Capacity and Redundancy Trade-offs in Multi-Task Learning

In multi-task learning (MTL) negative transfer is often considered as an optimization artifact, but it can also be viewed as a consequence of limited shared capacity and weak task redundancy. We investigate this effect through a Capacity--Redundancy (CR) identity that decomposes the sum of per-task predictive informations into joint predictive information that includes label redundancy defined via total correlation (TC), and a residual coupling term that quantifies interference left unresolved by the shared representation. Additionally, we show two key results: (i) a clustering-gap decomposition that gives a necessary and sufficient condition for clustered sharing to outperform global sharing, and (ii) a gradient--TC bridge in a Gaussian multi-task model that formally justifies gradient cosine similarity as a proxy for redundancy ordering. Empirically, we estimate the residual coupling $\Delta$ from validation residual correlations, showing that clustered LoRA substantially reduces $\widehat{\Delta}$, outperforms size-matched random partitions, and results in statistically significant gains with multi-seed confidence intervals.