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Subspace Optimization for Large Language Models with Convergence Guarantees

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arxiv 2410.11289 v2 pith:CQAHT5DK submitted 2024-10-15 cs.LG math.OC

Subspace Optimization for Large Language Models with Convergence Guarantees

classification cs.LG math.OC
keywords convergencegalorelargeoptimizationsubspacealgorithmsgoloregradient
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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Subspace optimization algorithms, such as GaLore (Zhao et al., 2024), have gained attention for pre-training and fine-tuning large language models (LLMs) due to their memory efficiency. However, their convergence guarantees remain unclear, particularly in stochastic settings. In this paper, we reveal that GaLore does not always converge to the optimal solution and provide an explicit counterexample to support this finding. We further explore the conditions under which GaLore achieves convergence, showing that it does so when either (i) a sufficiently large mini-batch size is used or (ii) the gradient noise is isotropic. More significantly, we introduce GoLore (Gradient random Low-rank projection), a novel variant of GaLore that provably converges in typical stochastic settings, even with standard batch sizes. Our convergence analysis extends naturally to other subspace optimization algorithms. Finally, we empirically validate our theoretical results and thoroughly test the proposed mechanisms. Codes are available at https://github.com/pkumelon/Golore.

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Forward citations

Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. No Subspace to Track: Non-Identifiability and Optimizer State in Low-Rank Training

    cs.LG 2026-07 conditional novelty 7.0

    The top-r gradient subspace in GaLore-family optimizers is statistically non-identifiable beyond ~39 of 128 directions, and the fix is to transport optimizer state across refreshes rather than stabilize the basis.

  2. BROS: Bias-Corrected Randomized Subspaces for Memory-Efficient Single-Loop Bilevel Optimization

    cs.LG 2026-05 unverdicted novelty 6.0

    BROS achieves the same O(ε^{-2}) sample complexity as exact single-loop SBO methods while cutting peak memory by up to 44.9% through randomized subspaces and bias-corrected Hessian estimation.

  3. BROS: Bias-Corrected Randomized Subspaces for Memory-Efficient Single-Loop Bilevel Optimization

    cs.LG 2026-05 unverdicted novelty 6.0

    BROS achieves memory-efficient single-loop stochastic bilevel optimization with O(ε^{-2}) sample complexity by performing updates in randomized subspaces and using Rademacher bi-probe correction for unbiased estimation.

  4. Pro-KLShampoo: Projected KL-Shampoo with Whitening Recovered by Orthogonalization

    cs.LG 2026-05 unverdicted novelty 6.0

    Pro-KLShampoo projects KL-Shampoo preconditioners to a spike-and-flat parametric form on an r-dimensional subspace and recovers the full algebraic preconditioner via orthogonalization, outperforming KL-Shampoo on GPT-...

  5. CR-Net: Scaling Parameter-Efficient Training with Cross-Layer Low-Rank Structure

    cs.LG 2025-09 unverdicted novelty 6.0

    CR-Net uses cross-layer low-rank residuals in a dual-path network plus specialized recomputation to outperform prior low-rank methods on 60M-7B model pre-training while using less compute and memory.

  6. Memory-Efficient Differentially Private Training with Gradient Random Projection

    cs.LG 2025-06 conditional novelty 6.0

    DP-GRAPE reduces memory in differentially private neural network training by using random Gaussian projections on gradients instead of SVD, achieving comparable privacy-utility tradeoffs to DP-SGD and scaling to 6.7B ...

  7. GWT: Scalable Optimizer State Compression for Large Language Model Training

    cs.LG 2025-01 unverdicted novelty 6.0

    GWT projects gradients into wavelet subspaces to compress optimizer states for memory-efficient LLM training while claiming performance parity with full-rank updates.