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 O( bar d + bar d1/2 ε-3/2) and O( bar d + bar d1/2 ε-1/2) for nonconvex and convex optimization, respectively, where bar d is the effective dimension and ε is the target precision. Very recently, Adil, Bullins, Sidford, and Zhang (NeurIPS 2025) improved the gradient complexity to O( bar d + bar d1/3 ε-3/2 ln18 ε-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 O( bar d + bar d1/3 ε-3/2) and O( (bar d + bar d13/21 ε-2/7) ln bar d) for nonconvex and convex optimization, respectively, improving best-known results for both setups.