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Convergence of Cubic Regularization for Nonconvex Optimization under KL Property
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use cubic-regularized Newton updates under the KŁ property instead of first-order gradient descent | Nonconvex optimization · KŁ parameter θ ∈ [1/3, 1] | function value convergence rate: gradient descent: linear or sub-linear; cubic-regularization: super-linear or (super)-linear | Helped |
| Use cubic-regularized Newton updates under the KŁ property instead of first-order gradient descent | Nonconvex optimization · KŁ parameter θ ∈ (0, 1/3) | function value convergence rate: gradient descent: O(k^(-1/(1-2θ)); cubic-regularization: O(k^(-2/(1-3θ))) | Helped |
| Use cubic-regularized Newton updates under the KŁ property instead of first-order gradient descent | Nonconvex optimization · KŁ parameter θ ∈ [1/3, 1] | iterate distance convergence rate: gradient descent: linear or sub-linear; cubic-regularization: super-linear or (super)-linear | Helped |
| Extend the convergence analysis from special geometrical conditions to the KŁ property | Cubic regularization (CR) · Nonconvex objectives satisfying the KŁ property | iterate convergence | Helped |
| Allow inexact gradient and Hessian information satisfying the stated inexact criterion | Inexact cubic regularization · Nonconvex objectives satisfying the KŁ property | convergence guarantees | No clear effect |
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