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Nowadays, one practical limitation of deep neural network (DNN) is its high degree of specialization to a single task or domain (e.g., one visual domain). It motivates researchers to develop algorithms that can adapt DNN model to multiple domains sequentially, while still performing well on the past domains, which is known as multi-domain learning. Almost all conventional methods only focus on improving accuracy with minimal parameter update, while ignoring high computing and memory cost during training, which makes it difficult to deploy multi-domain learning into more and more widely used resource-limited edge devices, like mobile phones, IoT, embedded systems, etc. We observe that large memory used for activation storage is the bottleneck that largely limits the training time and cost on edge devices. To reduce training memory usage, while keeping the domain adaption accuracy performance, we propose Dynamic Additive Attention Adaption ($DA^3$), a novel memory-efficient on-device multi-domain learning method. $DA^3$ learns a novel additive attention adaptor module, while freezing the weights of the pre-trained backbone model for each domain. Differentiating from prior works, such module not only mitigates activation memory buffering for reducing memory usage during training but also serves as a dynamic gating mechanism to reduce the computation cost for fast inference. We validate $DA^3$ on multiple datasets against state-of-the-art methods, which shows great improvement in both accuracy and training time. Moreover, we deployed $DA^3$ into the popular NIVDIA Jetson Nano edge GPU, where the measured experimental results show our proposed $DA^3$ reduces the on-device training memory consumption by 19-37x, and training time by 2x, in comparison to the baseline methods (e.g., standard fine-tuning, Parallel and Series Res. adaptor, and Piggyback).