{"ID":6536207,"CreatedAt":"2026-07-14T01:21:01.169441415Z","UpdatedAt":"2026-07-15T03:28:55.185153975Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.10741","arxiv_id":"2607.10741","title":"Parameter-Free Cubic-Regularized Newton Method: Sharp Complexity and Generalized Smoothness","abstract":"We analyze a variant of the cubic-regularized Newton method for nonconvex optimization. This variant is parameter-free in that it requires no prior knowledge of problem-dependent parameters. Under the generalized smoothness condition $\\|\\nabla^3 f(x)\\| \\leq L_0 + L_1 \\|\\nabla f(x)\\|$, we derive an oracle complexity bound for finding an $(\\varepsilon, δ)$-second-order stationary point. This assumption is weaker than the generalized smoothness conditions used in existing analyses of second-order methods, while the complexity bound improves upon existing guarantees for parameter-free second-order methods. In particular, when $L_1 = 0$, the bound matches the optimal dependence on $L_0$ as well as on $\\varepsilon$, $δ$, and the initial function value gap, up to additive logarithmic terms. To establish this bound, we derive Taylor-type inequalities and prove their equivalence to the generalized smoothness condition.","short_abstract":"We analyze a variant of the cubic-regularized Newton method for nonconvex optimization. This variant is parameter-free in that it requires no prior knowledge of problem-dependent parameters. Under the generalized smoothness condition $\\|\\nabla^3 f(x)\\| \\leq L_0 + L_1 \\|\\nabla f(x)\\|$, we derive an oracle complexity bou...","url_abs":"https://arxiv.org/abs/2607.10741","url_pdf":"https://arxiv.org/pdf/2607.10741v1","authors":"[\"Shaoying Fang\",\"Naoki Marumo\",\"Akiko Takeda\"]","published":"2026-07-12T12:49:31Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}