Bilevel optimization is an important formulation for many machine learning problems. Current bilevel optimization algorithms assume that the gradient of the upper-level function is Lipschitz. However, recent studies reveal that certain neural networks such as recurrent neural networks (RNNs) and long-short-term memory networks (LSTMs) exhibit potential unbounded smoothness, rendering conventional bilevel optimization algorithms unsuitable. In this paper, we design a new bilevel optimization algorithm, namely BO-REP, to address this challenge. This algorithm updates the upper-level variable using normalized momentum and incorporates two novel techniques for updating the lower-level variable: \textit{initialization refinement} and \textit{periodic updates}. Specifically, once the upper-level variable is initialized, a subroutine is invoked to obtain a refined estimate of the corresponding optimal lower-level variable, and the lower-level variable is updated only after every specific period instead of each iteration. When the upper-level problem is nonconvex and unbounded smooth, and the lower-level problem is strongly convex, we prove that our algorithm requires $\widetilde{\mathcal{O}}(1/ε^4)$ iterations to find an $ε$-stationary point in the stochastic setting, where each iteration involves calling a stochastic gradient or Hessian-vector product oracle. Notably, this result matches the state-of-the-art complexity results under the bounded smoothness setting and without mean-squared smoothness of the stochastic gradient, up to logarithmic factors. Our proof relies on novel technical lemmas for the periodically updated lower-level variable, which are of independent interest. Our experiments on hyper-representation learning, hyperparameter optimization, and data hyper-cleaning for text classification tasks demonstrate the effectiveness of our proposed algorithm.
Single-Timescale Multi-Sequence Stochastic Approximation Without Fixed Point Smoothness: Theories and Applications
School of Computer Science and Engineering, Sun Yat-Sen University（中山大学计算机科学与工程学院）；Department of Computational Applied Math and Operations Research, Rice University（莱斯大学计算应用数学与运筹学系）
Implicit differentiation for fast hyperparameter selection in non-smooth convex learning
Université Paris-Saclay（巴黎萨克雷大学）；Inria（法国国家信息与自动化研究所）；CEA（法国原子能和替代能源委员会）；Université de Bourgogne（勃艮第大学）；Institut Mathématique de Bourgogne（勃艮第数学研究所）；Università degli Studi di Genova（热那亚大学）；Google Research（谷歌研究院）；CNRS（法国国家科学研究中心）；Université de Montpellier（蒙彼利埃大学）
META-STORM: Generalized Fully-Adaptive Variance Reduced SGD for Unbounded Functions
New York University（纽约大学）；Stern School of Business（斯特恩商学院）；Boston University（波士顿大学）；Northeastern University（东北大学）；Khoury College of Computer and Information Science（胡里计算机与信息科学学院）
On Convergence of Adam for Stochastic Optimization under Relaxed Assumptions
Center for Data Science and School of Mathematical Sciences, Zhejiang University（浙江大学数据科学中心与数学科学学院）；Center for Data Science, Zhejiang University（浙江大学数据科学中心）
Stochastic Zeroth-Order Optimization under Strongly Convexity and Lipschitz Hessian: Minimax Sample Complexity
University of California, Santa Barbara（加利福尼亚大学圣巴巴拉分校）；University of Texas at Dallas（德克萨斯大学达拉斯分校）；University of California, Berkeley（加利福尼亚大学伯克利分校）；New York University（纽约大学）；Princeton University（普林斯顿大学）
