Jacobian Descent for Multi-Objective Optimization

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.