Large-Scale Methods for Distributionally Robust Optimization (2010.05893v2)

Abstract: We propose and analyze algorithms for distributionally robust optimization of convex losses with conditional value at risk (CVaR) and $\chi^2$ divergence uncertainty sets. We prove that our algorithms require a number of gradient evaluations independent of training set size and number of parameters, making them suitable for large-scale applications. For $\chi^2$ uncertainty sets these are the first such guarantees in the literature, and for CVaR our guarantees scale linearly in the uncertainty level rather than quadratically as in previous work. We also provide lower bounds proving the worst-case optimality of our algorithms for CVaR and a penalized version of the $\chi^2$ problem. Our primary technical contributions are novel bounds on the bias of batch robust risk estimation and the variance of a multilevel Monte Carlo gradient estimator due to [Blanchet & Glynn, 2015]. Experiments on MNIST and ImageNet confirm the theoretical scaling of our algorithms, which are 9--36 times more efficient than full-batch methods.

• The paper introduces large-scale methods for Distributionally Robust Optimization (DRO) with convex loss functions, particularly focusing on CVaR and \u03c7\u00b2 divergence uncertainty sets for large machine learning problems.
• Key findings include algorithms with gradient evaluation counts independent of training size and parameter count, showing significant efficiency gains (9-36x faster) over traditional methods.
• The research provides worst-case optimality guarantees for algorithms in CVaR and penalized \u03c7\u00b2 problems, incorporating novel gradient estimation strategies like adjusted batch and multi-level Monte Carlo methods.

Insights on Distributionally Robust Optimization with Large-Scale Methods

The paper presented in this paper targets the efficient optimization of Distributionally Robust Optimization (DRO) for convex loss functions. The focus is on constraints derived from Conditional Value at Risk (CVaR) and $\chi^2$ divergence uncertainty sets. This research is particularly relevant to large-scale machine learning problems, where conventional optimization approaches face scalability challenges.

One of the critical contributions is the demonstration that the proposed algorithms require a number of gradient evaluations independent of the training set size and parameter count. This property significantly enhances their applicability to large-scale problems, distinguishing these methods from existing approaches, particularly those dealing with $\chi^2$ uncertainty sets where such guarantees hadn’t been established previously. The presented algorithms also show enhanced scaling in CVaR uncertainty settings, transitioning from quadratic to linear dependency on the uncertainty level, reducing computational demand substantially.

The experimental validation, performed on datasets such as MNIST and ImageNet, substantiates the theoretical claims. The proposed methods exhibit a marked increase in efficiency, being 9 to 36 times more efficient than traditional full-batch methods, depending on the specific setup and dataset.

Key Technical Contributions

1. Algorithmic Efficiency: The paper provides guarantees for the computational efficiency of the DRO methods proposed, irrespective of sample size or parameter dimensionality. This is particularly crucial for high-dimensional data environments encountered in modern machine learning tasks.
2. Optimality and Novel Bounds: The research delivers lower bounds certifying the worst-case optimality of the algorithms for both CVaR and a penalized version of the $\chi^2$ problem. This includes novel bounds on the bias in batch robust risk estimation and the variance within gradient estimation.
3. Method Innovations:
   • Efficient Gradient Estimation: The introduction of new gradient estimation strategies, including adjusted batch methods using subsampling techniques and multi-level Monte Carlo estimations, reduces computation intensity while maintaining estimator precision.
   • Multi-level Monte Carlo (MLMC): This enhanced method addresses the computation-heavy evaluations by maintaining unbiasedness with a logarithmic sample requirement, thereby enhancing feasibility for large datasets.

Theoretical and Practical Implications

• Theoretical Implications: The theoretical advancements afford a robust framework for assessing DRO within broader loss function types beyond the studied cases. Through worst-case optimalities, the research establishes a foundational understanding of potential theoretical bounds achievable in robust optimization contexts.
• Practical Implications: Reduced computation costs via proposed methods mean practical scalability for real-world applications, such as in autonomous systems or financial risk assessments where DRO is being increasingly employed.

Future Directions

Potential future directions could focus on expanding these methodologies to non-convex optimization problems where Gradient Descent and Stochastic Gradient methods face limitations. Additionally, exploration into more varied uncertainty sets beyond $\chi^2$ and CVaR could further enhance the robustness and applicability of DRO in disparate fields like communications network optimization and large-scale simulations. Extending the insights to incorporate neural network training as a practical laboratory could also provide depth to the applicability of these methods in artificial intelligence and deep learning contexts.

Overall, this paper solidifies the DRO's footing in scalable environments, setting a stage for both deeper theoretical exploration and wide-ranging practical applications.