We study the problem of alleviating the instability issue in the GAN training procedure via new architecture design. The discrepancy between the minimax and maximin objective values could serve as a proxy for the difficulties that the alternating gradient descent encounters in the optimization of GANs. In this work, we give new results on the benefits of multi-generator architecture of GANs. We show that the minimax gap shrinks to $ε$ as the number of generators increases with rate $\widetilde{O}(1/ε)$. This improves over the best-known result of $\widetilde{O}(1/ε^2)$. At the core of our techniques is a novel application of Shapley-Folkman lemma to the generic minimax problem, where in the literature the technique was only known to work when the objective function is restricted to the Lagrangian function of a constraint optimization problem. Our proposed Stackelberg GAN performs well experimentally in both synthetic and real-world datasets, improving Fréchet Inception Distance by $14.61\%$ over the previous multi-generator GANs on the benchmark datasets.
The use of machine learning models in consequential decision making often exacerbates societal inequity, in particular yielding disparate impact on members of marginalized groups defined by race and gender. The area under the ROC curve (AUC) is widely used to evaluate the performance of a scoring function in machine learning, but is studied in algorithmic fairness less than other performance metrics. Due to the pairwise nature of the AUC, defining an AUC-based group fairness metric is pairwise-dependent and may involve both \emph{intra-group} and \emph{inter-group} AUCs. Importantly, consideri…
Sparse principal component analysis (PCA) involves nonconvex optimization for which the global solution is hard to obtain. To address this issue, one popular approach is convex relaxation. However, such an approach may produce suboptimal estimators due to the relaxation effect. To optimally estimate sparse principal subspaces, we propose a two-stage computational framework named "tighten after relax": Within the 'relax' stage, we approximately solve a convex relaxation of sparse PCA with early stopping to obtain a desired initial estimator; For the 'tighten' stage, we propose a novel algorithm…
University of Michigan（密歇根大学）；University of California, Riverside（加州大学河滨分校）；University of British Columbia（不列颠哥伦比亚大学）
Standard federated optimization methods successfully apply to stochastic problems with single-level structure. However, many contemporary ML problems -- including adversarial robustness, hyperparameter tuning, and actor-critic -- fall under nested bilevel programming that subsumes minimax and compositional optimization. In this work, we propose \fedblo: A federated alternating stochastic gradient method to address general nested problems. We establish provable convergence rates for \fedblo in the presence of heterogeneous data and introduce variations for bilevel, minimax, and compositional op…
Wide applications of differentiable two-player sequential games (e.g., image generation by GANs) have raised much interest and attention of researchers to study efficient and fast algorithms. Most of the existing algorithms are developed based on nice properties of simultaneous games, i.e., convex-concave payoff functions, but are not applicable in solving sequential games with different settings. Some conventional gradient descent ascent algorithms theoretically and numerically fail to find the local Nash equilibrium of the simultaneous game or the local minimax (i.e., local Stackelberg equil…
Provable Robust Overfitting Mitigation in Wasserstein Distributionally Robust Optimization
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Kaiyuan International Mathematical Sciences Institute（开元国际数学科学研究院）
University of Basel（巴塞尔大学）；ELLIS Institute Tübingen（埃利斯研究所蒂宾根分所）；MPI for Intelligent Systems Tübingen（蒂宾根智能系统马克斯·普朗克研究所）；AI Center（人工智能中心）；Yahoo! Research（雅虎研究院）；University of Oslo（奥斯陆大学）
Keio University（庆应义塾大学）；University of Southern California（南加州大学）
This paper presents a convex-analytic framework to learn sparse graphs from data. While our problem formulation is inspired by an extension of the graphical lasso using the so-called combinatorial graph Laplacian framework, a key difference is the use of a nonconvex alternative to the $\ell_1$ norm to attain graphs with better interpretability. Specifically, we use the weakly-convex minimax concave penalty (the difference between the $\ell_1$ norm and the Huber function) which is known to yield sparse solutions with lower estimation bias than $\ell_1$ for regression problems. In our framework,…
Michigan State University（密歇根州立大学）；Eastern Mediterranean University（东地中海大学）
Adversarial representation learning is a promising paradigm for obtaining data representations that are invariant to certain sensitive attributes while retaining the information necessary for predicting target attributes. Existing approaches solve this problem through iterative adversarial minimax optimization and lack theoretical guarantees. In this paper, we first study the "linear" form of this problem i.e., the setting where all the players are linear functions. We show that the resulting optimization problem is both non-convex and non-differentiable. We obtain an exact closed-form express…
Max-affine regression refers to a model where the unknown regression function is modeled as a maximum of $k$ unknown affine functions for a fixed $k \geq 1$. This generalizes linear regression and (real) phase retrieval, and is closely related to convex regression. Working within a non-asymptotic framework, we study this problem in the high-dimensional setting assuming that $k$ is a fixed constant, and focus on estimation of the unknown coefficients of the affine functions underlying the model. We analyze a natural alternating minimization (AM) algorithm for the non-convex least squares object…
