AI Insight
This paper introduces Jacobian Descent (JD), a new optimization algorithm for problems involving multiple conflicting objectives. Unlike existing gradient-based methods that struggle with conflicting objectives, JD uses the Jacobian matrix of vector-valued objective functions and projects gradients to fully resolve conflicts, with proven convergence to the Pareto front in smooth convex cases. The authors also propose instance-wise risk minimization (IWRM), where each training example's loss is treated as a separate objective, showing preliminary positive results in image classification tasks.
Why it matters
This approach could improve machine learning systems that must balance competing goals, such as accuracy versus fairness, or performance across different data subgroups. The IWRM paradigm offers a new framework for training models that could better handle diverse or imbalanced datasets by treating individual examples as distinct optimization targets.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: Many optimization problems require balancing multiple conflicting objectives. To generalize gradient descent, which is limited to single-objective optimization, we introduce Jacobian descent (JD) and its stochastic variants. This algorithm iteratively updates parameters using the Jacobian matrix of a vector-valued objective function, in which each row is the gradient of an individual objective. While several methods to combine gradients already exist in the literature, they generally struggle when the objectives conflict. In contrast, we propose projecting gradients to fully resolve conflict. We prove convergence to the Pareto front in the smooth convex case with this approach, a stronger guarantee than the usual convergence to a weakly Pareto stationary point. Our method also enables instance-wise risk minimization (IWRM), a novel learning paradigm in which the loss of each training example is considered a separate objective. Early experimentation on image classification shows promising results for IWRM. Lastly, we provide an efficient implementation of JD using the Gramian of the Jacobian matrix to drastically reduce memory requirements.