Problem Setup
Domain adaptation deals with training a classifier on a source distribution and generalizing to a slightly different target distribution. Instead of assuming identical distributions, the notes introduce representation functions that map instances to feature spaces and develop bounds on target generalization using domain divergence.
Domain Distances
Ben-David et al. define the dA-distance over hypothesis classes and show that minimizing empirical error plus the domain distance yields better adaptation. Mansour et al. extend this work by introducing the discrepancy distance, which generalizes to arbitrary loss functions and retains finite-sample estimability thanks to VC-dimension limits.
Rademacher Bounds
Rademacher complexity is used to quantify how rich the function class is. The notes restate bounds that relate empirical loss, complexity, and holding probability, providing sharper guarantees when distributions shift. The discrepancy distance is bounded using Rademacher complexity, ensuring it converges as sample size grows.
Generalization Guarantees
The final section presents a theorem that bounds target loss by a combination of source risk, discrepancy, and the distance between optimal source and target hypotheses. When the discrepancy is small and representations keep labeling functions close, adaptation succeeds.