Summary
Some mathematical problems involve finding the best solution within a curved shape, using only yes-or-no questions about points inside it. The authors showed that any computer method that tries to solve these problems or pick a random point inside the shape using such questions must take almost as many steps as the square of the number of dimensions. This means you can't do much better than current methods. They also used this idea to show the same limit applies when estimating the shape's volume.
What this means in practice
- •For optimization engineers: Know that linear optimization methods relying solely on membership queries cannot be substantially sped up beyond a near quadratic number of steps.
- •For data scientists: Understand limits on algorithms drawing uniform samples or estimating volumes of complex shapes when only membership checks are accessible.
A theory result. No direct application yet.
Abstract
We prove nearly quadratic lower bounds for randomized algorithms for linear optimization and uniform sampling over convex bodies in the membership oracle model. For linear optimization, this matches the known nearly quadratic upper bound up to a polylog factor in the dimension. For uniform sampling, this improves on the previous linear lower bound. Our construction also implies the same lower bound for volume estimation.