This research aims to develop statistical tools to improve the reliability of artificial intelligence (AI) that is widely used in real-world systems such as automated decision-making, financial forecasting, and neuroscience research. Modern AI often relies on efficient machine learning algorithms to process large-scale, sequentially arriving datasets. While these algorithms are powerful, understanding their behavior and measuring their uncertainty remains a major scientific challenge. To bridge this gap, the investigators will focus on establishing mathematically rigorous methods for uncertainty quantification to build trustworthy AI. Applications will include enhancing theoretical guarantees and interpretability of neural networks, providing robust estimation and inference for econometric and biomedical studies, and detecting real-time change-points in high-dimensional time series data. The projects will promote the progress of science through open-source software and graduate education, and will support the national interest by contributing to reliable, data-driven decision-making in fields important to economic resilience, public health and national security. This research will provide a comprehensive theoretical framework for online statistical inference in machine learning, focusing on constant learning-rate stochastic gradient descent (SGD) algorithms. It addresses fundamental challenges such as non-stationarity caused by arbitrarily fixed initialization and complex