Empirical Bernstein in smooth Banach spaces

Understand

Existing concentration bounds for bounded vector-valued random variables include extensions of the scalar Hoeffding and Bernstein inequalities.

• While the latter is typically tighter, it requires knowing a bound on the variance of the random variables.
• We derive a new vector-valued empirical Bernstein inequality, which makes use of an empirical estimator of the variance instead of the true variance.
• The bound holds in 2-smooth separable Banach spaces, which include finite dimensional Euclidean spaces and separable Hilbert spaces.