The Exponential Price of Determinism in Nonsmooth Nonconvex Optimization

This paper considers the nonconvex nonsmooth problem in which the objective function is Lipschitz continuous. We focus on the stochastic setting where the algorithm can access stochastic function value evaluations with heavy-tailed noise, which is prevalent in many popular machine learning applications. We propose a stochastic zeroth-order algorithm that refines the framework of online-to-nonconvex conversion by clipping the two-point gradient estimator. The theoretical analysis shows that our algorithm can find a $(δ, ε)$-Goldstein stationary point with zeroth-order oracle complexity of ${\mathcal O}(d^{\frac{p}{2(p-1)}}δ^{-1}ε^{-\frac{2p-1}{p-1}})$, where $d$ is the problem dimension and $p\in(1,2]$ is the order of bounded moments. Note that our dependence on dimension $d$ matches the best-known results of stochastic zeroth-order optimization for finding the sub-optimal solution of a stochastic convex nonsmooth problem. In addition, our dependence on accuracy parameters $δ$ and $ε$ is consistent with that of the best-known stochastic first-order algorithms for stochastic nonconvex nonsmooth problems. Finally, we conduct numerical experiments to demonstrate the effectiveness of the proposed method.

We study the deterministic first-order oracle complexity of finding $ε$-stationary points in smooth nonconvex optimization when the objective satisfies higher-order smoothness assumptions. While the classical $ε^{-2}$ rate is optimal under only Lipschitz gradients, higher-order smoothness leads to accelerated first-order upper bounds, most notably the $ε^{-7/4}$ rate under Lipschitz Hessians and the $ε^{-5/3}$ rate under Lipschitz third derivatives. The matching lower bounds, however, have remained open. We resolve this gap by proving a new dimension-free first-order lower bound for higher-order smooth nonconvex functions, valid for every finite smoothness order. In particular, our construction gives a matching $Ω(ε^{-7/4})$ lower bound in the Hessian-Lipschitz case and a matching $Ω(ε^{-5/3})$ lower bound in the third-order-smooth regime. The hard instance is based on a \emph{block-chain} mechanism that enforces blockwise oracle revelation while preserving the smoothness structure needed for the scalar hard instance. The lower-bound construction was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors.

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.