Line searching techniques

Techniques of line-searching to find a minimum value of a loss function l projected onto a line are disclosed, wherein the loss function l is continuous and piece-wise linear. One technique includes identifying points for at least one output derived from the loss function l at which the slope of the output changes, determining at least one derivative value for an initial point and at least one change in derivative value for at least some of the points using a processor, determining a cumulative reduction in loss for at least some of the points using at least one of the determined derivative values or at least one of the change in derivative values, and selecting a point corresponding to the minimum value of the loss function l based on the cumulative reduction in loss.