The diameter of a single proton is in the order of 1fm (10-15m). Electrons and positrons are much smaller. Therefore, it takes very powerful focusing indeed to ensure that two particles traveling in an accelerator at relativistic speeds collide.
Throughout most of the acceleration process, a particle bunch is spread across the cross sectional area of the beam pipe. This helps avoid premature collisions in the case of circular accelerators which use the same beam pipe to accelerate both bunches. On attaining the energy required for the experiment, the beams are focused using quadrupole magnets. These have the property of focusing charged particles in one plane and defocusing them in the orthogonal one. A succession of alternate quadrupole magnets have the net effect of focusing the particle beam in both directions, just as a succession of equally strong converging and diverging lenses focuses a beam of light.