Preference-Ratio Adjacency for Distributed Multi-Objective Training

Integration point: use multi-task federated learning, personalized learning, or distributed fine-tuning where every client shares public loss definitions but has a private task-versus-regularization preference. Represent the preference by r_i=a_i/b_i and communicate only q_i=r_i g_task,i+g_effort,i, not the separate weights. Pseudocode: compute g_task and g_effort; read private r_i; form q_i=r_i*g_task+g_effort; clip q_i to norm C; set sigma=C*sqrt(2*log(1.25/delta))/epsilon; transmit q_i+sigma*Normal(0,I); aggregate client messages. For a tighter adaptive implementation, estimate ||g_task|| and set sigma proportional to Gamma||g_task||, subject to a public clipping cap. The paper-derived component is the ratio adjacency relation; the Gaussian release and gradient construction are the neural-network adaptation. Estimate sensitivity by paired training on neighboring client datasets and compare it with full-vector sensitivity. First experiment: train a small CNN on CIFAR-10 with cross-entropy as L_task and L2 parameter cost as L_effort across 20 heterogeneous clients. Compare full-weight DP, ratio-only DP, and non-private baselines at matched epsilon and delta. Sweep Gamma in {0.01,0.05,0.1}. Prediction: ratio sensitivity should scale linearly with Gamma, and required variance should scale as Gamma squared; a successful implementation should obtain at least a 2x variance reduction at equal measured privacy loss.