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Math 281a – Problem Set # 5

Module 6

Consider Maximum Likelihood and Uniform Laws of Large Numbers (Rademacher’s com-plexity).

1. Compute the population risk

in the following cases:

(a) Bernoulli:

for x ∈ {0, 1};

(b) Poisson:

for x ∈ {0, 1, 2, · · · };

(c) Multivariate Gaussian N (θ, Σ) where the covariance matrix Σ is known and is fixed (also invertible).

2. For each of the cases above

(a) Let denote the maximum likelihood estimate. Give an explicit expression for the excess risk

(b) Give an upper bound on the excess risk in terms of an appropriate Rademacher complexity.