Tight Lower Bounds for the Bit and Inner Product Oracle for Constrained Convex Optimization
Amitabh Basu, Phillip Kerger, Marco Molinaro
TL;DR
This work establishes tight lower bounds on the information complexity for constrained, mixed-integer convex optimization under two bit-wise first-order oracles: a bit oracle and an inner-product oracle. The authors prove that solving feasibility requires at least $\Omega\big(2^n d^2 \log(R/\rho)\big)$ queries for the bit oracle and at least $\Omega\big(2^n d^2 \big(1 + \frac{1}{\log d}\log(R/\rho)\big)\big)$ queries for the inner-product oracle, up to logarithmic factors, even when the oracle may reveal full coordinate or inner-product data. These lower bounds match, up to logarithmic terms, natural discretizations of cutting-plane methods, demonstrating that standard bit representations of first-order information are essentially optimal for constrained convex optimization in this information-analytic sense. The results tie to broader communication-complexity insights and point to future work on unconstrained and zeroth-order settings, with conjectures suggesting similar $\Omega(d^2)$-type limits in broader contexts.
Abstract
We establish new lower-bounds for the information complexity of mixed-integer convex optimization under two "bit-wise" oracles. The first oracle provides bits of first-order information in the standard coordinate model, and the second oracle answers whether the inner product of a specified vector with the gradient of the function at a point or the normal vector of a separating hyperplane for the feasible region is positive or non-positive, thus also providing one bit of first-order information. The new contribution is that under such oracles, the complexity is quadratic in the number of continuous decision variables, which was not known before even for continuous convex optimization. These new lower-bounds are tight (up to a logarithmic term), matched by a natural discretization of standard cutting-plane methods for convex optimization. These reveal that using a standard bit-representation of the first-order information is, in general, the best one can do with respect to the number of bits of information needed to solve constrained convex optimization problems.