A bfloat16 transformer can train normally for many steps and then collapse abruptly. Distinct low-precision errors can trigger the same failure, leaving unclear whether each source needs its own repair or one shared route can be blocked. We isolate a reproduced GPT-2-class collapse to the streaming-softmax accumulator, where fp32 accumulation repairs it, and use the fault as an assay for moving controlled errors across sources. Errors placed outside attention still drive the same query-key (QK) spectral runaway, while correcting only QK keeps training stable with the source fault active. This source-channel dissociation shows that fault source is not failure channel. It holds across the tested architectures and scales and reproduces on a second GPU architecture. A causal probe projects each update off the current QK weights' leading three singular directions: the query projection's largest singular value stays at 11.1, whereas removing equal energy elsewhere leaves it at 237. The QK channel therefore drives the early runaway rather than merely tracking it. Entry depends on temporal sign-coherence across steps, not aggregate deviation. QK-Guard closes the channel with a dormant controller that switches on parameter-free QK normalization when attention-logit saturation begins. It contains every tested runaway and matches always-on QK normalization over 60k steps, while non-QK actions at the same trigger fail. The results support intervention at the shared QK locus rather than separate repair at each fault source.
Finite-precision arithmetic unavoidably introduces numerical approximation errors. Numerical computations may use insufficient precision or an improper formulation, which leads to numerical instability. In this paper, we introduce the first unified software tool that integrates CESTAC for detecting the numerical stability of deep learning operators. Our developed software not only enables numerical validation with a single computation pass but also detects the sources of numerical instability and provides numerical stability monitoring during deep learning training and inference. We verified its effectiveness on the detection of polluted operators with injected numerical instabilities across various tasks. We believe that our developed method and tool provide valuable insights into developing numerically stable computing kernels, which are particularly critical for numerically stable and efficient deep learning training and inference.
As Large Language Models (LLMs) deploy into mission-critical domains (e.g., finance, medicine, and law), output reproducibility has become a strict system requirement. While practitioners use greedy decoding to eliminate algorithmic stochasticity, empirical deployments with 16-bit precisions still exhibit catastrophic output divergence across heterogeneous GPUs. Through SASS-level profiling, we reveal that this inconsistency is fundamentally driven by truncation errors introduced during downcasting at kernel boundaries. However, achieving reproducibility via a global FP32 pipeline incurs prohibitive system penalties: bypassing 16-bit hardware accelerators hurts compute efficiency, while upcasting the KV cache doubles memory overhead. To bridge this gap, we propose Hybrid Error ALleviation (HEAL), a targeted intervention that approximates FP32 precision while resolving hardware constraints through two targeted mechanisms. First, recognizing that floating-point formats underutilize their bit-width for Q, K, V tensors, HEAL applies INT16 quantization that preserves numerical stability without expanding the KV cache footprint. Second, HEAL synthesizes high-precision matrix multiplications via an algebraic error compensation strategy, executing entirely on high-throughput 16-bit Tensor Cores. To evaluate our approach practically, we introduce MCR-Bench, a benchmark targeting reproducibility in mission-critical tasks. HEAL achieves the same level of reproducibility on downstream tasks as the FP32 baseline while reducing the performance overhead by up to 7.1x.
Aleksandros Sobczyk, Gioele Gottardo, Christos K. Matzoros +4cs.LG
Linear attention has emerged as a cornerstone for efficient long-context architectures, as evidenced by its integration into state-of-the-art open-source models including Qwen3.5/3.6, Kimi Linear, and RWKV-7. Models that incorporate linear attention layers with the so-called Delta-Rule involve the inversion of triangular matrices as a core sub-routine. This operation often forms a performance bottleneck, and, due to its high-sensitivity to numerical errors, it can significantly deteriorate end-to-end model accuracy if it is not carefully implemented. This work provides a systematic analysis of both direct and iterative triangular inversion algorithms, targeting methods that are rich in matrix products, and, therefore, have the potential to efficiently utilize modern hardware. To that end, our analysis covers a broad spectrum of mathematical and practical aspects, with a heavy focus on numerical stability, computational complexity, and, ultimately, hardware efficiency and practical considerations. We provide a rigorous experimental evaluation to verify these properties in practical scenarios, and in low-precision floating-point representations, highlighting the strengths and limitations of each method. Performance benchmarks on NPUs reveal up to $4.3\times$ speed-up against the state-of-the-art implementations of SGLang for triangular matrix inversion, leading to significant performance improvements on the entire layer level, while maintaining full end-to-end model accuracy.