The weighted mean trick – optimization strategies for robustness

We prove that minimizing a weighted mean results in optimizing the higher-order moments of the loss distribution such as the variance, skewness, and kurtosis. By optimizing the higher-order moments, one can tighten the upper bound of the loss mean deviating from the true expectation and improve the robustness against outliers. Such types of optimization problems often lead to non-convex objectives, therefore, we explore the extent to which the proposed weighted mean trick preserves convexity, albeit at times at a decrease in efficiency. Experimental results show that the weighted mean trick exhibits similar performance with other specialized robust loss functions when training on noisy datasets while providing a stronger theoretical background. The proposed weighted mean trick is a simple yet powerful optimization framework that is easy to integrate into existing works.