A note on the expected probability of constraint violation in sampled convex programs

In this note we derive an exact expression for the expected probability V of constraint violation in a sampled convex program (see [1], [2] for definitions and an introduction to this topic): V = expected number of support constraints/1 + number of constraints. This result (Theorem 1) is obtained using a simple technique based on cardinality count. In the note, we also use a Chernoff bounding technique on the upper tail violation probability expression derived in to obtain one of the tightest available explicit bounds on the sample complexity of sampled convex programs (Proposition 3).