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PSL: Rethinking and Improving Softmax Loss from Pairwise Perspective for Recommendation
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Proposes Pairwise Softmax Loss (PSL), which replaces the exponential in Softmax Loss with bounded surrogate activations (tanh, atan, ReLU) to get a tighter DCG surrogate, better weight balance against false negatives, and a DRO-enhanced BPR interpretation.
Problem
Softmax Loss (SL) in recommendation is a loose surrogate for DCG/MRR because the exponential overshoots the Heaviside step, and its exponential weighting makes it highly sensitive to false negative noise.
Approach
Reformulate SL pairwise and replace exp(.) with other surrogate activations approximating the Heaviside step function (tanh+1, arctan+1, ReLU(d+1)); the outside-temperature form is equivalent to a KL-DRO over BPR loss.
Key results
- Under the IID setting on MF, PSL-relu improves NDCG@20 over SL by 1.40% to 2.31% across datasets, with statistical significance.
- Under the OOD popularity-shift setting (MF), PSL improves NDCG@20 over SL by 2.02% to 5.02%.
- Under false-negative noise, PSL's gain over SL exceeds 10% at noise ratio p=0.5, per the paper's reported results.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace SL's exponential activation with ReLU in PSL | MF recommendation model · IID: Amazon-Book, Amazon-Electronic, Amazon-Movie, Gowalla | NDCG@20: +1.40%*; +2.31%*; +1.72%*; +1.42%* | Helped |
| Replace SL's exponential activation with ReLU in PSL | MF recommendation model · OOD popularity shift: Amazon-CD, Amazon-Electronic, Gowalla, Yelp2018 | NDCG@20: +3.01%*; +5.02%*; +2.02%*; +2.05%* | Helped |
| Use Tanh instead of ReLU as the PSL surrogate activation (ablation) | PSL on MF · IID: Amazon-Book, Amazon-Electronic, Amazon-Movie, Gowalla | NDCG@20: 0.1227 -> 0.1225; 0.0541 -> 0.0535; 0.0945 -> 0.0941; 0.1647 -> 0.1646 | Mixed |
| Use Atan instead of ReLU as the PSL surrogate activation (ablation) | PSL on MF · IID: Amazon-Book, Amazon-Electronic, Amazon-Movie, Gowalla | NDCG@20: 0.1227 -> 0.1226; 0.0541 -> 0.0535; 0.0945 -> 0.0941; 0.1647 -> 0.1646 | Mixed |
| Increase the false-negative sampling noise ratio | PSL and SL · Noise setting | NDCG@20 | Hurt |
| Use PSL instead of SL under false-negative noise | MF recommendation model · Noise setting, p=0.5 | NDCG@20: > 10% when p=0.5 | Helped |
| Use AdvInfoNCE instead of SL under false-negative noise | MF recommendation model · Noise setting | NDCG@20 | Hurt |
Surprising findings
- Carefully tuned SL is a strong baseline; improvements of prior SL enhancements (AdvInfoNCE, BSL) over SL are smaller than reported in their papers.
Limitations
- SL and PSL require sampling many negatives per iteration, making them inefficient.
- Gains over SL in IID settings are modest (about 1-3%).
Experimental setup
Datasets: Amazon-Book Amazon-Electronic Amazon-Movie Gowalla Amazon-CD Yelp2018
Benchmarks: Amazon-Book Amazon-Electronic Amazon-Movie Gowalla Amazon-CD Yelp2018
Metrics: Recall@20 NDCG@20
Baselines: BPR SL AdvInfoNCE BSL LLPAUC
Builds on: Softmax Loss BPR BSL AdvInfoNCE
Closest papers by meaning
- Advancing Loss Functions in Recommender Systems: A Comparative Study with a Rényi Divergence-Based Solution (AAAI 2025 · similarity 0.91)
- From Pairwise to Ranking: Climbing the Ladder to Ideal Collaborative Filtering with Pseudo-Ranking (AAAI 2025 · similarity 0.89)
- Learning-Efficient Yet Generalizable Collaborative Filtering for Item Recommendation (ICML 2024 · similarity 0.88)
- On Softmax Direct Preference Optimization for Recommendation (NeurIPS 2024 · similarity 0.87)
- Objective Shaping with Hard Negatives: Windowed Partial AUC Optimization for RL-based LLM Recommenders (NeurIPS 2026 · similarity 0.83)
- RRL: Recommendation Reverse Learning (AAAI 2024 · similarity 0.83)
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