Two-temperature logistic regression based on the Tsallis divergence

Abstract

We develop a variant of multiclass logistic regression that is significantly more robust to noise. The algorithm has one weight vector per class and the surrogate loss is a function of the linear activations (one per class). The surrogate loss of an example with linear activation vector $\mathbf{a}$ and class $c$ has the form $-\log_{t_1} \exp_{t_2} (a_c - G_{t_2}(\mathbf{a}))$ where the two temperatures $t_1$ and $t_2$ ''temper'' the $\log$ and $\exp$, respectively, and $G_{t_2}(\mathbf{a})$ is a scalar value that generalizes the log-partition function. We motivate this loss using the Tsallis divergence. Our method allows transitioning between non-convex and convex losses by the choice of the temperature parameters. As the temperature $t_1$ of the logarithm becomes smaller than the temperature $t_2$ of the exponential, the surrogate loss becomes ''quasi convex''. Various tunings of the temperatures recover previous methods and tuning the degree of non-convexity is crucial in the experiments. In particular, quasi-convexity and boundedness of the loss provide significant robustness to the outliers. We explain this by showing that $t_1 < 1$ caps the surrogate loss and $t_2 >1$ makes the predictive distribution have a heavy tail. We show that the surrogate loss is Bayes-consistent, even in the non-convex case. Additionally, we provide efficient iterative algorithms for calculating the log-partition value only in a few number of iterations. Our compelling experimental results on large real-world datasets show the advantage of using the two-temperature variant in the noisy as well as the noise free case.