Probable Domain Generalization via Quantile Risk Minimization (2207.09944v4)

Abstract: Domain generalization (DG) seeks predictors which perform well on unseen test distributions by leveraging data drawn from multiple related training distributions or domains. To achieve this, DG is commonly formulated as an average- or worst-case problem over the set of possible domains. However, predictors that perform well on average lack robustness while predictors that perform well in the worst case tend to be overly-conservative. To address this, we propose a new probabilistic framework for DG where the goal is to learn predictors that perform well with high probability. Our key idea is that distribution shifts seen during training should inform us of probable shifts at test time, which we realize by explicitly relating training and test domains as draws from the same underlying meta-distribution. To achieve probable DG, we propose a new optimization problem called Quantile Risk Minimization (QRM). By minimizing the $\alpha$-quantile of predictor's risk distribution over domains, QRM seeks predictors that perform well with probability $\alpha$. To solve QRM in practice, we propose the Empirical QRM (EQRM) algorithm and provide: (i) a generalization bound for EQRM; and (ii) the conditions under which EQRM recovers the causal predictor as $\alpha \to 1$. In our experiments, we introduce a more holistic quantile-focused evaluation protocol for DG and demonstrate that EQRM outperforms state-of-the-art baselines on datasets from WILDS and DomainBed.

• The paper introduces QRM that minimizes the α-quantile risk, ensuring predictors perform well with high probability across unseen domains.
• It employs kernel density estimation in the EQRM algorithm to smooth risk distributions and extrapolate beyond observed training risks.
• Empirical results confirm EQRM outperforms state-of-the-art baselines, offering a balanced trade-off between robustness and performance.

Probable Domain Generalization via Quantile Risk Minimization

This paper presents an innovative probabilistic framework for domain generalization (DG) called Quantile Risk Minimization (QRM), poised to address some fundamental limitations of existing strategies in the field. Traditional approaches to DG typically involve either average-case or worst-case scenarios, each coming with distinct drawbacks. The paper argues that neither encapsulates the complete essence of domain generalization. Instead, it proposes that a probabilistic interpretation, seeking to optimize performance with high probability, provides a more balanced and theoretically sound approach.

Key Contributions and Methodology

The authors introduce QRM, a novel optimization problem that minimizes the $\alpha$-quantile of the predictor's risk distribution over domains. This formulation aims to ensure predictors perform well with probability $\alpha$ across unseen domains, providing an interpretable trade-off between robustness and performance. The $\alpha$ parameter flexibly interpolates between average-case and worst-case scenarios, thus enabling the development of robust predictive models without being overly conservative.

The practical implementation of QRM is achieved through the Empirical QRM (EQRM) algorithm. Leveraging kernel density estimation (KDE), EQRM forms an estimated risk distribution, which is then minimized to meet the QRM objectives. By smoothing risk distributions, EQRM allows for risk extrapolation beyond the largest training risks, a critical step for achieving invariant prediction across domains.

Theoretical Insights

The paper rigorously explores the theoretical underpinnings of QRM, providing substantial insights into its practical implications. A key theoretical result is a generalization bound for EQRM, asserting that given enough domains and samples, the empirical $\alpha$-quantile risk approximates the population $\alpha$-quantile risk. Furthermore, the authors demonstrate the conditions under which EQRM recovers the causal predictor, linking it with invariant risk across domains—a significant step in understanding performance guarantees for unseen domains.

Empirical Evaluation

Empirically, EQRM is shown to outperform several state-of-the-art baselines across a variety of datasets, including those from WILDS and DomainBed. In particular, the effectiveness of the KDE-smoothed risk CDFs underscores EQRM's superior ability to balance predictive performance and robustness. The introduction of a holistic evaluation protocol concentrating on quantile risk rather than average performance marks a shift towards more nuanced assessments of DG models.

Implications and Future Directions

By setting a new benchmark for DG through the lens of QRM, the paper charts a pathway for future research to explore probabilistic approaches in AI. The interpretability of the $\alpha$ parameter as a robustness measure is particularly promising for applications where specific performance guarantees are essential. Additionally, the proposed framework encourages the reconsideration of domain data collection processes to better accommodate the assumptions of i.i.d. domains, potentially expanding the applicability of robust models in real-world scenarios.

While the requirement for numerous i.i.d.-sampled domains presents a practical challenge, the theoretical advancements presented are substantial. Future work may focus on relaxing domain independence assumptions or incorporating domain dependencies, such as temporal factors, to enhance the framework's applicability.

In summary, this paper provides a comprehensive and theoretically backed approach to domain generalization that balances robustness and performance, making significant contributions to both the theoretical and practical aspects of machine learning under distributional shifts. The insights presented are expected to stimulate further innovations in the field, particularly in understanding and managing uncertainty and variability in predictive modeling.