  This paper asks a basic question: how much training is required to beat a
universal source coder? Traditionally, there have been two types of source
coders: fixed, optimum coders such as Huffman coders; and universal source
coders, such as Lempel-Ziv The paper considers a third type of source coders:
learned coders. These are coders that are trained on data of a particular type,
and then used to encode new data of that type. This is a type of coder that has
recently become very popular for (lossy) image and video coding.
  The paper consider two criteria for performance of learned coders: the
average performance over training data, and a guaranteed performance over all
training except for some error probability $P_e$. In both cases the coders are
evaluated with respect to redundancy.
  The paper considers the IID binary case and binary Markov chains. In both
cases it is shown that the amount of training data required is very moderate:
to code sequences of length $l$ the amount of training data required to beat a
universal source coder is $m=K\frac{l}{\log l}$, where the constant in front
depends the case considered.
