state aggregation

The crudest, most intuitive form of function approximation: lump states into groups and give every state in a group the same value. It is like a low-resolution map — a whole neighborhood gets one label. You learn one number per group instead of one per state, so a huge or continuous space collapses to a handful of cells.

It is a special case of linear approximation with one-hot group features: x(s) has a single 1 for the group s belongs to. Updating any state in a group updates the shared value for all of them. Because the features are non-overlapping and the within-group weighting is set by the policy's state distribution, on-policy aggregation is stable and converges; it simply converges to a coarse, distribution-weighted average of the true values.

Resolution is capped by the grouping — values that vary within a group can never be represented, so the asymptotic error is set by how well your partition matches the true value's structure. It is a useful teaching tool, a fast baseline, and the conceptual ancestor of tile and coarse coding.