Gradient Aligned Regression via Pairwise Losses

Regression is a fundamental task in machine learning that has garnered extensive attention over the past decades. The conventional approach for regression involves employing loss functions that primarily concentrate on aligning model prediction with the ground truth for each individual data sample. Recent research endeavors have introduced novel perspectives by incorporating label similarity to regression via imposing extra pairwise regularization on the latent feature space and demonstrated the effectiveness. However, there are two drawbacks for those approaches: i) their pairwise operation in latent feature space is computationally more expensive than conventional regression losses; ii) it lacks of theoretical justifications behind such regularization. In this work, we propose GAR (Gradient Aligned Regression) as a competitive alternative method in label space, which is constituted by a conventional regression loss and two pairwise label difference losses for gradient alignment including magnitude and direction. GAR enjoys: i) the same level efficiency as conventional regression loss because the quadratic complexity for the proposed pairwise losses can be reduced to linear complexity; ii) theoretical insights from learning the pairwise label difference to learning the gradient of the ground truth function. We limit our current scope as regression on the clean data setting without noises, outliers or distributional shifts, etc. We demonstrate the effectiveness of the proposed method practically on two synthetic datasets and on eight extensive real-world tasks from six benchmark datasets with other eight competitive baselines. Running time experiments demonstrate the superior efficiency of the proposed GAR over existing methods with pairwise regularization in latent feature space and ablation studies demonstrate the effectiveness of each component for GAR.