BERT Rediscovers the Classical NLP Pipeline

3.2 Cumulative Scoring

1. What is cumulative scoring in Tenney et al.?
   Train probes \(P_\tau^{(\ell)}\) that may use layers 0 to \(\ell\) (layer 0 alone is a non-contextual baseline); the differential score \(\Delta_\tau^{(\ell)} = \text{Score}(P_\tau^{(\ell)}) - \text{Score}(P_\tau^{(\ell-1)})\) measures how much layer \(\ell\) adds, measured on evaluation data.
   • Tenney et al., 2019, BERT Rediscovers the Classical NLP Pipeline (Proceedings of the 57th Annual Meeting of ACL), §3.2, pp. 4595–4596
2. What is the expected layer statistic, and how does it differ from the mixing-weight center of gravity?
   \(\bar{E}_\Delta[\ell] = \frac{\sum_{\ell=1}^{L} \ell \cdot \Delta_\tau^{(\ell)}}{\sum_{\ell=1}^{L} \Delta_\tau^{(\ell)}}\), roughly the layer at which an example gets resolved (ignoring those resolved at layer 0). Mixing weights come from training data and say which layers the probe finds useful; cumulative scores come from evaluation data and say how many layers are needed.
   • Tenney et al., 2019, BERT Rediscovers the Classical NLP Pipeline (Proceedings of the 57th Annual Meeting of ACL), §3.2, pp. 4595–4596
3. Why can’t a probe on a single layer tell where a label becomes predictable?
   Information about a span may be spread across several layers, and the encoder may discard information in higher layers; cumulative probes see all layers up to \(\ell\), so performance can only grow with access to more information.
   • Tenney et al., 2019, BERT Rediscovers the Classical NLP Pipeline (Proceedings of the 57th Annual Meeting of ACL), §3.2, pp. 4595–4596