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This theoretical study provides an exact worst-case analysis of stochastic gradient descent (SGD) performance in smooth convex optimization problems. The researchers derived the minimax optimal stepsize schedule for SGD when gradient noise variance is sufficiently large, proving that these optimal stepsizes must decrease monotonically and ruling out acceleration techniques that work in deterministic settings. The analysis applies to any sequence of N stepsizes and identifies matching problem instances where the theoretical bounds are exactly achieved.
Why it matters
This work provides rigorous theoretical guidance for setting learning rates in machine learning optimization algorithms, particularly for training neural networks and other models using SGD. The findings clarify that certain acceleration strategies successful in deterministic optimization cannot improve expected performance in stochastic settings, helping practitioners avoid ineffective hyperparameter choices.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: This work considers the design of stepsize schedules for stochastic gradient descent in smooth convex optimization. Throughout, we assume that gradient noise is conditionally zero-mean with bounded variance. Among problems with dimension $dgeq N+1$, we provide an exact worst-case analysis of the expected suboptimality of stochastic gradient descent’s final iterate. Our analysis applies to any (potentially nonconstant) schedule of $N$ stepsizes in $[0,1.5]^N$. Here, exact means that we provide a matching objective function and noise distributions where our guarantee is attained. Whenever the variance bound is at least $Omega(1/sqrt{N})$, our theory provides the minimax optimal stepsize schedule for stochastic gradient descent and proves its uniqueness. These optimal stepsizes monotonically decrease, ruling out any accelerations in expected final suboptimality from fixed fractal schedules observed in the deterministic case.
Source: Exact Theory and Optimal Stepsizes for Stochastic Gradient Descent in Smooth Convex Minimization