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Flow-based Distributionally Robust Optimization
Machine Learning · Submitted 30 Oct 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We demonstrate its usage in adversarial learning, distributionally robust hypothesis testing, and a new mechanism for data-driven distribution perturbation differential privacy, where the proposed method gives strong empirical performance on high-dimensional real data.
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The abstract, sentence by sentence
We present a computationally efficient framework, called $\texttt{FlowDRO}$, for solving flow-based distributionally robust optimization (DRO) problems with Wasserstein uncertainty sets while aiming to find continuous worst-case distribution (also called the Least Favorable Distribution, LFD) and sample from it. The requirement for LFD to be continuous is so that the algorithm can be scalable to problems with larger sample sizes and achieve better generalization capability for the induced robust algorithms. To tackle the computationally challenging infinitely dimensional optimization problem, we leverage flow-based models and continuous-time invertible transport maps between the data distribution and the target distribution and develop a Wasserstein proximal gradient flow type algorithm. In theory, we establish the equivalence of the solution by optimal transport map to the original formulation, as well as the dual form of the problem through Wasserstein calculus and Brenier theorem. In practice, we parameterize the transport maps by a sequence of neural networks progressively trained in blocks by gradient descent. We demonstrate its usage in adversarial learning, distributionally robust hypothesis testing, and a new mechanism for data-driven distribution perturbation differential privacy, where the proposed method gives strong empirical performance on high-dimensional real data.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present a computationally efficient framework, called $\texttt{FlowDRO}$, for solving flow-based distributionally robust optimization (DRO) problems with Wasserstein uncertainty sets while aiming to find continuous worst-case distribution (also called the Least Favorable Distribution, LFD) and sample from it. | contribution 1.00 | 0.93 | 0.30 | asserted 0.99 |
| s1 | The requirement for LFD to be continuous is so that the algorithm can be scalable to problems with larger sample sizes and achieve better generalization capability for the induced robust algorithms. | interpretation 0.47 | 0.27 | 0.06 | asserted 0.99 |
| s2 | To tackle the computationally challenging infinitely dimensional optimization problem, we leverage flow-based models and continuous-time invertible transport maps between the data distribution and the target distribution and develop a Wasserstein proximal gradient flow type algorithm. | method 0.98 | 0.92 | 0.08 | asserted 1.00 |
| s3 | In theory, we establish the equivalence of the solution by optimal transport map to the original formulation, as well as the dual form of the problem through Wasserstein calculus and Brenier theorem. | contribution 0.57 | 0.89 | 0.11 | asserted 1.00 |
| s4 | In practice, we parameterize the transport maps by a sequence of neural networks progressively trained in blocks by gradient descent. | method 1.00 | 0.95 | 0.04 | asserted 1.00 |
| s5 | We demonstrate its usage in adversarial learning, distributionally robust hypothesis testing, and a new mechanism for data-driven distribution perturbation differential privacy, where the proposed method gives strong empirical performance on high-dimensional real data. | result 0.85 | 0.95 | 0.41 | asserted 1.00 |
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The paper
Original abstract
We present a computationally efficient framework, called $\texttt{FlowDRO}$, for solving flow-based distributionally robust optimization (DRO) problems with Wasserstein uncertainty sets while aiming to find continuous worst-case distribution (also called the Least Favorable Distribution, LFD) and sample from it. The requirement for LFD to be continuous is so that the algorithm can be scalable to problems with larger sample sizes and achieve better generalization capability for the induced robust algorithms. To tackle the computationally challenging infinitely dimensional optimization problem, we leverage flow-based models and continuous-time invertible transport maps between the data distribution and the target distribution and develop a Wasserstein proximal gradient flow type algorithm. In theory, we establish the equivalence of the solution by optimal transport map to the original formulation, as well as the dual form of the problem through Wasserstein calculus and Brenier theorem. In practice, we parameterize the transport maps by a sequence of neural networks progressively trained in blocks by gradient descent. We demonstrate its usage in adversarial learning, distributionally robust hypothesis testing, and a new mechanism for data-driven distribution perturbation differential privacy, where the proposed method gives strong empirical performance on high-dimensional real data.
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What this paper does with the arXiv papers it cites
What it cites · 6 sentences without a confident label
Uses the method or tool of 1 paper
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This connection enables us to adapt the recently developed computationally efficient Wasserstein proximal gradient flow [ref] to develop computationally efficient flow-based models parameterized by neural networks.
Convergence of flow-based generative models via proximal gradient descent in Wasserstein space · Proposed: Flow-DRO · 0.85
Compares with 2 papers
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We first compare the proposed in Algorithm [ref] against WDRO [ref] in finding LFD.
Robust Hypothesis Testing with Wasserstein Uncertainty Sets · Comparison with existing DRO methods · 0.99
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Compared with recent works on flow-based generative models [ref] , where only KL divergence was considered for the loss function, we consider general loss as motivated by various applications.
Convergence of flow-based generative models via proximal gradient descent in Wasserstein space · Flow-based generative models · 0.90
Cites as support 1 paper
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A useful observation that occurs in such infinite-dimensional optimization problem is that the worst-case distribution solution of the WDRO problem [ref] turns out to be discrete, as shown in the original paper [ref] and various follow-up works including [ref] for the distributionally robust hypothesis test.
Robust Hypothesis Testing with Wasserstein Uncertainty Sets · Motivating example: Why continuous density for LFD? · 0.98
Cites as background 5 papers
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DRO arises from various applications, including robust hypothesis testing [ref] , boosting [ref] , semi-supervised learning [ref] , fair classification [ref] , clustering [ref] , and so on; see [ref] for a more complete review.
A Distributionally Robust Approach to Fair Classification · Introduction · 0.98
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DRO arises from various applications, including robust hypothesis testing [ref] , boosting [ref] , semi-supervised learning [ref] , fair classification [ref] , clustering [ref] , and so on; see [ref] for a more complete review.
Robust Hypothesis Testing with Wasserstein Uncertainty Sets · Introduction · 0.98
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Besides, numerous variants of DP with rigorous definitions such as $f$-DP [ref] , Renyi DP [ref] , and Concentrated DP [ref] have been established and studied; for a comprehensive overview, see [ref] .
Concentrated Differential Privacy · Differential privacy · 0.99
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