SlowGuess/NAACL_2021_OCR huggingface dataset:
Hang Hua $^{1*†}$ , Xingjian Li $^{2,3*}$ , Dejing Dou $^{2\boxtimes}$ , Chengzhong Xu $^{3}$ , Jiebo Luo $^{1}$
<sup>1</sup>University of Rochester, Rochester, NY, USA
hhua2, jluo}@cs.ROchester.edu
$^{2}$ Big Data Lab, Baidu Research, Beijing, China
{lixingjian, doudejing}@baidu.com
$^{3}$ Department of Computer Science, University of Macau, Macau, China
czxu@um.edu.mo
Fine-tuning pre-trained language models such as BERT has become a common practice dominating leaderboards across various NLP tasks. Despite its recent success and wide adoption, this process is unstable when there are only a small number of training samples available. The brittleness of this process is often reflected by the sensitivity to random seeds. In this paper, we propose to tackle this problem based on the noise stability property of deep nets, which is investigated in recent literature (Arora et al., 2018; Sanyal et al., 2020). Specifically, we introduce a novel and effective regularization method to improve fine-tuning on NLP tasks, referred to as Layer-wise Noise Stability Regularization (LNSR). We extend the theories about adding noise to the input and prove that our method gives a stabler regularization effect. We provide supportive evidence by experimentally confirming that well-performing models show a low sensitivity to noise and fine-tuning with LNSR exhibits clearly higher generalizability and stability. Furthermore, our method also demonstrates advantages over other state-of-the-art algorithms including  $\mathbf{L}^2$ -SP (Li et al., 2018), Mixout (Lee et al., 2020) and SMART (Jiang et al., 2020).
Large-scale pre-trained language models such as BERT (Devlin et al., 2019) have been widely used in natural language processing tasks (Guu et al., 2020; Liu, 2019; Wadden et al., 2019; Zhu et al., 2020b). A typical process of training a supervised downstream dataset is to fine-tune a pre-trained model for a few epochs. In this process, most of the model's parameters are reused, while a random initialized task-specific layer is added to adapt the model to the new task.
Fine-tuning BERT has significantly boosted the state of the art performance on natural lan
guage understanding (NU) benchmarks such as GLUE (Wang et al., 2018) and SuperGLUE (Wang et al., 2019). However, despite the impressive empirical results, this process remains unstable due to the randomness involved by data shuffling and the initialization of the task-specific layer. The observed instability in fine-tuning BERT was first discovered by Devlin et al. (2019); Dodge et al. (2020), and several approaches have been proposed to solve this problem (Lee et al., 2020; Zhang et al., 2020; Mosbach et al., 2020).
In this study, we consider the fine-tuning stability of BERT from the perspective of the sensitivity to input perturbation. This is motivated by Arora et al. (2018) and Sanyal et al. (2020) who show that noise injected at the lower layers has very little effect on the higher layers for neural networks with good generalizability. However, for a well pre-trained BERT, we find that the higher layers are still very sensitive to the lower layer's perturbation (as shown in Figure 1), implying that the high level representations of the pre-trained BERT may not generalize well on downstreaming tasks and consequently lead to instability. This phenomenon coincides with the observation that transferring the top pre-trained layers of BERT slows down learning and hurts performance (Zhang et al., 2020). In addition, Yosinski et al. (2014) also point out that in transfer learning models for object recognition, the lower pre-trained layers learn more general features while the higher layers closer to the output specialize more to the pre-training tasks. We argue that this result also applies to BERT. Intuitively, if a trained model is insensitive to the perturbation of the lower layers' output, then the model is confident about the output, and vice versa. Based on the above theoretical and empirical results, we propose a simple and effective regularization method to reduce the noise sensitivity of BERT and thus improve the stability and performance of fine-tuned BERT.
$\dagger$ Contribution during internship at Baidu Research.
* Equal contributions.  $\boxtimes$  Correspondence.
Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 3229-3241 June 6-11, 2021. ©2021 Association for Computational Linguistics
Figure 1: Attenuation of injected noise on the BERT-Large-Uncased model on the MRPC task (X-axis: the layer index. Y-axis: the  $L^2$  norm between the original output and noise perturbed output). A curve starts at the layer where a scaled Gaussian noise is injected to its output whose  $l_{2}$  norm is set to  $5\%$  of the norm of its original output. As it propagates up, the injected noise has a rapidly decreasing effect on the lower layers but becomes volatile on the higher layers, which indicates the poor generalizability and brittleness of the BERT top layers. Moreover, models with higher accuracies (marked in the upper right) usually have lower error ratios or higher noise stability in top layers.
To verify our approach, we conduct extensive experiments on different few-sample (fewer than 10k training samples) NLP tasks, including CoLA (Warstadt et al., 2019), MRPC (Dolan and Brockett, 2005), RTE (Wang et al., 2018; Dagan et al., 2005; Bar-Haim et al., 2006; Giampiccolo et al., 2007), and STS-B (Cer et al., 2017). With the layerwise noise stability regularization, we obtain strong empirical performance. Compared with other state-of-the-art models, our approach not only improves the fine-tuning stability (with a smaller standard deviation) but also consistently improve the overall performance (with a larger mean, median and maximum).
- We propose a lightweight and effective regularization method, referred to as Layer-wise Noise Stability Regularization (LNSR) to improve the local Lipschitz continuity of each BERT layer and thus ensure the smoothness of the whole model. The empirical results show that the fine-tuned BERT models regularized with LNSR obtain significantly more accurate and stable results. LNSR also outperforms other state-of-the-art methods aiming at stabilizing fine-tuning such as  $\mathbf{L}^2$ -SP (Li et al., 2018), Mixout (Lee et al., 2020) and SMART (Jiang et al., 2020).
- We are the first to study the effect of noise stability in NLP tasks. We extend classic theories of adding noise to explicitly constraining the output consistency when adding noise to the input. We theoretically prove that our proposed layer-wise noise stability regularizer is equivalent to a special case of the Tikhonov regularizer, which serves as a stabler regularizer than simply adding noise to the input (Rifai et al., 2011).
- We investigate the relation of the noise stability property to the generalizability of BERT. We find that in general, models with good generalizability tend to be insensitive to noise perturbation; the lower layers of BERT show a better error resilience property but the higher layers of BERT remain sensitive to the lower layers' perturbation (as is depicted in Figure 1).
2.1 Pre-training
Pre-training has been well studied in machine learning and natural language processing (Erhan et al., 2009, 2010). Mikolov et al. (2013) and Pennington et al. (2014) proposed to use distributional representations (i.e., word embeddings) for indi
vidual words. Dai and Le (2015) proposed to train a language model or an auto-encoder with unlabeled data and then leveraged the obtained model to finetune downstream tasks. Recently, pretrained language models, like ELMo (Peters et al., 2018), GPT/GPT-2 (Radford, 2018; Radford et al., 2019), BERT (Devlin et al., 2019), cross-lingual language model (briefly, XLM) (Lample and Conneau, 2019), XLNet (Yang et al., 2019), RoBERTa (Liu et al., 2019) and ALBERT(Lan et al., 2020) have attracted more and more attention in natural language processing communities. The models are first pre-trained on large amount of unlabeled data to capture rich representations of the input, and then applied to the downstream tasks by either providing context-aware embeddings of an input sequence (Peters et al., 2018), or initializing the parameters of the downstream model (Devlin et al., 2019) for fine-tuning. Such pre-training approaches deliver decent performance on natural language understanding tasks.
2.2 Instability in Fine-tuning
Fine-tuning instability of BERT has been reported in various previous works. Devlin et al. (2019) report instabilities when fine-tuning BERT on small datasets and resort to performing multiple restarts of fine-tuning and selecting the model that performs best on the development set. Dodge et al. (2020) performs a large-scale empirical investigation of the fine-tuning instability of BERT. They found dramatic variations in fine-tuning accuracy across multiple restarts and argue how it might be related to the choice of random seed and the dataset size. Lee et al. (2020) propose a new regularization method named Mixout to improve the stability and performance of fine-tuning BERT. Zhang et al. (2020) evaluate the importance of debiasing step empirically by fine-tuning BERT with both BERTAdam and standard Adam optimizer (Kingma and Ba, 2015) and propose a re-initialization method to get a better initialization point for fine-tuning optimization. Mosbach et al. (2020) analyses the cause of fine-tuning instability and propose a simple but strong baseline (small learning rate combined with bias correction).
2.3 Regularization
There has been several regularization approaches to stabilizing the performance of models. Loshchilov and Hutter (2019) propose a decoupled weight decay regularizer integrated in Adam (Kingma and
Ba, 2015) optimizer to prevent neural networks from being too complicate. Gunel et al. (2020) use contrastive learning method to augment training set to improve the generalization performance. In addition, spectral norm (Yoshida and Miyato, 2017; Roth et al., 2019) serves as a general method can also be used to constrain the Lipschitz continuous of matrix, which can increase the stability of generalized neural networks.
There are also several noise-based methods have been proposed to improve the generalizability of pre-trained language models, including SMART (Jiang et al., 2020), FreeLB (Zhu et al., 2020a) and R3F (Aghajanyan et al., 2020). They achieve state of the art performance on GLUE, SNLI (Bowman et al., 2015), SciTail (Khot et al., 2018), and ANLI (Nie et al., 2020) NLU benchmarks. Most of these algorithms employ adversarial training method to improve the robustness of language model finetuning. SMART uses an adversarial methodology to encourage models to be smooth within a neighborhoods of all the inputs; FreeLB optimizes a direct adversarial loss through iterative gradient ascent steps; R3F removes the adversarial nature of SMART and optimize the smoothness of the whole model directly. Different from these methods, our proposed method does not adopt the adversarial training strategy, we optimize the smoothness of each layer of BERT directly and thus improve the stability of whole model.
3 Using Noise Stability as a Regularizer
One of the central issues in neural network training is to determine the optimal degree of complexity for the model. A model which is too limited will not sufficiently capture the structure in the data, while one which is too complex will model the noise on the data (the phenomenon of over-fitting). In either case, the performance on new data, that is the ability of the network to generalize, will be poor. The problem can be regarded as one of finding the optimal trade-off between the high bias of a model which is too inflexible and the high variance of a model with too much freedom (Geman et al., 1992; Bishop, 1995; Novak et al., 2018; Bishop, 1991). To control the trade-off of bias against variance of BERT models, we impose an explicit noise regularization method.
3.1 Introduction of Our Method
Denoting the training set as  $D$ , we give the general form of optimization objective for a BERT model  $f(\cdot ;\theta)$  with  $L$  layers, as following:
$$
\theta^ {*} = \arg \min  _ {\theta} \mathbb {E} _ {(x, y) \sim D} [ \mathcal {L} (f (x; \theta), y) + \hat {\mathcal {R}} (\theta) ]. \tag {1}
$$
To represent  $\hat{\mathcal{R}} (\theta)$ , we first define the injection position as the input of layer  $b$  which is denoted as  $x^{b}$ . If the regularization is operated at the output of layer  $r$ , we can further denote the function between layer  $b$  and  $r$  as  $f^{b,r}$ , satisfying that  $1 < = b < = r < = L$ . To implement the noise stability regularization, we inject a Gaussian-like noise vector  $\varepsilon$  to  $x^{b}$  and get a neighborhood  $x^{b} + \varepsilon$ . Specifically, each element  $\varepsilon_{i}$  is independently randomly sampled from a Gaussian distribution with the mean of zero and the standard deviation of  $\sigma$  as  $\varepsilon_{i} \sim \mathcal{N}(0,\sigma^{2})$ . The probability density function of the noise distribution can be written as  $p(\varepsilon_i) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{\varepsilon_i^2}{2\sigma^2}}$ . Our goal is to minimize the discrepancy between their outputs over  $f^{b,r}$  defined as  $||f^{b,r}(x^{b} + \varepsilon) - f^{b,r}(x^{b})||^{2}$ . In our framework, we use a fixed position  $b$  as the position of noise injection and constrain the output distance on all layers following layer  $b$ . Denoting the regularization weight corresponding to each  $f^{b,r}$  as  $\lambda^{b,r}$ , given a sample  $(x,y) \sim D$ , the regularization term is represented by the following formulas:
$$
\hat {\mathcal {R}} (\theta) = \sum_ {r = b} ^ {L} \lambda^ {b, r} | | f ^ {b, r} (x ^ {b} + \varepsilon) - f ^ {b, r} (x ^ {b}) | | ^ {2}. \tag {2}
$$
An overall algorithm is represented in Algorithm 1.
Regularization is a kind of commonly used techniques to reduce the function complexity and, as a result, to make the learned model generalize well on unseen examples. In this part, we theoretically prove that the proposed LNSR algorithm has the effects of encouraging the local Lipschitz continuity and imposing a Tikhonov regularizer under different assumptions. For simplicity, we omit the notations about the layer number in this part, denoting  $f$  as the target function and  $x$  as the input of  $f$  parameterized by  $\theta$ . Given a sample  $(x,y) \sim D$ , we discuss the general form of the noise stability defined as following:
$$
\mathcal {R} (\theta) = \underset {\varepsilon} {E} \{\| f (x + \varepsilon) - f (x) \| ^ {2} \}. \tag {3}
$$
Lipschitz continuity. The Lipschitz property reflects the degree of smoothness for a function. Recent theoretical studies on deep learning has revealed the close connection between Lipschitz property and generalization (Bartlett et al., 2017; Neyshabur et al., 2017).
Given a sampled  $\varepsilon$ , minimizing  $\|f(x + \varepsilon) - f(x)\|^2$  is equivalent to minimizing:
$$
\frac {\| f (x + \varepsilon) - f (x) \| ^ {2}}{\| x + \varepsilon - x \| ^ {2}}. \tag {4}
$$
Thus the noise stability regularization can be regarded as minimizing the Lipschitz constant in a local region around the input  $x$ .
Tikhonov regularizer. The Tikhonov regularizer (Willoughby, 1979) involves constraints on the derivatives of the objective function with respect to different orders. For the simplest first-order case, it can be regarded as imposing robustness and shaping a flatter loss surface at input, which makes the learned function smoother.
Assuming that the magnitude of  $\varepsilon$  is small, we can expand the first term as a Taylor approximation as:
$$
f (x + \varepsilon) = f (x) + \varepsilon \cdot J _ {f} (x) + \frac {1}{2} \varepsilon^ {T} \cdot H _ {f} (x) \cdot \varepsilon + O (\varepsilon^ {3}), \tag {5}
$$
where  $J_{f}(x)$  and  $H_{f}(x)$  refer to the Jacobian and Hessian of  $f$  with respect to the input  $x$  respectively.
Ignoring the higher order term  $O(\varepsilon^3)$  and denoting  $f_{k}$  as the k-th output of the function  $f$ , we can rewrite the regularizer by substituting Eq. 5 in Eq. 3 as:
$$
\begin{array}{l} \mathcal {R} (\theta) = \underset {\varepsilon} {E} \left\{\| \varepsilon \cdot J _ {f} (x) + \frac {1}{2} \varepsilon^ {T} \cdot H _ {f} (x) \cdot \varepsilon \| ^ {2} \right\} \\ = \int | | \varepsilon \cdot J _ {f} (x) + \frac {1}{2} \varepsilon^ {T} \cdot H _ {f} (x) \cdot \varepsilon | | ^ {2} p (\varepsilon) d \varepsilon \\ = \sum_ {k} \int \left\| \varepsilon \cdot J _ {f _ {k}} (x) + \frac {1}{2} \varepsilon^ {T} \cdot H _ {f _ {k}} (x) \cdot \varepsilon \right\| ^ {2} p (\varepsilon) d \varepsilon . \tag {6} \\ \end{array}
$$
We define the input vector  $x$  as  $(x_{1}, x_{2}, \dots, x_{d_{1}})$  and noise vector  $\varepsilon$  as  $\varepsilon = (\varepsilon_{1}, \varepsilon_{2}, \dots, \varepsilon_{d_{1}})$ . Assuming that distributions of the noise and the input are irrelevant, and the derivative of  $f$  with respect to different elements of the input vector is independent with each other, we expand the second order term
corresponding to the Jacobian as:
$$
\begin{array}{l} \Omega_ {J} (f _ {k}) = \int \| \varepsilon \cdot J _ {f _ {k}} (x) \| ^ {2} p (\varepsilon) d \varepsilon \\ = \int \sum_ {i} \left(\varepsilon_ {i} \frac {\partial f _ {k}}{\partial x _ {i}}\right) ^ {2} p (\varepsilon) d \varepsilon \tag {7} \\ = \sum_ {i} (\frac {\partial f _ {k}}{\partial x _ {i}}) ^ {2} \int \varepsilon_ {i} ^ {2} p (\varepsilon) d \varepsilon \\ = \sigma^ {2} \| J _ {f _ {k}} (x) \| ^ {2}. \\ \end{array}
$$
According to the characteristics of the Gaussian distribution, we also have
$$
\int \varepsilon_ {i} \varepsilon_ {j} p (\varepsilon) d \varepsilon = 0 \quad \text {f o r a n y} i \neq j. \tag {8}
$$
Thus, we can rewrite the second order term corresponding to the Hessian in Eq. 6 as:
$$
\begin{array}{l} \Omega_ {H} (f _ {k}) = \int \| \frac {1}{2} \varepsilon^ {T} \cdot H _ {f _ {k}} (x) \cdot \varepsilon \| ^ {2} p (\varepsilon) d \varepsilon \\ = \int \frac {1}{4} \sum_ {i} \left(\varepsilon_ {i} ^ {2} \frac {\partial^ {2} f _ {k}}{\partial x _ {i} ^ {2}}\right) ^ {2} p (\varepsilon) d \varepsilon \tag {9} \\ = \frac {1}{4} \sum_ {i} (\frac {\partial^ {2} f _ {k}}{\partial x _ {i} ^ {2}}) ^ {2} \int \varepsilon_ {i} ^ {4} p (\varepsilon) d \varepsilon \\ = C \| \operatorname {T r} \left(H _ {f _ {k}} (x) ^ {T} H _ {f _ {k}} (x)\right) \| ^ {2}, \\ \end{array}
$$
Where  $C$  is a constant independent of the input  $x$ . The third term generated from the expansion of Eq. 6 is zero as we have  $\int \varepsilon^3 p(\varepsilon)d\varepsilon = 0$ . Thus we get
$$
\mathcal {R} (\theta) = \sum_ {k} \left\{\Omega_ {J} \left(f _ {k}\right) + \Omega_ {H} \left(f _ {k}\right) \right\}. \tag {10}
$$
Considering that the input and output of the function  $f$  are both scalar variables, the Tikhonov regularization (Willoughby, 1979) takes the general form as:
$$
\mathcal {R} _ {\mathcal {T}} (\theta) = \sum_ {r} \int h _ {r} (x) \left(\frac {\partial^ {r} f}{\partial x ^ {r}}\right) ^ {2} d x. \tag {11}
$$
Eq. 10 shows that our proposed regularizer ensuring the noise stability is equivalent to a special case of the Tikhonov regularizer, where we involve the first and second order derivatives of the objective function  $f$ .
An alternative for improving the robustness is to directly add noise to the input, without explicitly constraining the output stability. (Rifai et al., 2011) has derived that adding noise to the input has the effect of penalizing both the  $L_{2}$ -norm of
the Jacobian  $\| J_{f}(x)\|^{2}$  and the trace of the Hessian  $\mathrm{Tr}(H_f(x))$ , whereas the Hessian term is not constrained to be positive. While the regularizer brought by our proposed LNSR is guaranteed to be positive by involving the sum of squares of the first and second order derivatives. Moreover, our work relaxes the assumption of MSE regression loss required by (Rifai et al., 2011). By imposing the explicit constraint of noise stability on middle layer representations, we extend the theoretical understanding of noise stability into deep learning algorithms.
In this section, we experimentally demonstrate the effectiveness of LNSR method on text classification tasks over other regularization methods, and confirm that the insensitivity to noise promotes the generalizability and stability of BERT.
We conduct experiments on four few-sample (less than 10k training samples) text classification tasks
RTE	MRPC	CoLA	STS-B
mean	std	max	mean	std	max	mean	std	max	mean	std	max
FT (Devlin et al., 2019)	70.13	1.84	72.56	87.57	0.92	89.16	60.54	1.49	62.59	89.38	0.53	90.23
L2-SP (Li et al., 2018)	70.58	1.29	73.28	87.74	0.86	88.95	60.19	1.42	63.89	89.25	0.62	90.14
Mixout(Lee et al., 2020)	71.35	1.66	74.36	87.63	0.62	88.91	63.12	1.68	65.12	89.58	0.35	90.11
SMART(Jiang et al., 2020)	72.23	2.41	75.45	87.86	0.63	89.09	63.16	1.17	65.21	90.11	0.33	90.83
LNSR(ours)	73.31	1.55	76.17	88.50	0.56	90.02	63.35	1.05	65.99	90.23	0.31	90.97
Table 1: The mean, standard deviation, and maximum performance on the development set of RTE, MRPC, CoLA, and STS-B tsak across 25 random seeds when fine-tuning the BERT-Large model with various regularization methods. FT refers to the standard BERT fine-tuning. Standard deviation: lower is better.
Figure 2: Performance distribution box plot of each model on the four tasks from 25 random seeds.
of GLUE<sup>1</sup>, the datasets are described below and summarized in Appendix A Table 4.
Corpus of Linguistic Acceptability (CoLA (Warstadt et al., 2019)) consists of English acceptability judgments drawn from books and journal articles on linguistic theory. Each example is a sequence of words annotated with whether it is a grammatical English sentence. This is a binary classification task and Matthews correlation coefficient (MCC) (Matthews, 1975) is used to evaluate the performance.
Microsoft Research Paraphrase Corpus (MRPC (Dolan and Brockett, 2005)) is a corpus of sentence pairs with human annotations for whether the sentences in the pair are semantically equivalent. The evaluation metrics is the average of F1 and Accuracy.
Recognizing Textual Entailment (RTE (Wang et al., 2018)) (Dagan et al., 2005) (Bar-Haim et al., 2006) (Giampiccolo et al., 2007) is a corpus of textual entailment, and each example is a sentence pair annotated whether the first entails the second. The evaluation metrics is Accuracy.
Semantic Textual Similarity Benchmark (STS-B (Cer et al., 2017)) is a regression task. Each example is a sentence pair and is human-annotated with a similarity score from 1 to 5; the task is to predict these scores. The evaluation metrics is the average of Pearson and Spearman correlation coef
ficients.
We use BERT (Devlin et al., 2019), a large-scale bidirectional pre-trained language model as the base model in all experiments. We adopt pytorch edition implemented by Wolf et al. (2019).
Fine-tuning. We use the standard BERT fine-tuning method described in Devlin et al. (2019).
$\mathbf{L}^2$ -SP (Li et al., 2018) is a regularization scheme that explicitly promotes the similarity of the final solution with the initial model. It is usually used for preventing pre-trained models from catastrophic forgetting. We adopt the form of  $\Omega(w) = \frac{\alpha}{2} ||w_s - w_s^0|| + \frac{\beta}{2} ||w_{\bar{s}}||$ .
Mixout (Lee et al., 2020) is a stochastic regularization technique motivated by Dropout (Srivastava et al., 2014) and DropConnect (Wan et al., 2013). At each training iteration, each model parameter is replaced with its pre-trained value with probability  $p$ . The goal is to improve the generalizability of pre-trained language models.
SMART (Jiang et al., 2020) imposes an smoothness regularizer inducing an adversarial manner to control the model complexity at the fine-tuning stage. It also employs a class of Bregman proximal point optimization methods to prevent the model from aggressively updating during fine-tuning.
<sup>1</sup>https://gluebenchmark.com/