Min-Max Optimization Requires Exponentially Many Queries

We prove two lower bounds for the first order oracle complexity of minimizing a $d$-dimensional $1$-Lipschitz convex function over the unit ball with $m$ bits of memory. We first show that any such (possibly randomized) algorithm must make $\tildeΩ(\frac{d^2}{\sqrt{m}})$ oracle queries. For deterministic optimization algorithms, we show that $\tildeΩ(\min\{d^{1.6},\frac{d^{8/3}}{m^{2/3}}\})$ queries are required. For all memory regimes of interest, these improves upon the previous best known lower bounds of $\tildeΩ(\max\{\frac{d^{8/3}}{m^{4/3}},\frac{d^{4/3}}{m^{1/6}}\})$ and $\tildeΩ(\frac{d^{5/3}}{m^{1/3}})$ for randomized and deterministic algorithms respectively. Notably, due to existing upper bounds, our lower bound for deterministic algorithms is the first to show a sharp oracle complexity phase transition around $m\approx d^2$, where a polylogarithmic change in memory leads to a $\mathsf{poly}(d)$ change in the number of required oracle calls. Further, when the suboptimality is polynomially small in $d$, our lower bound randomized algorithms is the first to show that $\tildeΩ(d^2)$ memory is necessary to nearly match the optimal query complexity among algorithms without memory constraints. Previously, such a result was only known for the regime where the suboptimality is quasipolynomially small in $d$.

We study the stochastic first-order oracle complexity for constrained or regularized convex-concave min-max optimization and stochastic monotone variational inequalities. We focus on the case when suboptimality is measured in terms of the gradient mapping, also known as, forward-backward or natural residual, an optimality notion that generalizes the gradient norm for unconstrained problems. In this setting, under standard unbiased oracle access with now-standard variance assumptions, the best-known complexity for making the norm of the gradient mapping less than $\varepsilon$ is $\widetilde{O}(\varepsilon^{-4})$, compared to the near-optimal $\widetilde{O}(\varepsilon^{-2})$ that is established in the unconstrained case. We bridge this gap to improve the gradient mapping complexity for constrained convex-concave min-max problems to $\widetilde{O}(\varepsilon^{-2})$. We then extend to prove the same complexity for problems without the bounded variance, by using the Blum-Gladyshev assumption.

We study the running time, in terms of first order oracle queries, of differentially private empirical/population risk minimization of Lipschitz convex losses. We first consider the setting where the loss is non-smooth and the optimizer interacts with a private proxy oracle, which sends only private messages about a minibatch of gradients. In this setting, we show that expected running time $Ω(\min\{\frac{\sqrt{d}}{α^2}, \frac{d}{\log(1/α)}\})$ is necessary to achieve $α$ excess risk on problems of dimension $d$ when $d \geq 1/α^2$. Upper bounds via DP-SGD show these results are tight when $d>\tildeΩ(1/α^4)$. We further show our lower bound can be strengthened to $Ω(\min\{\frac{d}{\bar{m}α^2}, \frac{d}{\log(1/α)} \})$ for algorithms which use minibatches of size at most $\bar{m} < \sqrt{d}$. We next consider smooth losses, where we relax the private oracle assumption and give lower bounds under only the condition that the optimizer is private. Here, we lower bound the expected number of first order oracle calls by $\tildeΩ\big(\frac{\sqrt{d}}α + \min\{\frac{1}{α^2}, n\}\big)$, where $n$ is the size of the dataset. Modifications to existing algorithms show this bound is nearly tight. Compared to non-private lower bounds, our results show that differentially private optimizers pay a dimension dependent runtime penalty. Finally, as a natural extension of our proof technique, we show lower bounds in the non-smooth setting for optimizers interacting with information limited oracles. Specifically, if the proxy oracle transmits at most $Γ$-bits of information about the gradients in the minibatch, then $Ω\big(\min\{\frac{d}{α^2Γ}, \frac{d}{\log(1/α)}\}\big)$ oracle calls are needed. This result shows fundamental limitations of gradient quantization techniques in optimization.