Robustness Starts with the Objective: Designing and Certifying Robust LLM-Generated Heuristics
LLMs are increasingly used to design heuristics for computer systems. However, most synthesis frameworks evaluate candidates by their average performance, which can mask severe regressions on specific workloads. Even adding adversarial training inputs doesn't eliminate these regressions, and percentile objectives (e.g., optimizing for the 95th percentile) overlook how severe regressions can be beyond the selected threshold. We present TAILOR, a framework that guides LLM synthesis using the Conditional Value at Risk (CVaR) of the cost difference between each candidate and an existing baseline on the same workload. This paired CVaR measures the average difference over the worst fraction of workloads. We show that a negative paired CVaR guarantees improvements in both average performance and tail optimality gap relative to the unknown optimum, without ever computing that optimum. We find that TAILOR achieves the best mean--tail trade-off among the synthesis objectives we evaluate across three case studies: Azure GPU batch admission, Azure VM packing, and server placement. Tested across seven diverse optimization benchmarks, TAILOR produces heuristics that are certified with 99% confidence to improve both average performance and tail optimality gaps over the baseline in six of them. On these six benchmarks, TAILOR also outperforms known robust, human-designed alternatives in the tail while achieving better or comparable average-performance improvement over the baseline.