Research map / Optimization / Nonconvex and smooth optimization
Sharpened Quasi-Newton Methods: Faster Superlinear Rate and Larger Local Convergence Neighborhood
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add a second, greedy BFGS update (direction chosen by max u'Gu/u'Au over coordinate vectors) after the standard BFGS update, giving Sharpened-BFGS | BFGS · Logistic regression experiments (Figure 2) | Newton decrement ratio convergence | Helped |
| Use the greedy Hessian-approximation direction after BFGS (Sharpened-BFGS) versus standard BFGS alone | BFGS · Theory: local superlinear rate | convergence rate exponent | Helped |
| Sharpened-BFGS versus Greedy-BFGS: reaches superlinear regime after dκ iterations instead of dκ ln(dκ) | Greedy-BFGS · Theory comparison (Section 4, Table 1) | iterations to superlinear convergence | Helped |
| Sharpened-BFGS versus standard BFGS: superlinear onset at dκ iterations instead of d ln κ | BFGS · Theory comparison (Section 4, Table 1) | iterations to superlinear convergence | Hurt |
| Skip the correction term r_t (set Ĝ_t = Ḡ_t) in the hybrid and greedy methods | Sharpened-BFGS (practical implementation) · Logistic regression experiments | convergence in practice | No clear effect |
| Randomized greedy direction (random Gaussian vector via Cholesky factor) in place of greedy coordinate direction | Sharpened-BFGS · Theory (Appendix H) | superlinear rate exponent | No clear effect |
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