Jacobian Descent for Multi-Objective Optimization

Abstract: 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.

• The paper introduces Jacobian Descent (JD), an extension of gradient descent that handles multi-objective optimization by directly using the Jacobian matrix of gradients for each objective.
• JD unifies existing multi-gradient methods and proposes novel aggregators, like $\upgrad$, designed to manage inter-objective conflicts and scale updates effectively.
• The authors demonstrate JD's practical use in Instance-Wise Risk Minimization (IWRM), showing improved performance in experiments by addressing instance-level conflicts and theoretically reaching the Pareto front in convex settings.

Jacobian Descent for Multi-Objective Optimization: An Expert Overview

This paper introduces Jacobian Descent (JD), an extension of the conventional gradient descent algorithm, aimed explicitly at addressing multi-objective optimization problems by leveraging vector-valued objective functions. Classical optimization methods typically minimize scalar loss functions, which can undermine the performance on individual objectives when their relative importances are unknown. The authors propose JD as an approach that foregoes scalarization altogether, instead utilizing a Jacobian matrix made up of gradients for each objective, dynamically aggregated into an actionable update vector.

One key achievement of the paper is the formal introduction of JD, which encapsulates existing multi-gradient descent mechanisms under a unified framework. This formalization includes novel contributions to the theory of aggregators, specifically the introduction of the "Unconflicting Projection of Gradients" aggregator, or $\upgrad$, designed to mitigate inter-objective conflicts and scale updates proportionally to gradient magnitudes. This proposal sets new theoretical benchmarks for aggregator properties—namely, non-conflicting behavior, linear scaling properties, and weighting attributes.

JD's practical implications are significant across several domains within machine learning. In particular, the authors present instance-wise risk minimization (IWRM) as a novel paradigm that uses JD to consider each training instance as an individual objective, contrasted with the empirical risk minimization (ERM) which focuses on the average loss. Experiments on simple image classification tasks validate their approach, demonstrating improved outcomes when JD's aggregators address inter-instance conflicts.

The paper's results indicate that JD can conservatively reach the Pareto front in smooth, convex optimization landscapes, theoretically underpinning its applicability in a wide range of machine learning tasks that naturally feature multiple, potentially conflicting objectives. By addressing speed limitations inherent to JD with prospects for implementing efficient computations, the authors pave the way for JD's integration into mainstream machine learning model training, particularly for settings where multi-objective conflicts are substantial.

Future research directions are suggested, including deriving fast algorithms for computing the Gramian of the Jacobian to unlock more efficient JD implementations. Emphasizing optimizing vector objectives indicates a paradigm shift away from the conventional reductions to scalar problems, thereby suggesting JD's potential to enhance convergence speed and model performance in complex task environments.

In conclusion, the paper contributes significantly to the mathematical and computational toolkit available for multi-objective optimization, with JD and its associated theoretical insights offering promising prospects for expanding the frontier of machine learning capabilities. The implications of JD reach beyond simple optimization efficiencies, touching on foundational aspects of how we consider and solve multi-faceted learning problems in artificial intelligence.