Random search avoids repeated projections on irrelevant axes

Learning goal: a balanced grid with budget B over d dimensions has roughly B^1/d distinct values per axis, while continuous random draws usually give B distinct values on each axis. If only k dimensions matter, those projected coverages differ; the grid's per-axis scale is B ^1/d/B.

First two search coordinates; background: objective; edge ticks: projections onto each axis.

Best loss versus budget on log axes; median over 40 repeated searches.

Budget B Nominal dimension d Effective dimension k

B = — Grid values per axis = — Random values per axis = — Best grid loss = — Best random-search loss = — Ratio = —

Observe: increasing nominal d reduces the grid's distinct values per axis from B toward B ^1/5, while random projections remain numerous. Setting k=d can make the curves closer in this experiment. This does not prove equal performance for every fully relevant objective: geometry, scale, and budget still matter. Inspect which hyperparameters affect the objective.