Preconditioned Inexact Stochastic ADMM for Deep Model

Sign-based optimization algorithms, such as SignSGD, have garnered significant attention for their remarkable performance in distributed learning and training large foundation models. Despite their empirical superiority, SignSGD is known to diverge on non-smooth objectives, which are ubiquitous in modern machine learning due to ReLUs, max-pools, and mixture-of-experts. To overcome this fundamental limitation, we propose \textbf{StoSignSGD}, an algorithm that injects structural stochasticity into the sign operator while maintaining an unbiased update step. In the regime of (online) convex optimization, our theoretical analysis shows that StoSignSGD rigorously resolves the non-convergence issues of SignSGD, achieving a sharp convergence rate matching the lower bound. For the more challenging non-convex non-smooth optimization, we introduce generalized stationary measures that encompass prior definitions, proving that StoSignSGD improves upon the best-known complexity bounds by dimensional factors. Empirically, StoSignSGD exhibits robust stability and superior efficiency across diverse large language model (LLM) training regimes. Notably, in low-precision FP8 pretraining -- a setting where AdamW fails catastrophically -- StoSignSGD remains highly stable and yields a remarkable 1.44$\times$ to 2.14$\times$ speedup relative to established baselines. Furthermore, when fine-tuning 7B LLMs on mathematical reasoning tasks, StoSignSGD delivers substantial performance gains over both AdamW and SignSGD. Finally, to dissect the mechanisms driving its success, we develop a sign conversion framework capable of transforming any general optimizer into its unbiased, sign-based counterpart. Utilizing this framework, we deconstruct the core components of StoSignSGD and present a comprehensive ablation study to empirically validate our algorithmic design choices.

Adaptive optimizers such as Adam have achieved great success in training large-scale models like large language models and diffusion models. However, they often generalize worse than non-adaptive methods, such as SGD on classical architectures like CNNs. We identify a key cause of this performance gap: adaptivity in pre-conditioners, which limits the optimizer's ability to adapt to diverse optimization landscapes. To address this, we propose Anon (Adaptivity Non-restricted Optimizer with Novel convergence technique), a novel optimizer with continuously tunable adaptivity in R, allowing it to interpolate between SGD-like and Adam-like behaviors and even extrapolate beyond both. To ensure convergence across the entire adaptivity spectrum, we introduce incremental delay update (IDU), a novel mechanism that is more flexible than AMSGrad's hard max-tracking strategy and enhances robustness to gradient noise. We theoretically establish convergence guarantees under both convex and non-convex settings. Empirically, Anon consistently outperforms state-of-the-art optimizers on representative image classification, diffusion, and language modeling tasks. These results demonstrate that adaptivity can serve as a valuable tunable design principle, and Anon provides the first unified and reliable framework capable of bridging the gap between classical and modern optimizers and surpassing their advantageous properties.

Deep learning optimization relies heavily on the assumption of smooth loss landscapes, a condition systematically violated by modern architectures due to non-smooth components such as ReLU activations and quantization operators. In such non-smooth regimes, adaptive optimizers such as Adam suffer from gradient chattering, violent oscillations caused by conflicting signals within the Clarke subdifferential, leading to poor convergence and suboptimal generalization. To address this, we introduce Singularity-aware Adam (S-Adam), a novel optimizer that stabilizes training by dynamically modulating step sizes based on local geometric instability. Our key contribution is the Local Geometric Instability (LGI) metric, a computationally efficient estimator of the Clarke subdifferential diameter derived from the variance of randomized directional derivatives. S-Adam incorporates an adaptive damping mechanism exp(-λ$$ρ) that decelerates updates in high-instability regions while preserving fast convergence in smooth basins. We provide a rigorous convergence analysis using differential inclusions, proving that S-Adam converges almost surely to ($δ$,$ε$)-Clarke stationary points at the optimal O(1/$\sqrt(T)$) rate. Empirical evaluations on Quantization-Aware Training (QAT) and high-noise small-batch learning demonstrate that S-Adam consistently outperforms AdamW and Prox-SGD, achieving accuracy gains of up to 6 percent on CIFAR-100 and 3 percent on TinyImageNet while effectively mitigating gradient oscillations.