Healthcare is dynamic as demographics, diseases, and therapeutics constantly evolve. This dynamic nature induces inevitable distribution shifts in populations targeted by clinical AI models, often rendering them ineffective. Incremental learning provides an effective method of adapting deployed clinical models to accommodate these contemporary distribution shifts. However, since incremental learning involves modifying a deployed or in-use model, it can be considered unreliable as any adverse modification due to maliciously compromised or incorrectly labelled data can make the model unsuitable for the targeted application. This paper introduces self-aware stochastic gradient descent (SGD), an incremental deep learning algorithm that utilises a contextual bandit-like sanity check to only allow reliable modifications to a model. The contextual bandit analyses incremental gradient updates to isolate and filter unreliable gradients. This behaviour allows self-aware SGD to balance incremental training and integrity of a deployed model. Experimental evaluations on the Oxford University Hospital datasets highlight that self-aware SGD can provide reliable incremental updates for overcoming distribution shifts in challenging conditions induced by label noise