Learning No-Regret Sparse Generalized Linear Models with Varying Observation(s)

Generalized Linear Models (GLMs) encompass a wide array of regression and classification models, where prediction is a function of a linear combination of the input variables. Often in real-world scenarios, a number of observations would be added into or removed from the existing training dataset, necessitating the development of learning systems that can efficiently train optimal models with varying observations in an online (sequential) manner instead of retraining from scratch. Despite the significance of data-varying scenarios, most existing approaches to sparse GLMs concentrate on offline batch updates, leaving online solutions largely underexplored. In this work, we present the first algorithm without compromising accuracy for GLMs regularized by sparsity-enforcing penalties trained on varying observations. Our methodology is capable of handling the addition and deletion of observations simultaneously, while adaptively updating data-dependent regularization parameters to ensure the best statistical performance. Specifically, we recast sparse GLMs as a bilevel optimization objective upon varying observations and characterize it as an explicit gradient flow in the underlying space for the inner and outer subproblems we are optimizing over, respectively. We further derive a set of rules to ensure a proper transition at regions of non-smoothness, and establish the guarantees of theoretical consistency and finite convergence. Encouraging results are exhibited on real-world benchmarks.