Functional Bilevel Optimization for Machine Learning (2403.20233v4)

Abstract: In this paper, we introduce a new functional point of view on bilevel optimization problems for machine learning, where the inner objective is minimized over a function space. These types of problems are most often solved by using methods developed in the parametric setting, where the inner objective is strongly convex with respect to the parameters of the prediction function. The functional point of view does not rely on this assumption and notably allows using over-parameterized neural networks as the inner prediction function. We propose scalable and efficient algorithms for the functional bilevel optimization problem and illustrate the benefits of our approach on instrumental regression and reinforcement learning tasks.

• The paper’s main contribution is introducing a functional perspective to bilevel optimization, which improves adaptability for over-parameterized neural networks.
• It develops scalable algorithms using functional implicit differentiation, bypassing the need for strong convexity in traditional parameter-based methods.
• The approach is validated through applications in instrumental regression and reinforcement learning, demonstrating enhanced efficiency and reduced over-fitting.

An Overview of Functional Bilevel Optimization for Machine Learning

In their research, Petrulionyte, Mairal, and Arbel introduce a novel approach to bilevel optimization (BO) that focuses on optimizing functions rather than parameters, which is particularly relevant in the context of machine learning models such as over-parameterized neural networks. This functional point of view offers advantages over traditional parametric methods by avoiding the strong convexity assumption with respect to the model parameters, making the approach more adaptable to modern machine learning scenarios. The paper proposes a scalable method for functional bilevel optimization and demonstrates its effectiveness in areas such as instrumental regression and reinforcement learning.

Key Concepts and Contributions

1. Functional Bilevel Optimization (FBO): The paper redefines the BO problem to optimize over a space of functions rather than parameter spaces, effectively transforming the inner problem into a functional optimization challenge. This approach is particularly beneficial when dealing with neural networks, which may not satisfy strong convexity with respect to their parameters.
2. Scalable Algorithms: The authors develop efficient algorithms that solve the FBO by leveraging the functional implicit differentiation framework. These methods are shown to be scalable and applicable to large-scale problems such as deep learning tasks.
3. Illustrative Applications: The efficacy of the FBO approach is demonstrated through instrumental regression and reinforcement learning tasks. These applications illustrate the natural hierarchical structure present in many machine learning problems and how FBO can be effectively utilized.
4. Theoretical Foundations: The work includes a thorough theoretical examination of FBO using tools such as the functional version of the implicit function theorem. The authors derive expressions for the Jacobian and total gradient in the context of functional spaces, addressing both practical and theoretical challenges.

Numerical Results and Claims

• Improved Flexibility: The proposed method does not require the strong convexity of the inner objective with respect to parameters, thus accommodating over-parameterized models and leading to potentially better solutions in practice.
• Avoidance of Over-Fitting: By using function space rather than parameter space for optimization, the approach helps mitigate the risk of over-fitting, especially when complex models are used for the inner-level problem.
• Convincing Experiments: In experimental settings, the proposed algorithms showed substantial accuracy and efficiency, substantiating the theoretical advantages of the FBO framework.

Implications and Future Directions

The introduction of FBO offers a promising direction for optimizing nested problems in machine learning that involve complex function approximations like deep neural networks. By moving away from parameter-centric approaches, FBO allows practitioners to employ richer model approximations without violating convexity assumptions, potentially leading to improved performance and generalizability of the resulting models.

The implications of this work are significant as they pave the way for further exploration into more sophisticated function-based approaches for optimization in machine learning, especially within areas such as meta-learning, inverse problems, and reinforcement learning. Future research could look into extending the theoretical framework of FBO to other types of function spaces or exploring its applications in more diverse machine learning settings. Additionally, investigating the implications of FBO in terms of computational complexity and convergence in more depth could further validate its utility and efficiency over traditional methods.