Regularized loss minimization, where a statistical model is obtained from minimizing the sum of a loss function and weighted regularization terms, is still in widespread use in machine learning. The statistical performance of the resulting models depends on the choice of weights (regularization parameters) that are typically tuned by cross-validation. For finding the best regularization parameters, the regularized minimization problem needs to be solved for the whole parameter domain. A practically more feasible approach is covering the parameter domain with approximate solutions of the loss minimization problem for some prescribed approximation accuracy. The problem of computing such a covering is known as the approximate solution gamut problem. Existing algorithms for the solution gamut problem suffer from several problems. For instance, they require a grid on the parameter domain whose spacing is difficult to determine in practice, and they are not generic in the sense that they rely on problem specific plug-in functions. Here, we show that a well-known algorithm from vector optimization, namely the Benson algorithm, can be used directly for computing approximate solution gamuts while avoiding the problems of existing algorithms. Experiments for the Elastic Net on real world data sets demonstrate the effectiveness of Benson’s algorithm for regularization parameter tracking.