🗂️ How this fits into κ-budgeting

$\kappa$-budgeting is still useful as a static safety rail $\|W_\ell\|\leq \kappa_\ell$, since it bounds:

• fixed-point dynamic range,
• layer gain,
• representational Lipschitzness,
• overflow risk,
• static hardware safety.

On the other hand, the throttle controls:

$$\theta_{t+1} = \theta_t-\alpha_t\eta G_t.$$

It bounds:

• learning speed,
• closed-loop adaptation stability,
• overshoot under drift,
• optimizer stiffness.

So:

$$\boxed{ \text{$\kappa$-budgeting controls the admissible region.} }$$ $$\boxed{ \text{global throttling controls the trajectory inside that region.} }$$

A very clean combined formulation is:

$$\theta_{t+1}^{\text{ctrl}} = \theta_t-\alpha_t\eta G_t.$$

Then:

$$\theta_{t+1} = \Pi_{\kappa} \left( \theta_{t+1}^{\text{ctrl}} \right).$$

But the key is that $\Pi_\kappa$ should be a loose guard rail, not the primary stabilizer applied aggressively every step.

Key Insight: The throttle should prevent the system from hitting the $\kappa$ rails too violently. The $\kappa$ rails are still there as final protection. So if we want, we can actually disable the $\kappa$ projection to spare hardware and computation and keep the global throttle as the main stabilizer.