Global Before Local: Search and Selection in Flow-Based Inverse Problems

When more inference computation is available, should an inverse solver refine one reconstruction longer or explore alternatives? We study this through global-before-local (GBL) inference, which advances multiple candidates, selects a subset, and completes the retained candidates with the same solver. Broader search can find adequate candidates yet still lose when selection discards them. We separate best-candidate advantage from selection loss, quantify the recovery needed to beat equal-work refinement, and use a Bayes benchmark to distinguish information limits from selector error. GBL with measurement-based selection gains 3.88 dB in mean peak signal-to-noise ratio (PSNR) over refinement in CelebA phase retrieval. Predictive selection improves reconstruction quality in held-out inpainting and exceeds refinement in AFHQ irregular masking. On CelebA 4× point-decimation super-resolution, development data show that even the best searched candidate trails the same-budget single trajectory and forecast a 1.30 dB mean-PSNR deficit for predictive selection; held-out evaluation confirms the ordering with a 1.15 dB deficit. These results show that the value of broader search depends not only on the candidates it finds, but also on how much of that advantage selection preserves.