Despite recent advances, developing general-purpose universal denoising and artifact-removal networks remains largely an open problem: Given fixed network weights, one inherently trades-off specialization at one task (e.g.,~removing Poisson noise) for performance at another (e.g.,~removing speckle noise). In addition, training such a network is challenging due to the curse of dimensionality: As one increases the dimensions of the specification-space (i.e.,~the number of parameters needed to describe the noise distribution) the number of unique specifications one needs to train for grows exponentially. Uniformly sampling this space will result in a network that does well at very challenging problem specifications but poorly at easy problem specifications, where even large errors will have a small effect on the overall mean squared error. In this work we propose training denoising networks using an adaptive-sampling/active-learning strategy. Our work improves upon a recently proposed universal denoiser training strategy by extending these results to higher dimensions and by incorporating a polynomial approximation of the true specification-loss landscape. This approximation allows us to reduce training times by almost two orders of magnitude. We test our method on simulated joint Poisson-Gaussian-Speckle noise and demonstrate that with our proposed training strategy, a single blind, generalist denoiser network can achieve peak signal-to-noise ratios within a uniform bound of specialized denoiser networks across a large range of operating conditions. We also capture a small dataset of images with varying amounts of joint Poisson-Gaussian-Speckle noise and demonstrate that a universal denoiser trained using our adaptive-sampling strategy outperforms uniformly trained baselines.