Neural networks have been widely studied and used in recent years due to its high

classification accuracy and training efficiency. With the increase of network depth, however,

the models become worse calibrated, meaning they cannot reflect the true probabilities. On

the other hand, in many applications such as medical diagnosis, facial recognition and selfdriving cars, the calibrated output probabilities are of critical importance. Therefore, the

understanding of the cause of deep neural network uncalibration is of much concern.

The influence of model structures on the output calibration has been explored.

However, the impact of the training dataset quality and heterogeneity, such as dataset size

and label noise remains unclear. In this thesis, the impact of data quality and heterogeneity

on the output calibration is investigated theoretically and experimentally. Afterwards, the

defect of calibration methods using single global parameter are discussed. To overcome

the calibration issues resulting from the dataset heterogeneity, we propose an improved

calibration technique that can give better performance.

Machine learning has emerged as a leading force in revolutionizing technology, education, and almost every aspect of our lives. Reinforcement learning is a sub-field of machine learning that deals with the effects of dynamic feedback and systems that interact with the environment. In these settings, classic statistical and algorithmic guarantees often do not hold because of non i.i.d. data, dynamic feedback, and distribution shift.

We develop a framework for single trajectory learning of nonlinear dynamical systems using mixing arguments. Our main result studies the landscape of empirical risk minimization for learning nonlinear dynamical systems from a single trajectory, and provides uniform gradient convergence guarantee, which is combined with novel one-point convexity to facilitate the learning of nonlinear dynamical systems. Our proposed framework allows for non-convex loss landscape and our sample complexity and statistical error rates are optimal in terms of the trajectory length, dimensions of the system and input/noise strength.

Next, we study the problem of learning bilinear dynamical systems from a single trajectory of the system’s states and inputs. Our main contribution is the application of martingale small-ball arguments to derive learning guarantees for non-mixing bilinear dynamical systems. We further extend our analysis to time varying dynamical systems by studying the problem of learning non-mixing Markov jump systems. Specifically, we learn the dynamics in each mode and the Markov transition matrix, underlying the evolution of the mode switches, from a single trajectory of the system’s states, inputs, and modes. Our sample complexity and statistical error rates are optimal in terms of the trajectory length, the dimensions of the system and the input/noise strength.

Lastly, as a preliminary to the problem of finding the best LTI dynamical system that can minimize least-squares loss given a single trajectory of an unknown dynamical system, we study the simpler problem of finding the best linear model in high dimensions, given a dataset. Specifically, we analyze projected gradient descent algorithm to estimate the population minimizer in the finite sample regime. We show that the nonlinearity of the problem can be treated as uncorrelated noise and establish linear convergence rate and data-dependent estimation error bounds for the projected gradient descent algorithm.

Geographically Weighted Regression (GWR) is a seminal technique with rich applications in geospatial data analysis. However, it has critical drawbacks in the age of big data in terms of expressiveness,i.e., predictive power, and scalability. This work proposes Augmented GWR (A-GWR) that alleviates these drawbacks. A-GWRadapts a novel technique, Stateless-MGWR or S-MGWR, that en-riches the predictive power by allowing distinct bandwidths for individual features of the training data. S-MGWR uses a customized black-box optimization approach for discovering optimal band widths in a fast and efficient way. In addition, A-GWR modularly combines S-MGWR or GWR with versatile models such as random forest models. Moreover, A-GWR enables scalability by operating on flexible partitions of the data that can adapt to the computational budget. Our extensive evaluations on various real and synthetic datasets demonstrate the scalability and accuracy benefits of the proposed techniques over state-of-the-art competitors.