Research map / Optimization / Nonconvex and smooth optimization
Acceleration via Symplectic Discretization of High-Resolution Differential Equations
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Discretize the high-resolution NAG-SC ODE with symplectic Euler rather than explicit Euler (ablation) | NAG-SC ODE · Strongly convex smooth functions | objective gap convergence rate: 1 + \frac{1}{9} \sqrt{ \frac{\mu}{L} } -> 1 - \frac{\mu}{80L} | Helped |
| Discretize the high-resolution NAG-SC ODE with implicit Euler rather than symplectic Euler (ablation) | NAG-SC ODE · Strongly convex smooth functions | objective gap convergence rate: 1 + \frac{1}{9} \sqrt{ \frac{\mu}{L} } -> 1 + \frac{1}{4} \sqrt{\frac{\mu}{L}} | No clear effect |
| Retain the high-resolution gradient-correction term when using symplectic Euler (ablation) | Strongly convex NAG-SC discretization · Strongly convex smooth functions | objective gap convergence rate: 1 + \frac{1}{9} \sqrt{ \frac{\mu}{L} } -> 1 + \frac{\mu}{16L} | Helped |
| Discretize the heavy-ball ODE with implicit Euler instead of symplectic Euler (ablation) | Heavy-ball ODE · Strongly convex smooth functions | objective gap convergence rate: 1 + \frac{\mu}{16L} -> 1 + \frac{1}{4} \sqrt{\frac{\mu}{L}} | Helped |
| Discretize the modified high-resolution NAG-C ODE with symplectic Euler | NAG-C ODE · Smooth convex functions | objective gap convergence rate: O(1/k^2) | Helped |
| Use explicit Euler on the modified high-resolution NAG-C ODE (ablation) | NAG-C ODE · Smooth convex functions | convergence | Hurt |
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