Aug 2026· 5 citations· ⚡ 1 influential· 15 references
Mathematics
TL;DR
It is proved that every deterministic first-order method requires $\Omega(\ell\Delta\kappa/\epsilon^2)$ oracle queries in the worst case to find $x$ satisfying $\Phi(0)-\inf_x\Phi(x)$ and that the linear dependence on $\kappa$ is unavoidable for deterministic first-order methods.
Abstract
We study the deterministic first-order oracle complexity of finding stationary points of the value function in smooth nonconvex-Polyak-{\L}ojasiewicz (NC-P{\L}) minimax optimization. We assume that the objective is jointly $\ell$-smooth and satisfies the $\mu$-P{\L} condition in the dual variable, and that its value function $\Phi(x):=\max_y f(x;y)$ satisfies $\Phi(0)-\inf_x\Phi(x)\leq\Delta$. When $\kappa:=\ell/\mu\gtrsim 1$ and $0<\epsilon^2\lesssim\ell\Delta$, we prove that every deterministic first-order method requires $\Omega(\ell\Delta\kappa/\epsilon^2)$ oracle queries in the worst case to find $x$ satisfying $\|\nabla\Phi(x)\|\leq\epsilon$. This rate matches the known upper bound in its dependence on $(\ell,\Delta,\kappa,\epsilon)$ [Yang et al., 2022] and shows that the linear dependence on $\kappa$ is unavoidable for deterministic first-order methods.
It is proved that every deterministic first-order algorithm requires a first-order oracle that returns both the function value and the full subdifferential at every query point, and establishes the optimal deterministic oracle complexity.
We study the simultaneous approximation of constant-degree polynomials over convex sets. For any family of $m$ degree-$d$ polynomials and any convex set ${H} \subseteq \mathbb{R}_{\ge0}^n$, we construct an $\epsilon$-Cover of the joint value set $\{(f_1(x), \dots, f_m(x)) : x \in {H}\}$ in the $\ell_\infty$-norm. This...
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We study a deterministic family of sharpness-aware minimization methods for smooth nonconvex functions. The perturbation is $$ y_k=x_k+\rho\, \frac{\nabla f(x_k)}{\norm{\nabla f(x_k)}^\alpha}, \qquad 0\leq\alpha\leq 1, $$ so that its effective radius is $\rho\norm{\nabla f(x_k)}^{1-\alpha}$. For $0<\alpha\leq1$, we giv...
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Let $X\subset\mathbb{R}$ be finite and let $\gamma(t)=(t,f(t))$, where $f$ is strictly convex. We show that \[ J_3(\gamma(X)) =\#\{(x_1,\ldots,x_6)\in X^6:\sum_{i=1}^3\gamma(x_i)=\sum_{i=4}^6\gamma(x_i)\} \ll_{\epsilon}|X|^{3+\epsilon}. \] When specialized to the parabola, our result implies near-optimal estimates for...
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