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Jacobian Descent for Multi-Objective Optimization

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arxiv 2406.16232 v3 pith:TIAI4T4M submitted 2024-06-23 cs.LG cs.AImath.OC

Jacobian Descent for Multi-Objective Optimization

classification cs.LG cs.AImath.OC
keywords jacobiandescentobjectiveoptimizationconflictdirectgradientgradients
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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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.

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