Rademacher complexity
In computational learning theory (machine learning and theory of computation), Rademacher complexity, named after Hans Rademacher, measures richness of a class of real-valued functions with respect to a probability distribution.
Given a training sample , and a class
of real-valued functions defined on a domain space
, the empirical Rademacher complexity of
is defined as:
where are independent random variables drawn from the Rademacher distribution i.e.
for
.
Let be a probability distribution over
.
The Rademacher complexity of the function class
with respect to
for sample size
is:
where the above expectation is taken over an identically independently distributed (i.i.d.) sample generated according to
.
One can show, for example, that there exists a constant , such that any class of
-indicator functions with Vapnik-Chervonenkis dimension
has Rademacher complexity upper-bounded by
.
Gaussian complexity
Gaussian complexity is a similar complexity with similar physical meanings, and can be obtained from the previous complexity using the random variables instead of
, where
are Gaussian i.i.d. random variables with zero-mean and variance 1, i.e.
.
References
- Peter L. Bartlett, Shahar Mendelson (2002) Rademacher and Gaussian Complexities: Risk Bounds and Structural Results. Journal of Machine Learning Research 3 463-482
- Giorgio Gnecco, Marcello Sanguineti (2008) Approximation Error Bounds via Rademacher's Complexity. Applied Mathematical Sciences, Vol. 2, 2008, no. 4, 153 - 176