Fixing Weight Decay Regularization in Adam

TLDR

Common implementations of adaptive gradient algorithms such as Adam limit the benefit of weight decay because weights decay additively rather than multiplicatively. We propose a simple way to resolve this issue by decoupling weight decay from the optimization steps taken with respect to the loss function. The authors decouple weight decay from the gradient updates, applying it independently of the loss‑based optimization. Empirical results show that this decoupling allows Adam to match or surpass SGD with momentum on image classification, reduces dependence on weight decay magnitude over longer runs, and yields a warm‑restart variant (AdamWR) that achieves state‑of‑the‑art performance on CIFAR‑10 and ImageNet32×32. Source code will be released after review.

Abstract

We note that common implementations of adaptive gradient algorithms, such as Adam, limit the potential benefit of weight decay regularization, because the weights do not decay multiplicatively (as would be expected for standard weight decay) but by an additive constant factor. We propose a simple way to resolve this issue by decoupling weight decay and the optimization steps taken w.r.t. The loss function. We provide empirical evidence that our proposed modification (i) decouples the optimal choice of weight decay factor from the setting of the learning rate for both standard SGD and Adam, and (ii) substantially improves Adam's generalization performance, allowing it to compete with SGD with momentum on image classification datasets (on which it was previously typically outperformed by the latter). We also demonstrate that longer optimization runs require smaller weight decay values for optimal results and introduce a normalized variant of weight decay to reduce this dependence. Finally, we propose a version of Adam with warm restarts (AdamWR) that has strong anytime performance while achieving state-of-the-art results on CIFAR-10 and ImageNet32x32. Our source code will become available after the review process.