Sketch-and-Solve: Optimized Overdetermined Least-Squares Using Randomized Numerical Linear Algebra

Sketch-and-solve is a powerful paradigm for tackling large-scale computational problems by reducing their dimensionality using sketching matrices. This paper focuses on applying sketch-and-solve algorithms to efficiently solve the overdetermined least squares problem, which is fundamental in various domains such as machine learning, signal processing, and numerical optimization. We provide a comprehensive overview of the sketch-and-solve paradigm and analyze different sketching operators, including dense and sparse variants. We introduce the Sketch-and-Apply (SAA-SAS) algorithm, which leverages randomized numerical linear algebra techniques to compute approximate solutions efficiently. Through extensive experiments on large-scale least squares problems, we demonstrate that our proposed approach significantly outperforms the traditional Least-Squares QR (LSQR) algorithm in terms of runtime while maintaining comparable accuracy. Our results highlight the potential of sketch-and-solve techniques in efficiently handling large-scale numerical linear algebra problems.

Uncertainty-aware Language Modeling for Selective Question Answering

We present an automatic large language model (LLM) conversion approach that produces uncertainty-aware LLMs capable of estimating uncertainty with every prediction. Our approach is model- and data-agnostic, is computationally-efficient, and does not rely on external models or systems. We evaluate converted models on the selective question answering setting -- to answer as many questions as possible while maintaining a given accuracy, forgoing providing predictions when necessary. As part of our results, we test BERT and Llama 2 model variants on the SQuAD extractive QA task and the TruthfulQA generative QA task. We show that using the uncertainty estimates provided by our approach to selectively answer questions leads to significantly higher accuracy over directly using model probabilities.