A Tunable Loss Function for Robust Classification: Calibration, Landscape, and Generalization

Abstract

We introduce a tunable loss function called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -loss, parameterized by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (0,\infty]$ </tex-math></inline-formula> , which interpolates between the exponential loss ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha = 1/2$ </tex-math></inline-formula> ), the log-loss ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha = 1$ </tex-math></inline-formula> ), and the 0–1 loss ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha = \infty $ </tex-math></inline-formula> ), for the machine learning setting of classification. Theoretically, we illustrate a fundamental connection between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -loss and Arimoto conditional entropy, verify the classification-calibration of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -loss in order to demonstrate asymptotic optimality via Rademacher complexity generalization techniques, and build-upon a notion called strictly local quasi-convexity in order to quantitatively characterize the optimization landscape of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -loss. Practically, we perform class imbalance, robustness, and classification experiments on benchmark image datasets using convolutional-neural-networks. Our main practical conclusion is that certain tasks may benefit from tuning <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -loss away from log-loss ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha = 1$ </tex-math></inline-formula> ), and to this end we provide simple heuristics for the practitioner. In particular, navigating the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> hyperparameter can readily provide superior model robustness to label flips ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha &gt; 1$ </tex-math></inline-formula> ) and sensitivity to imbalanced classes ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha &lt; 1$ </tex-math></inline-formula> ).