Many optimization problems require balancing multiple conflicting objectives. As gradient descent is limited to single-objective optimization, we introduce its direct generalization: Jacobian descent (JD). 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 are generally hindered when the objectives conflict. In contrast, we propose projecting gradients to fully resolve conflict while ensuring that they preserve an influence proportional to their norm. We prove significantly stronger convergence guarantees with this approach, supported by our empirical results. 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. Applied to simple image classification tasks, IWRM exhibits promising results compared to the direct minimization of the average loss. Additionally, we outline an efficient implementation of JD using the Gramian of the Jacobian matrix to reduce time and memory requirements.

翻译：许多优化问题需要平衡多个相互冲突的目标。由于梯度下降仅限于单目标优化，我们引入其直接推广：雅可比下降法。该算法利用向量值目标函数的雅可比矩阵迭代更新参数，其中每一行对应单个目标的梯度。虽然现有文献已提出多种梯度组合方法，但这些方法在目标冲突时通常存在局限。相比之下，我们提出通过梯度投影来完全解决冲突，同时确保各梯度保持与其范数成比例的影响力。我们证明了该方法具有显著更强的收敛保证，并通过实证结果予以验证。本方法还实现了实例级风险最小化，这是一种新颖的学习范式，其中每个训练样本的损失被视为独立目标。在简单图像分类任务中的应用表明，与直接最小化平均损失相比，实例级风险最小化展现出有前景的结果。此外，我们通过雅可比矩阵的格拉姆矩阵概述了雅可比下降法的高效实现方案，以降低时间和内存需求。