Discrete distributions, particularly in high-dimensional deep models, are often highly multimodal due to inherent discontinuities. While gradient-based discrete sampling has proven effective, it is susceptible to becoming trapped in local modes due to the gradient information. To tackle this challenge, we propose an automatic cyclical scheduling, designed for efficient and accurate sampling in multimodal discrete distributions. Our method contains three key components: (1) a cyclical step size schedule where large steps discover new modes and small steps exploit each mode; (2) a cyclical balancing schedule, ensuring "balanced" proposals for given step sizes and high efficiency of the Markov chain; and (3) an automatic tuning scheme for adjusting the hyperparameters in the cyclical schedules, allowing adaptability across diverse datasets with minimal tuning. We prove the non-asymptotic convergence and inference guarantee for our method in general discrete distributions. Extensive experiments demonstrate the superiority of our method in sampling complex multimodal discrete distributions.

Ensuring fair and stable allocation of scarce resources is a fundamental challenge in a wide range of applications. Examples of domains where these challenges manifest include applications where geographical and time constraints impede information collection, such as distributing resources to food banks and providing humanitarian aid to disaster areas and war zones [Aleksandrov et al., 2015, Aleksandrov and Walsh, 2020]. Even in online marketplaces devoid of physical constraints, such as dating services and job matching, evaluating information and collecting data presents a formidable challenge. Recent literature bridges this gap partially by learning noisy preferences as allocation decisions are made. This approach makes allocation processes more adaptable and efficient when the information is incomplete or dynamically changing. However, the current research typically assumes that input from all participants is available at each time-epoch of the allocation process [Bistritz et al., 2020, Cen and Shah, 2022, Leshem, 2024, Liu et al., 2020, Yamada et al., 2023]. Since gathering information is costly and often practical considerations make it infeasible, assuming its availability overlooks the possibility of designing efficient algorithms that operate with limited feedback and the accompanying analysis fails to illuminate which feedback is crucial for efficient design.