A representation-learning game for classes of prediction tasks

We propose a game-based formulation for learning dimensionality-reducing representations of feature vectors, when only a prior knowledge on future prediction tasks is available. In this game, the first player chooses a representation, and then the second player adversarially chooses a prediction task from a given class, representing the prior knowledge. The first player aims to minimize, and the second player to maximize, the regret: The minimal prediction loss using the representation, compared to the same loss using the original features. We consider the canonical setting in which the representation, the response to predict and the predictors are all linear functions, and the loss function is the mean squared error. We derive the theoretically optimal representation in pure strategies, which shows the effectiveness of the prior knowledge, and the optimal regret in mixed strategies, which shows the usefulness of randomizing the representation. For general representation, prediction and loss functions, we propose an efficient algorithm to optimize a randomized representation. The algorithm only requires the gradients of the loss function, and is based on incrementally adding a representation rule to a mixture of such rules.