Temporal Resolution and Effect Flexibility: The $K\times M$ Geometry of Survival Learning

Discrete-time survival models can be highly sensitive to interval count $K$, usually selected empirically without revealing the underlying mechanism. We trace this sensitivity to a larger model-optimisation problem over temporal resolution $K$ and effect flexibility $M$, which governs how freely covariate effects vary across intervals. We make $M$ explicit through a unified temporal basis model family, recovering the effect structures of two existing boundaries: time-invariant models such as proportional hazards at $M=1$, and interval-specific models such as Nnet-survival and DeepHit at $M=K$. Under baseline and effect regularities $r$ and $s$, our theory explains $K$-sensitivity through separate approximation–estimation balances $K^{-2r}+K/n$ for temporal resolution and $M^{-2s}+Md/n$ for effect flexibility, yielding distinct oracle scales $K^{\star }\asymp n^{1/(2r+1)}$ and $M^{\star }\asymp (n/d{)}^{1/(2s+1)}$, where $n$ is the sample size and $d$ the representation dimension. Since these scales generally differ, tuning existing $M=1$ or $M=K$ models explores only fixed paths and cannot guarantee the joint optimum. We prove that held-out joint validation over $(K,M)$ attains the oracle rate without knowing $r$ or $s$, making optimisation over the full geometry practical. Controlled studies confirm the predicted geometry. Across six real-world datasets and four deep-survival heads, joint selection lowers mean test loss on held-out outer folds relative to $M=K$ in 20 of 24 dataset–head strata and favours $M<K$ in 353 of 360 evaluations. Thus, $K$-sensitivity fundamentally reflects a larger joint approximation–estimation geometry of temporal resolution and effect flexibility: existing models trace only fixed paths through it, while reaching the joint optimum requires selecting $(K,M)$, not $K$ alone.