As the population ages, the number of older adults experiencing mild cognitive impairment (MCI), Alzheimer's disease, and other forms of dementia will increase dramatically over the next few decades. Unfortunately, cognitive changes associated with these conditions threaten independence and quality of life. To address this, researchers have developed promising cognitive training interventions to help prevent or reverse cognitive decline and cognitive impairment. However, the promise of these interventions will not be realized unless older adults regularly engage with them over the long term, and like many health behaviors, adherence to cognitive training interventions can often be poor. To maximize training benefits, it would be useful to be able to predict when adherence lapses for each individual, so that support systems can be personalized to bolster adherence and intervention engagement at optimal time points. The current research uses data from a technology-based cognitive intervention study to recognize patterns in participants' adherence levels and predict their future adherence to the training program. We leveraged the feature learning capabilities of deep neural networks to predict patterns of adherence for a given participant, based on their past behavior. A separate, personalized model was trained for each participant to capture individualistic features of adherence. We posed the adherence prediction as a binary classification problem and exploited multivariate time series analysis using an adaptive window size for model training. Further, data augmentation techniques were used to overcome the challenge of limited training data and enhance the size of the dataset. To the best of our knowledge, this is the first research effort to use advanced machine learning techniques to predict older adults' daily adherence to cognitive training programs. Experimental evaluations corroborated the promise and potential of deep learning models for adherence prediction, which furnished highest mean F-scores of 75.5, 75.5, and 74.6% for the Convolution Neural Network (CNN), Long Short-Term Memory (LSTM) network, and CNN-LSTM models respectively.