Algorithms and matching lower bounds for approximately-convex optimization

In recent years, a rapidly increasing number of applications in practice requires solving non-convex objectives, like training neural networks, learning graphical models, maximum likelihood estimation etc. Though simple heuristics such as gradient descent with very few modifications tend to work well, theoretical understanding is very weak. We consider possibly the most natural class of non-convex functions where one could hope to obtain provable guarantees: functions that are ``approximately convex'', i.e. functions $\tf: \Real^d \to \Real$ for which there exists a \emph{convex function} $f$ such that for all $x$, $|\tf(x) - f(x)| \le \errnoise$ for a fixed value $\errnoise$. We then want to minimize $\tf$, i.e. output a point $\tx$ such that $\tf(\tx) \le \min_{x} \tf(x) + \err$. It is quite natural to conjecture that for fixed $\err$, the problem gets harder for larger $\errnoise$, however, the exact dependency of $\err$ and $\errnoise$ is not known. In this paper, we strengthen the known \emph{information theoretic} lower bounds on the trade-off between $\err$ and $\errnoise$ substantially, and exhibit an algorithm that matches these lower bounds for a large class of convex bodies.