Mitigating the negative impact of noisy labels has been a perennial issue in supervised learning. Robust loss functions have emerged as a prevalent solution to this problem. In this work, we introduce the Variation Ratio as a novel property related to the robustness of loss functions, and propose a new family of robust loss functions, termed Variation-Bounded Loss (VBL), which is characterized by a bounded variation ratio. We provide theoretical analyses of the variation radio, proving that a smaller variation ratio would lead to better robustness. Furthermore, we reveal that the variation ratio provides a feasible method to relax the symmetric condition and offers a more concise path to achieve the asymmetric condition. Based on the variation ratio, we reformulate several commonly used loss functions into a variation-bounded form for pract ical applications. Positive experiments on various datasets exhibit the effectiveness and flexibility of our approach.

Learning with noisy labels is a crucial task for training accurate deep neural networks. To mitigate label noise, prior studies have proposed various robust loss functions, particularly symmetric losses. Nevertheless, symmetric losses usually suffer from the underfitting issue due to the overly strict constraint. To address this problem, the Active Passive Loss (APL) jointly optimizes an active and a passive loss to mutually enhance the overall fitting ability. Within APL, symmetric losses have been successfully extended, yielding advanced robust loss functions. Despite these advancements, emerging theoretical analyses indicate that asymmetric losses, a new class of robust loss functions, possess superior properties compared to symmetric losses. However, existing asymmetric losses are not compatible with advanced optimization frameworks such as APL, limiting their potential and applicability. Motivated by this theoretical gap and the prospect of asymmetric losses, we extend the asymmetric loss to the more complex passive loss scenario and propose the Asymetric Mean Square Error (AMSE), a novel asymmetric loss. We rigorously establish the necessary and sufficient condition under which AMSE satisfies the asymmetric condition. By substituting the traditional symmetric passive loss in APL with our proposed AMSE, we introduce a novel robust loss framework termed Joint Asymmetric Loss (JAL). Extensive experiments demonstrate the effectiveness of our method in mitigating label noise. Code available at: https://github.com/cswjl/joint-asymmetric-loss