Lower Bounds for Smooth Nonconvex Finite-Sum Optimization

Dongruo Zhou; Quanquan Gu

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Lower Bounds for Smooth Nonconvex Finite-Sum Optimization

Dongruo Zhou, Quanquan Gu

Optimization and Control · Submitted 31 Jan 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove tight lower bounds for the complexity of finding $\epsilon$-suboptimal point and $\epsilon$-approximate stationary point in different settings, for a wide regime of the smallest eigenvalue of the Hessian of the objective function (or each component function).

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Smooth finite-sum optimization has been widely studied in both convex and nonconvex settings. However, existing lower bounds for finite-sum optimization are mostly limited to the setting where each component function is (strongly) convex, while the lower bounds for nonconvex finite-sum optimization remain largely unsolved. In this paper, we study the lower bounds for smooth nonconvex finite-sum optimization, where the objective function is the average of $n$ nonconvex component functions. We prove tight lower bounds for the complexity of finding $\epsilon$-suboptimal point and $\epsilon$-approximate stationary point in different settings, for a wide regime of the smallest eigenvalue of the Hessian of the objective function (or each component function). Given our lower bounds, we can show that existing algorithms including KatyushaX (Allen-Zhu, 2018), Natasha (Allen-Zhu, 2017), RapGrad (Lan and Yang, 2018) and StagewiseKatyusha (Chen and Yang, 2018) have achieved optimal Incremental First-order Oracle (IFO) complexity (i.e., number of IFO calls) up to logarithm factors for nonconvex finite-sum optimization. We also point out potential ways to further improve these complexity results, in terms of making stronger assumptions or by a different convergence analysis.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Smooth finite-sum optimization has been widely studied in both convex and nonconvex settings.	background 1.00	0.03	0.01	asserted 1.00
s1	However, existing lower bounds for finite-sum optimization are mostly limited to the setting where each component function is (strongly) convex, while the lower bounds for nonconvex finite-sum optimization remain largely unsolved.	gap 1.00	0.04	0.03	asserted 0.99
s2	In this paper, we study the lower bounds for smooth nonconvex finite-sum optimization, where the objective function is the average of $n$ nonconvex component functions.	contribution 0.98	0.77	0.05	asserted 0.97
s3	We prove tight lower bounds for the complexity of finding $\epsilon$-suboptimal point and $\epsilon$-approximate stationary point in different settings, for a wide regime of the smallest eigenvalue of the Hessian of the objective function (or each component function).	result 0.87	0.96	0.72	asserted 1.00
s4	Given our lower bounds, we can show that existing algorithms including KatyushaX (Allen-Zhu, 2018), Natasha (Allen-Zhu, 2017), RapGrad (Lan and Yang, 2018) and StagewiseKatyusha (Chen and Yang, 2018) have achieved optimal Incremental First-order Oracle (IFO) complexity (i.e., number of IFO calls) up to logarithm factors for nonconvex finite-sum optimization.	interpretation 0.67	0.78	0.41	asserted 0.98
s5	We also point out potential ways to further improve these complexity results, in terms of making stronger assumptions or by a different convergence analysis.	future work 1.00	0.79	0.04	hedged possibility 0.91

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Original abstract

Smooth finite-sum optimization has been widely studied in both convex and nonconvex settings. However, existing lower bounds for finite-sum optimization are mostly limited to the setting where each component function is (strongly) convex, while the lower bounds for nonconvex finite-sum optimization remain largely unsolved. In this paper, we study the lower bounds for smooth nonconvex finite-sum optimization, where the objective function is the average of $n$ nonconvex component functions. We prove tight lower bounds for the complexity of finding $\epsilon$-suboptimal point and $\epsilon$-approximate stationary point in different settings, for a wide regime of the smallest eigenvalue of the Hessian of the objective function (or each component function). Given our lower bounds, we can show that existing algorithms including KatyushaX (Allen-Zhu, 2018), Natasha (Allen-Zhu, 2017), RapGrad (Lan and Yang, 2018) and StagewiseKatyusha (Chen and Yang, 2018) have achieved optimal Incremental First-order Oracle (IFO) complexity (i.e., number of IFO calls) up to logarithm factors for nonconvex finite-sum optimization. We also point out potential ways to further improve these complexity results, in terms of making stronger assumptions or by a different convergence analysis.

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4 arXiv papers cite this in 7 sentences

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This complexity matches the lower-bound worst-case complexity in [ref] up to a constant factor when $n ≤ε^-4$.

ProxSARAH: An Efficient Algorithmic Framework for Stochastic Composite Nonconvex Optimization · Introduction · 0.93

Cites it as background 1 paper

The convergence and the optimality of SGD have been thoroughly studied [ref] .

Stability of SGD: Tightness Analysis and Improved Bounds · Introduction · 0.97

What it cites · 31 sentences without a confident label

Cites as background 7 papers

The first category of work [ref] makes the assumption that each individual component function $f_i()$ is $L$-smooth and $F()$ is $(l,L)$ smooth.

Natasha: Faster Non-Convex Stochastic Optimization Via Strongly Non-Convex Parameter · Introduction · 0.85
The first category of work [ref] makes the assumption that each individual component function $f_i()$ is $L$-smooth and $F()$ is $(l,L)$ smooth.

Katyusha X: Practical Momentum Method for Stochastic Sum-of-Nonconvex Optimization · Introduction · 0.85
[ref] proved the lower bounds for both deterministic and randomized algorithms on nonconvex optimization with high-order smoothness assumption.

Lower Bounds for Finding Stationary Points II: First-Order Methods · Additional Related Work · 0.81

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