Production machine-learning models are derived artifacts of time-bounded training snapshots: a deployed model is a materialized view over a training cut that ages the instant it is built. A common response is to replace the fixed retraining cadence with an adaptive trigger -- a weighted staleness score that retrains when accumulated source risk crosses a threshold. We show this is the wrong lever, and identify the right one. First, an equivalence limit: any refresh trigger that is a static, strictly monotone function of a single shared global training-data age is operationally equivalent to a calibrated uniform age timer, so a global staleness budget, however elaborately it weights segments, sources, and sensitivities, carries no scheduling information a clock does not. The limit also shows how to escape it: refresh segments differentially, giving each its own age and refresh interval, which is meaningful when refresh cost is separable across segments (incremental training or per-segment models). We solve the resulting budget-allocation problem. In the frequent-refresh regime each segment's optimal refresh rate is proportional to the square root of its risk wjλj (weight times change rate), and the optimal policy never costs more than the uniform timer, beating it by a closed-form Cauchy-Schwarz "price of uniformity" that is zero for homogeneous workloads and grows with heterogeneity. In a discrete-event simulation with real Poisson change events, the optimal policy lowers realized weighted stale exposure by 8-29% relative to the uniform timer at matched refresh budget, winning on 86-100% of seeds; a naive exposure-threshold policy does not, showing the allocation is what helps; and the advantage survives 50% rate-estimation noise. The leverage in model refresh is not a better score but a better action.
Figures & tables
Figure 1: The equivalence limit (Theorem 1 ). NWSE(a) for a heterogeneous six-segment set is strictly monotone in the shared age, so each budget β maps through F−1 to a unique age τβ : thresholding the score is thresholding age in different units.
Figure 2: The optimal allocation (Theorem 2 ). In the frequent-refresh regime the exact numerical optimum obeys the closed-form square-root law xj⋆∝wjλj (log–log slope 1/2 ); shown for a 40-segment moderate-heterogeneity profile.
Heterogeneity
Sparse
Moderate
Frequent
Low
0.2
0.2
0.3
Moderate
7.5
7.1
8.4
High
11.4
12.9
14.0
Extreme
21.0
24.1
29.5
Table 1: Analytical cost saving of optimal differential refresh over the uniform timer (mean over 50 profiles, %). Uniform is optimal at low heterogeneity; the gap grows with dispersion of segment risk.
Heterogeneity
K
Opt. win (%)
Naive (%)
Opt. seeds (%)
Moderate
10
0 8.6
− 0.2
0 98
Moderate
20
0 7.7
−4.3
0 86
Moderate
40
0 8.6
−8.1
0 94
High
10
11.0
− 4.1
100
High
20
12.7
− 3.0
100
High
40
16.0
− 2.5
100
Table 2: Realized weighted stale exposure saving over the uniform timer in the discrete-event simulation, at matched refresh count (50 seeds). K is the uniform timer’s number of full refreshes. “Opt. win” is the optimal differential policy; “Naive” is the exposure-threshold policy; “seeds” is the fraction on which the optimal policy wins.
Figure 3: Realized saving over the uniform timer at matched refresh count (mean over the three budgets of Table 2 , error bars ± s.d.). The optimal differential policy wins and grows with heterogeneity; the naive exposure-threshold policy is near break-even or worse.
Heterogeneity
noise 0.00
0.10
0.25
0.50
Moderate
7.0
7.0
6.6
5.5
High
13.0
12.9
12.6
11.3
Extreme
24.3
24.3
24.0
23.2
Table 3: Analytical win (%) under multiplicative λ -estimation noise at the moderate budget (50 profiles, mean).