L2hforadaptivity Ef F1 F3 F5 (Browser)

We define three local error estimators for each element K:

Purpose: Evaluates how gracefully the system reshuffles its L2-H mapping when computational or energy resources are limited.

Unlike F1 (accuracy of mapping), F3 focuses on adaptivity overhead. It measures:

EF-F3 = (Throughput_adaptive / Throughput_non-adaptive) × (1 - Latency_overhead / Latency_baseline) l2hforadaptivity ef f1 f3 f5

A score of 1.0 indicates no negative impact from adaptivity. Scores below 0.5 suggest the hierarchy reconfiguration consumes more resources than it saves. L2HforAdaptivity uses EF-F3 to trigger a “lazy hierarchy” mode where L2 operates semi-autonomously without continuous H updates.

Purpose: Assesses the system’s ability to maintain effective adaptivity over a rolling horizon of five decision steps.

The number 5 in F5 is not arbitrary. L2H’s designers found that most adaptive control problems exhibit Markov-like properties up to 5 steps; beyond that, environmental noise dominates. EF-F5 is computed as: We define three local error estimators for each

EF-F5 = (1/5) Σ_t=1 to 5 [ Stability(t) × Adaptation_Gain(t) ]

Where:

If EF-F5 drops below a threshold (typically 0.7), the system triggers a full hierarchy recomputation rather than incremental updates. If EF-F5 drops below a threshold (typically 0

Standard adaptivity refines elements where the estimated error exceeds a threshold. However, using only the L² norm can miss high‑frequency oscillations, while pure H¹ adaptivity over‑refines smooth regions. The hybrid L²‑H¹ (often written as l2hforadaptivity) balances these:

Despite its promise, L2HforAdaptivity is not turnkey. Key challenges include:

In a standard convolutional or transformer backbone, features evolve as they deepen. In the L2H4A context, we categorize these into three distinct functional domains.