. The algorithm works by randomly selecting k + 1 vectors, forming an affine sub‑cube of size 2^{k+1}, and querying all points of the cube except the target x. If none of the queried points have been altered by the ε‑fraction of noise, the parity of the queried values determines fσ(x). The probability that any of the 2^{k+1} − 1 queried points is corrupted is bounded by (2^{k+1} − 1)·ε, which is < 1/4 for ε < 2^{−k−3}, guaranteeing success probability ≥ 3/4.

To show that the exponential upper bound is tight for some juntas, the paper employs Yao’s minimax principle. Two distributions D₀ and D₁ over functions that are o(1)‑close to an isomorphic copy of the AND‑type k‑junta are constructed. In D₀ the relevant k variables lie in the first half of the coordinates, and the function outputs 0 on a balanced input x; in D₁ the relevant variables are placed in the second half, making the same x evaluate to 1. Both distributions modify only an o(1) fraction of the truth table, so they satisfy the ε‑closeness requirement. However, any single query returns 1 with probability at most 2^{−Ω(k)} under either distribution. Consequently, any algorithm that makes fewer than 2^{Ω(k)} queries cannot distinguish D₀ from D₁ with noticeable advantage, implying an Ω(2^k) lower bound on the query complexity for these particular juntas. This demonstrates that the exponential bound cannot be improved in the worst case.

The paper then argues that such worst‑case juntas are atypical. For a random k‑junta, each influencing variable has influence Inf_i(f) = Pr_x