SlowGuess/NAACL_2022_OCR huggingface dataset:
Time Complexity. For each sentence, the inside-outside algorithm Eq. (22) and CKY algorithm Eq. (23) reduced to  $\mathcal{O}(|\mathbf{w}|)$  semiring matrix multiplications. Sentence-level parallelism can also be applied to all the sentences.
4.2 Semantic Role Labeling
The goal of SRL is to classify the semantic role of every argument  $\sigma_{ij}$  with respect to a given predicate. Following the notation style in §4.1, we use  $\sigma_{ij} \stackrel{l}{\rightsquigarrow} v$  to denote that  $\sigma_{ij}$  has the semantic role  $l$
$^{8}$ Following (Lee et al., 2017) the dummy antecedent  $\epsilon$  represents two possible scenarios: (1) the span is not an entity mention or (2) the span is an entity mention but it is not coreferent with any previous span.
in the frame of predicate  $v$ . The joint probability  $p\left(\sigma_{ij}, \sigma_{ij} \stackrel{\ell}{\sim} v\right)$  can be written as:
$$
\begin{array}{c c}\underbrace {p (\sigma_ {i j})} _ {\text {p r .} \sigma_ {i j} \text {i s a n a r g u m e n t}} \times&\underbrace {p \left(\sigma_ {i j} \stackrel {{\ell}} {{\rightsquigarrow}} v \mid \sigma_ {i j}\right)} _ {\text {p r .} \sigma_ {i j} \text {, s r o l e i s} \ell \text {w . r . t . p r e d i c a t e} v}\end{array}
$$
He et al. (2018) as a Role Classifier. As this work focuses on span selection, we choose the role classifier to be the popular and effective one from He et al. (2018), and use gold predicates  $v$  during training and evaluation. The semantic role label  $l$  takes its values from a discrete label space  $\mathcal{L}$  which contains all the semantic roles plus the null relation  $\epsilon$ . The classifier then models the following probability distribution:
$$
\begin{array}{l} p \left(\sigma_ {i j} \stackrel {\ell} {\rightsquigarrow} v \mid \sigma_ {i j}\right) = \frac {\exp s \left(\sigma_ {i j} , v , \ell\right)}{\sum_ {\ell \in \mathcal {L}} \exp s \left(\sigma_ {i j} , v , \ell\right)} \\ s \left(\sigma_ {i j}, v, \ell\right) = s _ {m} \left(\sigma_ {i j}\right) + s _ {r} \left(\sigma_ {i j}, v, \ell\right) \tag {16} \\ \end{array}
$$
Similar to the coreference model of Lee et al. (2017),  $s_m(\sigma_{ij})$  is the score for span  $[i,j]$  to be an argument and  $s_r(\sigma_{ij},v,\ell)$  for  $[i,j]$  to play the role  $l$  for predicate  $v$ . Note that the score for the  $\epsilon$  label (i.e., no relation),  $s(\sigma_{ij},v,\epsilon)$ , is set to constant 0, similar to the dummy antecedent case in coreference.
Training Objective. He et al.'s model also suffers from the challenge of tuning the hyperparameter  $\lambda$ . The training objective for SRL with our structured span selection model is then:
$$
L _ {1} = \sum_ {\sigma_ {i j} \in \mathcal {G} _ {\mathbf {w}}} \sum_ {v} \log p \left(\sigma_ {i j}, \sigma_ {i j} \stackrel {l} {\rightsquigarrow} v\right) \tag {17}
$$
where  $l$  is the correct semantic label of  $\sigma_{ij}$  with regard to predicate  $v$ . Similar to coreference resolution, we handle the issue of partial annotation for the span selection model by adding the log-likelihood that  $\sigma_{ij}$  may not be an annotated argument:
$$
\begin{array}{l} L _ {2} = \sum_ {\sigma_ {i j} \in \overline {{\mathcal {G}}} _ {\mathbf {w}}} \sum_ {v} \log \left(p \left(\sigma_ {i j} \stackrel {\epsilon} {\rightsquigarrow} v \mid \sigma_ {i j}\right) p \left(\sigma_ {i j}\right)\right. \\ \left. + \left(1 - p \left(\sigma_ {i j}\right)\right)\right) \tag {18} \\ \end{array}
$$
And the final objective function is  $L = L_{1} + L_{2}$
5.1 The Greedy Baseline
Previous work has mostly considered a greedy procedure for span selection as opposed to Eq. (8). The
approach produces a score  $s_g(\sigma_{ik})$  independently for each span  $\sigma_{ik}$ . As the number of spans  $\sigma_{ik}$  is potentially very large, the set of spans is greedily pruned to a set of size  $K$ . For instance, in SRL, spans are selected for each sentence:
$$
\begin{array}{l} M _ {\mathrm {t o p k}} ^ {*} \tag {19} \\ = \operatorname {a t o p} _ {K} \left(\left\{s _ {g} \left(\sigma_ {i k}\right) \mid 1 \leq i <   k \leq | \mathbf {w} | \right\}\right) \\ \end{array}
$$
However, in coreference resolution, a set of spans is selected for the entire document:
$$
\begin{array}{l} M _ {\text {t o p k}} ^ {*} \tag {20} \\ = \operatorname {a t o p} _ {K} \Big (\bigcup_ {\mathbf {w} \in D} \left\{s _ {g} \left(\sigma_ {i k}\right) \mid 1 \leq i <   k \leq | \mathbf {w} | \right\} \Big) \\ \end{array}
$$
where  $\operatorname{atop}_K$  is shorthand for  $\operatorname{argtop}_K$ . We will see in our experiments that tuning  $K$  can be quite challenging (see Fig. 3). Moreover, as the greedy approach scores each span independently, it ignores the structure of the provided span annotation.
Coreference. We experiment on the CoNLL-2012 English shared task dataset (OntoNotes) (Pradhan et al., 2012) and LitBank (Bamman et al., 2020) in our experiments. As a part of our experiments on OntoNotes, we apply the speaker encoding in Wu et al. (2020), that is using special tokens (<speaker>, </speaker>) to denote the speaker's name, as opposed to the original binary features used by Lee et al. (2017). This simple change brings a consistent boost to the performance by 0.2 F1. A major difference between these two datasets is that LitBank has singleton mention annotations while OntoNotes does not. For LitBank, we use the standard 10-fold cross-validation setup, as is the standard practice.
SRL. We use the CoNLL-2012 SRL dataset. Gold predicates are provided to the model.
5.3 Coreference Resolution
We report the average precision, recall, and F1 scores of the standard MUC,  $\mathbf{B}^3$ ,  $\mathrm{CEAF}_{\phi_4}$ , and the average CoNLL F1 score on the OntoNotes test set in Tab. 1. The average F1 scores on LitBank are shown in Tab. 2. For OntoNotes, we run the experiments with 5 random initializations and the improvements reported are significant under the two-tailed paired  $t$ -test.
We compare our models with several representative previous works. In order to focus on comparing
MUC	B3	CEAFφ4	Avg. F1
P	R	F	P	R	F	P	R	F
Lee et al. (2017)	78.4	73.4	75.8	68.6	61.8	65.0	62.7	59.0	60.8	67.2
Lee et al. (2018)	81.4	79.5	80.4	72.2	69.5	70.8	68.2	67.1	67.6	73.0
Fei et al.	85.4	77.9	81.4	77.9	66.4	71.7	70.6	66.3	68.4	73.8
Kantor and Globerson	82.6	84.1	83.4	73.3	76.2	74.7	72.4	71.1	71.8	76.6
Joshi et al. (2019)	84.7	82.4	83.5	76.5	74.0	75.3	74.1	69.8	71.9	76.9
Joshi et al. (2020)	85.8	84.8	85.3	78.3	77.9	78.1	76.4	74.2	75.3	79.6
Xu and Choi	85.7	85.3	85.5	78.6	78.6	78.6	76.8	74.8	75.8	79.9
Joshi et al. (S)	86.6	84.5	85.6	80.4	77.3	78.8	77.8	74.0	75.8	80.1
Ours	86.1	85.5	85.8	79.8	78.8	79.3	77.4	75.4	76.4	80.5
the impact of mention detection, we do not consider higher-order inference techniques in our models and report the non-higher order result from Xu and Choi (2020). Joshi et al. (S) is the major baseline that uses SpanBERT (Joshi et al., 2020) and is trained with the speaker encoding discussed in §5.2. This encoding yields an F1 score improvement of 0.2 over the result reported by Xu and Choi (2020).
Our model with the structured mention detector achieves an F1 score of 80.5, an improvement of 0.4 F1 over the baseline. While on LitBank, our model achieves an F1 score of 76.3, which is an improvement of a 0.7 F1 over (Joshi et al., 2020). It can also be observed that this gain mainly comes from improved recall, which is because we have a superior mention detector that can retrieve mentions with better accuracy. We further analyze this result in the following section.
Table 1: Results on the CoNLL-2012 English shared task test set. Avg. F1 in the last column denotes the average F1 of MUC,  $\mathbf{B}^3$ , and  $\mathrm{CEAF}_{\phi_4}$ . Joshi et al. (S) refers to the original end-to-end model with SpanBERT and trained with speaker encoding. The improvements shown in the table are significant under a two-tailed paired  $t$ -test.
Avg. P	Avg. R	Avg. F1
Bamman et al.	-	-	68.1
LB-MEM	-	-	75.7
U-MEM	-	-	75.9
Joshi et al.	78.7	72.9	75.6
Ours	77.4	75.3	76.3
Table 2: Results on the test set of LitBank. The results are averaged over 10 train/dev/test splits. LB-MEM and U-MEM are reported in Toshiwal et al. (2020).
5.3.1 Analysis of Mention Detector
Next, we examine the performance of our proposed mention detection scheme. As shown in Fig. 3, compared with Joshi et al. (S), our model predicts
Figure 3: Recall of gold mentions as we vary the ratio of spans kept. Ratio refers to the number of predicted mentions divided by  $|D|$ . Our mention detector significantly outperforms Joshi et al. (S)'s with a ratio of 0.26 and a recall of  $97.0\%$ . Yet Joshi et al. (S) only achieves  $96.2\%$  recall with a ratio of 0.40.
mentions more accurately with a higher recall. In contrast to Joshi et al. (S) who select  $0.4|D|$  mention spans, our method on average selects  $0.26|D|$ . The smaller span set makes the coreference model more efficient as well.
5.3.2 Analysis of Structured Modeling
To see how our structured modeling benefits coreference, we further compare our approach with a baseline Sigmoid which replaces the  $p(\sigma_{ij})$  in Eq. (10) with a simple non-structured estimator:
$$
p _ {\text {s i g m}} \left(\sigma_ {i j}\right) = \text {s i g m o i d} \left(s _ {p} \left(\sigma_ {i j}\right)\right) \tag {21}
$$
Nested Depth	1	2	3+
Greedy	96.5	87.1	86.2
Ours	97.8	93.2	93.9
where  $\operatorname{sigmoid}(x) = \frac{1}{1 + \exp(-x)}$ . The loss function used for Sigmoid is the same as the structured model given in §4.1. Through this comparison, we aim to show the effectiveness of structured modeling. We also build a multi-task learning baseline MTL similar to Swayamdipta et al. (2018). The baseline adds an auxiliary classifier that classifies spans into noun phrases, other syntactic constituents, or non-constituents. The coefficient of the multi-task loss is set to 0.1 as in Swayamdipta et al. (2018).
This comparison is shown in Tab. 4. We find that replacing  $p(\sigma_{ij})$  with unstructured  $p_{\mathrm{sign}}(\sigma_{ij})$  degrades the performance by an F1 score of 0.3. Thus, we conclude that the structured probability function is more expressive than  $p_{\mathrm{sign}}(\sigma_{ij})$  as it models the global annotation for each sentence. In contrast, the mention detectors in Joshi et al. (S) and the Sigmoid model each span independently.
5.3.3 Analysing the source of improvement
Next, we try to explore where the gains of our model come from.
Nested Mentions. We first investigate the capability of our structured model in handling nested mentions. Tab. 3 shows the recall rate of mentions of different nested depths. Here, nested depth refers to the level of nesting in the mentions. E.g., in the first example given in Fig. 1, The president is of depth 1, while he and his wife, now a New York senator is of depth 2. As shown in Tab. 3, the gains of our method are larger for deeply nested mentions, which highlight the capability of our structured span detector to handle more difficult nested mentions that cannot be handled by the greedy selector.
Widths of Mentions. We also compare the recall rate for mentions of different widths in Tab. 5. We show that our model can detect longer spans better, which are usually more difficult to detect. For spans with 5–12 words, our structured model still maintains a recall rate of  $96.5\%$ , compared to a sharp drop for the greedy unstructured model.
Table 3: Recall rate of mentions of different nested depth on CoNLL-2012 dev set. There are 16873, 2100, and 182 mentions respectively of each nested depth.
Avg. P	Avg. R	Avg. F1
Joshi et al. (S)	81.6	78.6	80.1
Sigmoid	80.8	79.6	80.2
MTL	80.8	78.9	79.8
Ours	81.1	79.9	80.5
Table 4: Comparison with three constructed baselines.
Span Width	1-4	5-12	12+
Greedy	96.2	92.7	82.5
Ours	97.8	96.5	85.2
5.4 Semantic Role Labeling
For semantic role labeling, we report the precision, recall, and F1 score on the CoNLL-2012 SRL dataset. The gold predicates are provided during both training and evaluation. Therefore, the model has to focus on extracting the correct arguments and classifying their roles for each predicate. The results are shown in Tab. 6 in comparison with previous span-based models. He et al.  $\mathrm{SpanBERT}$  refers to the model of He et al. (2018) with  $\mathrm{SpanBERT}_{\mathrm{large}}$  (Joshi et al., 2020) as a sentence encoder.
Next, we report the performance of our span selector on the SRL task. Following the same trend as coreference resolution, we find that our structured model is able to extract much more accurate arguments and thus, significantly reduce the memory consumption for the downstream task. While keeping a comparable recall rate of gold arguments (96.5% for the greedy selector and 96.2% for ours), our span selector reduces
Table 5: Recall rate of mentions of different width on the CoNLL-2012 dev set. There are 16356, 2180, and 619 mentions respectively of each width interval.
Avg. P	Avg. R	Avg. F1
He et al. (2018)	-	-	85.5
Ouchi et al. (2018)	87.1	85.3	86.2
Li et al. (2019)	85.7	86.3	86.0
Shi and Lin (2019)	85.9	87.0	86.5
He et al.SpanBERT	88.3	85.9	87.1
Ours	88.1	86.9	87.5
Table 6: Results on the test set of the CoNLL-2012 semantic role labeling task. The precision, recall, and F1 scores are averaged over all semantic roles.
P	R	F1
BIO	90.2	91.0	90.6
Greedy	92.8	90.2	91.5
Ours	92.8	91.2	92.0
Table 7: Comparison of unlabeled span accuracy.
Coref	SRL
Greedy	11.5	12.2
Ours	8.5	9.3
the number of enumerated spans by  $21.2\%$ . We compare the accuracy of retrieving unlabeled argument spans in Tab. 7. BIO refers to the tagger-style SRL model using the same text encoder. Our model outperforms both baselines.
5.5 Qualitative Examples
In this section, we show a qualitative example to illustrate the grammar learned by our span selector for coreference resolution and SRL. Two sets of extracted spans for the same input sentence are shown in Tab. 9. For coreference resolution, our model selects maximal NPs (containing all modifiers) and verbs. While in SRL, the parse tree consists of much denser and syntactically heterogeneous spans of NPs, PPs, modal verbs, adverbs, etc. This comparison empirically shows that our model is capable and robust enough to learn a complex underlying grammar from partial annotation.
5.6 Memory Efficiency
We further analyze the memory efficiency of our model. We evaluate the peak GPU memory usage on the development set of OntoNotes. For both tasks, we see a significant reduction in memory usage of  $27\%$  for coreference and  $24\%$  for SRL.
In this paper, we proposed a novel structured model for span selection. In contrast to prior span selection methods, the model is structured, which allows it to model spans better. Instead of a greedy span selection procedure, the span selector uses partial span annotations provided in data to directly obtain the set of optimal spans. We evaluated our span se
Table 8: Comparison of peak GPU memory usage in GBs at inference time on the development set.
Coref	[[The world's] fifth [Disney] park]
will soon [open] to [the public] here .
SRL	[[The world's] [fifth] [Disney] [park]]
[will] [soon] [ [open] [to the public]]
[here] .
Table 9: A qualitative example of the grammar learned by the structured span selector.
lector on two typical span prediction tasks, namely coreference resolution and semantic role labeling, and achieved consistent gains in terms of accuracy as well as efficiency over greedy span selection.
To the best of our knowledge, the datasets used in our work do not contain sensitive information, and we foresee no further ethical concerns with the work.
Mrinmaya Sachan acknowledges support from an ETH Zürich Research grant (ETH-19 21-1) and a grant from the Swiss National Science Foundation (project #201009) for this work.
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