  We construct a self-interacting dark matter model that could simultaneously
explain the observed muon anomalous magnetic moment. It is based on a gauged
U$(1)_{L_{\mu} - L_{\tau}}$ extension of the standard model, where we introduce
a vector-like pair of fermions as the dark matter candidate and a new Higgs
boson to break the symmetry. The new gauge boson has a sizable contribution to
muon $(g-2)$, while being consistent with other experimental constraints. The
U$(1)_{L_{\mu} - L_{\tau}}$ Higgs boson acts as a light force carrier,
mediating dark matter self-interactions with a velocity-dependent cross
section. It is large enough in galaxies to thermalize the inner halo and
explain the diverse rotation curves and diminishes towards galaxy clusters.
Since the light mediator dominantly decays to the U$(1)_{L_{\mu} - L_{\tau}}$
gauge boson and neutrinos, the astrophysical and cosmological constraints are
weak. We study the thermal evolution of the model in the early Universe and
derive a lower bound on the gauge boson mass.
