Apply one gradient-descent update to the Python lists values and gradients:

Here $i$ indexes coordinates, $\theta_i$ is the original value, $g_i$ is its gradient, $\eta$ is learning_rate, and $\theta'_i$ is the updated value. The predicted first-order objective change is

Here $\Delta L_{\mathrm{pred}}$ is a local prediction, not a measured change in loss. Empty input lists produce an empty updated list and 0.0. Return a two-item tuple: a fresh list of updated Python floats, followed by the predicted change as a Python float.