This paper proposes a novel graph-based regularized regression estimator - the hierarchical feature regression (HFR) -, which mobilizes insights from the domains of machine learning and graph theory to estimate robust parameters for a linear regression. The estimator constructs a supervised feature graph that decomposes parameters along its edges, adjusting first for common variation and successively incorporating idiosyncratic patterns into the fitting process. The graph structure has the effect of shrinking parameters towards group targets, where the extent of shrinkage is governed by a hyperparamter, and group compositions as well as shrinkage targets are determined endogenously. The method offers rich resources for the visual exploration of the latent effect structure in the data, and demonstrates good predictive accuracy and versatility when compared to a panel of commonly used regularization techniques across a range of empirical and simulated regression tasks.