Before revision:  This models practical scenarios in which the online algorithm has some history to learn from. This way of modeling a history was formalized recently by the authors (SODA 20) and was called the AOS model (for Adversarial Online with a Sample). It allows developing online algorithms for the secretary problem that compete even when the secretaries arrive in an adversarial order . Here we use the same model to attack the much more challenging matching problem. We analyze a natural algorithmic framework that decides how to match an arriving vertex v by applying an offline matching algorithm to v and the sample.
After revision:  This models practical scenarios in which the online player is provided with a sample, and the sample.