Spectra: Rethinking Optimizers for LLMs Under Spectral Anisotropy

Gradient signals in LLM training are highly anisotropic: recurrent linguistic structure concentrates energy into a small set of dominant spectral directions, while context-specific information resides in a long tail. We show that this spike–tail separation persists throughout training, with the spike occupying only about 1.5% of directions yet dominating optimizer statistics. This dominance suppresses tail learning by contracting tail updates through second-moment normalization and tightening the globally stable learning-rate bound. Motivated by this analysis, we propose \textit{Spectra}, a spike-aware optimizer that suppresses the dominant low-rank spike subspace without amplifying the noise-sensitive spectral tail. Spectra tracks the spike subspace via cached, warm-started power iteration and applies low-rank spectral shaping with negligible overhead and substantially reduced optimizer-state memory. On LLaMA3-8B trained on 50B tokens, Spectra reaches the same target loss 30% faster than AdamW, reduces per-step end-to-end overhead by 0.7%, cutting optimizer-state memory by 49.25%, and improves average downstream accuracy by 1.62%. Compared to Muon, Spectra is $5.1\times$ faster in optimizer processing time, achieves a lower final loss, and improves average accuracy by 0.66%. Spectra's Megatron integration is released publicly (https://tinyurl.com/29n4vv5f).