Multi-Task Score Fusion

Background

A production ranker exposes multiple heads - predicted click, watch time, completion, like. The serving stack has to fuse these into one comparable score. The canonical combiner is a weighted geometric mean (i.e. weighted product in score space): $\text{score} = \prod_i p_i^{w_i}$. The weights are not learned - they're tuned via online A/B tests, expressing product trade-offs.

Problem statement

Implement multi_task_fusion(head_outputs, weights). head_outputs is a 2-D array of shape $(N, K)$ where each row is one item's $K$ head outputs in $[0, 1]$ (typically click-prob, watch-time-frac, etc.). weights is a length-$K$ array of non-negative floats. Return a length-$N$ array of fused scores:

$$\text{score}_n \;=\; \prod_{k=1}^{K} (p_{n,k} + \epsilon)^{w_k}$$

with a small $\epsilon$ on each factor to avoid $0^0$ / $\log 0$ pathologies.

Input

• head_outputs - np.ndarray shape $(N, K)$, values in $[0, 1]$.
• weights - array-like length $K$, non-negative.
• eps - small float (default $10^{-9}$).

Output

• np.ndarray length $N$, fused scores.

Examples

Example 1 - equal weights

Input:  head_outputs = [[0.5, 0.5], [0.8, 0.2]], weights = [1, 1]
Output: [0.25, 0.16]

Example 2 - weight a head to zero -> it drops out

Input:  head_outputs = [[0.5, 0.9], [0.5, 0.1]], weights = [1, 0]
Output: ~[0.5, 0.5]    # second head ignored

Constraints

• Use the numerically stable form: compute in log space then exponentiate. exp(sum(w * log(p + eps))).
• Negative weights raise ValueError.
• Mismatched lengths raise ValueError.

Notes

• Why geometric, not arithmetic. Geometric (product / log-sum-exp-style) penalises items that are weak on any single head; arithmetic mean lets one strong head mask a weak one. Production rankers want a balanced item.
• Tuning weights. They're searched online via A/B factorial designs, typically in log-space (a 5x5 grid of exponents is a real experimentation plan).