The (gradient-based) bilevel programming framework is widely used in hyperparameter optimization and has achieved excellent performance empirically. Previous theoretical work mainly focuses on its optimization properties, while leaving the analysis on generalization largely open. This paper attempts to address the issue by presenting an expectation bound w.r.t. the validation set based on uniform stability. Our results can explain some mysterious behaviours of the bilevel programming in practice, for instance, overfitting to the validation set. We also present an expectation bound for the classical cross-validation algorithm. Our results suggest that gradient-based algorithms can be better than cross-validation under certain conditions in a theoretical perspective. Furthermore, we prove that regularization terms in both the outer and inner levels can relieve the overfitting problem in gradient-based algorithms. In experiments on feature learning and data reweighting for noisy labels, we corroborate our theoretical findings.

The (gradient-based) bilevel programming framework is widely used in hyperparameter optimization and has achieved excellent performance empirically. Previous theoretical work mainly focuses on its optimization properties, while leaving the analysis on generalization largely open. This paper attempts to address the issue by presenting an expectation bound w.r.t. the validation set based on uniform stability. Our results can explain some mysterious behaviours of the bilevel programming in practice, for instance, overfitting to the validation set. We also present an expectation bound for the classical cross-validation algorithm.

The Bilevel Optimization Problem is a hierarchical optimization problem with two agents, a leader and a follower. The leader make their own decisions first, and the followers make the best choices accordingly. The leader knows the information of the followers, and the goal of the problem is to find the optimal solution by considering the reactions of the followers from the leader's point of view. For the Bilevel Optimization Problem, there are no general and efficient algorithms or commercial solvers to get an optimal solution, and it is very difficult to get a good solution even for a simple problem. In this paper, we propose a deep learning approach using Graph Neural Networks to solve the bilevel knapsack problem. We train the model to predict the leader's solution and use it to transform the hierarchical optimization problem into a single-level optimization problem to get the solution. Our model found the feasible solution that was about 500 times faster than the exact algorithm with $1.7\%$ optimal gap. Also, our model performed well on problems of different size from the size it was trained on.

Due to the hierarchical structure of many machine learning problems, bilevel programming is becoming more and more important recently, however, the complicated correlation between the inner and outer problem makes it extremely challenging to solve. Although several intuitive algorithms based on the automatic differentiation have been proposed and obtained success in some applications, not much attention has been paid to finding the optimal formulation of the bilevel model. Whether there exists a better formulation is still an open problem. In this paper, we propose an improved bilevel model which converges faster and better compared to the current formulation. We provide theoretical guarantee and evaluation results over two tasks: Data Hyper-Cleaning and Hyper Representation Learning. The empirical results show that our model outperforms the current bilevel model with a great margin. This is a concurrent work with Liu et al. [20] and we submitted to ICML 2020. Now we put it on the arxiv for record.

We introduce a framework based on bilevel programming that unifies gradient-based hyperparameter optimization and meta-learning. We show that an approximate version of the bilevel problem can be solved by taking into explicit account the optimization dynamics for the inner objective. Depending on the specific setting, the outer variables take either the meaning of hyperparameters in a supervised learning problem or parameters of a meta-learner. We provide sufficient conditions under which solutions of the approximate problem converge to those of the exact problem. We instantiate our approach for meta-learning in the case of deep learning where representation layers are treated as hyperparameters shared across a set of training episodes. In experiments, we confirm our theoretical findings, present encouraging results for few-shot learning and contrast the bilevel approach against classical approaches for learning-to-learn.