An All-Reduce Compatible Top-K Compressor for Communication-Efficient Distributed Learning

Communication remains a central bottleneck in large-scale distributed machine learning, and gradient sparsification has emerged as a promising strategy to alleviate this challenge. However, existing gradient compressors face notable limitations: Rand-$K$\ discards structural information and performs poorly in practice, while Top-$K$\ preserves informative entries but loses the contraction property and requires costly All-Gather operations. In this paper, we propose ARC-Top-$K$, an {All-Reduce}-Compatible Top-$K$ compressor that aligns sparsity patterns across nodes using a lightweight sketch of the gradient, enabling index-free All-Reduce while preserving globally significant information. ARC-Top-$K$\ is provably contractive and, when combined with momentum error feedback (EF21M), achieves linear speedup and sharper convergence rates than the original EF21M under standard assumptions. Empirically, ARC-Top-$K$\ matches the accuracy of Top-$K$\ while reducing wall-clock training time by up to 60.7\%, offering an efficient and scalable solution that combines the robustness of Rand-$K$\ with the strong performance of Top-$K$.

Subspace Optimization for Large Language Models with Convergence Guarantees

Subspace optimization algorithms, with GaLore (Zhao et al., 2024) as a representative method, have gained popularity for pre-training or 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 unexpectedly discover that GaLore does not always converge to the optimal solution and substantiate this finding with an explicit counterexample. We then investigate the conditions under which GaLore can achieve convergence, demonstrating that it does so either in deterministic scenarios or when using a sufficiently large mini-batch size. More significantly, we introduce GoLore (Gradient random Low-rank projection), a novel variant of GaLore that provably converges in stochastic settings, even with standard batch sizes. Our convergence analysis can be readily extended to other sparse subspace optimization algorithms. Finally, we conduct numerical experiments to validate our theoretical results and empirically explore the proposed mechanisms. Codes are available at https://github.com/pkumelon/Golore.