Efficient Stochastic Algorithm for Large-Scale Non-convex Constrained Distributionally Robust Optimization

Core Concepts

This paper develops a stochastic algorithm and its performance analysis for non-convex constrained distributionally robust optimization (DRO) problems. The computational complexity of the proposed algorithm is independent of the overall dataset size, making it suitable for large-scale applications.

Abstract

The paper focuses on constrained DRO problems with non-convex loss functions, which is a practical yet challenging case. Existing studies on constrained DRO mostly focus on convex loss functions.
Key highlights:

Develops a stochastic algorithm for large-scale non-convex constrained DRO problems with the general Cressie-Read family divergence.
Constructs a smooth and Lipschitz approximation of the original non-smooth and non-Lipschitz objective function to overcome the challenges in the constrained formulation.
Proves that the proposed algorithm finds an ε-stationary point with a computational complexity of O(ε^-3k^^-5), where k^ is the parameter of the Cressie-Read divergence.
Extends the analysis to the smoothed conditional value at risk (CVaR) DRO problem.
Numerical results show the proposed algorithm outperforms existing methods.

Stats

The loss function is bounded: 0 ≤ ℓ(x; s) ≤ B for some B > 0, ∀x ∈ ℝ^d, s ∈ S.
The loss function is G-Lipschitz and L-smooth.

Quotes

None.

Key Insights Distilled From

Large-Scale Non-convex Stochastic Constrained Distributionally Robust Optimization

by Qi Zhang,Yi ... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.01200.pdf

Deeper Inquiries

How can the proposed algorithm be extended to handle other types of uncertainty sets beyond the Cressie-Read family

The proposed algorithm can be extended to handle other types of uncertainty sets beyond the Cressie-Read family by adapting the formulation of the objective function and the constraints to accommodate different divergence functions. For instance, for uncertainty sets defined by other divergence metrics such as KL divergence, χ2 divergence, or Sinkhorn distance, the algorithm can be modified to incorporate the specific properties of these divergence functions. By adjusting the dual form and the optimization process accordingly, the algorithm can be tailored to address a wide range of uncertainty sets.

What are the potential applications of the developed non-convex constrained DRO framework beyond machine learning

The developed non-convex constrained DRO framework has potential applications beyond machine learning in various domains such as finance, operations research, and healthcare. In finance, it can be utilized for portfolio optimization under uncertain market conditions. In operations research, it can aid in decision-making processes involving risk management and resource allocation. In healthcare, the framework can be applied to optimize treatment plans considering uncertain patient outcomes. The robustness and flexibility of the algorithm make it suitable for addressing complex optimization problems in diverse fields where uncertainty plays a significant role.

Can the algorithm be further improved to achieve a better computational complexity, e.g., by leveraging additional problem structures

To improve the computational complexity of the algorithm further, several strategies can be considered. One approach could involve incorporating problem-specific structures or constraints to exploit additional information and reduce the search space. By leveraging problem characteristics such as sparsity, symmetry, or separability, the algorithm can be optimized to achieve faster convergence and lower computational costs. Additionally, exploring advanced optimization techniques like parallel computing, adaptive learning rates, or stochastic variance reduction methods can enhance the efficiency of the algorithm and lead to better scalability for large-scale applications.