As deep learning methods increasingly utilize sensitive data on a widespread scale, differential privacy (DP) offers formal guarantees to protect against information leakage during model training. A significant challenge remains in implementing DP optimizers that retain strong performance while preserving privacy. Recent advances introduced ever more efficient optimizers, with AdamW being a popular choice for training deep learning models because of strong empirical performance. We study \emph{DP-AdamW} and introduce \emph{DP-AdamW-BC}, a differentially private variant of the AdamW optimizer with DP bias correction for the second moment estimator. We start by showing theoretical results for privacy and convergence guarantees of DP-AdamW and DP-AdamW-BC. Then, we empirically analyze the behavior of both optimizers across multiple privacy budgets ($ε= 1, 3, 7$). We find that DP-AdamW outperforms existing state-of-the-art differentially private optimizers like DP-SGD, DP-Adam, and DP-AdamBC, scoring over 15\% higher on text classification, up to 5\% higher on image classification, and consistently 1\% higher on graph node classification. Moreover, we empirically show that incorporating bias correction in DP-AdamW (DP-AdamW-BC) consistently decreases accuracy, in contrast to the improvement of DP-AdamBC improvement over DP-Adam.

We introduce Cautious Weight Decay (CWD), a one-line, optimizer-agnostic modification that applies weight decay only to parameter coordinates whose signs align with the optimizer update. Unlike standard decoupled decay, which implicitly optimizes a regularized or constrained objective, CWD preserves the original loss and admits a bilevel interpretation: it induces sliding-mode behavior upon reaching the stationary manifold, allowing it to search for locally Pareto-optimal stationary points of the unmodified objective. In practice, CWD is a drop-in change for optimizers such as AdamW, Lion, and Muon, requiring no new hyperparameters or additional tuning. For language model pre-training and ImageNet classification, CWD consistently improves final loss and accuracy at million- to billion-parameter scales.

The pursuit of faster optimization algorithms remains an active and important research direction in deep learning. Recently, the Muon optimizer [JJB+24] has demonstrated promising empirical performance, but its theoretical foundation remains less understood. In this paper, we bridge this gap and provide a theoretical analysis of Muon by placing it within the Lion-$\mathcal{K}$ family of optimizers [CLLL24]. Specifically, we show that Muon corresponds to Lion-$\mathcal{K}$ when equipped with the nuclear norm, and we leverage the theoretical results of Lion-$\mathcal{K}$ to establish that Muon (with decoupled weight decay) implicitly solves an optimization problem that enforces a constraint on the spectral norm of weight matrices. This perspective not only demystifies the implicit regularization effects of Muon but also leads to natural generalizations through varying the choice of convex map $\mathcal{K}$, allowing for the exploration of a broader class of implicitly regularized and constrained optimization algorithms.

With the success of deep neural networks (NNs) in a variety of domains, the computational and storage requirements for training and deploying large NNs have become a bottleneck for further improvements. Sparsification has consequently emerged as a leading approach to tackle these issues. In this work, we consider a simple yet effective approach to sparsification, based on the Bridge, or $L_p$ regularization during training. We introduce a novel weight decay scheme, which generalizes the standard $L_2$ weight decay to any $p$ norm. We show that this scheme is compatible with adaptive optimizers, and avoids the gradient divergence associated with $0

Weight decay is a popular regularization technique for training of deep neural networks. Modern deep learning libraries mainly use $L_{2}$ regularization as the default implementation of weight decay. \citet{loshchilov2018decoupled} demonstrated that $L_{2}$ regularization is not identical to weight decay for adaptive gradient methods, such as Adaptive Momentum Estimation (Adam), and proposed Adam with Decoupled Weight Decay (AdamW). However, we found that the popular implementations of weight decay, including $L_{2}$ regularization and decoupled weight decay, in modern deep learning libraries usually damage performance. First, the $L_{2}$ regularization is unstable weight decay for all optimizers that use Momentum, such as stochastic gradient descent (SGD). Second, decoupled weight decay is highly unstable for all adaptive gradient methods. We further propose the Stable Weight Decay (SWD) method to fix the unstable weight decay problem from a dynamical perspective. The proposed SWD method makes significant improvements over $L_{2}$ regularization and decoupled weight decay in our experiments. Simply fixing weight decay in Adam by SWD, with no extra hyperparameter, can usually outperform complex Adam variants, which have more hyperparameters.

In the next release of adabelief-pytorch, we will modify the default of several arguments, in order to fit the needs of for general tasks such as GAN and Transformer. Please check if you specify these arguments or use the default when upgrade from version 0.0.5 to higher. AdaBelief uses a different denominator from Adam, and is orthogonal to other techniques such as recification, decoupled weight decay, weight averaging et.al. This implies when you use some techniques with Adam, to get a good result with AdaBelief you might still need those techniques. The default value epsilon 1e-8 is not a good option in many cases, will modify it later to 1e-12 or 1e-16 later. If you task needs a "non-adaptive" optimizer, which means SGD performs much better than Adam(W), such as on image recognition, you need to set a large epsilon(e.g.