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Bilevel optimization is one of the fundamental problems in machine learning and optimization. Recent theoretical developments in bilevel optimization focus on finding the firstorder stationary points for nonconvex-strongly-convex cases. In this paper, we analyze algorithms that can escape saddle points in nonconvex-strongly-convex bilevel optimization. Specifically, we show that the perturbed approximate implicit differentiation (AID) with a warm start strategy finds an -approximate local minimum of bilevel optimization in ̃O(−2) iterations with high probability. Moreover, we propose an inexact NEgativecurvature-Originated-from-Noise Algorithm (iNEON), an algorithm that can escape saddle point and find local minimum of stochastic bilevel optimization. As a by-product, we provide the first nonasymptotic analysis of perturbed multi-step gradient descent ascent (GDmax) algorithm that converges to local minimax point for minimax problems.
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FEDNEST: Federated Bilevel, Minimax, and Compositional Optimization
Standard federated optimization methods successfully apply to stochastic problems with singlelevel structure. However, many contemporary ML problems – including adversarial robustness, hyperparameter tuning, actor-critic – fall under nested bilevel programming that subsumes minimax and compositional optimization. In this work, we propose FEDNEST: A federated alternating stochastic gradient method to address general nested problems. We establish provable convergence rates for FEDNEST in the presence of heterogeneous data and introduce variations for bilevel, minimax, and compositional optimization. FEDNEST introduces multiple innovations including federated hypergradient computation and variance reduction to address inner-level heterogeneity. We complement our theory with experiments on hyperparameter & hyper-representation learning and minimax optimization that demonstrate the benefits of our method in practice.
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- Award ID(s):
- 1845076
- PAR ID:
- 10408770
- Date Published:
- Journal Name:
- Proceedings of the 39th International Conference on Machine Learning
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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