The Exponential Price of Determinism in Nonsmooth Nonconvex Optimization
We study the complexity of finding $(δ,ε)$-Goldstein stationary points of nonsmooth nonconvex Lipschitz functions. By now, it is known that randomized first-order algorithms can solve this task with a dimension-free oracle complexity [Zhang et al., 2020], whereas deterministic algorithms cannot, as their complexity must scale at least linearly with the dimension $d$ [Jordan et al., 2023, Tian and So, 2024]. This leaves open whether deterministic algorithms can nevertheless solve the problem with oracle complexity polynomial in $d$. We answer this question negatively by proving a lower bound of order $(1/ε)^{Ω(d)}$ for deterministic algorithm, closing the exponential gap between the previo...
| Authors | Guy Kornowski |
|---|---|
| Primary category | math.OC |
| Published | 20 September 2026 |
| Last updated at source | 20 September 2026 |
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