AdamHD: Decoupled Huber Decay Regularization for Language Model Pre-Training

Adaptive optimizers with decoupled weight decay, such as AdamW, are the de facto standard for pre-training large transformer-based generative models. Yet the quadratic nature of the ℓ2 penalty embedded in weight decay drives all parameters toward the origin at the same rate, making the update vulnerable to rare but extreme gradient directions and often over-penalizing well-conditioned coordinates. We propose AdamHuberDecay, a drop-in replacement for AdamW that substitutes the ℓ2 penalty with a decoupled smooth Huber regularizer. The resulting update decays parameters quadratically while their magnitude remains below a threshold δ, and linearly (ℓ1-like) once they exceed δ, yielding (i) bounded regularization gradients, (ii) invariance to per-coordinate second-moment rescaling, and (iii) stronger sparsity pressure on overgrown weights. We derive the closed-form decoupled Huber decay step and show how to integrate it with any Adam-family optimizer at O(1) extra cost. Extensive experiments on GPT-2 and GPT-3 pre-training demonstrate that AdamHuberDecay (a) converges 10-15% faster in wall-clock time, (b) reduces validation perplexity by up to 4 points, (c) delivers performance improvements of 2.5-4.7% across downstream tasks, and (d) yields visibly sparser weight histograms that translate into 20-30% memory savings after magnitude pruning, without tuning the decay coefficient beyond the default grid used for AdamW. Ablations confirm robustness to outlier gradients and large-batch regimes, together with theoretical analyses that bound the expected parameter norm under noisy updates. AdamHuberDecay therefore provides a simple, principled path toward more efficient and resilient training of next-generation foundational generative transformers.