Societal and real-world considerations such as robustness, fairness, social welfare and multi-agent tradeoffs have given rise to multi-distribution learning paradigms, such as collaborative [Blum et al. 2017], group distributionally robust [Sagawa et al. 2019], and fair federated learning [Mohri et al. 2019]. In each of these settings, a learner seeks to minimize its worstcase loss over a set of $n$ predefined distributions, while using as few samples as possible. In this paper, we establish the optimal sample complexity of these learning paradigms and give algorithms that meet this sample complexity. Importantly, our sample complexity bounds exceed that of the sample complexity of learning a single distribution only by an additive factor of $\frac{n\log(n)}{\epsilon^2}$. These improve upon the best known sample complexity of agnostic federated learning by Mohri et al. 2019 by a multiplicative factor of $n$, the sample complexity of collaborative learning by Nguyen and Zakynthinou 2018 by a multiplicative factor $\frac{\log(n)}{\epsilon^3}$, and give the first sample complexity bounds for the group DRO objective of Sagawa et al. 2019. To achieve optimal sample complexity, our algorithms learn to sample and learn from distributions on demand. Our algorithm design and analysis extends stochastic optimization techniques to solve zero-sum games in a new stochastic setting.

Collective action against algorithmic systems provides an opportunity for a small group of individuals to strategically manipulate their data to get specific outcomes, from classification to recommendation models. This effectiveness will invite more growth of this type of coordinated actions, both in the size and the number of distinct collectives. With a small group, however, coordination is key. Currently, there is no formal analysis of how coordination challenges within a collective can impact downstream outcomes, or how multiple collectives may affect each other's success. In this work, we aim to provide guarantees on the success of collective action in the presence of both coordination noise and multiple groups. Our insight is that data generated by either multiple collectives or by coordination noise can be viewed as originating from multiple data distributions. Using this framing, we derive bounds on the success of collective action. We conduct experiments to study the effects of noise on collective action. We find that sufficiently high levels of noise can reduce the success of collective action. In certain scenarios, large noise can sink a collective success rate from $100\%$ to just under $60\%$. We identify potential trade-offs between collective size and coordination noise; for example, a collective that is twice as big but with four times more noise experiencing worse outcomes than the smaller, more coordinated one. This work highlights the importance of understanding nuanced dynamics of strategic behavior in algorithmic systems.

We study a typical optimization model where the optimization variable is composed of multiple probability distributions. Though the model appears frequently in practice, such as for policy problems, it lacks specific analysis in the general setting. For this optimization problem, we propose a new structural condition/landscape description named generalized quasar-convexity (GQC) beyond the realms of convexity. In contrast to original quasar-convexity \citep{hinder2020near}, GQC allows an individual quasar-convex parameter \gamma_i for each variable block i and the smaller of \gamma_i implies less block-convexity. To minimize the objective function, we consider a generalized oracle termed as the internal function that includes the standard gradient oracle as a special case.

Societal and real-world considerations such as robustness, fairness, social welfare and multi-agent tradeoffs have given rise to multi-distribution learning paradigms, such as collaborative [Blum et al. 2017], group distributionally robust [Sagawa et al. 2019], and fair federated learning [Mohri et al. 2019]. In each of these settings, a learner seeks to minimize its worstcase loss over a set of n predefined distributions, while using as few samples as possible. In this paper, we establish the optimal sample complexity of these learning paradigms and give algorithms that meet this sample complexity. Importantly, our sample complexity bounds exceed that of the sample complexity of learning a single distribution only by an additive factor of \frac{n\log(n)}{\epsilon 2} . These improve upon the best known sample complexity of agnostic federated learning by Mohri et al. 2019 by a multiplicative factor of n, the sample complexity of collaborative learning by Nguyen and Zakynthinou 2018 by a multiplicative factor \frac{\log(n)}{\epsilon 3}, and give the first sample complexity bounds for the group DRO objective of Sagawa et al. 2019.

Modern challenges of robustness, fairness, and decision-making in machine learning have led to the formulation of multi-distribution learning (MDL) frameworks in which a predictor is optimized across multiple distributions. We study the calibration properties of MDL to better understand how the predictor performs uniformly across the multiple distributions. Through classical results on decomposing proper scoring losses, we first derive the Bayes optimal rule for MDL, demonstrating that it maximizes the generalized entropy of the associated loss function. Our analysis reveals that while this approach ensures minimal worst-case loss, it can lead to non-uniform calibration errors across the multiple distributions and there is an inherent calibration-refinement trade-off, even at Bayes optimality. Our results highlight a critical limitation: despite the promise of MDL, one must use caution when designing predictors tailored to multiple distributions so as to minimize disparity.

Societal and real-world considerations such as robustness, fairness, social welfare and multi-agent tradeoffs have given rise to multi-distribution learning paradigms, such as collaborative [Blum et al. 2017], group distributionally robust [Sagawa et al. 2019], and fair federated learning [Mohri et al. 2019]. In each of these settings, a learner seeks to minimize its worstcase loss over a set of n predefined distributions, while using as few samples as possible. In this paper, we establish the optimal sample complexity of these learning paradigms and give algorithms that meet this sample complexity. Importantly, our sample complexity bounds exceed that of the sample complexity of learning a single distribution only by an additive factor of \frac{n\log(n)}{\epsilon 2} . These improve upon the best known sample complexity of agnostic federated learning by Mohri et al. 2019 by a multiplicative factor of n, the sample complexity of collaborative learning by Nguyen and Zakynthinou 2018 by a multiplicative factor \frac{\log(n)}{\epsilon 3}, and give the first sample complexity bounds for the group DRO objective of Sagawa et al. 2019.

Divergence measures play a central role and become increasingly essential in deep learning, yet efficient measures for multiple (more than two) distributions are rarely explored. This becomes particularly crucial in areas where the simultaneous management of multiple distributions is both inevitable and essential. Examples include clustering, multi-source domain adaptation or generalization, and multi-view learning, among others. While computing the mean of pairwise distances between any two distributions is a prevalent method to quantify the total divergence among multiple distributions, it is imperative to acknowledge that this approach is not straightforward and necessitates significant computational resources. In this study, we introduce a new divergence measure tailored for multiple distributions named the generalized Cauchy-Schwarz divergence (GCSD). Additionally, we furnish a kernel-based closed-form sample estimator, making it convenient and straightforward to use in various machine-learning applications. Finally, we explore its profound implications in the realm of deep learning by applying it to tackle two thoughtfully chosen machine-learning tasks: deep clustering and multi-source domain adaptation. Our extensive experimental investigations confirm the robustness and effectiveness of GCSD in both scenarios. The findings also underscore the innovative potential of GCSD and its capability to significantly propel machine learning methodologies that necessitate the quantification of multiple distributions.

In many problems, the measured variables (e.g., image pixels) are just mathematical functions of the hidden causal variables (e.g., the underlying concepts or objects). For the purpose of making predictions in changing environments or making proper changes to the system, it is helpful to recover the hidden causal variables $Z_i$ and their causal relations represented by graph $\mathcal{G}_Z$. This problem has recently been known as causal representation learning. This paper is concerned with a general, completely nonparametric setting of causal representation learning from multiple distributions (arising from heterogeneous data or nonstationary time series), without assuming hard interventions behind distribution changes. We aim to develop general solutions in this fundamental case; as a by product, this helps see the unique benefit offered by other assumptions such as parametric causal models or hard interventions. We show that under the sparsity constraint on the recovered graph over the latent variables and suitable sufficient change conditions on the causal influences, interestingly, one can recover the moralized graph of the underlying directed acyclic graph, and the recovered latent variables and their relations are related to the underlying causal model in a specific, nontrivial way. In some cases, each latent variable can even be recovered up to component-wise transformations. Experimental results verify our theoretical claims.