Randomized Query Composition and Product Distributions

Abstract

Let 𝖱_ε denote randomized query complexity for error probability ε, and R: = 𝖱_{1/3}. In this work we investigate whether a perfect composition theorem 𝖱(f∘gⁿ) = Ω(𝖱(f)⋅ 𝖱(g)) holds for a relation f ⊆ {0,1}ⁿ × 𝒮 and a total inner function g:{0,1}^m → {0,1}.
Composition theorems of the form 𝖱(f∘gⁿ) = Ω(𝖱(f)⋅ 𝖬(g)) are known for various measures 𝖬. Such measures include the sabotage complexity RS defined by Ben-David and Kothari (ICALP 2015), the max-conflict complexity defined by Gavinsky, Lee, Santha and Sanyal (ICALP 2019), and the linearized complexity measure defined by Ben-David, Blais, Göös and Maystre (FOCS 2022). The above measures are asymptotically non-decreasing in the above order. However, for total Boolean functions no asymptotic separation is known between any two of them.
Let 𝖣^{prod} denote the maximum distributional query complexity with respect to any product (over variables) distribution . In this work we show that for any total Boolean function g, the sabotage complexity RS(g) = Ω̃(𝖣^{prod}(g)). This gives the composition theorem 𝖱(f∘gⁿ) = Ω̃(𝖱(f)⋅ 𝖣^{prod}(g)). In light of the minimax theorem which states that 𝖱(g) is the maximum distributional complexity of g over any distribution, our result makes progress towards answering the composition question.
We prove our result by means of a complexity measure 𝖱_ε^{prod} that we define for total Boolean functions. Informally, 𝖱_ε^{prod}(g) is the minimum complexity of any randomized decision tree with unlabelled leaves with the property that, for every product distribution μ over the inputs, the average bias of its leaves is at least ((1-ε)-ε)/2 = 1/2-ε. It follows by standard arguments that 𝖱_{1/3}^{prod}(g) = Ω(𝖣^{prod}(g)). We show that 𝖱_{1/3}^{prod} is equivalent to the sabotage complexity up to a logarithmic factor.
Ben-David and Kothari asked whether RS(g) = Θ(𝖱(g)) for total functions g. We generalize their question and ask if for any error ε, 𝖱_ε^{prod}(g) = Θ̃(𝖱_ε(g)). We observe that the work by Ben-David, Blais, Göös and Maystre (FOCS 2022) implies that for a perfect composition theorem 𝖱_{1/3}(f∘gⁿ) = Ω(𝖱_{1/3}(f)⋅𝖱_{1/3}(g)) to hold for any relation f and total function g, a necessary condition is that 𝖱_{1/3}(g) = O(1/(ε)⋅ 𝖱_{1/2-ε}(g)) holds for any total function g. We show that 𝖱_ε^{prod}(g) admits a similar error-reduction 𝖱_{1/3}^{prod}(g) = Õ(1/(ε)⋅𝖱_{1/2-ε}^{prod}(g)). Note that from the definition of 𝖱_ε^{prod} it is not immediately clear that 𝖱_ε^{prod} admits any error-reduction at all.
We ask if our bound RS(g) = Ω̃(𝖣^{prod}(g)) is tight. We answer this question in the negative, by showing that for the NAND tree function, sabotage complexity is polynomially larger than 𝖣^{prod}. Our proof yields an alternative and different derivation of the tight lower bound on the bounded error randomized query complexity of the NAND tree function (originally proved by Santha in 1985), which may be of independent interest. Our result shows that sometimes, 𝖱_{1/3}^{prod} and sabotage complexity may be useful in producing an asymptotically larger lower bound on 𝖱(f∘gⁿ) than Ω̃(𝖱(f)⋅ 𝖣^{prod}(g)). In addition, this gives an explicit polynomial separation between 𝖱 and 𝖣^{prod} which, to our knowledge, was not known prior to our work.