Faster Newton Methods for Convex and Nonconvex Optimization in Gradient Complexity

Second-order optimization methods are computationally expensive for large-scale problems. Recently, Doikov, Chayti, and Jaggi (ICML 2023) proposed the LazyCRN method that reduces computation by studying the gradient complexity of second-order methods. Their method can achieve a gradient complexity of $\mathcal{O}(d+d^{1\mathrm{/}2}{\epsilon }^{-3\mathrm{/}2})$ and $\mathcal{O}(d+d^{1\mathrm{/}2}{\epsilon }^{-1\mathrm{/}2})$ for nonconvex and convex optimization, respectively, where $d$ is the effective dimension and $\epsilon$ is the target precision. Very recently, Adil, Bullins, Sidford, and Zhang (NeurIPS 2025) improved the gradient complexity to $\mathcal{O}(d+d^{1\mathrm{/}3}{\epsilon }^{-3\mathrm{/}2}{\ln }^{18}{\epsilon }^{-1})$ for nonconvex optimization. However, the tightness of these methods remains open. In this work, we propose new methods that achieve an improved complexity of $\mathcal{O}(d+d^{1\mathrm{/}3}{\epsilon }^{-3\mathrm{/}2})$ and $\mathcal{O}((d+d^{13\mathrm{/}21}{\epsilon }^{-2\mathrm{/}7})\ln d)$ for nonconvex and convex optimization, respectively, improving best-known results for both setups.