Adaptive (Re)Utility Index — ARUI

ARUI is an experimental framework for estimating how suitable the learned behaviour of an adaptive gating system may be for recreation using static gate weights. ARUI can be used to measure how suitable an adaptive-model seed may be for static recreation, while also providing the suggested fixed gate weights.

Status: Experimental / pre-validation.
ARUI v0.1.0 defines the proposed methodology. Its ability to predict successful static recreation is currently being evaluated experimentally.

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Motivation

Adaptive gating systems can dynamically change the contribution of different information sources. This flexibility may improve predictive performance, but it can also increase architectural and computational complexity.

In some cases, an adaptive gate may learn a useful allocation that changes only slightly over time. If so, its behaviour may potentially be approximated using a simpler static gate.

A seed with high adaptive gain and low gate displacement receives a higher ARUI value. The ARUI value cannot currently go beyond its adaptive gain value.

ARUI may also help avoid using large computational power whilst being uncertain about the use of the adaptive weights for fixed models.

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Methodology

Assume an adaptive gate contains (K) data types and is evaluated across (T) observations.

At observation (t), the gate produces:

[ \mathbf{w}t = (w{t,1}, w_{t,2}, \ldots, w_{t,K}) ]

For a normalized gate:

[ \sum_{j=1}^{K} w_{t,j} = 1 ]

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1. Mean gate weight

For each data type (j), ARUI calculates its mean gate weight:

[ \bar{w}j = \frac{1}{T} \sum{t=1}^{T} w_{t,j} ]

The vector

[ \bar{\mathbf{w}}

(\bar{w}_1,\bar{w}_2,\ldots,\bar{w}_K) ]

forms the proposed static recreation of the adaptive gate.

For normalized gate outputs, the mean weights also sum to 1, apart from floating-point error.

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2. Mean variation

For each data type, ARUI measures its mean absolute variation around its mean gate weight:

[ d_j = \frac{1}{T} \sum_{t=1}^{T} |w_{t,j}-\bar{w}_j| ]

This measures the typical distance between the dynamic gate allocation and the constant weight that would be used during static recreation.

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3. Seed displacement

The overall seed displacement is defined as the mean variation across all (K) data types:

[ D_s = \frac{1}{K} \sum_{j=1}^{K} d_j ]

A lower (D_s) indicates that the adaptive gate remains closer to its mean configuration.

A perfectly static gate has:

[ D_s = 0 ]

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4. Adaptive gain

ARUI defines signed adaptive gain so that a positive value always indicates better adaptive performance.

For a metric where higher_is_better is True:

[ G_s = A_s - F_s ]

For a metric where higher_is_better is False:

[ G_s = F_s - A_s ]

where:

• (A_s) is the adaptive performance for seed (s)
• (F_s) is the corresponding fixed-model performance

Through this calculation:

• (G_s > 0): adaptive model performed better
• (G_s = 0): equal performance
• (G_s < 0): adaptive model underperformed

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5. Adaptive (Re)Utility Index

ARUI combines adaptive gain and weight displacement:

[ \boxed{ ARUI_s = \frac{G_s}{1+D_s} } ]

A higher ARUI therefore favours solutions that combine:

• greater adaptive performance gain;
• lower average gate variation.

The addition of 1 also ensures the denominator remains defined when:

[ D_s=0 ]

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