Forecast Projection and Decision Regret: Theory and Evidence from Battery Arbitrage

Decision-focused learning trains forecasts for the decisions they lead to, but forecasts are often projected onto known bounds before use. We study the resulting mismatch in battery arbitrage. When a battery starts and ends the day at the same charge level, a common shift in prices affects its schedule only through round-trip losses. As a result, decision-focused losses identify the forecast level only weakly. For the SPO+ and perturbed Fenchel-Young losses, we characterize the preferred level by a throughput-matching condition. Projecting forecasts onto a price cap can then remove profitable spreads and increase regret, even as prediction error falls. We give bounds on the change in regret, a dual-threshold condition under which a raw-optimal schedule remains optimal after clipping, and a general version of the mechanism for other problems with a conservation constraint. On 1,158 holdout days of the Indian day-ahead market, an equal-weight average of three statistical forecasts outperforms the five decision-focused variants tested. On 53 later days, with 62% of price blocks at the cap, no variant outperforms the regression or the equal-weight average, although most differences from the average are not significant. The results suggest that forecast post-processing should be treated as part of the predict-then-optimize pipeline.