We investigate the estimation properties of the mixture of experts (MoE)\nmodel in a high-dimensional setting, where the number of predictors is much\nlarger than the sample size, and for which the literature is particularly\nlacking in theoretical results. We consider the class of softmax-gated Gaussian\nMoE (SGMoE) models, defined as MoE models with softmax gating functions and\nGaussian experts, and focus on the theoretical properties of their\n$l_1$-regularized estimation via the Lasso. To the best of our knowledge, we\nare the first to investigate the $l_1$-regularization properties of SGMoE\nmodels from a non-asymptotic perspective, under the mildest assumptions, namely\nthe boundedness of the parameter space. We provide a lower bound on the\nregularization parameter of the Lasso penalty that ensures non-asymptotic\ntheoretical control of the Kullback--Leibler loss of the Lasso estimator for\nSGMoE models. Finally, we carry out a simulation study to empirically validate\nour theoretical findings.\n