Accelerated Fully First-Order Methods for Solving Nonconvex-Strongly-Convex Bilevel Optimization and Nonconvex-Strongly-Concave Minimax Optimization Problems

Core Concepts

This paper presents new accelerated fully first-order algorithms, (P)RAF2BA, that can efficiently solve nonconvex-strongly-convex bilevel optimization problems and nonconvex-strongly-concave minimax optimization problems, achieving state-of-the-art convergence rates without requiring Hessian-vector product or Jacobian-vector product oracles.

Abstract

The paper introduces the (Perturbed) Restarted Accelerated Fully First-order methods for Bilevel Approximation, or (P)RAF2BA, for solving nonconvex-strongly-convex bilevel optimization problems. The key contributions are:

The (P)RAF2BA algorithm leverages fully first-order oracles and seeks approximate stationary points, enhancing oracle complexity for efficient optimization.
Theoretical guarantees are established for finding approximate first-order and second-order stationary points at state-of-the-art query complexities.
For the special case of nonconvex-strongly-concave minimax optimization, the (P)RAF2BA algorithm is shown to be equivalent to the perturbed restarted accelerated gradient descent ascent (PRAGDA) algorithm, achieving improved oracle complexity bounds.
Empirical studies on hyperparameter optimization, data hyper-cleaning, and synthetic minimax problems validate the outperformance of the proposed algorithms compared to existing baselines.

Stats

The upper-level function f(x, y) is M-Lipschitz continuous and ℓ-Lipschitz gradient continuous with respect to y.
The lower-level function g(x, y) is μ-strongly convex and ℓ-Lipschitz gradient continuous with respect to y.
The Jacobians and Hessians of f and g are ρ-Lipschitz continuous.
The third-order derivatives of g are ν-Lipschitz continuous.

Quotes

"This paper presents a new algorithm member for accelerating first-order methods for bilevel optimization, namely the (Perturbed) Restarted Accelerated Fully First-order methods for Bilevel Approximation, abbreviated as (P)RAF2BA."
"The significance of (P)RAF2BA in optimizing nonconvex-strongly-convex bilevel optimization problems is underscored by its state-of-the-art convergence rates and computational efficiency."

Key Insights Distilled From

Accelerated Fully First-Order Methods for Bilevel and Minimax Optimization

by Chris Junchi... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.00914.pdf

Deeper Inquiries

How can the (P)RAF2BA algorithm be extended to handle stochastic or online settings of bilevel and minimax optimization problems

To extend the (P)RAF2BA algorithm to handle stochastic or online settings of bilevel and minimax optimization problems, we can introduce stochastic gradients in the optimization process. In stochastic settings, instead of computing the gradients using the entire dataset, we can estimate the gradients using mini-batches of data. This approach reduces the computational burden and allows the algorithm to adapt to changing data distributions in online scenarios.
For bilevel optimization, we can incorporate stochastic gradients in the computation of the gradients of the upper-level and lower-level functions. This involves updating the gradients based on randomly sampled subsets of the data, making the algorithm more scalable and suitable for large datasets. In the case of minimax optimization, stochastic gradients can be used to approximate the gradients of the objective function with respect to the inner maximization and outer minimization variables.
By introducing stochastic gradients, the (P)RAF2BA algorithm can efficiently handle stochastic or online settings of bilevel and minimax optimization problems, enabling it to tackle dynamic and evolving optimization tasks effectively.

What are the potential applications of the (P)RAF2BA algorithm beyond the examples provided in the paper, and how can it be adapted to those domains

The (P)RAF2BA algorithm has a wide range of potential applications beyond the examples provided in the paper. Some potential applications include:

Deep Learning: The (P)RAF2BA algorithm can be applied to optimize hyperparameters in deep learning models, such as neural networks. By efficiently searching for optimal hyperparameters, the algorithm can enhance the performance and efficiency of deep learning models.

Portfolio Optimization: In finance, the algorithm can be used for portfolio optimization to maximize returns while minimizing risks. By formulating the problem as a bilevel or minimax optimization task, the algorithm can help investors make informed decisions in complex financial markets.

Supply Chain Management: The algorithm can optimize supply chain operations by finding the best allocation of resources and minimizing costs. It can handle the complexities of supply chain networks and provide optimal solutions for inventory management, production planning, and distribution.

Game Theory: In game theory, the algorithm can be applied to solve competitive games and strategic interactions. By formulating games as minimax optimization problems, the algorithm can find equilibrium solutions and strategies for players in various game scenarios.

To adapt the (P)RAF2BA algorithm to these domains, specific problem formulations and constraints need to be considered. Customized objective functions and oracle complexities can be defined to address the unique requirements of each application, ensuring optimal performance and convergence in diverse real-world scenarios.

Can the ideas behind (P)RAF2BA be applied to other classes of structured optimization problems beyond bilevel and minimax optimization

The ideas behind the (P)RAF2BA algorithm can be applied to other classes of structured optimization problems beyond bilevel and minimax optimization. Some classes of structured optimization problems where these ideas can be beneficial include:

Constrained Optimization: The algorithm can be extended to handle constrained optimization problems by incorporating constraints into the objective function. By considering additional constraints, such as linear or nonlinear constraints, the algorithm can find optimal solutions while satisfying the given constraints.

Sparse Optimization: For problems involving sparse data or sparse models, the algorithm can be adapted to promote sparsity in the solutions. By incorporating sparsity-inducing norms or penalties, the algorithm can encourage solutions with fewer non-zero components, leading to more interpretable and efficient models.

Robust Optimization: In scenarios with uncertain or noisy data, the algorithm can be modified to account for robustness. By introducing robust optimization techniques, such as uncertainty sets or robust constraints, the algorithm can find solutions that are resilient to variations in the data, ensuring stability and reliability in uncertain environments.

By applying the principles and methodologies of the (P)RAF2BA algorithm to these classes of structured optimization problems, it is possible to address a wide range of complex optimization challenges and achieve optimal solutions in diverse application domains.