(A) The effects of incomplete gradient ascent on the relation between projected bound distance and local curvature (that is, second derivative of the reward rate at estimated bound) are illustrated for a fictional maximization problem with only two bounds. The grey trajectory shows a sequence of gradient ascent steps on the reward rate function, whose shape is illustrated by two iso-reward rate contours (black) around its maximum (cross). Stopping this gradient ascent procedure (large grey filled circle) before it reaches the optimum causes this stopping point to be close to the optimal bound in directions of large curvature (red), and farther away from the optimum in directions of shallow curvature (blue). (B) Curvature at the estimated bound location is plotted against the distance between the estimated and optimal bound (see text for details). This plot includes 7 (3 coherence condition) or 13 (6 coherence condition) data points per subject, one for each modality/coherence combination. Data for the visual, vestibular and combined conditions are shown in shades of blue/cyan, green, and red/yellow, respectively, and motion coherence is indicated by color saturation. (C) The reward rate loss (i.e., optimal model reward rate minus subject's reward rate) as estimated from the model (abscissa) is plotted against the loss predicted by the quadratic approximation used in the analysis in (A)–(B), for each subject (ordinate). If the reward rate has a quadratic dependence on the bounds, then all the data points would lie along the diagonal. Small deviations from the diagonal indicate that the reward rate is indeed close-to-quadratic in these bounds.