Unified-regret Decomposition for Coupled Constraint-Preference Learning, Counterfactual-regret Minimization, and GAN Training

Abstract: Multi-objective Optimization in multi-agent Settings requires jointly-solving three intertwined Subproblems: learning latent-human Preferences, computing strategic Equilibria, and generating diverse Pareto-optimal Solutions. While mature Tools exist for each Subproblem in isolation (contrastive-preference Learning, counterfactual-regret Minimization, generative-adversarial Networks), no prior Work provides formal Guarantees for their joint Operation, proves whether all three are necessary, or gives principled Guidance on resource Allocation across Components. We propose CPCFR-GAN, a Framework that unifies these three Components under a common regret-minimization Lens. Our key Insight is that CPL, CFR, and GAN all operate as regret Minimizers in different Domains (preference, strategy, generation), enabling a unified Analysis of their Interaction. We establish four main Results: (1) a unified-regret Decomposition (Theorem 9) showing that total-system Regret decomposes additively into three independently-controllable Terms, with the CPL approximation Floor as the asymptotic Bottleneck; (2) impossibility Results (Theorems 17 to 19) proving that removing any single Component causes Omega (1) Degradation, establishing the three-component Architecture as structurally necessary; (3) a resource-allocation Theorem (Theorem 14) deriving the optimal budget Split xᵢ* proportional to (aᵢ/cᵢ) ^ (2/3) in closed Form; (4) a matching lower Bound (Theorem 15) proving Omega (1/sqrt (T) ), establishing that the convergence Rate is minimax-optimal. To our Knowledge, these are the first formal design Principles for multi-component multi-objective optimization Systems.