Classification problems in the small data regime (with small data statistic $T$ and relatively large feature space dimension $D$) impose challenges for the common machine learning (ML) and deep learning (DL) tools. The standard learning methods from these areas tend to show a lack of robustness when applied to data sets with significantly fewer data points than dimensions and quickly reach the overfitting bound, thus leading to poor performance beyond the training set. To tackle this issue, we propose eSPA$+$, a significant extension of the recently formulated entropy-optimal scalable probabilistic approximation algorithm (eSPA). Specifically, we propose to change the order of the optimization steps and replace the most computationally expensive subproblem of eSPA with its closed-form solution. We prove that with these two enhancements, eSPA$+$ moves from the polynomial to the linear class of complexity scaling algorithms. On several small data learning benchmarks, we show that the eSPA$+$ algorithm achieves a many-fold speed-up with respect to eSPA and even better performance results when compared to a wide array of ML and DL tools. In particular, we benchmark eSPA$+$ against the standard eSPA and the main classes of common learning algorithms in the small data regime: various forms of support vector machines, random forests, and long short-term memory algorithms. In all the considered applications, the common learning methods and eSPA are markedly outperformed by eSPA$+$, which achieves significantly higher prediction accuracy with an orders-of-magnitude lower computational cost.