Warm-start or cold-start? A comparison of generalizability in gradient-based hyperparameter tuning

Bilevel optimization (BO) has garnered increasing attention in hyperparameter tuning. BO methods are commonly employed with two distinct strategies for the inner-level: cold-start, which uses a fixed initialization, and warm-start, which uses the last inner approximation solution as the starting point for the inner solver each time, respectively. Previous studies mainly stated that warm-start exhibits better convergence properties, while we provide a detailed comparison of these two strategies from a generalization perspective. Our findings indicate that, compared to the cold-start strategy, warm-start strategy exhibits worse generalization performance, such as more severe overfitting on the validation set. To explain this, we establish generalization bounds for the two strategies. We reveal that warm-start strategy produces a worse generalization upper bound due to its closer interaction with the inner-level dynamics, naturally leading to poor generalization performance. Inspired by the theoretical results, we propose several approaches to enhance the generalization capability of warm-start strategy and narrow its gap with cold-start, especially a novel random perturbation initialization method. Experiments validate the soundness of our theoretical analysis and the effectiveness of the proposed approaches.