Robust Regression via Model Based Methods
This work addresses robust regression for applications like autoencoders and multi-target regression, offering an incremental improvement by adapting existing model-based optimization methods to handle non-differentiable l_p norms more efficiently.
The paper tackles the problem of robust regression by addressing the non-differentiability of l_p norms, which are robust to outliers but challenging to optimize with methods like stochastic gradient descent. It proposes SADM, a stochastic variant of OADM, achieving a convergence rate of O(log T/T) and demonstrating improved efficiency over gradient methods in experiments on autoencoders and multi-target regression.
The mean squared error loss is widely used in many applications, including auto-encoders, multi-target regression, and matrix factorization, to name a few. Despite computational advantages due to its differentiability, it is not robust to outliers. In contrast, l_p norms are known to be robust, but cannot be optimized via, e.g., stochastic gradient descent, as they are non-differentiable. We propose an algorithm inspired by so-called model-based optimization (MBO) [35, 36], which replaces a non-convex objective with a convex model function and alternates between optimizing the model function and updating the solution. We apply this to robust regression, proposing SADM, a stochastic variant of the Online Alternating Direction Method of Multipliers (OADM) [50] to solve the inner optimization in MBO. We show that SADM converges with the rate O(log T/T). Finally, we demonstrate experimentally (a) the robustness of l_p norms to outliers and (b) the efficiency of our proposed model-based algorithms in comparison with gradient methods on autoencoders and multi-target regression.