While deep learning has achieved great success in various fields, a large amount of memory is necessary to train deep neural networks, which hinders the development of massive state-of-the-art models. The reason is the conventional learning rule, backpropagation, should temporarily store input activations of all the layers in the network. To overcome this, recent studies suggested various memory-efficient implementations of backpropagation. However, those approaches incur computational overhead due to the recomputation of activations, slowing down neural network training. In this work, we propose a new learning rule which significantly reduces memory requirements while closely matching the performance of backpropagation. The algorithm combines auxiliary activation with output activation during forward propagation, while only auxiliary activation is used during backward propagation instead of actual input activation to reduce the amount of data to be temporarily stored. We mathematically show that our learning rule can reliably train the networks whose loss landscape is convex if the auxiliary activation satisfies certain conditions. Based on this observation, we suggest candidates of auxiliary activation that satisfy those conditions. Experimental results confirm that the proposed learning rule achieves competitive performance compared to backpropagation in various models such as ResNet, Transformer, BERT, ViT, and MLP-Mixer.