We provide some background on the rate of escape of random walks and its relation to compression exponents. We then show that any simple symmetric random walk on a Cayley graph of a polycyclic group has the same rate of escape as a random walk on the integer lattice so long as the Fitting subgroup has uniform exponential distortion. The ideas behind this proof can be generalized to metabelian groups which contain non-finitely generated subgroups, where a similar result is obtained.