Research map / Optimization / Nonconvex and smooth optimization
Tighter Lower Bounds for Shuffling SGD: Random Permutations and Beyond
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Tighten the Random Reshuffling final-iterate lower bound's condition-number dependence | Random Reshuffling · Strongly-convex smooth finite-sum, final iterate, K ≥ c₂κ | expected objective gap lower bound: \Omega \left( \frac{\nu^2}{\mu n K^2} \right) -> \Omega \left( \frac{L \nu^2}{\mu^2 n K^2} \right) | Helped |
| Use arbitrary weighted averaging of end-of-epoch iterates | Random Reshuffling · Strongly-convex smooth finite-sum, K ≥ c₂κ | expected objective gap lower bound: \Omega \left( \frac{L \nu^2}{\mu^2 n K^2} \right) | Helped |
| Average iterates on a convex objective | Random Reshuffling · Convex smooth finite-sum | expected objective gap: \Omega \left( \frac{L^{1/3} \nu^{2/3} D^{4/3}}{n^{1/3} K^{2/3}} \right); matching upper bound \gO \left( \frac{L^{1/3} \ | Mixed |
| Apply a lower bound uniformly to any manually chosen permutation | Permutation-based SGD · Strongly-convex smooth finite-sum with component convexity | objective gap lower bound: \Omega \bigopen{\frac{\nu^2}{L n^3 K^2}} -> \Omega \left( \frac{L \nu^2}{\mu^2 n^2 K^2} \right) | Helped |
| Allow nonconvex component functions in the lower-bound construction | Permutation-based SGD · PŁ finite-sum, τ = κ ≥ 8n, sufficiently large K | objective gap lower bound: \Omega \left( \frac{L \nu^2}{\mu^2 n^2 K^2} \right) -> \Omega \left( \frac{L^2 \nu^2}{\mu^3 n^2 K^2} \right) | Helped |
| Remove component convexity from the lower-bound assumptions | Permutation-based SGD · Ill-conditioned PŁ finite-sum | objective gap lower bound: \Omega \left( \frac{L^2 \nu^2}{\mu^3 n^2 K^2} \right) | Helped |
| Use Offline GraB as the matching permutation-selection upper-bound method | Permutation-based SGD · PŁ finite-sum, fixed dimension | objective gap: \Omega \left( \frac{L^2 \nu^2}{\mu^3 n^2 K^2} \right) versus \tilde{\gO} \bigopen{\frac{H^2 L^2 \nu^2}{\mu^3 n^2 K^2}} | No clear effect |
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