In this empirical article, we introduce INNAprop, an optimization algorithm that combines the INNA method with the RMSprop adaptive gradient scaling. It leverages second-order information and rescaling while keeping the memory requirements of standard DL methods as AdamW or SGD with momentum. After giving geometrical insights, we evaluate INNAprop on CIFAR-10, Food101, and ImageNet with ResNets, VGG, DenseNet, and ViT, and on GPT-2 (OpenWebText) train from scratch and with LoRA fine-tuning (E2E). INNAprop consistently matches or outperforms AdamW both in training speed and accuracy, with minimal hyperparameter tuning in large-scale settings. Our code is publicly available at \url{https://github.com/innaprop/innaprop}.