This chapter presents the new family of soft diamond synaptic regularizers based on thick-tailed symmetric alpha stable $SαS$ probability bell curves. These new parametrized weight priors improved deep-learning performance on image and language-translation test sets and increased the sparsity of the trained weights. They outperformed the state-of-the-art hard-diamond Laplacian regularizer of sparse lasso regression and classification. The $SαS$ synaptic weight priors have power-law bell-curve tails that are thicker than the thin exponential tails of Gaussian bell curves that underly ridge regularizers. Their tails get thicker as the $α$ parameter decreases. These thicker tails model more impulsive behavior and allow for occasional distant search in synaptic weight spaces of extremely high dimension. The geometry of their constraint sets has a diamond shape. The shape varies from a circle to a star or diamond that depends on the $α$ tail thickness and dispersion of the $SαS$ weight prior. These $SαS$ bell curves lack a closed form in general and this makes direct training computationally intensive. We removed this computational bottleneck by using a precomputed look-up table. We tested the soft diamond regularizers with deep neural classifiers on both image test sets and German-to-English language translation. The image simulations used the three datasets CIFAR-10, CIFAR-100, and Caltech-256. The regularizers improved the accuracy and sparsity of the classifiers. We also tested with deep neural machine-translation models on the IWSLT-2016 Evaluation dataset for German-to-English text translation. They also outperformed ridge regularizers and lasso regularizers. These findings recommend the sub-Cauchy $α= 0.5$ soft diamond regularizer as a competitive and sparse regularizer for large-scale machine learning.