  Linear and geometric mixtures are two methods to combine arbitrary models in
data compression. Geometric mixtures generalize the empirically well-performing
PAQ7 mixture. Both mixture schemes rely on weight vectors, which heavily
determine their performance. Typically weight vectors are identified via Online
Gradient Descent. In this work we show that one can obtain strong code length
bounds for such a weight estimation scheme. These bounds hold for arbitrary
input sequences. For this purpose we introduce the class of nice mixtures and
analyze how Online Gradient Descent with a fixed step size combined with a nice
mixture performs. These results translate to linear and geometric mixtures,
which are nice, as we show. The results hold for PAQ7 mixtures as well, thus we
provide the first theoretical analysis of PAQ7.
