Nonsmooth Nonconvex Optimization from Noisy Comparisons: Unbiased Gap Estimation and the Price of Preference Feedback
We study the minimization of $L$-Lipschitz, possibly nonsmooth and nonconvex $f:R^d\to R$ from noisy pairwise comparisons: comparing $x$ and $y$ returns a bit equal to $1$ with probability $σ((f(y)-f(x))/τ)$, for a known link $σ$ and temperature $τ$, and the goal is a $(δ,ε)$-Goldstein stationary point. We give an exactly unbiased estimator of the gap $f(y)-f(x)$ whenever $σ^{-1}$ is analytic on a lens in the complex plane, as for the logistic, probit and cauchit links: compare the pair until the outcome changes and return $\pmτ$ times a partial sum of the expansion coefficients of the inverse link, the harmonic number of the run length for the logistic link. It uses at most $1+1/(1-α)$ comparisons in expectation, $α$ the largest queried success probability, and has second moment of order the gap squared plus $τ^2$. With randomized smoothing and the online-to-nonconvex conversion this gives $O\big(d\,(1+d\,C_{link}/κ^2)\,L^2Δ_f/(δε^3)\big)$ comparisons when $Δ_f\ge Lδ$, where $κ$ is the ratio of the largest queried gap to $τ$, which the smoothing radius caps at a constant; relative to the optimal rate $O(dL^2Δ_f/(δε^3))$ for noisy value oracles, comparisons cost a factor $d/κ^2$. We prove that any comparison-based method needs $Ω(d^2τ^2/(ε^2δ^2))$ comparisons on a smooth convex subclass once $d\gtrsim(εδ/τ)^2$: in the comparison-noise-dominated regime the bounds agree in $d$ and $τ$, and a factor $Δ_f/(δε)$ remains open. Experiments illustrate the estimator's cost and variance and the $d^2$ growth of the gradient-estimator second moment.