Research map / Optimization / Nonconvex and smooth optimization
Two Sides of One Coin: the Limits of Untuned SGD and the Power of Adaptive Methods
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use untuned decaying SGD with arbitrary eta rather than a stepsize tuned to smoothness | SGD · Smooth nonconvex stochastic optimization | epsilon-stationarity iteration complexity: $\widetilde{\cO}(T^{-1/4})$ gradient-norm rate; complexity includes $(4e)^{2\eta^2\ell^2}$ | Mixed |
| Use large initial eta with decaying SGD | SGD · Smooth nonconvex deterministic optimization | gradient norm / epsilon-stationarity complexity: $\Omega\left(\eta^{-4}\ell^{-2}(8e)^{\eta^2\ell^2/8}\epsilon^{-4}\right)$ | Hurt |
| Normalize each gradient to set the step length | NGD · Smooth nonconvex deterministic optimization | epsilon-stationarity iteration complexity: $\widetilde{\cO}((\gamma^{-2}+\gamma^2\ell^2)\epsilon^{-2})$ | Helped |
| Normalize stochastic gradients without momentum | NSGD · Smooth nonconvex stochastic optimization | convergence: Nonconvergent | Hurt |
| Add momentum to normalized SGD | NSGD · Smooth nonconvex stochastic optimization | epsilon-stationarity iteration complexity: $\widetilde{\cO}((\gamma^{-4}+\gamma^4\ell^4)\epsilon^{-4})$ | Helped |
| Use AMSGrad-norm with the default decaying step under unbounded stochastic gradients | AMSGrad-norm · Smooth nonconvex stochastic optimization | gradient norm convergence rate: $\Omega\left(\epsilon^{-\frac{2}{1-\zeta}}\right)$ | Hurt |
| Use AMSGrad-norm in the deterministic setting without tuning | AMSGrad-norm · Smooth nonconvex deterministic optimization | epsilon-stationarity iteration complexity: $\widetilde{\cO}((\ell^4\gamma^4+\ell^2+\ell^3\gamma^2+\ell\gamma^{-2})\epsilon^{-4})$ | Helped |
| Accumulate squared gradient norms to adapt the stepsize | AdaGrad-norm · Smooth nonconvex stochastic optimization | epsilon-stationarity iteration complexity: $\widetilde{\cO}\left((\eta^{-2}+\sigma^2+\eta^2\ell^2)(\epsilon^{-2}+\sigma^2\epsilon^{-4})\right)$ | Helped |
| Choose eta = 1/sqrt(ell) for AdaGrad-norm (ablation) | AdaGrad-norm · Smooth nonconvex stochastic optimization | epsilon-stationarity iteration complexity: $\widetilde{\cO}(\epsilon^{-4})$ | Helped |
| Use a constant stepsize for AMSGrad-norm instead of a decaying schedule (ablation) | AMSGrad-norm · Smooth nonconvex optimization with unknown problem parameters | convergence: Nonconvergent, $\alpha = 0$ | Hurt |
| Use a faster polynomial decay exponent alpha > 1/2 for SGD (ablation) | SGD · Smooth nonconvex optimization | epsilon-stationarity iteration complexity: $\cO\left((4e^{2\alpha})^{\frac{(\eta\ell)^{1/\alpha}}{1-\alpha}}(\eta\ell)^{\frac{1}{\alpha(1-\alpha)}}\cdot\epsilon^{\ | Mixed |
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