Memory-Constrained Algorithms for Convex Optimization via Recursive Cutting-Planes

We propose a family of recursive cutting-plane algorithms to solve feasibility problems with constrained memory, which can also be used for first-order convex optimization. Precisely, in order to find a point within a ball of radius $ε$ with a separation oracle in dimension $d$ -- or to minimize $1$-Lipschitz convex functions to accuracy $ε$ over the unit ball -- our algorithms use $\mathcal O(\frac{d^2}{p}\ln \frac{1}ε)$ bits of memory, and make $\mathcal O((C\frac{d}{p}\ln \frac{1}ε)^p)$ oracle calls, for some universal constant $C \geq 1$. The family is parametrized by $p\in[d]$ and provides an oracle-complexity/memory trade-off in the sub-polynomial regime $\ln\frac{1}ε\gg\ln d$. While several works gave lower-bound trade-offs (impossibility results) -- we explicit here their dependence with $\ln\frac{1}ε$, showing that these also hold in any sub-polynomial regime -- to the best of our knowledge this is the first class of algorithms that provides a positive trade-off between gradient descent and cutting-plane methods in any regime with $ε\leq 1/\sqrt d$. The algorithms divide the $d$ variables into $p$ blocks and optimize over blocks sequentially, with approximate separation vectors constructed using a variant of Vaidya's method. In the regime $ε\leq d^{-Ω(d)}$, our algorithm with $p=d$ achieves the information-theoretic optimal memory usage and improves the oracle-complexity of gradient descent.