Adaptive optimizers are commonly parameterized by a fixed power of the second-moment estimate.
Existing partially adaptive methods study exponents between momentum-like updates and the standard Adam square root, while the interaction between this exponent and the global learning rate is less understood.
We perform a controlled cross-environment study using a paired four-environment classification problem with stable sparse features, environment-dependent spurious sparse features, dense features, and high-dimensional noise.
Across \NumRuns source-training runs covering 21 preconditioning exponents $p\in[-0.5,0.5]$ and five learning rates $η\in[10^{-4},10^{-2}]$
We find that the exponent maximizing cross-environment accuracy decreases almost linearly with $\log_{10}η$.
The fitted slopes range from $-0.270$ to $-0.300$, with $R^2$ between $0.972$ and $0.996$.
At $η=10^{-2}$
source-validation selection still prefers positive exponents in all four environments, whereas cross-environment and worst-environment criteria prefer negative exponents.
Checkpoint decomposition shows that lower $p$ reduces the learned spurious-to-stable and noise-to-stable weight ratios; under reversed correlation, it also reduces the magnitude of the harmful spurious margin.
Negative $p$ is therefore not a universally optimal setting.
It is a high-step-size allocation regime produced by the joint action of learning rate and preconditioning.
The study also exposes a model-selection conflict:
source-domain validation systematically selects a different preconditioning regime from the one that maximizes robustness to environmental change.
The results are a single-seed, finite-budget mechanism study rather than a broad benchmark claim.