Solving Hidden Monotone Variational Inequalities with Surrogate Losses

2024-11-07

International Conference on Learning Representations

AI Summary

Gradient descent methods for implicitly monotone variational inequalities (VIs) often suffer from divergence and oscillation. Method: This paper proposes a deep-learning-compatible solving framework based on a surrogate loss function. Theoretically, it establishes the first convergence guarantee for the surrogate loss under practical assumptions—including implicit monotonicity. Methodologically, it unifies existing VI algorithms and introduces a novel paradigm compatible with modern optimizers (e.g., Adam), integrating projected Bellman error minimization with an improved TD(0) policy. Results: Experiments demonstrate substantial improvements in stability and convergence speed for min-max optimization and Bellman error minimization tasks. In deep reinforcement learning, the framework simultaneously enhances computational efficiency and sample efficiency.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Imitation Learning & Inverse Reinforcement Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Deep learning has proven to be effective in a wide variety of loss minimization problems. However, many applications of interest, like minimizing projected Bellman error and min-max optimization, cannot be modelled as minimizing a scalar loss function but instead correspond to solving a variational inequality (VI) problem. This difference in setting has caused many practical challenges as naive gradient-based approaches from supervised learning tend to diverge and cycle in the VI case. In this work, we propose a principled surrogate-based approach compatible with deep learning to solve VIs. We show that our surrogate-based approach has three main benefits: (1) under assumptions that are realistic in practice (when hidden monotone structure is present, interpolation, and sufficient optimization of the surrogates), it guarantees convergence, (2) it provides a unifying perspective of existing methods, and (3) is amenable to existing deep learning optimizers like ADAM. Experimentally, we demonstrate our surrogate-based approach is effective in min-max optimization and minimizing projected Bellman error. Furthermore, in the deep reinforcement learning case, we propose a novel variant of TD(0) which is more compute and sample efficient.

Problem

• Solving variational inequalities with surrogate losses in deep learning
• Addressing divergence in gradient-based VI methods via surrogate approach
• Improving efficiency in min-max optimization and Bellman error minimization

Innovation

• Surrogate-based approach for variational inequalities
• Guarantees convergence under practical assumptions
• Compatible with deep learning optimizers like ADAM