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.