Artificial Intelligence Theory and Application

  PAC-Bayes Theory: A Bridge Between Statistical Learning and Bayesian Methods PAC-Bayes ( Probably Approximately Correct Bayesian ) theory provides  generalization guarantees  for machine learning models by combining  PAC learning  (Valiant, 1984) and  Bayesian inference . Unlike classical VC theory, PAC-Bayes accounts for  prior knowledge  and  posterior distributions  over hypotheses, making it especially useful for modern ML (e.g., neural networks, stochastic algorithms). 1. Core Idea PAC-Bayes bounds quantify how well a  randomized predictor  (e.g., a Bayesian model or stochastic neural network) generalizes from training data to unseen data. The key insight: Prior ( P P ) : A fixed distribution over hypotheses  before  seeing data (e.g., initial neural network weights). Posterior ( Q Q ) : A learned distribution over hypotheses  after  training (e.g., noisy/approximate Bayesian inference). Bound : The g...

VC dimension (Vapnik-Chervonenkis dimension)  is based on the foundational work by  Vladimir Vapnik and Alexey Chervonenkis  (1971, with key refinements in Vapnik's 1998 book  "Statistical Learning Theory" ): 1. What is VC Dimension? The  VC dimension  is a measure of the  capacity  (complexity) of a hypothesis class (e.g., neural networks, decision trees). It quantifies the  largest set of points  that a model can  shatter  (classify correctly for all possible labelings). Key Idea : A model with high VC dimension can fit more complex patterns but risks overfitting. Formal Definition : A hypothesis class  H H  has VC dimension  d d  if there exists a set of  d d  points that can be shattered by  H H , but no set of  d + 1 d + 1  points can be shattered. 2. Intuitive Example Example: Linear Classifiers in 2D VC dimension = 3 : Can shatter any 3 non-collinear points (all  2 3 = 8 ...