Tejes Srivastava

Jiaxiang Li, Shiqian Ma, Tejes Srivastava

We consider a class of Riemannian optimization problems where the objective is the sum of a smooth function and a nonsmooth function, considered in the ambient space. This class of problems finds important applications in machine learning and statistics such as the sparse principal component analysis, sparse spectral clustering, and orthogonal dictionary learning. We propose a Riemannian alternating direction method of multipliers (ADMM) to solve this class of problems. Our algorithm adopts easily computable steps in each iteration. The iteration complexity of the proposed algorithm for obtaining an $ε$-stationary point is analyzed under mild assumptions. Existing ADMM for solving nonconvex problems either does not allow nonconvex constraint set, or does not allow nonsmooth objective function. Our algorithm is the first ADMM type algorithm that minimizes a nonsmooth objective over manifold -- a particular nonconvex set. Numerical experiments are conducted to demonstrate the advantage of the proposed method.

Yangqiaoyu Zhou, Haokun Liu, Tejes Srivastava et al.

Effective generation of novel hypotheses is instrumental to scientific progress. So far, researchers have been the main powerhouse behind hypothesis generation by painstaking data analysis and thinking (also known as the Eureka moment). In this paper, we examine the potential of large language models (LLMs) to generate hypotheses. We focus on hypothesis generation based on data (i.e., labeled examples). To enable LLMs to handle arbitrarily long contexts, we generate initial hypotheses from a small number of examples and then update them iteratively to improve the quality of hypotheses. Inspired by multi-armed bandits, we design a reward function to inform the exploitation-exploration tradeoff in the update process. Our algorithm is able to generate hypotheses that enable much better predictive performance than few-shot prompting in classification tasks, improving accuracy by 31.7% on a synthetic dataset and by 13.9%, 3.3% and, 24.9% on three real-world datasets. We also outperform supervised learning by 12.8% and 11.2% on two challenging real-world datasets. Furthermore, we find that the generated hypotheses not only corroborate human-verified theories but also uncover new insights for the tasks.