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Title: Robustness Verification for Transformers

Paper content:
Robustness verification that aims to formally certify the prediction behavior of  neural networks has become an important tool for understanding the behavior of a given model and for obtaining safety guarantees.
However, previous methods are usually limited to relatively simple neural networks.
In this paper, we consider the robustness verification problem for Transformers.
Transformers have complex self-attention layers that pose many challenges for verification, including cross-nonlinearity and cross-position dependency, which have not been discussed in previous work.
We resolve these challenges and develop the first verification algorithm for Transformers.
The certified robustness bounds computed by our method are significantly tighter than those by naive Interval Bound Propagation.
These bounds also shed light on interpreting Transformers as they consistently reflect the importance of words in sentiment analysis.
Deep neural networks have been successfully applied to many domains.
However, a major criticism is that these black box models are difficult to analyze and their behavior is not guaranteed.
Moreover, it has been shown that the predictions of deep networks become unreliable and unstable when tested in unseen situations, e.g., in the presence of small and adversarial perturbation to the input (Szegedy et al., 2013; Goodfellow et al., 2014; Lin et al., 2019) .
Therefore, neural network verification has become an important tool for analyzing and understanding the behavior of neural networks, with applications in safety-critical applications (Katz et al., 2017; Julian et al., 2019; Lin et al., 2019) , model explanation (Shih et al., 2018) and robustness analysis (Tjeng et al., 2019; Wang et al., 2018c; Gehr et al., 2018; Wong & Kolter, 2018; Singh et al., 2018; Weng et al., 2018; Zhang et al., 2018) .
Formally, a neural network verification algorithm aims to provably characterize the prediction of a network within some input space.
For example, given a K-way classification model f : R d → R K , we can verify some linear specification (defined by a vector c) as below:
where S is a predefined input space.
For example, in the robustness verification problem that we are going to focus on in this paper, S = {x | x−x 0 p ≤ } is defined as some small p -ball around the original example x 0 , and setting up c = 1 y0 − 1 y can verify whether the logit output of class y 0 is always greater than another class y within S. This is a nonconvex optimization problem which makes computing the exact solution challenging, and thus algorithms are recently proposed to find lower bounds of Eq. (1) in order to efficiently obtain a safety guarantee (Gehr et al., 2018; Weng et al., 2018; Zhang et al., 2018; Singh et al., 2019) .
Moreover, extension of these algorithms can be used for verifying some properties beyond robustness, such as rotation or shift invariant (Singh et al., 2019) , conservation of energy (Qin et al., 2019) and model correctness (Yang & Rinard, 2019) .
However, most of existing verification methods focus on relatively simple neural network architectures, such as feed-forward and recurrent neural networks, and cannot handle complex structures.
In this paper, we develop the first robustness verification algorithm for Transformers (Vaswani et al., 2017) with self-attention layers.
Transformers have been widely used in natural language processing (Devlin et al., 2018; Yang et al., 2019; Liu et al., 2019) and many other domains (Parmar et al., 2018; Kang & McAuley, 2018; Li et al., 2019b; Su et al., 2019; Li et al., 2019a) .
For frames under perturbation in the input sequence, we aim to compute a lower bound such that when these frames are perturbed within p -balls centered at the original frames respectively and with a radius of , the model prediction is certified to be unchanged.
To compute such a bound efficiently, we adopt the linear-relaxation framework (Weng et al., 2018; Zhang et al., 2018 ) -we recursively propagate and compute linear lower bound and upper bound for each neuron with respect to the input within perturbation set S.
We resolve several particular challenges in verifying Transformers.
First, Transformers with selfattention layers have a complicated architecture.
Unlike simpler networks, they cannot be written as multiple layers of linear transformations or element-wise operations.
Therefore, we need to propagate linear bounds differently for self-attention layers.
Second, dot products, softmax, and weighted summation in self-attention layers involve multiplication or division of two variables under perturbation, namely cross-nonlinearity, which is not present in feed-forward networks.
Ko et al. (2019) proposed a gradient descent based approach to find linear bounds, however it is inefficient and poses a computational challenge for transformer verification as self-attention is the core of transformers.
In contrast, we derive closed-form linear bounds that can be computed in O(1) complexity.
Third, neurons in each position after a self-attention layer depend on all neurons in different positions before the self-attention (namely cross-position dependency), unlike the case in recurrent neural networks where outputs depend on only the hidden features from the previous position and the current input.
Previous works (Zhang et al., 2018; Weng et al., 2018; Ko et al., 2019) have to track all such dependency and thus is costly in time and memory.
To tackle this, we introduce an efficient bound propagating process in a forward manner specially for self-attention layers, enabling the tighter backward bounding process for other layers to utilize bounds computed by the forward process.
In this way, we avoid cross-position dependency in the backward process which is relatively slower but produces tighter bounds.
Combined with the forward process, the complexity of the backward process is reduced by O(n) for input length n, while the computed bounds remain comparably tight.
Our contributions are summarized below:
• We propose an effective and efficient algorithm for verifying the robustness of Transformers with self-attention layers.
To our best knowledge, this is the first method for verifying Transformers.
• We resolve key challenges in verifying Transformers, including cross-nonlinearity and crossposition dependency.
Our bounds are significantly tighter than those by adapting Interval Bound Propagation (IBP) (Mirman et al., 2018; .
• We quantitatively and qualitatively show that the certified lower bounds consistently reflect the importance of input words in sentiment analysis, which justifies that the computed bounds are meaningful in practice.
We propose the first robustness verification method for Transformers, and tackle key challenges in verifying Transformers, including cross-nonlinearity and cross-position dependency, for efficient and effective verification.
Our method computes certified lower bounds that are significantly tighter than those by IBP.
Quantitative and qualitative analyses further show that our bounds are meaningful and can reflect the importance of different words in sentiment analysis.
A ILLUSTRATION OF DIFFERENT BOUNDING PROCESSES Figure 1 : Illustration of three different bounding processes: Fully-Forward
(a), Fully-Backward
(b), and Backward&Forward
(c).
We show an example of a 2-layer Transformer, where operations can be divided into two kinds of blocks, "Feed-forward" and "Self-attention".
"Self-attention" contains operations in the self-attention mechanism starting from queries, keys, and values, and "Feed-forward" contains all the other operations including linear transformations and unary nonlinear functions.
Arrows with solid lines indicate the propagation of linear bounds in a forward manner.
Each backward arrow A k → B k with a dashed line for blocks A k , B k indicates that there is a backward bound propagation to block B k when computing bounds for block A k .
Blocks with blue rectangles have forward processes inside the blocks, while those with green rounded rectangles have backward processes inside.
Backward & Forward algorithm, we use backward processes for the feed-forward parts and forward processes for self-attention layers, and for layers after self-attention layers, they no longer need backward bound propagation to layers prior to self-attention layers.
In this way, we resolve the cross-position dependency in verifying Transformers while still keeping bounds comparably tight as those by using fully backward processes.
Empirical comparison of the three frameworks are presented in Sec. 4.3.

assistant: text: We propose the first algorithm for verifying the robustness of Transformers.