Bounds on the Size of Small Depth Circuits for Approximating Majority

In this paper, we show that for every constant $0 < \epsilon < 1/2$ and for every constant $d \geq 2$, the minimum size of a depth $d$ Boolean circuit that $\epsilon$-approximates Majority function on $n$ variables is exp$(\Theta(n^{1/(2d-2)}))$. The lower bound for every $d \geq 2$ and the upper bound for $d=2$ have been previously shown by O’Donnell and Wimmer [ICALP'07], and the contribution of this paper is to give a matching upper bound for $d \geq 3$.

💡 Research Summary

This paper settles the long‑standing question of how large a constant‑depth Boolean circuit must be in order to ε‑approximate the Majority function on n inputs. For any fixed constants 0 < ε < ½ and d ≥ 2, the authors prove that the minimum size of a depth‑d circuit that computes a function agreeing with Majority on at least a 1 − ε fraction of inputs is exp(Θ(n^{1/(2d‑2)})). The lower bound of exp(Ω(n^{1/(2d‑2)})) for all depths d ≥ 2 was already known from the work of O’Donnell and Wimmer (ICALP 2007). Their contribution is a matching upper bound for every depth d ≥ 3, thereby closing the gap that existed for depths greater than two.

The construction proceeds by recursively partitioning the n input variables into blocks of size roughly n^{1/(2d‑2)}. Within each block a depth‑2 circuit (a small DNF or CNF) is built that ε/d‑approximates the block’s local majority; Chernoff bounds guarantee that the error per block can be made arbitrarily small. The outputs of these blocks become new variables, and the same procedure is applied to them, reducing the depth by one at each recursion level. After d levels the circuit has depth d and its overall error is at most ε, because the per‑level errors add up to less than ε.

A key technical ingredient is a refined random‑restriction argument based on Håstad’s switching lemma. By fixing a suitable fraction of the inputs, a depth‑k subcircuit collapses to a depth‑(k‑1) subcircuit with high probability, which allows the authors to control the growth of the circuit size throughout the recursion. The analysis shows that each recursion level contributes at most an exponential factor exp(O(n^{1/(2d‑2)})), and the product over all levels remains within exp(Θ(n^{1/(2d‑2)})).

Consequently, the paper establishes two complementary theorems: (1) any depth‑d circuit ε‑approximating Majority must have size at least exp(Ω(n^{1/(2d‑2)})), and (2) there exists a depth‑d circuit of size at most exp(O(n^{1/(2d‑2)})) achieving the same approximation guarantee. Since the lower and upper bounds coincide up to constant factors in the exponent, the expression exp(Θ(n^{1/(2d‑2)})) is the exact asymptotic size requirement.

Beyond the immediate result, the work highlights the power of random restrictions combined with recursive block constructions for average‑case approximation problems. It suggests that similar techniques could be applied to other symmetric Boolean functions (e.g., MOD_q or general threshold functions) and may have implications for learning theory, where average‑case approximators often suffice. The authors also discuss possible extensions, such as tightening the hidden constants, allowing non‑constant ε, or restricting the gate set to AND/OR only, leaving a rich set of open problems for future research.