Research map / Optimization / Nonconvex and smooth optimization
The Complexity of Finding Stationary Points with Stochastic Gradient Descent
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Measure only the gradient norm at a fixed continuous deterministic aggregate | Deterministic first-order methods on smooth nonconvex functions · Smooth nonconvex functions | gradient norm at output: \norm{\nabla f(\bxout)}\geq \frac{1}{2} | Hurt |
| Allow a non-negative linear aggregation of iterates | SGD on Lipschitz-gradient functions · Lipschitz-gradient functions | squared gradient norm: \|\boldsymbol{\gamma}\|^2 = \frac{\sigma}{16(T-1)} \left(\sqrt{64 L \Delta (T-1)+9 \sigma ^2}-3 \sigma \right) | Hurt |
| Omit the aggregation step | SGD on Lipschitz-Hessian functions · Lipschitz-Hessian functions | squared gradient norm at every iterate: \|\boldsymbol{\gamma}\|^2 = \frac{\sigma}{2}\left(\frac{\rho \Delta ^2}{ (T-1)^2}\right)^{1/3} | Hurt |
| Use SGD without an aggregation step and a stated reasonable step-size schedule | SGD on convex quadratic functions with Gaussian noise · Convex quadratic functions | minimum squared gradient norm across iterates: c_0 \frac{\min\{L \Delta, \sigma^2\}}{\sqrt{T}} | Hurt |
| Use a constant step size | SGD on smooth nonconvex functions · Smooth nonconvex functions | minimum squared gradient norm across iterates: \sigma \sqrt{\frac{2 L \Delta}{T-1}} | Helped |
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