Tree-Projected Gradient Descent for Estimating Gradient-Sparse Parameters on Graphs

Abstract: We study estimation of a gradient-sparse parameter vector $$\boldsymbol{\theta}^* \in \mathbb{R}<sup>p$$, having strong gradient-sparsity $$s^*:=|\nabla_G \boldsymbol{\theta}<sup>*|_0$$ on an underlying graph $G$. Given observations $Z_1,\ldots,Z_n$ and a smooth, convex loss function $\mathcal{L}$ for which $\boldsymbol{\theta}<sup>*$ minimizes the population risk $$\mathbb{E}[\mathcal{L}(\boldsymbol{\theta};Z_1,\ldots,Z_n)]$$, we propose to estimate $\boldsymbol{\theta}<sup>*$ by a projected gradient descent algorithm that iteratively and approximately projects gradient steps onto spaces of vectors having small gradient-sparsity over low-degree spanning trees of $G$. We show that, under suitable restricted strong convexity and smoothness assumptions for the loss, the resulting estimator achieves the squared-error risk $\frac{s^*}{n} \log (1+\frac{p}{s<sup>*})$ up to a multiplicative constant that is independent of $G$. In contrast, previous polynomial-time algorithms have only been shown to achieve this guarantee in more specialized settings, or under additional assumptions for $G$ and/or the sparsity pattern of $\nabla_G \boldsymbol{\theta}<sup>*$. As applications of our general framework, we apply our results to the examples of linear models and generalized linear models with random design.