LESSER: Post-Training Data Selection with Output-Layer Gradients

The choice of post-training data for large language models substantially affects downstream performance. Gradient-based data selection is a popular approach that ranks training data by how well their gradients align with those of a small validation set. However, ranking with full-parameter gradients requires an expensive backward pass on every sample, making computation intractable for large candidate pools. This raises a natural question: can we approximate full-gradient features at a fraction of the cost? Conveniently, we find that output-layer gradients suffice for effective data selection, yet require only the cheaper forward pass. We implement this as LESSER, a drop-in wrapper for selection methods that reduces the feature-extraction FLOP cost by 9.7 times for SFT and 3.0 times for RL benchmarks, while tracking full-gradient performance on downstream tasks. Empirically, we find that even when output-layer and full gradients rank individual samples differently, they select batches with aligned gradients.