Nick Polson

Nick Polson, Vadim Sokolov, Jianeng Xu

Quantum Bayesian Computation (QBC) is an emerging field that levers the computational gains available from quantum computers to provide an exponential speed-up in Bayesian computation. Our paper adds to the literature in two ways. First, we show how von Neumann quantum measurement can be used to simulate machine learning algorithms such as Markov chain Monte Carlo (MCMC) and Deep Learning (DL) that are fundamental to Bayesian learning. Second, we describe data encoding methods needed to implement quantum machine learning including the counterparts to traditional feature extraction and kernel embeddings methods. Our goal then is to show how to apply quantum algorithms directly to statistical machine learning problems. On the theoretical side, we provide quantum versions of high dimensional regression, Gaussian processes (Q-GP) and stochastic gradient descent (Q-SGD). On the empirical side, we apply a Quantum FFT model to Chicago housing data. Finally, we conclude with directions for future research.

Maria Nareklishvili, Nick Polson, Vadim Sokolov

Generative methods (Gen-AI) are reviewed with a particular goal of solving tasks in machine learning and Bayesian inference. Generative models require one to simulate a large training dataset and to use deep neural networks to solve a supervised learning problem. To do this, we require high-dimensional regression methods and tools for dimensionality reduction (a.k.a. feature selection). The main advantage of Gen-AI methods is their ability to be model-free and to use deep neural networks to estimate conditional densities or posterior quintiles of interest. To illustrate generative methods , we analyze the well-known Ebola data set. Finally, we conclude with directions for future research.

Nick Polson, Vadim Sokolov

Double descent is a phenomenon of over-parameterized statistical models such as deep neural networks which have a re-descending property in their risk function. As the complexity of the model increases, risk exhibits a U-shaped region due to the traditional bias-variance trade-off, then as the number of parameters equals the number of observations and the model becomes one of interpolation where the risk can be unbounded and finally, in the over-parameterized region, it re-descends -- the double descent effect. Our goal is to show that this has a natural Bayesian interpretation. We also show that this is not in conflict with the traditional Occam's razor -- simpler models are preferred to complex ones, all else being equal. Our theoretical foundations use Bayesian model selection, the Dickey-Savage density ratio, and connect generalized ridge regression and global-local shrinkage methods with double descent. We illustrate our approach for high dimensional neural networks and provide detailed treatments of infinite Gaussian means models and non-parametric regression. Finally, we conclude with directions for future research.

Nick Polson, Vadim Sokolov

Our goal is to provide a review of deep learning methods which provide insight into structured high-dimensional data. Rather than using shallow additive architectures common to most statistical models, deep learning uses layers of semi-affine input transformations to provide a predictive rule. Applying these layers of transformations leads to a set of attributes (or, features) to which probabilistic statistical methods can be applied. Thus, the best of both worlds can be achieved: scalable prediction rules fortified with uncertainty quantification, where sparse regularization finds the features.

Shiva Maharaj, Nick Polson, Alex Turk

Endgame studies have long served as a tool for testing human creativity and intelligence. We find that they can serve as a tool for testing machine ability as well. Two of the leading chess engines, Stockfish and Leela Chess Zero (LCZero), employ significantly different methods during play. We use Plaskett's Puzzle, a famous endgame study from the late 1970s, to compare the two engines. Our experiments show that Stockfish outperforms LCZero on the puzzle. We examine the algorithmic differences between the engines and use our observations as a basis for carefully interpreting the test results. Drawing inspiration from how humans solve chess problems, we ask whether machines can possess a form of imagination. On the theoretical side, we describe how Bellman's equation may be applied to optimize the probability of winning. To conclude, we discuss the implications of our work on artificial intelligence (AI) and artificial general intelligence (AGI), suggesting possible avenues for future research.

Shiva Maharaj, Nick Polson

Anatoly Karpov's Queen sacrifices are analyzed. Stockfish 14 NNUE -- an AI chess engine -- evaluates how efficient Karpov's sacrifices are. For comparative purposes, we provide a dataset on Karpov's Rook and Knight sacrifices to test whether Karpov achieves a similar level of accuracy. Our study has implications for human-AI interaction and how humans can better understand the strategies employed by black-box AI algorithms. Finally, we conclude with implications for human study in. chess with computer engines.