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+# SWITCHING LINEAR DYNAMICS FOR VARIATIONAL BAYES FILTERING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+System identification of complex and nonlinear systems is a central problem for model predictive control and model-based reinforcement learning. Despite their complexity, such systems can often be approximated well by a set of linear dynamical systems if broken into appropriate subsequences. This mechanism not only helps us find good approximations of dynamics, but also gives us deeper insight into the underlying system. Leveraging Bayesian inference and Variational Autoencoders, we show how to learn a richer and more meaningful state space, e.g. encoding joint constraints and collisions with walls in a maze, from partial and high-dimensional observations. This representation translates into a gain of accuracy of the learned dynamics which we showcase on various simulated tasks.
+
+# 1 INTRODUCTION
+
+Learning dynamics from raw data (also known as system identification) is a key component of model predictive control and model-based reinforcement learning. Problematically, environments of interest often give rise to very complex and highly nonlinear dynamics which are seemingly difficult to approximate. However, switching linear dynamical systems (SLDS) approaches claim that those environments can often be broken down into simpler units made up of areas of equal and linear dynamics (Ackerson & Fu, 1970; Chang & Athans, 1978). Not only are those approaches capable of good predictive performance, which often is the sole goal of learning a system’s dynamics, they also encode valuable information into so called switching variables which determine the dynamics of the next transition. For example, when looking at the movement of an arm, one is intuitively aware of certain restrictions of possible movements, e.g. constraints to the movement due to joint constraints or obstacles. The knowledge is present without the need to simulate; it’s explicit. Exactly this kind of information will be encoded when successfully learning switching dynamics. Our goal in this work will therefore entail the search for richer representations in the form of latent state space models which encode knowledge about the underlying system dynamics. In turn, we expect this to improve the accuracy of our simulation as well. Such a representation alone could then be used in a reinforcement learning approach that possibly only takes advantage of the learned latent features but not necessarily its learned dynamics.
+
+To learn richer representations, we identify one common problem with prevalent recurrent Variational Autoencoder models (Karl et al., 2017a; Krishnan et al., 2015; Chung et al., 2015; Fraccaro et al., 2016): the non-probabilistic treatment of the transition dynamics often modeled by a powerful nonlinear function approximator. From the history of the Autoencoder to the Variational Autoencoder, we know that in order to detect features in an unsupervised manner, probabilistic treatment of the latent space is paramount. As our starting point, we will build on previously proposed approaches by Krishnan et al. (2017) and Karl et al. (2017a). The latter already made use of locally linear dynamics, but only in a deterministic fashion. We extend their approaches by a stochastic switching LDS model and show that such treatment is vital for learning richer representations and simulation accuracy.
+
+# 2 BACKGROUND
+
+We consider discretized time-series data consisting of continuous observations $x _ { t } \in \mathcal { X } \subset \mathbb { R } ^ { n _ { x } }$ and control inputs $u _ { t } \in \mathcal { U } \subset \mathbb { R } ^ { n _ { u } }$ that we would like to model by corresponding latent states $z _ { t } \in \mathcal { Z } \subset \mathbb { R } ^ { n _ { z } }$ . We’ll denote sequences of variables by $x _ { 1 : T } = ( x _ { 1 } , x _ { 2 } , . . . , x _ { T } )$ .
+
+![](images/da18197c9717341f6caf03dbb955c01e82140562065970885fef03304c9f77aa.jpg)  
+Figure 1: (a) $s _ { t }$ denote discrete switch variables, $z _ { t }$ are continuous latent variables, $x _ { t }$ continuous observed variables, $u _ { t }$ are (optional) continuous control inputs. (b) By introducing a special latent variable $w$ used for initial state inference, we want to make explicit that the first step is treated differently from the rest of the sequence.
+
+# 2.1 SWITCHING LINEAR DYNAMICAL SYSTEMS
+
+Switching Linear Dynamical System models (SLDS) enable us to model nonlinear time series data by splitting it into sequences of linear dynamical models. At each time $t = 1 , 2 , . . . , T$ , a discrete switch variable $s _ { t } \in { 1 , . . . , M }$ chooses of a set LDSs a system which is to be used to transform our continuous latent state $z _ { t }$ to the next time step (Barber, 2012).
+
+$$
+\begin{array} { r l r l } & { z _ { t } = A ( s _ { t } ) z _ { t - 1 } + B ( s _ { t } ) u _ { t - 1 } + \epsilon ( s _ { t } ) \quad } & & { \epsilon ( s _ { t } ) \sim { \mathcal N } ( 0 , Q ( s _ { t } ) ) } \\ & { x _ { t } = H ( s _ { t } ) z _ { t } + \eta ( s _ { t } ) \quad } & & { \eta ( s _ { t } ) \sim { \mathcal N } ( 0 , R ( s _ { t } ) ) } \end{array}
+$$
+
+Here $A \in \mathbb { R } ^ { n _ { z } \times n _ { z } }$ is the state matrix, $B \in \mathbb { R } ^ { n _ { z } \times n _ { u } }$ control matrix, $\epsilon$ the transition noise with covariance matrix $Q$ and $\eta$ the emission/sensor noise with covariance matrix $R$ . Finally, the observation matrix $H \in \mathbb { R } ^ { n _ { x } \times n _ { z } }$ defines a linear mapping from latent to observation space which we will replace by a nonlinear transformation parameterized by a neural net. These equations imply the following joint distribution:
+
+$$
+p \left( { x _ { 1 : T } , z _ { 1 : T } , s _ { 1 : T } } \mid { u _ { 1 : T } } \right) = \prod _ { t = 1 } ^ { T } p \left( { x _ { t } } \mid { z _ { t } } \right) p \left( { z _ { t } } \mid { z _ { t - 1 } , u _ { t - 1 } , s _ { t } } \right) p \left( { s _ { t } } \mid { z _ { t - 1 } , u _ { t - 1 } , s _ { t - 1 } } \right)
+$$
+
+with $p ( z _ { 1 } \mid z _ { 0 } , u _ { 0 } , s _ { 1 } ) = p ( z _ { 1 } )$ being the initial state distribution. The corresponding graphical model is shown in figure 1a.
+
+# 2.2 STOCHASTIC GRADIENT VARIATIONAL BAYES
+
+$$
+p ( x ) = \int p ( x , z ) \mathrm { d } z = \int p ( x \mid z ) p ( z ) \mathrm { d } z
+$$
+
+Given the simple graphical model in equation (3), Kingma & Welling (2014) and Rezende et al. (2014) introduced the Variational Autoencoder (VAE) which overcomes the intractability of posterior inference of $q ( z \mid x )$ by maximizing the evidence lower bound (ELBO) of the model log-likelihood.
+
+$$
+\mathcal { L } _ { \mathrm { E L B O } } ( x ; \theta , \phi ) = \mathbb { E } _ { q _ { \phi } ( z | x ) } [ \ln p _ { \theta } ( x \mid z ) ] - D _ { \mathrm { K L } } ( q _ { \phi } ( z \mid x ) \mid | p ( z ) ) \le \log p ( x )
+$$
+
+Their main innovation was to approximate the intractable posterior distribution by a recognition network $q _ { \phi } ( z | x )$ from which they can sample via the reparameterization trick to allow for stochastic backpropagation through both the recognition and generative model at once. Assuming that the latent state is normally distributed, a simple transformation allows us to obtain a Monte Carlo gradient estimate of $\mathbb { E } _ { q _ { \phi } ( z | x ) } \left[ \ln p _ { \theta } ( x | z ) \right]$ w.r.t. to $\phi$ . Given that $z \sim \mathcal { N } ( \mu , \sigma ^ { 2 } )$ , we can generate samples by drawing from an auxiliary variable $\epsilon \sim \mathcal { N } ( 0 , 1 )$ and applying the deterministic and differentiable transformation $z = \mu + \sigma \epsilon$ .
+
+# 2.3 THE CONCRETE DISTRIBUTION
+
+One simple and efficient way to obtain samples $d$ from a $k$ -dimensional categorical distribution with class probabilities $\alpha$ is the Gumbel-Max trick:
+
+$$
+d = \mathrm { o n e \_ h o t } \left( \mathrm { a r g m a x } \big [ g _ { i } + \log \alpha _ { i } \big ] \right) , \quad \mathrm { w i t h } \ g _ { 1 } , \dots , g _ { k } \sim \mathrm { G u m b e l } ( 0 , 1 )
+$$
+
+However, since the derivative of the argmax is 0 everywhere except at the boundary of state changes, where it is undefined, we can’t learn a parameterization by backpropagation. The Gumbel-Softmax trick approximates the argmax by a softmax which gives us a probability vector (Maddison et al., 2017; Jang et al., 2017). We can then draw samples via
+
+$$
+d _ { k } = \frac { \exp ( ( \log \alpha _ { k } + g _ { k } ) / \lambda ) } { \sum _ { i = 1 } ^ { n } \exp ( ( \log \alpha _ { i } + g _ { i } ) / \lambda ) } , \quad \mathrm { w i t h ~ } g _ { 1 } , \dots , g _ { k } \sim \mathrm { G u m b e l } ( 0 , 1 )
+$$
+
+This softmax computation approaches the discrete argmax as temperature $\lambda  0$ , for $\lambda \to \infty$ it approaches a uniform distribution.
+
+# 3 RELATED WORK
+
+Our model can be viewed as a Deep Kalman Filter (Krishnan et al., 2015) with structured inference (Krishnan et al., 2017). In our case, structured inference entails another stochastic variable model with parameter sharing inspired by Karl et al. (2017b) and Karl et al. (2017a) which pointed out the importance of backpropagating the reconstruction error through the transition. We are different to a number of stochastic sequential models like Bayer & Osendorfer (2014); Chung et al. (2015); Shabanian et al. (2017); Goyal et al. (2017) by directly transitioning the stochastic latent variable over time instead of having an RNN augmented by stochastic inputs. Fraccaro et al. (2016) has a transition over both a deterministic and a stochastic latent state sequence, wanting to combine the best of both worlds.
+
+Previous models (Watter et al., 2015; Karl et al., 2017a; Fraccaro et al., 2017) have already combined locally linear models with recurrent Variational Autoencoders, however they provide a weaker structural incentive for learning latent variables determining the transition function. Van Steenkiste et al. (2018) approach a similar multi bouncing ball problem (see section 5.1) by first distributing the representation of different balls into their own entities without supervision and then structurally hardwiring a transition function with interactions based on an attention mechanism.
+
+Recurrent switching linear dynamical systems (Linderman et al., 2016) uses message passing for approximate inference, but has restricted itself to low-dimensional observations and a multi-stage training process. Johnson et al. (2016) propose a similar model to ours but combine message passing for discrete switching variables with a neural network encoder for observations learned by stochastic backpropagation. Tackling the problem of propagating state uncertainty over time, various combinations of neural networks for inference and Gaussian processes for transition dynamics have been proposed (Eleftheriadis et al., 2017; Doerr et al., 2018). However, these models have not been demonstrated to work with high-dimensional observation spaces like images. One feature a switching LDS model may learn are interactions which have recently been approached by employing Graph Neural Networks (Battaglia et al., 2016; Kipf et al., 2018). These methods are similar in that they predict edges which encode interactions between components of the state space (nodes).
+
+# 4 PROPOSED APPROACH
+
+Our goal is to fit a series of continuous state $z _ { 1 : T }$ and switching variables $s _ { 2 : T }$ to a given sequence of observations $x _ { 1 : T }$ . We assume a nonlinear mapping between observations and latent space which we generally approximate by neural networks, apart from the transition which is modeled by a locally linear function. Our generative model is shown in figure 1b an our inference model in figure 2a.
+
+# 4.1 GENERATIVE MODEL
+
+Our generative model for a single $x _ { t }$ is described by
+
+$$
+p ( x _ { t } ) = \int _ { s \leq t } \int _ { z \leq t } p ( x _ { t } \mid z _ { t } ) p ( z _ { t } \mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } ) p ( s _ { t } \mid s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } ) p ( z _ { t - 1 } , s _ { t - 1 } ) d s _ { t } d t .
+$$
+
+which is close to the one of the original SLDS model (see figure 1a). Latent states $z _ { t }$ are continuous and represent the state of the system while states $s _ { t }$ are the switching variables determining the transition. We approximate the discrete switching variables by a continuous relaxation, namely the Concrete distribution.1 Differently to the original model, we do not condition the likelihood of the current observation $p _ { \theta } ( x _ { t } \mid z _ { t } )$ directly on the switching variables. This limits the influence of the switching variables to choosing a proper transition dynamic for the continuous latent space. The likelihood model is parameterized by a neural network with either a Gaussian or a Bernoulli distribution as output depending on the data.
+
+There is both a transition on the continuous states $z _ { t }$ and discrete latent states $s _ { t }$ . For the continuous state transition $p ( z _ { t } \mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } )$ we follow (1) and maintain a set of $M$ base matrices $\{ \left( A ^ { ( i ) } , B ^ { ( i ) } , Q ^ { ( i ) } \right) | \forall i . 0 < i < M \}$ as our linear dynamical systems to choose from. For the transition on discrete latent states $p ( s _ { t } \mid s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } )$ , we usually require the learning of a Markov transition matrix. However, since we approximate our discrete switching variables by a continuous relaxation, we can parameterize this transition by a neural network. Therefore, our entire generative model can be learned end-to-end by (stochastic) backpropagation. Finally, the resulting dynamics matrices are computed through a linear combination of the base matrices:
+
+$$
+A _ { t } ( s _ { t } ) = \sum _ { i = 1 } ^ { M } s _ { t } ^ { ( i ) } A ^ { ( i ) } , \qquad B ( s _ { t } ) = \sum _ { i = 1 } ^ { M } s _ { t } ^ { ( i ) } B ^ { ( i ) } , \qquad Q ( s _ { t } ) = \sum _ { i = 1 } ^ { M } s _ { t } ^ { ( i ) } Q ^ { ( i ) }
+$$
+
+Both transition models – the continuous state transition $p _ { \theta } ( z _ { t } \mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } )$ and concrete switching variables transition $p _ { \theta } \big ( s _ { t } \ | \ s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } \big ) -$ are shared with the inference model which is key for good performance.
+
+$$
+\begin{array} { r l } & { \quad p _ { \theta } ( z _ { t } \mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } ) = \mathcal { N } \big ( \mu , \sigma ^ { 2 } \big ) \qquad \mathrm { w h e r e } \ [ \mu , \sigma ^ { 2 } ] = f _ { \theta } ( z _ { t - 1 } , s _ { t } , u _ { t - 1 } ) } \\ & { \quad p _ { \theta } ( s _ { t } \mid s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } ) = \mathrm { C o n c r e t e } ( \alpha , \lambda _ { \mathrm { p r i o r } } ) \qquad \mathrm { w h e r e } \ \alpha = g _ { \theta } ( z _ { t - 1 } , s _ { t - 1 } , u _ { t - 1 } ) } \end{array}
+$$
+
+# 4.2 INFERENCE
+
+# 4.2.1 STRUCTURED INFERENCE OF CONTINUOUS LATENT STATE
+
+We split our inference model $q _ { \phi } ( z _ { t } \mid z _ { t - 1 } , s _ { t } , x _ { \geq t } , u _ { \geq t - 1 } )$ into two parts: 1) transition model $q _ { \mathrm { t r a n s } } \bar { ( } z _ { t } \mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } )$ and 2) inverse measurement model $q _ { \mathrm { m e a s } } ( z _ { t } \mid x _ { \geq t } , u _ { \geq t } )$ as previously proposed in Karl et al. (2017b). This split allows us to reuse our generative transition model in place of $q _ { \mathrm { t r a n s } } ( z _ { t } \mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } )$ . This sharing of variables is essential for good performance as it forces the reconstruction error to be backpropagated throughwe only share the computation of the transition mean $\mu _ { \mathrm { t r a n s } }$ nsition model. For prbut not the variance $\sigma _ { \mathrm { t r a n s } } ^ { 2 }$ l reasons,between inference and generative model. Both parts, and $q _ { \mathrm { t r a n s } }$ , will give us independent predictions about the new state $z _ { t }$ which will be combined in a manner akin to a Bayesian update in a Kalman Filter.
+
+$$
+\begin{array} { r } { l \phi \left( z _ { t } \mid z _ { t - 1 } , s _ { t } , x _ { \geq t } , u _ { \geq t - 1 } \right) \propto q _ { \operatorname* { m e a s } } \left( z _ { t } \mid x _ { \geq t } , u _ { \geq t } \right) \times q _ { \mathrm { t r a n s } } \left( z _ { t } \mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } \right) = \mathcal { N } \left( \mu _ { q } , \sigma _ { q } ^ { 2 } \right) } \\ { q _ { \operatorname* { m e a s } } \left( z _ { t } \mid x _ { \geq t } , u _ { \geq t } \right) = \mathcal { N } \left( \mu _ { \operatorname* { m e a s } } , \sigma _ { \operatorname* { m e a s } } ^ { 2 } \right) \mathrm { ~ w h e r e ~ } \left[ \mu _ { \operatorname* { m e a s } } , \sigma _ { \operatorname* { m e a s } } ^ { 2 } \right] = h _ { \phi } \left( x _ { \geq t } , u _ { \geq t } \right) \quad \mathrm { ( } 1 \mathrm { ~ t ~ } } \\ { q _ { \operatorname { t r a n s } } \left( z _ { t } \mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } \right) = \mathcal { N } \left( \mu _ { \operatorname { t r a n s } } , \sigma _ { \operatorname* { t r a n s } } ^ { 2 } \right) \mathrm { ~ w h e r e ~ } \left[ \mu _ { \operatorname { t r a n s } } , \sigma _ { \operatorname { t r a n s } } ^ { 2 } \right] = f _ { \theta } \left( z _ { t - 1 } , s _ { t } , u _ { t - 1 } \right) } \end{array}
+$$
+
+The densities of $q _ { \mathrm { m e a s } }$ and $q _ { \mathrm { t r a n s } }$ are multiplied resulting in another Gaussian density:
+
+$$
+\mu _ { q } = \frac { \mu _ { \mathrm { t r a n s } } \sigma _ { \mathrm { m e a s } } ^ { 2 } + \mu _ { \mathrm { m e a s } } \sigma _ { \mathrm { t r a n s } } ^ { 2 } } { \sigma _ { \mathrm { m e a s } } ^ { 2 } + \sigma _ { \mathrm { t r a n s } } ^ { 2 } } , \qquad \sigma _ { q } ^ { 2 } = \frac { \sigma _ { \mathrm { m e a s } } ^ { 2 } \sigma _ { \mathrm { t r a n s } } ^ { 2 } } { \sigma _ { \mathrm { m e a s } } ^ { 2 } + \sigma _ { \mathrm { t r a n s } } ^ { 2 } }
+$$
+
+This update scheme is highlighted in figure 2b.
+
+We found empirically that conditioning the inverse measurement model $q _ { \mathrm { m e a s } } ( z _ { t } \mid x _ { \geq t } , u _ { \geq t } )$ solely on the current observation $x _ { t }$ instead of the entire remaining trajectory to lead to better results. We hypothesize that the recurrent model needlessly introduces very high-dimensional and complicated dynamics which are harder to approximate with our locally linear transition model.
+
+For the initial state $z _ { 1 }$ we do not have a conditional prior from the transition model as in the rest of the sequence. Other methods (Krishnan et al., 2015) have used a standard normal prior, however this is not a good fit. We therefore decided that instead of predicting $z _ { 1 }$ directly to predict an auxiliary variable $w$ that is then mapped deterministically to a starting state $z _ { 1 }$ . A standard Gaussian prior is then applied to $w$ . Alternatively, we could specify a more complex or learned prior for the initial state like the VampPrior (Tomczak & Welling, 2017). Empirically, this has lead to worse results.
+
+$$
+\begin{array} { r l } { q _ { \phi } ( w \mid x _ { 1 : T } , u _ { 1 : T } ) = { \mathcal { N } } { \big ( } w ; \mu _ { w } , \sigma _ { w } ^ { 2 } { \big ) } } & { { } { \mathrm { w h e r e } } \quad [ \mu _ { w } , \sigma _ { w } ^ { 2 } ] = i _ { \phi } ( x _ { 1 : T } , u _ { 1 : T } ) } \\ { z _ { 1 } = f _ { \phi } ( w ) } & { { } } \end{array}
+$$
+
+While we could condition on the entire sequence, we restrict it to just the first couple of observations.
+
+![](images/77ec647f77f9f545bed126f45f7c914263e5c545ff3e46593cee575823abc31e.jpg)  
+Figure 2: (a) Depicts the inference model. $b _ { t }$ is the hidden state of the backward RNN of $q _ { \phi } \mathbf { \bar { ( } } s _ { t } \mid x _ { \geq t } , u _ { \geq t } )$ . Initial inference of $w$ may be conditioned on the entire sequence of observations, or just a subsequence. We��ve omitted the arrows for sake of clarity for the rest of the graph. (b) Shows schematically how we combine the transition with the inverse measurement model in the inference network. Transitions (in blue) are (partially) shared with the generative model.
+
+# 4.2.2 INFERENCE OF SWITCHING VARIABLES
+
+Following Maddison et al. (2017) and Jang et al. (2017), we can reparameterize a discrete latent variable with the Gumbel-softmax trick. Again, we split our inference network $q _ { \phi } ( s _ { t } \mid s _ { t - 1 } , z _ { t - 1 } , x _ { \geq t } , u _ { \geq t - 1 } )$ in an identical fashion into two components: 1) Transition model $q _ { \mathrm { t r a n s } } ( s _ { t } \mid s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } )$ and 2) inverse measurement model $q _ { \mathrm { m e a s } } ( s _ { t } \mid x _ { \geq t } , u _ { \geq t } )$ . The transition model is again shared with the generative model and is implemented via a neural network as we potentially require quick changes to chosen dynamics. The inverse measurement model is parametrized by a backward LSTM. However, for the case of concrete variables, we cannot do the same Gauss multiplication as in the previous case. Therefore, we let each network predict the logits of a Concrete distribution and our inverse measurement model $q _ { \phi } ( s _ { t } \mid x _ { \geq t } , u _ { \geq t } )$ produces an additional vector $\gamma$ , which determines the value of a gate deciding how the two predictions are to be weighted:
+
+$$
+\begin{array} { r l } & { q _ { \phi } \big ( s _ { t } \mid s _ { t - 1 } , z _ { t - 1 } , x _ { \geq t } , u _ { \geq t - 1 } \big ) = \mathrm { C o n c r e t e } \big ( \alpha , \lambda _ { \mathrm { p o s t e r i o r } } \big ) \quad \mathrm { w i t h } \quad \alpha = \gamma \alpha _ { \mathrm { t r a n s } } + ( 1 - \gamma ) \alpha _ { \mathrm { m e a s } } } \\ & { q _ { \operatorname* { m e a s } } \big ( s _ { t } \mid x _ { \geq t } , u _ { \geq t } \big ) = \mathrm { C o n c r e t e } \big ( \alpha _ { \mathrm { m e a s } } , \lambda _ { \mathrm { p o s t e r i o r } } \big ) \quad \mathrm { w h e r e } \quad \big [ \alpha _ { \mathrm { m e a s } } , \gamma \big ] = k _ { \phi } \big ( x _ { \geq t } , u _ { \geq t } \big ) \qquad ( 1 3 ) } \\ & { q _ { \mathrm { t r a n s } } \big ( s _ { t } \mid s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } \big ) = \mathrm { C o n c r e t e } \big ( \alpha _ { \mathrm { t r a n s } } , \lambda _ { \mathrm { p r i o r } } \big ) \quad \mathrm { w h e r e } \quad \alpha = g _ { \theta } \big ( z _ { t - 1 } , s _ { t - 1 } , u _ { t - 1 } \big ) } \end{array}
+$$
+
+The temperatures $\lambda _ { \mathrm { p o s t e r i o r } }$ and $\lambda _ { \mathrm { p r i o r } }$ are set as a hyperparameter and can be set differently for the prior and approximate posterior. The gating mechanism gives the model the option to balance between prior and approximate posterior. If the prior is good enough to explain the next observation, $\gamma$ will be pushed to 1 which ignores the measurement and minimizes the KL between prior and posterior by only propagating the prior. If the prior is not sufficient, information from the inverse measurement model can flow by decreasing $\gamma$ and incurring a KL penalty.
+
+Since the concrete distribution is a relaxation of the categorical, our sample will not be a one-hot vector, but a vector whose elements sum up to 1. We face two options here: we could take a categorical sample by choosing the linear system corresponding to the highest value in the sample (hard forward pass) and only use the relaxation for our backward pass. This, however, means that we will follow a biased gradient. Alternatively, we can use the relaxed version for our forward pass and aggregate the linear systems based on their corresponding weighting (see (8)). Here, we lose the discrete switching of linear systems, but maintain a valid lower bound. We note that the hard forward pass has led to worse results and focus on the soft forward pass for this paper.
+
+Lastly, we could go further away from the theory and instead treat the switching variables also as normally distributed. If this worked better than the approach with Concrete variables, it would highlight still existing optimization problems of discrete random variables. As such, it will act as an ablation study for our model. The mixing coefficients for linear systems would then be determined by a linear combination of these latent variables:
+
+$$
+\alpha = \operatorname { s o f t m a x } ( W s _ { t } + b ) \in \mathbb { R } ^ { M }
+$$
+
+Our inference scheme for normally distributed switching variables is then identical to the one described in the previous section. We compare both approaches throughout our experimental section.
+
+# 4.3 TRAINING
+
+Our objective function is the commonly used evidence lower bound for our hierarchical model.
+
+$$
+\begin{array} { r l } { \mathcal { L } _ { \theta , \phi } \big ( x _ { 1 : T } \ \big | \ u _ { 1 : T } \big ) \geq } & { \mathbb { E } _ { q _ { \phi } ( z _ { 1 : T } , s _ { 1 : T } \mid x _ { 1 : T } ) } \big [ \log p _ { \theta } \big ( x _ { 1 : T } \ \big | \ z _ { 1 : T } , s _ { 1 : T } , u _ { 1 : T } \big ) \big ] } \\ & { - D _ { \mathrm { K L } } \big ( q _ { \phi } \big ( z _ { 1 : T } , s _ { 1 : T } \ \big | \ x _ { 1 : T } , u _ { 1 : T } \big ) \ \big | \ \big | \ p \big ( z _ { 1 : T } , s _ { 1 : T } \ \big | \ u _ { 1 : T } \big ) \big ) } \end{array}
+$$
+
+We choose to factorize over time, so the loss for a single observation $x _ { t }$ becomes:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \theta , \phi } ( x _ { t } \mid u _ { 1 : T } ) = \mathbb { E } _ { q _ { \phi } \left( s _ { t } \mid s _ { t - 1 } , z _ { t - 1 } , x _ { \geq t } , u _ { \geq t - 1 } \right) } \left[ \mathbb { E } _ { q _ { \phi } \left( z _ { t } \mid s _ { t } , z _ { t - 1 } , x _ { \geq t } , u _ { \geq t - 1 } \right) } \left[ \log p _ { \theta } ( x _ { t } \mid z _ { t } ) \right] \right] \qquad ( 1 6 ) } \\ & { \qquad - \mathbb { E } _ { s _ { t - 1 } } \left[ \mathbb { E } _ { z _ { t - 1 } } \left[ D _ { \mathrm { K L } } \left( q _ { \phi } \left( s _ { t } \mid s _ { t - 1 } , z _ { t - 1 } , x _ { \geq t } , u _ { \geq t - 1 } \right) \mid \mid p _ { \theta } \left( s _ { t } \mid s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } \right) \right) \right] \right] } \\ & { \qquad - \mathbb { E } _ { z _ { t - 1 } } \left[ \mathbb { E } _ { s _ { t } } \left[ D _ { \mathrm { K L } } \left( q _ { \phi } \left( z _ { t } \mid z _ { t - 1 } , s _ { t } , x _ { \geq t } , u _ { \geq t - 1 } \right) \mid \mid p _ { \theta } \left( z _ { t } \mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } \right) \right) \right] \right] } \end{array}
+$$
+
+The full derivation can be found in appendix A. We learn the parameters of our model by backpropagation through time and we (generally) approximate the expectations with one sample by using the reparametrization trick. The exception is the KL between two Concrete random variables in which case we take 10 samples for the approximation. For the KL on the switching variables, we further introduce a scaling factor $\beta < 1$ (as first suggested in Higgins et al. (2016), although they suggested increasing the KL term) to down weigh its importance. More details on the training procedure can be found in appendix B.2.
+
+# 5 EXPERIMENTS
+
+In this section, we evaluate our approach on a diverse set of physics and robotics simulations based on partially observable system states or high-dimensional images as observations. We show that our model outperforms previous models and that our switching variables learn meaningful representations.
+
+Models we compare to are Deep Variational Bayes Filter (DVBF) (Karl et al., 2017a), DVBF Fusion (Karl et al., 2017b) (called fusion as they do the same Gauss multiplication in the inference network) which is closest to our model but doesn’t have a stochastic treatment of the transition, the Kalman VAE (KVAE) (Fraccaro et al., 2017) and a LSTM (Hochreiter & Schmidhuber, 1997).
+
+![](images/e0194ebf04f7331684f7436c765cd1d05be1f6ea391f318e881011b153ff9194.jpg)  
+(a) Multi agent maze envi- (b) Variable encoding free (c) Variable encoding walls (d) System activation for ronment. space for agent 2. for agent 1. deterministic transition.
+
+Figure 3: Figures (b) and (c) depict an agent’s position colored by the average value of a single latent variable $s$ marginalized over all control inputs $u$ and velocities. Figure (d) highlights a representative activation for a single transition system for the deterministic treatment of the transition dynamics. It doesn’t generalize to the entire maze and stays fairly active in proximity to the wall.
+
+# 5.1 MULTIPLE BOUNCING BALLS IN A MAZE
+
+Our first experiment is a custom 3-agent maze environment simulated with Box2D. Each agent is fully described by its $x$ and $y$ coordinates and its current velocity and has the capability to accelerate in either direction. We learn in a partially observable setting and limit the observations to the agents’ positions, therefore $x \in \mathbb { R } ^ { 6 }$ while the true state space is in $\mathbb { R } ^ { 1 2 }$ and $u \in \mathbb { R } ^ { 6 }$ . First, we train a linear regression model on the latent space $z$ to see if we have recovered a linear encoding of the unobserved velocities. We achieve an R2 score of 0.92 averaged over all agents and velocity directions.
+
+Our focus shifts now to our switching variables which we expect to encode interactions with walls. We provide a visual confirmation of that in figure 3 where we see switching variables encoding all space where there is no interaction in the next time step, and variables which encode walls, distinguishing between vertical and horizontal ones. In figure 3d one can see show that if the choice of locally linear transition is treated deterministically, we don’t learn global features of the same kind. To confirm our visual inspection, we train a simple decision tree based on latent space $s$ in order to predict interaction with a wall. Here, we achieve an F1 score of 0.46. It is difficult to say what a good value should look like as collisions with low velocity are virtually indistinguishable from no collision.
+
+We compare our prediction quality to several other methods in table 1 where we outperform all of our chosen baselines. Also, modeling switching variables by a Normal distribution outperforms the Concrete distribution in all of our experiments. Aside from known practical issues with training a discrete variable via backpropagation, we explore one reason why that may be in section 5.4, which is the greater susceptibility to the scale of temporal discretization. We provide plots of predicted trajectories in appendix D. Transitioning multiple agents with a single transition matrix comes with scalability issues with regards to switching dynamics which we explore further in appendix C.
+
+Table 1: Mean squared error (MSE) on predicting future observations. Static refers to constantly predicting the first observation of the sequence.   
+
+<table><tr><td></td><td colspan="3">REACHER</td><td colspan="3">3-BALL MAZE</td></tr><tr><td>PREDICTION STEPS</td><td>1</td><td>5</td><td>10</td><td>1</td><td>5</td><td>10</td></tr><tr><td>STATIC</td><td>5.80E-02</td><td>5.36E-01</td><td>1.25E+00</td><td>1.40E-02</td><td>5.74E-01</td><td>2.65E+00</td></tr><tr><td>LSTM</td><td>3.07E-01</td><td>7.76E-01</td><td>1.22E+00</td><td>7.20E-02</td><td>1.58E-01</td><td>2.60E-01</td></tr><tr><td>DVBF</td><td>1.10E-01</td><td>3.08E-01</td><td>6.07E-01</td><td>6.20E-02</td><td>1.36E-01</td><td>1.82E-01</td></tr><tr><td>DVBFFUSION</td><td>4.90E-03</td><td>2.97E-02</td><td>8.25E-02</td><td>4.33E-03</td><td>2.03E-02</td><td>4.88E-02</td></tr><tr><td>OURS (CONCRETE)</td><td>1.06E-02</td><td>5.73E-02</td><td>1.56E-01</td><td>2.28E-03</td><td>1.22E-02</td><td>3.40E-02</td></tr><tr><td>OURS (NORMAL)</td><td>3.39E-03</td><td>1.85E-02</td><td>4.97E-02</td><td>1.30E-03</td><td>5.52E-03</td><td>1.38E-02</td></tr></table>
+
+# 5.2 REACHER
+
+We then evaluate our model on the Roboschool reacher environment. To make things more interesting, we learn only on partial observations, removing time derivative information (velocities), leaving us with just the positions or angles of various joints as observations. Table 1 shows a comparison of various methods on predicting the next couple of time steps. One critical point is the possible collision2 between lower and upper joint which is one we’d like our model to capture. We again learn a linear classifier based on latent space $s$ to see if this is successfully encoded and reach an F1 score of 0.46.
+
+# 5.3 BALL IN A BOX ON IMAGE DATA
+
+Finally, we evaluate our method on high-dimensional image observations using the single bouncing ball environment used by Fraccaro et al. (2017). They simulated 5000 sequences of 20 time steps each of a ball moving in a two-dimensional box, where each video frame is a $3 2 \times 3 2$ binary image. There are no forces applied to the ball, except for the fully elastic collisions with the walls. Initial position and velocity are randomly sampled.
+
+![](images/e336e1bf737c0b1609bca209428a9329da2814fbc7a23379990da437d282bb0e.jpg)  
+Figure 4: First row: data, second row: filtered reconstructions, third row: predictions. The first 4 steps are used to find a stable starting state, predictions start with step 5.
+
+In figure 5a we compare our model to both the smoothed and generative version of the KVAE. The smoothed version receives the final state of the trajectory after the $n$ predicted steps which is fed into the smoothing capability of the KVAE. One can see that our model learns a better transition model, even outperforming the smoothed KVAE for longer sequences. For short sequences, KVAE performs better which highlights the value of it disentangling the latent space into separate object and dynamics representation. A sample trajectory is plotted in figure 4.
+
+# 5.4 SUSCEPTIBILITY TO THE SCALE OF TEMPORAL DISCRETIZATION
+
+In this section, we’d like to explore how the choice of $\Delta t$ when discretizing a system influences our results. In particular, we’d expect our model with discrete (concrete) switching latent variables to be more susceptible to it than when modeled by a continuous distribution. This is because in the latter case the switching variables can scale the various matrices more freely, while in the former scaling up one system necessitates scaling down another. For empirical comparison, we go back to our custom maze environment (this time with only one agent as this is not pertinent to our question at hand) and learn the dynamics on various discretization scales. Then we compare the absolute error’s growth for both approaches in figure 5b which supports our hypothesis. While the discrete approximation even outperforms for small $\Delta t$ , there is a point where it rapidly becomes worse and gets overtaken by the continuous approximation. This suggests that $\Delta t$ was simply chosen to be too large in both the reacher and the ball in a box with image observations experiment.
+
+![](images/0e452e2e6f69830aa478e0687ad1ffd2d7d2264958af6ea33ad244daeec98297.jpg)  
+Figure 5: (a) Our dynamics model is outperforming even the smoothed KVAE for longer trajectories. (b) Modeling switching variables as Concrete random variables scales less favorably.
+
+# 6 DISCUSSION
+
+We want to emphasize some subtle differences to previously proposed architectures that make an empirical difference, in particular for the case when $s _ { t }$ is chosen to be continuous. In Watter et al. (2015) and Karl et al. (2017a), the latent space is already used to draw transition matrices, however they do not extract features such as walls or joint constraints. There are a few key differences from our approach. First, our latent switching variables $s _ { t }$ are only involved in predicting the current observation $x _ { t }$ through the transition selection process. The likelihood model therefore doesn’t need to learn to ignore some input dimensions which are only helpful for reconstructing future observations but not the current one. There is also a clearer restriction on how $s _ { t }$ and $z _ { t }$ may interact: $s _ { t }$ may now only influence $z _ { t }$ by determining the dynamics, while previously $z _ { t }$ influenced both the choice of transition function as well as acted inside the transition. These two opposing roles lead to conflicting gradients as to what should be improved. Furthermore, the learning signal for $s _ { t }$ is rather weak so that scaling down the KL-regularization was necessary to detect good features. Lastly, a (locally) linear transition may not be a good fit for variables determining dynamics as such variables may change very abruptly.
+
+# 7 CONCLUSION
+
+We have shown that our construction of using switching variables encourages learning a richer and more interpretable latent space. In turn, the richer representation led to an improvement of simulation accuracy in various tasks. In the future, we’d like to look at other ways to approximate the discrete switching variables and exploit this approach for model-based control on real hardware systems. Furthermore, addressing the open problem of disentangling latent spaces is essential to fitting simple dynamics and would lead to significant improvements of this approach.
+
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+
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+
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+
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+
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+
+Diederik P Kingma and Max Welling. Auto-encoding variational Bayes. In Proceedings of the 2nd International Conference on Learning Representations (ICLR), 2014.
+
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+
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+Chris J. Maddison, Andriy Mnih, and Yee Whye Teh. The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables. In Proceedings of the International Conference on Learning Representations (ICLR), pp. 1–17, 2017. ISBN 0780365402. URL http://arxiv. org/abs/1611.00712.
+
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+
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+
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+
+# A LOWER BOUND DERIVATION
+
+For brevity we omit conditioning on control inputs $u _ { 1 : T }$ .
+
+$$
+\begin{array} { l } { \displaystyle \log p ( x _ { T } ) = \log \int _ { z _ { 1 : T } } \int _ { s _ { 1 : T } } q _ { \phi } ( s _ { 1 : T } , z _ { 1 : T } \mid x _ { 1 : T } ) \frac { p _ { \theta } ( x _ { 1 : T } \mid z _ { 1 : T } ) p _ { \theta } ( z _ { 1 : T } , s _ { 1 : T } ) } { q _ { \phi } ( s _ { 1 : T } , z _ { 1 : T } \mid x _ { 1 : T } ) } } \\ { \displaystyle \geq \int _ { z _ { 1 : T } } \int _ { s _ { 1 : T } } q _ { \phi } ( s _ { 1 : T } , z _ { 1 : T } \mid x _ { 1 : T } ) \log \frac { p _ { \theta } ( x _ { 1 : T } \mid z _ { 1 : T } ) p _ { \theta } ( z _ { 1 : T } , s _ { 1 : T } ) } { q _ { \phi } ( s _ { 1 : T } , z _ { 1 : T } \mid x _ { 1 : T } ) } } \\ { \displaystyle = \sum _ { t = 1 } ^ { T } \mathbb { E } _ { s _ { t } } [ \mathbb { E } _ { z _ { t } } [ p ( x _ { t } \mid z _ { t } , s _ { t } ) ] ] - D _ { \mathrm { K L } } ( q ( z _ { 1 : T } , s _ { 1 : T } \mid x _ { 1 : T } ) \mid | p ( z _ { 1 : T } , s _ { 1 : T } ) ) } \end{array}
+$$
+
+# A.1 FACTORIZATION OF THE KL DIVERGENCE
+
+The dependencies on data $x _ { T }$ and $u _ { T }$ as well as parameters $\phi$ and $\theta$ are omitted in the following for convenience.
+
+$$
+D _ { \mathrm { K L } } ( q ( z _ { 1 } , s _ { 2 } , \ldots , s _ { T } , z _ { T } ) \parallel p ( z _ { 1 } , s _ { 2 } , \ldots , s _ { T } , z _ { T } ) )
+$$
+
+$$
+\begin{array} { r l } { { } } & { { = \displaystyle \int _ { z _ { 1 } } \int _ { s _ { 2 } } \cdots \int _ { s _ { T } } \int _ { z _ { T } } q ( z _ { 1 } ) q ( s _ { 2 } \mid z _ { 1 } ) \ldots q ( s _ { T } \mid z _ { T - 1 } , s _ { T - 1 } ) q ( z _ { T } \mid z _ { T - 1 } , s _ { T } ) } } \\ { { } } & { { \phantom { = \displaystyle \int _ { z _ { 1 } } \int _ { s _ { 2 } } \cdots \int _ { s _ { T } } \int _ { z _ { T } } q ( z _ { 1 } ) q ( s _ { 2 } \mid z _ { 1 } ) \ldots q ( s _ { T } \mid z _ { T - 1 } , s _ { T - 1 } ) q ( z _ { T } \mid z _ { T - 1 } , s _ { T } ) } } } \\ { { } } & { { \phantom { = \displaystyle \int _ { z _ { 1 } } \int _ { s _ { 2 } } \cdots d \int _ { z _ { T - 1 } } \int _ { z _ { T - 1 } } s _ { T - 1 } \int _ { z _ { T } } } } } \end{array}
+$$
+
+(Factorization of the prior)
+
+$$
+\begin{array} { r l } { { } } & { { = { \displaystyle \int _ { z _ { 1 } } \int _ { s _ { 2 } } \cdots \int _ { s _ { T } } \int _ { z _ { T } } q ( z _ { 1 } ) q ( s _ { 2 } \mid z _ { 1 } ) \ldots q ( s _ { T } \mid z _ { T - 1 } , s _ { T - 1 } ) q ( z _ { T } \mid z _ { T - 1 } , s _ { T } ) } } } \\ { { } } & { { { \log \frac { q ( z _ { 1 } ) q ( s _ { 2 } \mid z _ { 1 } ) \ldots q ( s _ { T } \mid z _ { T - 1 } , s _ { T - 1 } ) q ( z _ { T } \mid z _ { T - 1 } , s _ { T } ) } { p ( z _ { 1 } ) p ( s _ { 2 } \mid z _ { 1 } ) \ldots p ( s _ { T } \mid z _ { T - 1 } , s _ { T - 1 } ) p ( z _ { T } \mid z _ { T - 1 } , s _ { T } ) } } } } \end{array}
+$$
+
+(Expanding the logarithm by the product rule)
+
+$$
+\begin{array} { l } { { = \displaystyle \int _ { z _ { 1 } } q ( z _ { 1 } ) \log \frac { q ( z _ { 1 } ) } { p ( z _ { 1 } ) } + \displaystyle \int _ { z _ { 1 } } \int _ { s _ { 1 } } q ( z _ { 1 } ) q ( s _ { 1 } \mid z _ { 1 } ) \log \frac { q ( s _ { 1 } \mid z _ { 1 } ) } { p ( s _ { 1 } \mid z _ { 1 } ) } } } \\ { { + \displaystyle \sum _ { t = 2 } ^ { T } \int _ { z _ { 1 } } \int _ { s _ { 2 } } \cdots \int _ { s _ { T } } \int _ { z _ { T } } q ( z _ { 1 } ) q ( s _ { 2 } \mid z _ { 1 } ) \ldots q ( z _ { T } \mid z _ { T - 1 } , s _ { T } ) \log \frac { q ( z _ { t } \mid z _ { t - 1 } , s _ { t } ) } { p ( z _ { t } \mid z _ { t - 1 } , s _ { t } ) } } } \\ { { + \displaystyle \sum _ { t = 3 } ^ { T } \int _ { z _ { 1 } } \int _ { s _ { 2 } } \cdots \int _ { s _ { T } } \int _ { z _ { T } } q ( z _ { 1 } ) q ( s _ { 2 } \mid z _ { 1 } ) \ldots q ( z _ { T } \mid z _ { T - 1 } , s _ { T } ) \log \frac { q ( s _ { t } \mid z _ { t - 1 } , s _ { t - 1 } ) } { p ( s _ { t } \mid z _ { t - 1 } , s _ { t - 1 } ) } } } \end{array}
+$$
+
+(Ignoring constants)
+
+$$
+\begin{array} { l } { { \displaystyle = D _ { \mathrm { K L } } ( q ( z _ { 1 } ) \mid | \ p ( z _ { 1 } ) ) + \mathbb { E } _ { z _ { 1 } \sim q ( z _ { 1 } ) } [ D _ { \mathrm { K L } } ( q ( s _ { 2 } \mid z _ { 1 } ) \mid | \ p ( s _ { 2 } \mid z _ { 1 } ) ) ] } } \\ { { \displaystyle ~ + \sum _ { t = 2 } ^ { T - 1 } \mathbb { E } _ { s _ { t } , z _ { t - 1 } } [ D _ { \mathrm { K L } } ( q ( z _ { t } \mid z _ { t - 1 } , s _ { t } ) \mid | \ p ( z _ { t } \mid z _ { t - 1 } , s _ { t } ) ) ] } } \\ { { \displaystyle ~ + \sum _ { t = 3 } ^ { T - 1 } \mathbb { E } _ { s _ { t - 1 } , z _ { t - 1 } } [ D _ { \mathrm { K L } } ( q ( s _ { t } \mid z _ { t - 1 } , s _ { t - 1 } ) \mid | \ p ( s _ { t } \mid z _ { t - 1 } , s _ { t - 1 } ) ) ] } } \end{array}
+$$
+
+Table 2: Dimensionality of environments.   
+
+<table><tr><td>Dimensionality of（</td><td>Observation Space</td><td>Control Input Space</td><td>Ground Truth State Space</td></tr><tr><td>Reacher</td><td>7</td><td>2</td><td>9</td></tr><tr><td>Hopper</td><td>8</td><td>3</td><td>15</td></tr><tr><td>Multi Agent Maze</td><td>4</td><td>6</td><td>12</td></tr><tr><td>Image Ball in BoX</td><td>32 × 32</td><td>0</td><td>4</td></tr></table>
+
+# B DETAILS OF THE EXPERIMENTAL SETUP
+
+# B.1 ENVIRONMENTS
+
+# B.1.1 ROBOSCHOOL REACHER
+
+To generate data, we follow a Uniform distribution $\mathcal { U } \sim [ - 1 , 1 ]$ as the exploration policy. Before we record data, we take 20 warm-up steps in the environment to randomize our starting state. We take the data as is without any other preprocessing.
+
+# B.1.2 MULTI AGENT MAZE
+
+Observations are normalized to be in $[ - 1 , 1 ]$ . Both position and velocity is randomized for the starting state. We again follow a Uniform distribution $\mathcal { U } \sim [ - 1 , 1 ]$ as the exploration policy.
+
+# B.2 TRAINING
+
+Overall, training the Concrete distribution has given us the biggest challenge as it was very susceptible to various hyperparameters. We made use of the fact that we can use a different temperature for the prior and approximate posterior (Maddison et al., 2017) and we do independent hyperparameter search over both. For us, the best values were 0.75 for the posterior and 2 for the prior. Additionally, we employ an exponential annealing scheme for the temperature hyperparameter of the Concrete distribution. This leads to a more uniform combination of base matrices early in training which has two desirable effects. First, all matrices are scaled to a similar magnitude, making initialization less critical. Second, the model initially tries to fit a globally linear model, leading to a good starting state for optimization. We also tried increasing the number of samples taken (up to 100) to approximate the KL between the Concrete distributions, however we have not observed an improvement of performance. We therefore restrict ourselves to 10 samples for all experiments.
+
+In all experiments, we train everything end-to-end with the ADAM optimizer.(Kingma & Ba, 2015) We start with learning rate of $5 \mathrm { e } { - 4 }$ and use an exponential decay schedule with rate 0.97 every 2000 iterations.
+
+# B.3 NETWORK ARCHITECTURE
+
+For most networks, we use MLPs implemented as residual nets (He et al., 2016) with ReLU activations.
+
+Networks used for the reacher and maze experiments.
+
+• $q _ { \mathrm { m e a s } } ( z _ { t } \mid \cdot )$ : MLP consisting of two residual blocks with 256 neurons each. We only condition on the current observation $x _ { t }$ although we could condition on the entire sequence. This decision was taken based on empirical results.   
+） $q _ { \mathrm { t r a n s } } ( z _ { t } \mid \cdot )$ : In the case of Concrete random variables, we just combine the base matrices and apply the transition dynamics to $z _ { t - 1 }$ . For the Normal case, the combination of matrices is preceded by a linear combination with softmax activation. (see equation 14) $q _ { \mathrm { m e a s } } ( s _ { t } \mid \cdot )$ : is implemented by a backward LSTM with 256 hidden units. We reuse the preprocessing of $q _ { m e a s } ( z _ { t } \mid x _ { t } )$ and take the last hidden layer of that network as the input to the LSTM.   
+• $q _ { \mathrm { t r a n s } } ( s _ { t } \mid \cdot )$ : MLP consisting of one residual block with 256 neurons.   
+• $q _ { \mathrm { i n i t i a l } } ( w \mid \cdot )$ : MLP consisting of two residual block with 256 neurons optionally followed by a backward LSTM. We only condition on the first 3 or 4 observations for our experiments.   
+• $q _ { \mathrm { i n i t i a l } } ( s _ { 2 } )$ : The first switching variable in the sequence has no predecessor. We therefore require a replacement for $q _ { t r a n s } ( s _ { t } \mid \cdot )$ in the first time step, which we achieve by independently parameterizing another MLP.   
+• $p ( x _ { t } \mid z _ { t } )$ : MLP consisting of two residual block with 256 neurons.   
+• $p ( \boldsymbol { z } _ { t } \mid \cdot )$ : Shared parameters with $q _ { t r a n s } ( z _ { t } \mid \cdot )$ .   
+• $p ( s _ { t } \mid \cdot )$ : Shared parameters with $q _ { t r a n s } ( s _ { t } \mid \cdot )$ .
+
+We use the same architecture for the image ball in a box experiment, however we increase number of neurons of $q _ { \mathrm { m e a s } } ( z _ { t } \mid \cdot )$ to 1024.
+
+# B.4 HYPERPARAMETERS
+
+Table 3: Overview of hyperparameters.   
+
+<table><tr><td></td><td>Multi Agent Maze</td><td>Reacher</td><td>Image Ball in Box</td></tr><tr><td># episodes</td><td>50000</td><td>20000</td><td>5000</td></tr><tr><td>episode length</td><td>20</td><td>30</td><td>20</td></tr><tr><td>batch size</td><td>256</td><td>128</td><td>256</td></tr><tr><td>dimension of z</td><td>32</td><td>16</td><td>8</td></tr><tr><td>dimension of s</td><td>16</td><td>8</td><td>8</td></tr><tr><td>posterior temperature</td><td>0.75</td><td>0.75</td><td>0.67</td></tr><tr><td>prior temperature</td><td>2</td><td>2</td><td>2</td></tr><tr><td>temperature annealing steps</td><td>100</td><td>100</td><td>100</td></tr><tr><td>temperature annealing rate</td><td>0.97</td><td>0.97</td><td>0.98</td></tr><tr><td>β (KL-scaling of switching variables)</td><td>0.1</td><td>0.1</td><td>0.1</td></tr></table>
+
+# C ON SCALING ISSUES OF SWITCHING LINEAR DYNAMICAL SYSTEMS
+
+Let’s consider a simple representation of a ball in a rectangular box where its state is represented by its position and velocity. Given a small enough $\Delta t$ , we can approximate the dynamics decently by just 3 systems: no interaction with the wall, interaction with a vertical or horizontal wall (ignoring the corner case of interacting with two walls at the same time). Now consider the growth of required base systems if we increase the number of balls in the box (even if these balls cannot interact with each other). We would require a system for all combinations of a single ball’s possible states: $3 ^ { 2 }$ This will grow exponentially with the number of balls in the environment.
+
+One way to alleviate this problem that requires only a linear growth in base systems is to independently turn individual systems on and off and let the resulting system the sum of all activated systems. A base system may then represent solely the transition for a single ball being in specific state, while the complete system is then a combination of $N$ such systems where $N$ is the number of balls. Practically, this can be achieved by replacing the softmax by a sigmoid activation function or by replacing the categorical variable $s$ of dimension $M$ by $M$ Bernoulli variables indicating whether a single system is active or not. We do this for our multiple agents in a maze environment.
+
+Theoretically, a preferred approach would be to disentangle multiple systems (like balls, joints) and apply transitions only to their respective states. This, however, would require a proper and unsupervised separation of (mostly) independent components. We defer this to future work.
+
+# D FURTHER RESULTS
+
+# D.1 3-AGENT MAZE
+
+![](images/a6cbeb7cfa725ccd9d3588936bebbdaf170a4b93f96ae105d0b31e890791f724.jpg)  
+Figure 6: Comparison of actual and predicted 20 step trajectories. The diamond marker denotes the starting position of a trajectory.
+
+# D.2 IMAGE BALL IN A BOX
+
+![](images/591c3c42a554e02486eb4fda365270cf8b98b397e40d966b285967c2ff546d0a.jpg)
+
+Figure 7: First row: data, second row: reconstructions, third row: predictions. The first 4 steps are used to find a stable starting state, predictions start with step 5.

parse/train/BJe932EYwS/BJe932EYwS_content_list.json

@@ -0,0 +1,1714 @@
+[
+    {
+        "type": "text",
+        "text": "PNAT: NON-AUTOREGRESSIVE TRANSFORMER BY POSITION LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
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+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            236,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Non-autoregressive models are promising on various text generation tasks. Previous work hardly considers to explicitly model the positions of generated words. However, the position modeling is an essential problem in non-autoregressive text generation. In this study, we propose PNAT, which incorporates positions as a latent variable into the text generative process. Experimental results show that PNAT achieves top results on machine translation and paraphrase generation tasks, outperforming several strong baselines. ",
+        "bbox": [
+            233,
+            267,
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+            364
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            390,
+            336,
+            407
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Transformer (Vaswani et al., 2017) has been widely used in many text generation tasks, which is first proposed in neural machine translation, achieving great success for its promising performance. Nevertheless, the auto-regressive property of Transformer has been a bottleneck. Specifically, the decoder of Transformer generates words sequentially, and the latter words are conditioned on previous ones in a sentence. Such bottleneck prevents the decoder from higher efficiency in parallel computation, and imposes strong constrains in text generation, with which the generation order has to be left to right (or right to left) (Shaw et al., 2018; Vaswani et al., 2017). ",
+        "bbox": [
+            174,
+            421,
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+            520
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, many researches (Gu et al., 2018; Lee et al., 2018; Wang et al., 2019; Wei et al., 2019) are devoted to break the auto-regressive bottleneck by introducing non-autoregressive Transformer (NAT) for neural machine translation, where the decoder generates all words simultaneously instead of sequentially. Intuitively, NAT abandons feeding previous predicted words into decoder state at the next time step, but directly copy encoded representation at source side to the decoder inputs (Gu et al., 2018). However, without the auto-regressive constrain, the search space of the output sentence becomes larger (Wei et al., 2019), which brings the performance gap (Lee et al., 2018) between NAT and auto-regressive Transformer (AT). Related works propose to include some inductive priors or learning techniques to boost the performance of NAT. But most of previous work hardly consider explicitly modeling the position of output words during text generation. ",
+        "bbox": [
+            174,
+            526,
+            825,
+            666
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We argue that position prediction is an essential problem of NAT. Current NAT approaches do not explicitly model the position of output words, and may ignore the reordering issue in generating output sentences. Compared to machine translation, the reorder problem is much more severe in tasks such as table-to-text (Liu et al., 2018) and dialog generations (Shen et al., 2017). Additionally, it is straightforward to explicitly model word positions in output sentences, as position embeddings are used in Transformer, which is natively non-autoregressive, to include the order information. Intuitively, if output positions are explicitly modeled, the predicted position combined with Transformer to realize non-autoregressive generation would become more natural. ",
+        "bbox": [
+            174,
+            672,
+            825,
+            784
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose non-autoregressive transformer by position learning (PNAT). PNAT is simple yet effective, which explicitly models positions of output words as latent variables in the text generation. Specifically, we introduce a heuristic search process to guide the position learning, and max sampling is adopted to inference the latent model. The proposed PNAT is motivated by learning syntax position (also called syntax distance). Shen et al. (2018) show that syntax position of words in a sentence could be predicted by neural networks in a non-autoregressive fashion, which even obtains top parsing accuracy among strong parser baselines. Given the observations above, we try to directly predict the positions of output words to build a NAT model for text generation. ",
+        "bbox": [
+            174,
+            791,
+            825,
+            902
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our proposed PNAT takes following advantages: ",
+        "bbox": [
+            174,
+            910,
+            496,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• We propose PNAT, which first includes positions of output words as latent variables for text generation. Experiments show that PNAT achieves very top results in non-autoregressive NMT, outperforming many strong baselines. PNAT also obtains better results than AT in paraphrase generation task. Further analysis shows that PNAT has great potentials. With the increase of position prediction accuracy, performances of PNAT could increase significantly. The observations may shed light on the future direction of NAT. Thanks to the explicitly modeling of position, we could control the generation by facilitating the position latent variable, which may enable interesting applications such as controlling one special word left to another one. We leave this as future work. ",
+        "bbox": [
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+            242
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            262,
+            326,
+            280
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 AUTOREGRESSIVE DECODING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            295,
+            423,
+            309
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A target sequence $Y { = } y _ { 1 : M }$ is decomposed into a series of conditional probabilities autoregressively, each of which is parameterized using neural networks. This approach has become a de facto standard in language modeling(Sundermeyer et al., 2012), and has been also applied to conditional sequence modeling $p ( Y | X )$ by introducing an additional conditional variable $X { = } x _ { 1 : N }$ : ",
+        "bbox": [
+            173,
+            320,
+            825,
+            377
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bc2e6d38ae4d16ffc7573395110f0fb888db3e357530a1f3b8c49f45ef44dfa4.jpg",
+        "text": "$$\np ( \\boldsymbol { Y } | \\boldsymbol { X } ) = \\prod _ { t = 1 } ^ { M } p ( y _ { t } | y _ { < t } , \\boldsymbol { X } ; \\theta )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            383,
+            601,
+            428
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "With different choices of neural network architectures such as recurrent neural networks (RNNs) (Bahdanau et al., 2014; Cho et al., 2014), convolutional neural networks (CNNs) (Krizhevsky et al., 2012; Gehring et al., 2017), as well as self-attention based transformer (Vaswani et al., 2017), the autoregressive decoding has achieved great success in tasks such as machine translation (Bahdanau et al., 2014), paraphrase generation (Gupta et al., 2018), speech recognition (Graves et al., 2013), etc. ",
+        "bbox": [
+            173,
+            433,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 NON-AUTOREGRESSIVE DECODING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            520,
+            460,
+            535
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Autoregressive model suffers from the issue of slow decoding in inference, because tokens are generated sequentially and each of them depends on previous ones. As a solution to this issue, Gu et al. (2018) proposed Non-Autoregressive Transformer (denoted as NAT) for machine translation, breaking the dependency among the target tokens through time by decoding all the tokens simultaneously. Put simply, NAT (Gu et al., 2018) factorizes the conditional distribution over a target sequence into a series of conditionally independent distributions with respect to time: ",
+        "bbox": [
+            173,
+            546,
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+            631
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4294655b2e30c6c987b2f44cff34a959cbf065d351823d9b560b301f1c435043.jpg",
+        "text": "$$\np ( { \\cal Y } | { \\cal X } ) = p _ { L } ( { \\cal M } | { \\cal X } : \\theta ) \\cdot \\prod _ { t = 1 } ^ { M } p ( y _ { t } | { \\cal X } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            637,
+            627,
+            681
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "which allows trivially finding the most likely target sequence by arg $\\operatorname* { m a x } _ { Y }$ $p ( Y | X )$ for each timestep $t$ , effectively bypassing computational overhead and sub-optimality in decoding from an autoregressive model. ",
+        "bbox": [
+            173,
+            688,
+            825,
+            729
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Although non-autoregressive models achieves $1 5 \\times$ speedup in machine translation compared with autoregressive models, it comes at the expense of potential performance degradation (Gu et al., 2018). The degradation results from the removal of conditional dependencies within the decoding sentence $y _ { t }$ depend on $y _ { < t , }$ ). Without such dependencies, the decoder is hard to leverage the inherent sentence structure in prediction. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.3 LATENT VARIABLES FOR NON-AUTOREGRESSIVE DECODING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            823,
+            640,
+            838
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A non-autoregressive model could be incorporated with conditional dependency as latent variable to alleviate the degradation resulted from the absence of dependency: ",
+        "bbox": [
+            174,
+            849,
+            825,
+            878
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/caec75431086406ffd827fbdcee7b645930358d62501dde92abcc4fe71ccf9c7.jpg",
+        "text": "$$\nP ( Y | X ) = \\int _ { z } P ( z | X ) \\prod _ { t = 1 } ^ { M } P ( y _ { t } | z , X ) d z\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            886,
+            637,
+            928
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "For example, NAT-FT (Gu et al., 2018) models the inherent sentence structure with a latent fertility variable, which represents how many target tokens that a source token would translate to. Lee et al. (2018) introduces $L$ intermediate predictions $Y ^ { 1 : L }$ as random variables , and to refine the predictions from $Y ^ { 1 }$ to $Y ^ { L }$ in a iterative manner. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            159
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PNAT: POSITION-BASED NON-AUTOREGRESSIVE TRANSFORMER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            179,
+            753,
+            195
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose position-based non-autoregressive transformer (PNAT), an extension to transformer incorporated with non auto-regressive decoding and position learning. ",
+        "bbox": [
+            174,
+            209,
+            825,
+            238
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 MODELING POSITION WITH LATENT VARIABLES ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            253,
+            552,
+            267
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Languages are usually inconsistent with each other in word order. Thus reordering is usually required when translating a sentence from a language to another. In NAT family, words representations or encoder states at source side are copied to the target side to feed into decoder as its input. Previously, Gu et al. (2018) utilizes positional attention which incorporates positional encoding into decoder attention to perform local reordering. But such implicitly reordering mechanism by position attention may cause a repeated generation problem, because position learning module is not optimized directly, and is likely to be misguided by target supervision. ",
+        "bbox": [
+            173,
+            279,
+            825,
+            377
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To tackle with this problem, we propose to explicitly model the position as a latent variable. We rewrite the target sequence $Y$ with its corresponding position latent variable $z = z _ { 1 : M }$ as a set $Y _ { z } =$ $y _ { z _ { 1 } : z _ { M } }$ . The conditional probability $P ( { Y \\vert { X } } )$ is factorized with respect to the position latent variable: ",
+        "bbox": [
+            174,
+            383,
+            825,
+            426
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/52337457f69dd44f203ff4a919a92362ea2b23e7cbcadaf135ada6f7e25342d4.jpg",
+        "text": "$$\nP ( Y | X ) = \\sum _ { z \\in \\pi ( M ) } P ( z | X ) \\cdot P ( Y | z , X )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            428,
+            637,
+            463
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\pi ( M )$ is a set consisting of permutations with $M$ elements. At decoding time, the factorization allows us to decode sentences in parallel by pre-predicting the corresponding position variables $z$ . ",
+        "bbox": [
+            174,
+            464,
+            825,
+            493
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 MODEL ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            508,
+            383,
+            522
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/3eb7123eaf3819f4b5d53c6ee9e3f531727f2a7ab5d030eac7c5c99744ffb6d5.jpg",
+        "image_caption": [
+            "attention over the output of the encoder stack. Similar to the encoder, we employ residual connectionsaround each of the sub-layers, followed by layer normalization. We also modify the self-attention               around each of the sub-layers, followed by layer normalization. We also modify the self-attentionFigure 1: Illustration of the proposed model, where the black solid arrows represent differentiable sub-layer in the decoder stack to prevent positions from attending to subsequent positions. Tmasking, combined with fact that the output embeddings are offset by one position, ensures thatsub-layer in the decoder stack to prevent positions frommasking, combined with fact that the output embeddings aconnections and the dashed arrows are non-differentiable operations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            349,
+            537,
+            643,
+            820
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 1, PNAT is composed of four modules: an encoder stack, a bridge block, a 3.2 Attention 3.2 Attentionposition predictor as well as a decoder stack. Before detailing each component of PNAT model, we overview the architecture for a brief understanding. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Like most sequence-to-sequence models, PNAT first encodes a source sequence $X { = } x _ { 1 : N }$ into its contextual word representations $\\scriptstyle { E = e _ { 1 : N } }$ with the encoder stack. With generated contextual word representation $E$ at source side, the bridge block is leveraged to computed the target length $M$ as well as the corresponding features $D { = } d _ { 1 : { M } }$ , which is fed into the decoder as its input. It is worth noting that the decoder inputs $D$ is computed without reordering. Thus the position predictor is introduced to deal with this issue by predicting a permutation $z { = } z _ { 1 : { M } }$ over $D$ . Finally, PNAT generates the target sequence from the decoder input $D$ and its permutation $_ z$ . ",
+        "bbox": [
+            173,
+            103,
+            825,
+            202
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Encoder and Decoder Given a source sentence $X$ with length $N$ , PNAT encoder produces its contextual word representations $E$ . The contextual word representations $E$ are further used in computing target length $M$ and decoder initial states $D$ , and are also used as memory of attention at decoder side. ",
+        "bbox": [
+            173,
+            217,
+            825,
+            273
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Generally, PNAT decoder can be considered as a transformer with a broader vision, because it leverages future word information that is blind to the autoregressive transformer. Intuitively, we use relative position encoding in self-attention(Shaw et al., 2018), rather than absolute one that is more likely to cause position errors. Following Shaw et al. (2018) with a clipping distance $d$ (usually $d \\geq 2 \\AA$ ) set for relative positions, we preserve $d = 4$ relations. ",
+        "bbox": [
+            173,
+            280,
+            825,
+            351
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Bridge The bridge module predicts the target length $M$ , and initializes the decoder inputs $D$ from the source representations $E$ . The target length $M$ could be estimated from the source encoder representation: ",
+        "bbox": [
+            174,
+            366,
+            823,
+            407
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f99e2fc92f3dadb8437adc75d4aa8d6927bb2337852691f94b6093b428a217f0.jpg",
+        "text": "$$\nM = N + \\arg \\operatorname* { m a x } \\phi ( E )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            407,
+            584,
+            424
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\phi ( \\cdot )$ produces a categorical distribution ranged in $[ - B , B ]$ $B = 2 0$ ). It is notable that we use the predicted length at inference stage, although during training, we simply use the length of each reference target sequence. Then, we adopt the method proposed by Li et al. (2019) to compute $D$ . Given the source representation $E$ and the estimated target length $M$ , we linearly combine the embeddings of the neighboring source tokens to generate $D$ as follows: ",
+        "bbox": [
+            173,
+            428,
+            825,
+            500
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c1091016c6f57f1b9ec1948ccf69dd55e19b353bf330cae879ad668bc6edd87e.jpg",
+        "text": "$$\nd _ { j } = \\sum _ { i } w _ { j i } \\cdot e _ { i }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/63ce5bd6bbed11cf290cc1e07a89162d614b967ca7a02a7c4e6e169e01fe6272.jpg",
+        "text": "$$\nw _ { j i } = \\mathrm { s o f t m a x } ( - | j - i | / \\tau )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $w _ { j i }$ is a normalized weight that reflects the contribution of $e _ { i }$ to $d _ { j }$ , and $\\tau$ is a hyperparameter indicating the sharpness of the weight distribution. ",
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+    {
+        "type": "text",
+        "text": "Position Predictor For the proposed PNAT, we model position permutations with a position predictor. As shown in Figure 1, the position predictor takes the decoder inputs $D$ and the source representation $E$ to predict a permutation $_ z$ . The position predictor has a sub-encoder which stacks multiple layers of encoder units to predict its predicted input $R { = } r _ { 1 : { M } }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "With the predicted inputs $R$ , we conduct an autoregressive position predictor, denoted as AR-Predictor. The AR-Predictor searches a permutation $_ { z }$ with: ",
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+    {
+        "type": "equation",
+        "img_path": "images/4f06baed3002c6f0f56ba6095b92e170fd3fb179da0595ce8c0d0c13490d7282.jpg",
+        "text": "$$\nP ( \\boldsymbol { z } | D , E ) = \\prod _ { t = 1 } ^ { M } p _ { ( } z _ { t } | \\boldsymbol { z } _ { < t } , D , E ; \\boldsymbol { \\theta } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\theta$ is the parameter of AR-Predictor, which includes a RNN-based model incorporated with a pointer network (Vinyals et al., 2015). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To purse the efficiency of decoding, we also explore a non-autoregressive version for the position predictor, denoted as NAR-Predictor, to model the position permutation probabilities with: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/ad23a173b73b2d6aac124cd7b05df6aad86bdbfc475f7fef5e9982d3e3c2ac3f.jpg",
+        "text": "$$\nP ( z | D , E ) = \\prod _ { t = 1 } ^ { M } p ( z _ { t } | D , E ; \\theta )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To obtain the permutation $_ { z }$ , AR-Predictor performs greedy search whereas NAR-Predictor performs direct arg max. We chose the AR-Predictor as our mainly position module in PNAT, and we also analyze the effectiveness of position modeling in Sec. 4.4. ",
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+    {
+        "type": "text",
+        "text": "3.3 TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Training requires maximizing the marginalized likelihood in Eqn. 4. However, this is intractable since we need to enumerate all the $M !$ permutations of tokens. We therefore optimize this objective by Monte Carlo sampling method with a heuristic search algorithm. ",
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+    {
+        "type": "text",
+        "text": "Heuristic Search for Positions Intuitively, each target token should have a corresponding decoder input, and meanwhile each decoder input should be assigned to a target token. Based on this idea, we design a heuristic search algorithm to allocate positions. Given the decoder inputs and its target tokens, we first estimate the similarity between each pair of the decoder input $d _ { i }$ and the target token embedding $y _ { j }$ , which is also the weights of the target word classifier: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/f6608df04c7399e26dee1fd397c9966d86491391ee515485b9c44b4409790b25.jpg",
+        "text": "$$\n\\mathrm { s i m } _ { i , j } = \\mathrm { c o s i n e } \\left( d _ { i } , y _ { j } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Based on the cosine similarity matrix, HSP is designed to find a perfect matching between decoder inputs and target tokens: ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d0f6ef78e28399c29d2c4525e54c0389ff9648e4d42b4ac32779f9bd85f50b47.jpg",
+        "text": "$$\n\\mathrm { H S P } ( z ) = \\underset { z } { \\arg \\operatorname* { m a x } } \\sum _ { i = 0 } ^ { M } ( \\ s i \\mathrm { m } _ { i , z _ { i } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Here we apply a greedy algorithm to select the pair with the highest similarity score iteratively until a permutation $z _ { \\mathrm { r e f } }$ is generated. More details are provided in Appendix A. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The intuition behind is that, if the decoder input $d _ { i }$ is already the most similar one to a target word, it would be easier to keep and even reinforce this association in learning the model. We also analyze the effectiveness of the HSP in the Sec. 4.4. ",
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+    {
+        "type": "text",
+        "text": "Objective Function With the heuristically discovered positions as reference positions $z _ { \\mathrm { r e f } }$ , the position predictor could be trained with a position loss: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/07f4be13d5fb186fb3fc5189a34789b29000b531a00250c1bb4f910ad8d6fa16.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { p } } = - \\log P ( z _ { \\mathrm { r e f } } | D , E )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Grounding on the referenced positions, the generative process of target sequences is optimized by: ",
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+    {
+        "type": "equation",
+        "img_path": "images/d778ca903e7298128e8178b9d4126b4727e51ab80491f86aec9cc643cc7b56f2.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { g } } = - \\sum _ { t = 1 } ^ { M } \\log P ( Y | z _ { \\mathrm { r e f } } ; X )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Finally, combining two loss functions mentioned above, a full-fledged loss is derived as ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8236c49e2f45d15093d1e140dde491213c35e96a43ff6dd033948b2283bd96a7.jpg",
+        "text": "$$\n{ \\mathcal { L } } = { \\mathcal { L } } _ { \\mathrm { g } } + \\alpha { \\mathcal { L } } _ { \\mathrm { p } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The length predictor is a classifier that follows the previous settings. We also follow the previous practice (Gu et al., 2018; Wei et al., 2019) and perform an extra training process for the length predictor after the model trained and do not tune the parameter of the encoder. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.4 INFERENCE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We follow the common choice of approximating decoding algorithms (Gu et al., 2018; Lee et al., 2018) to reduce the search space of latent variable model. ",
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+    },
+    {
+        "type": "text",
+        "text": "Argmax Decoding Following Gu et al. (2018), one simple and effective method is to select the best sequence by choosing the highest-probability latent sequence $z$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/86e175d639e44b663b2e48c3a905029f63cd114a80f495343ee43c49e8c03699.jpg",
+        "text": "$$\n\\begin{array} { c } { { z ^ { * } = \\arg \\operatorname* { m a x } _ { z } P ( z | D , E ) } } \\\\ { {  } } \\\\ { { Y ^ { * } = \\arg \\operatorname* { m a x } _ { y } P ( Y | z ^ { * } , X ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where identifying $Y ^ { * }$ only requires independently maximizing the local probability for each output position. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Length Parallel Decoding We also consider the common practice of noisy parallel decoding (Gu et al., 2018), which generates a number of decoding candidates in parallel and selects the best via re-scoring using a pre-trained autoregressive model. For PNAT, we first predict the target length as $\\hat { M }$ , then generate output sequence with argmax decoding for each target length candidate $M \\in$ $[ \\hat { M } - \\Delta M , \\hat { M } + \\Delta M ]$ ( $M = 4$ in our experiments), which was called length parallel decoding (LPD). Then we use the pre-trained autoregressive model to rank these sequences and identify the best overall output as the final output. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We test PNAT on several benchmark sequence generation tasks. We first describe the experimental setting and implementation details and then present the main results, followed by some deep studies. ",
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETTING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To show the generation ability of PNAT, we conduct experiments on the popular machine translation and paraphrase generation tasks. These sequence generation task evaluation models from different perspectives. Translation tasks test the ability of semantic transforming across bilingual corpus. While paraphrase task focuses on substitution between the same languages while keeping the semantics. ",
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+    {
+        "type": "text",
+        "text": "Machine Translation We valid the effectiveness of PNAT on the most widely used benchmarks for machine translation — WMT14 EN-DE(4.5M pairs) and IWSLT16 DE-EN(196K pairs). The dataset is processed with Moses script (Koehn et al., 2007), and the words are segmented into subword units using byte-pair encoding (Sennrich et al., 2016, BPE). For both WMT datasets, the source and target languages share the same set of subword embeddings while for IWSLT we use separate embeddings. ",
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+    {
+        "type": "text",
+        "text": "Paraphrase Generation We conduct experiments following previous work (Miao et al., 2019) for paraphrase generation. We make use of the established Quora dataset 1 to evaluate on the paraphrase generation task. We consider the supervised paraphrase generation and split the Quora dataset in the standard setting. We sample $1 0 0 \\mathrm { k }$ pairs sentence as training data, and holds out 3k, 30k for validation and testing, respectively. ",
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+    {
+        "type": "text",
+        "text": "4.2 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Module Setting For machine translation, we follow the settings from Gu et al. (2018). In the case of IWSLT task, we use a small setting $( d _ { \\mathrm { m o d e l } } = 2 7 8$ , $d _ { \\mathrm { h i d d e n } } = 5 0 7$ , $p _ { \\mathrm { d r o p o u t } } = 0 . 1$ , $n _ { \\mathrm { l a y e r } } = 5$ and nhead $= 2$ ) suggested by Gu et al. (2018) for Transformer and NAT models. For WMT task, we use the base setting of the Vaswani et al. (2017) $\\dot { d } _ { \\mathrm { m o d e l } } = 5 1 2$ , $d _ { \\mathrm { h i d d e n } } = 5 1 2$ , $p _ { \\mathrm { d r o p o u t } } = 0 . 1$ , $n _ { \\mathrm { l a y e r } } = 6 $ ). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For paraphrase generation, we follow the settings from Miao et al. (2019), and set the 300-dimensional GRU with 2 layer for Seq-to-Seq (GRU). We empirically select a Transformer and NAT models with hyperparameters $\\dot { d } _ { \\mathrm { m o d e l } } = 4 0 0$ , $d _ { \\mathrm { h i d d e n } } = 8 0 0$ , $p _ { \\mathrm { d r o p o u t } } = 0 . 1$ , $n _ { \\mathrm { l a y e r } } = 3$ and $n _ { \\mathrm { h e a d } } = 4$ ). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Optimization We optimize the parameter with the Adam optimizer (Kingma & Ba, 2014). The hyperparameter $\\alpha$ used in Eqn. 14 was be set to 1.0 for WMT, 0.3 for IWSLT and Quora. We also use inverse square root learning rate scheduling (Vaswani et al., 2017) for the WMT, and using linear annealing (from $3 e - 4$ to $1 e - 5$ , suggested by Lee et al. (2018)) for the IWSLT and Quora. Each mini-batch consists of approximately 2K tokens for IWSLT and Quora, 32K tokens for WMT. ",
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+    },
+    {
+        "type": "text",
+        "text": "Knowledge Distillation Sequence-level knowledge distillation is applied to alleviate multimodality problem while training, using Transformer as a teacher (Hinton et al., 2015). Previous studies on non-autoregressive generation (Gu et al., 2018; Lee et al., 2018; Wei et al., 2019) have used translations produced by a pre-trained Transformer model as the training data, which significantly improves the performance. We follow this setting in translation tasks. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/9567ae4c56a4dc8c97de4e1bdef340fe986407d3d6cf8b28e3136c32090807c8.jpg",
+        "table_caption": [
+            "Table 1: Performance on the newstest-2014 for WMT14 EN-DE and test2013 for IWSLT EN-DE. ‘-’ denotes same numbers as above. ‘\\*’ indicates our implementation. The decoding speed is measured sentence-by-sentence and the speedup is computed by comparing with Transformer. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>WMT14 EN-DE DE-EN</td><td>IWSLT16 DE-EN</td><td></td><td>Speedup</td></tr><tr><td colspan=\"3\">Autoregressive Methods</td><td></td><td></td></tr><tr><td>Transformer-base (Vaswani et al., 2017) *Transformer(Beam=4)</td><td>27.30 27.40</td><td>1 31.33</td><td>34.81</td><td>1.0×</td></tr><tr><td colspan=\"3\">Non-Autoregressive Methods</td><td></td><td></td></tr><tr><td>Flowseq (Ma et al., 2019)</td><td>18.55</td><td>23.36</td><td></td><td></td></tr><tr><td>*NAT-base</td><td>/</td><td>11.02</td><td></td><td>/</td></tr><tr><td>*PNAT</td><td>19.73 NAT w/ Knowledge Distillation</td><td>24.04</td><td></td><td>/</td></tr><tr><td>NAT-FT (Gu et al., 2018) LT (Kaiser et al., 2018) IR-NAT (Lee et al., 2018)</td><td>17.69 19.80 13.91</td><td>21.47 /</td><td>/ / 27.68</td><td>15.6× 5.8×</td></tr><tr><td>ENAT (Guo et al., 2019) NAT-REG (Wang et al., 2019) imitate-NAT (Wei et al.,2019) Flowseq (Ma et al.,2019)</td><td>20.65 20.65 22.44</td><td>16.77 23.02 24.77 25.67</td><td>/ / 1</td><td>9.0× 24.3× 1 18.6×</td></tr><tr><td>*NAT-base</td><td>21.45 1 23.05</td><td>26.16 16.69</td><td>一</td><td>1.1× 13.5x</td></tr><tr><td colspan=\"3\">*PNAT 27.18 NATw/Reranking orIterative Refinments</td><td>31.23</td><td>7.3×</td></tr><tr><td>NAT-FT (rescoring 10 candidates) LT (rescoring 10 candidates) IR-NAT (refinement 10)</td><td>18.66 22.50</td><td>22.42 /</td><td>/ 一</td><td>7.7× /</td></tr><tr><td>ENAT (rescoring 9 candidates)</td><td>21.61 24.28</td><td>25.48 26.10</td><td>32.31 /</td><td>1.3× 12.4×</td></tr><tr><td>NAT-REG (rescoring 9 candidates) imitate-NAT (rescoring 9 candidates)</td><td>24.61 24.15</td><td>28.90 27.28</td><td>/ /</td><td>1 9.7×</td></tr><tr><td>Flowseq (rescoring 30 candidates) *PNAT (LPD n=9,△M=4)</td><td>23.48 24.48</td><td>28.40 29.16</td><td>/ 32.60</td><td>/ 3.7x</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 MAIN RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Machine Translation We compare the PNAT with strong NAT baselines, including the NAT with fertility (Gu et al., 2018, NAT-FT), the NAT with iterative refinement (Lee et al., 2018, IR-NAT), the NAT with regularization (Wang et al., 2019, NAT-REG), the NAT with enhanced decoder input (Guo et al., 2019, ENAT), the NAT with learning from auto-regressive model (Wei et al., 2019, imitateNAT), the NAT build on latent variables (Kaiser et al., 2018, LT), and the flow-based NAT model (Ma et al., 2019, Flowseq). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The results are shown in Table 1. We basically compare the proposed PNAT against the autoregressive counterpart both in terms of generation quality, which is measured with BLEU (Papineni et al., 2002) and inference speedup. For all our tasks, we obtain the performance of competitors by either directly using the performance figures reported in the previous works if they are available or producing them by using the open source implementation of baseline algorithms on our datasets.2 Clearly, PNAT achieves a comparable or better result to previous NAT models on both WMT and IWSLT tasks. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We list the result of the NAT models trained without using knowledge distillation in the second block of the Table 1. The PNAT achieves significant improvements (more than 13.0 BLEU points) over the naive baselines, which indicate that position learning greatly contributes to improve the model capability of NAT model. The PNAT also achieves a better result than the Flowseq around 1.0 BLEU, which demonstrates the effectiveness of PNAT in modeling dependencies between the target outputs. ",
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+    {
+        "type": "text",
+        "text": "As shown in the third block of the Table 1, without using reranking techniques, the PNAT outperforms all the competitors with a large margin, achieves a balance between performance and efficiency. In particular, the previous state-of-the-art(WMT14 DE-EN) Flowseq achieves good performance with the slow speed $( 1 . 1 \\times )$ , while PNAT goes beyond Flowseq in both respects. ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "Our best results are obtained with length parallel decoding which employ autoregressive model to rerank the multiple generation candidates of different target length. Specifically, on the large scale WMT14 DE-EN task, PNAT $( + \\mathrm { L P D } )$ surpass the NAT-REG by 0.76 BLEU score. Without reranking, the gap has increased to 2.4 BLEU score (27.18 v.s. 24.77). The experiments shows the power of explicitly position modeling which reduces the gap between non-autoregressive and the autoregressive models. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Paraphrase Generation Given a sentence, paraphrase generation aims to synthesize another sentence that is different from the given one, but conveys the same meaning. Comparing with translation task, paraphrase generation prefers a more similar order between source and target sentence, which possibly learn a trivial position model. PNAT can potentially yield better results with the position model to infer the relatively ordered alignment relationship. ",
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+    {
+        "type": "table",
+        "img_path": "images/ff36246f5b4911e4cf6f9e4b270b983e6a33049fc339a77de81902969bcf117f.jpg",
+        "table_caption": [
+            "Table 2: Results on validation set and test set of Quora. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model Valid</td><td>Paraphrase(BLEU) Test</td></tr><tr><td>Seq-to-seq(GRU) Transformer</td><td>24.68 24.75 25.46</td></tr><tr><td>NAT-base</td><td>25.88 19.80</td></tr><tr><td>PNAT 29.30</td><td>20.34 29.00</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The results of the paraphrase generation are shown in Table 2. In consist with our intuition, PNAT achieves the best result on this task and even surpass Transformer around 3.5 BLEU. The NAT model is not powerful enough to capture the latent position relationship. The comparison between NAT-base and PNAT shows that explicit position modeling in PNAT plays a crucial role in generating sentences. ",
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+    {
+        "type": "text",
+        "text": "4.4 ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Effectiveness of Heuristic Searched Position First, we analyze whether the position derived from the heuristic search is suitable for use as supervision to the position predictor. We evaluate the effectiveness of the searched position by training a PNAT as before and testing with the heuristic searched position instead of the predicted position. As shown in the second block of the Table 3, it is easier noticed that as PNAT w/ HSP achieves a significant improvement over the NAT-base and the Transformer, which demonstrates that the heuristic search for the position is effective. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/cb5275a6fd5b7d07cc8464ee8c697245b56a8e8f0006675c162ae15fff9110a3.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td colspan=\"2\">Position Accuracy(%) permutation-acc relative-acc(r=4)</td><td>WMT14DE-EN BLEU</td><td>Speed Up</td></tr><tr><td>Transformer(beam=4)</td><td>/</td><td>1</td><td>30.68</td><td>1.0×</td></tr><tr><td>NAT-base</td><td>/</td><td>/</td><td>16.71</td><td>13.5×</td></tr><tr><td>PNATw/HSP</td><td>100.00</td><td>100.00</td><td>46.03</td><td>12.5×</td></tr><tr><td>PNATw/AR-Predictor</td><td>25.30</td><td>59.27</td><td>27.11</td><td>7.3×</td></tr><tr><td>PNAT w/NAR-Predictor</td><td>23.11</td><td>55.57</td><td>20.81</td><td>11.7×</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 3: Results on validation set of WMT14 DE-EN with different position strategy. “HSP” means the reference position sequence derived from the heuristic position searching. ",
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+    {
+        "type": "text",
+        "text": "Effectiveness and Efficiency of Position Modeling We are also analysis the accuracy of our position modeling and its influence on the quality of generation on the WMT14 DE-EN task. For evaluating the position accuracy, we adopt the heuristic searched position as the position reference (denoted as “HSP”), which is the training target of the position predictor. PNAT requires the position information at two places. The first is the mutual relative relationship between the states that will be used during decoding. And the second is to reorder the decoded output after decoding. We then propose the corresponding metrics for evaluation, which is the relative position accuracy (with relation threshold $r = 4$ ) and the permutation accuracy. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "As shown in Table 3, better position accuracy always yields better generation performance. The non-autoregressive position model is less effective than the current autoregressive position model, both in the accuracy of the permutation and the relative position. Even though the current PNAT with a simple AR-Predictor has surpassed the previous NAT model, the position accuracy is still less desirable (say, less than $30 \\%$ ) and has a great exploration space. We provide a few examples in Appendix B. There is also a trade-off between the effectiveness and efficiency, the choice of the non-autoregressive means the efficiency and the choice of autoregressive means the effectiveness. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Repeated Generation Analysis Previous NAT often suffers from the repeated generation problem due to the lack of sequential position information. NAT is less effective to distinguish adjacent decoder hidden states, which is copied from the adjacent source representation. To further study this problem, we proposed to evaluate the gains of simply remove the repeated tokens. As shown in Table 4, we perform the repeated generation analysis on the paraphrase generation tasks. Removing repeated tokens has little impact for PNAT model, with only 0.05 BLEU differences. However for the NAT-base model, the gap comes with almost 1 BLEU (0.89). The results clearly demonstrate that the explicitly position model essentially learns the sequential information for sequence generation. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/46eaaabc9007fa1d6bd7c5b8ab5eb128c2175aeb4529a778c43dd2cbcc289761.jpg",
+        "table_caption": [
+            "Table 4: Results on test set of Quora. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">Paraphrase(Test-BLEU)</td><td></td></tr><tr><td>w/ remove repeats</td><td>w/o remove repeats</td><td>△BLEU</td></tr><tr><td>NAT-base</td><td>20.34</td><td>19.45</td><td>0.89</td></tr><tr><td>PNAT</td><td>29.00</td><td>28.95</td><td>0.05</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Convergence Efficiency We also perform the training efficiency analysis in IWSLT16 DE-EN Translation task. The learning curves are shown in 2. The curve of the PNAT is on the top-left corner. Remarkably, PNAT has the best convergence speed compared with the NAT competitors and even a strong autoregressive model. The results are in line with our intuition, that the position learning brings meaningful information of position relationship and benefits the generation of the target sentence. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/6dd652e56a43a853cee829ec482235cea4c60efa2b686c527044c7b9d5ba62ab.jpg",
+        "image_caption": [
+            "Figure 2: The learning curves from training of models on evaluation set of IWSLT-16 DE-EN. Mini-batch size is 2048 tokens. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Gu et al. (2018) first develops a non-autoregressive Transformer for neural machine translation (NMT) tasks, which produces the outputs in parallel and the inference speed is thus significantly boosted. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Due to the removal of the dependencies between the target outputs, it comes at the cost that the translation quality is largely sacrificed. A line of work has been proposed to mitigate such performance degradation. Some previous work is focused on enhancing the decoder inputs by replacing the target words as inputs, such as Guo et al. (2019) and Lee et al. (2018). Lee et al. (2018) proposed a method of iterative refinement based on the latent variable model and denoising autoencoder. Guo et al. (2019) enhances decoder input by introducing the phrase table in statistical machine translation and embedding transformation. Another part of previous work focuses on improving the supervision of NAT’s decoder states, including imitation learning from autoregressive models (Wei et al., 2019) or regularizing the decoder state with backward reconstruction error (Wang et al., 2019). There is also a line studies build upon latent variables, such as Kaiser et al. (2018) and Roy et al. (2018) utilize discrete latent variables for making decoding more parallelizable. Moreover, Shao et al. (2019) also proposed a method to retrieve the target sequential information for NAT models. Unlike previous work, we explicitly model the position, which has shown its importance to the autoregressive model and can well model the dependence between states. To the best of our knowledge, PNAT is the first work to explicitly model position information for non-autoregressive text generation. ",
+        "bbox": [
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We proposed PNAT, a non-autoregressive transformer by explicitly modeled positions, which bridge the performance gap between the non-autoregressive decoding and autoregressive decoding. Specifically, we model the position as latent variables, and training with heuristic searched positions with MC algorithms. As a result, PNAT leads to significant improvement and move more close to the performance gap between the NAT and AT on machine translation tasks. Besides, the experimental results of the paraphrase generation task show that the performance of the PNAT can exceed that of the autoregressive model, and at the same time, it also has a large improvement space. According to our further analysis on effectiveness of position modeling, in future work, we can still enhance the performance of the NAT model by strengthening position learning. ",
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+        "type": "text",
+        "text": "A HEURISTIC SEARCH FOR POSITIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Algorithm 1 Heuristic Search for Positions ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            140,
+            459,
+            154
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Input: The candidates set of decoder inputs: $D = \\{ d _ { 1 } , \\cdots , d _ { M } \\}$ and target embeddings: $Y =$ $\\{ y _ { 1 } , \\cdots , y _ { M } \\}$ ;   \nOutput: The position of the decoder inputs $\\hat { z }$ .   \n1: initial $A { \\stackrel { \\cdot } { = } } \\{ \\} , { \\hat { D } } = D , { \\hat { Y } } = Y$ ;   \n2: compute the similarity matrix $\\mathrm { S i m } _ { D , Y }$ : the $\\mathrm { S i m } [ i , j ]$ in the matrix is the the similarity between the $d _ { i }$ and $y _ { j }$ computing with $\\mathrm { s i m } _ { i , j } =$ cosine $( d _ { i } , y _ { j } )$ ;   \n3: repeat   \n4: extract the similarity matrix $\\mathrm { S i m } _ { \\hat { D } , \\hat { Y } }$ from the $\\mathrm { S i m } _ { D , Y }$ ;   \n5: select: (i, j) = arg max(i,j) SimD,ˆ Yˆ   \n6: update: $A  A \\cup \\{ ( i , j ) \\}$ , $\\hat { D }  \\hat { D } \\setminus \\{ d _ { i } \\} , \\hat { Y }  \\hat { Y } \\setminus \\{ y _ { j } \\} ;$   \n7: until $\\hat { D } = \\left\\{ \\begin{array} { r l r } \\end{array} \\right\\}$ and $\\hat { Y } = \\{ \\}$   \n8: for each pair $( i , j )$ in $A$ do set $\\hat { z } _ { i } = j$   \n9: end for;   \n10: return $\\hat { z }$ ",
+        "bbox": [
+            171,
+            156,
+            826,
+            381
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As shown in Algorithm 1, we perform a greedy algorithm to select the pair with the highest similarity score iteratively until the permutation $\\hat { z }$ is generated. ",
+        "bbox": [
+            176,
+            397,
+            821,
+            426
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The complexity of this algorithm is $o ( M ^ { 3 } )$ ( $M$ is the length of output sentence). Specifically, the complexity to select the maximum from the similarity matrix is $o ( \\dot { M } ^ { 2 } )$ for each loop. We need $M$ loops of greedy search to allocate positions for all decoder inputs. ",
+        "bbox": [
+            174,
+            433,
+            825,
+            476
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B CASE STUDY OF PREDICTED POSITIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            496,
+            544,
+            512
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We also provide a few examples in Table 5. For each source sentence, we first analyze the generation quality of the PNAT with a heuristic searched position. Besides, we also show the translation with the predicted position. We have the following observations: First, the output generated by the PNAT using the heuristic searched position always keeps the high consistency with the reference, shows the effectiveness of the heuristic searched position. Second, better position accuracy always yields better generation performance (Case 1,2 against Case 3). Third, as we can see in case 4, though the permutation accuracy is lower, it still generates a good result, the reason why we chose to use the relative self-attention instead of absolute self-attention. ",
+        "bbox": [
+            173,
+            526,
+            825,
+            638
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/4afaa02cdfdc17a0aa3a216dcf2a89230e664a56ce40782fef7249a3e66e60d6.jpg",
+        "table_caption": [
+            "Table 5: Examples of translation outputs from PNAT with different setting on WMT14 DE-EN. It is should be noted that the length is different between the position sequence and the output sequence because we keep the origin position output and combine the BPE sequence to word sequence. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Source</td><td>bei dem deutschen Gesetz geht es um die Zuweisung bei der Geburt.</td></tr><tr><td>Reference Heuristic Searched Position(HSP) PNATw/HSP</td><td>German law is about assigning it at birth . 3,6, 1, 2, 10, 0, 5,4, 7, 8, 9 German law is about assigning them at birth .</td></tr><tr><td>PredictedPosition PNAT w/Predicted Postion</td><td>3, 6,1, 2, 10, 0, 5, 4, 7, 8, 9 German law is about assigning them at birth .</td></tr><tr><td>Source</td><td>weiB er über das Telefon @-@ Hacking Bescheid ?</td></tr><tr><td>Reference Heuristic Searched Position(HSP) PNATw/HSP</td><td>does he know about phone hacking ? 2, 1, 3,4, 8,5,6, 0, 7,9 does he know the telephone hacking ?</td></tr><tr><td>PredictedPosition PNAT w/Predicted Postion</td><td>1, 0, 3,4, 8,5, 6,2, 7, 9 he know about the telephone hacking ?</td></tr><tr><td>Source</td><td>was CCAAbedeutet,mochte eineBesucherin wissen.</td></tr><tr><td>Reference Heuristic Searched Position(HSP)</td><td>one visitor wants to know what CCAA means . 5,6, 7,8,9,3,2, 0,1, 11,4, 10</td></tr><tr><td>PNATw/HSP Predicted Position</td><td>a visitor wants to know what CCAA means . 5,0, 1, 2,3, 7,4, 8, 9,11,6, 10</td></tr><tr><td>PNAT w/Predicted Postion Source</td><td>CCAA means wants to know to a visitor . eines von 2O Kindern in den Vereinigten</td></tr><tr><td></td><td>Staaten hat inzwischen eine Lebensmittelal-</td></tr><tr><td>Reference</td><td>lergie . one in 20 children in the United States now</td></tr><tr><td>Heuristic Searched Position(HSP)</td><td>have food allergies . 14,1,2, 3,4,5,6,7,9,8,0,10,11, 12,13</td></tr><tr><td>PNAT w/HSP Predicted Position</td><td>one of 2O children in theUnited Statesnowhas food allergy . 14, 0, 1, 2, 3,4,5, 6, 8,7, 9, 10, 11, 12, 13</td></tr></table>",
+        "bbox": [
+            215,
+            261,
+            781,
+            709
+        ],
+        "page_idx": 12
+    }
+]

parse/train/HkgaETNtDB/HkgaETNtDB.md

@@ -0,0 +1,391 @@
+# MIXOUT: EFFECTIVE REGULARIZATION TO FINETUNE LARGE-SCALE PRETRAINED LANGUAGE MODELS
+
+Cheolhyoung Lee∗
+
+cheolhyoung.lee@kaist.ac.kr
+
+Kyunghyun Cho† ‡ § kyunghyun.cho@nyu.edu
+
+Wanmo Kang∗
+
+wanmo.kang@kaist.ac.kr
+
+# ABSTRACT
+
+In natural language processing, it has been observed recently that generalization could be greatly improved by finetuning a large-scale language model pretrained on a large unlabeled corpus. Despite its recent success and wide adoption, finetuning a large pretrained language model on a downstream task is prone to degenerate performance when there are only a small number of training instances available. In this paper, we introduce a new regularization technique, to which we refer as “mixout”, motivated by dropout. Mixout stochastically mixes the parameters of two models. We show that our mixout technique regularizes learning to minimize the deviation from one of the two models and that the strength of regularization adapts along the optimization trajectory. We empirically evaluate the proposed mixout and its variants on finetuning a pretrained language model on downstream tasks. More specifically, we demonstrate that the stability of finetuning and the average accuracy greatly increase when we use the proposed approach to regularize finetuning of BERT on downstream tasks in GLUE.
+
+# 1 INTRODUCTION
+
+Transfer learning has been widely used for the tasks in natural language processing (NLP) (Collobert et al., 2011; Devlin et al., 2018; Yang et al., 2019; Liu et al., 2019; Phang et al., 2018). In particular, Devlin et al. (2018) recently demonstrated the effectiveness of finetuning a large-scale language model pretrained on a large, unannotated corpus on a wide range of NLP tasks including question answering and language inference. They have designed two variants of models, BERTLARGE (340M parameters) and BERTBASE (110M parameters). Although BERTLARGE outperforms BERTBASE generally, it was observed that finetuning sometimes fails when a target dataset has fewer than 10,000 training instances (Devlin et al., 2018; Phang et al., 2018).
+
+When finetuning a big, pretrained language model, dropout (Srivastava et al., 2014) has been used as a regularization technique to prevent co-adaptation of neurons (Vaswani et al., 2017; Devlin et al., 2018; Yang et al., 2019). We provide a theoretical understanding of dropout and its variants, such as Gaussian dropout (Wang & Manning, 2013), variational dropout (Kingma et al., 2015), and dropconnect (Wan et al., 2013), as an adaptive $L ^ { 2 }$ -penalty toward the origin (all zero parameters 0) and generalize dropout by considering a target model parameter $\textbf { \em u }$ (instead of the origin), to which we refer as $\mathtt { m i x o u t } ( { \pmb u } )$ . We illustrate mixout $( { \pmb u } )$ in Figure 1. To be specific, $\mathtt { m i x o u t } ( { \pmb u } )$ replaces all outgoing parameters from a randomly selected neuron to the corresponding parameters of $\textbf { \em u }$ . mixout $( { \pmb u } )$ avoids optimization from diverging away from $\textbf { \em u }$ through an adaptive $L ^ { 2 }$ -penalty toward $\textbf { \em u }$ . Unlike mixout $( { \pmb u } )$ , dropout encourages a move toward the origin which deviates away from $\textbf { \em u }$ since dropout is equivalent to mixout(0).
+
+We conduct experiments empirically validating the effectiveness of the proposed $\mathtt { m i x o u t } ( w _ { \mathrm { p r e } } )$ where ${ \pmb w } _ { \mathrm { p r e } }$ denotes a pretrained model parameter. To validate our theoretical findings, we train a fully connected network on EMNIST Digits (Cohen et al., 2017) and finetune it on MNIST. We observe that a finetuning solution of $\mathtt { m i x o u t } ( w _ { \mathrm { p r e } } )$ deviates less from ${ \pmb w } _ { \mathrm { p r e } }$ in the $L ^ { 2 }$ -sense than that of dropout. In the main experiment, we finetune $\mathrm { B E R T _ { L A R G E } }$ with mixout $( w _ { \mathrm { p r e } } )$ on small training sets of GLUE (Wang et al., 2018). We observe that $\mathtt { m i x o u t } ( w _ { \mathrm { p r e } } )$ reduces the number of unusable models that fail with the chance-level accuracy and increases the average development (dev) scores for all tasks. In the ablation studies, we perform the following three experiments for finetuning BERTLARGE with mixout $( { \pmb w } _ { \mathrm { p r e } } )$ : (i) the effect of $\mathtt { m i x o u t } ( w _ { \mathrm { p r e } } )$ on a sufficient number of training examples, (ii) the effect of a regularization technique for an additional output layer which is not pretrained, and (iii) the effect of probability of $\mathtt { m i x o u t } ( w _ { \mathrm { p r e } } )$ compared to dropout. From these ablation studies, we observe that three characteristics of mixout $( w _ { \mathrm { p r e } } )$ : (i) finetuning with mixout $( { \pmb w } _ { \mathrm { p r e } } )$ does not harm model performance even with a sufficient number of training examples; (ii) It is beneficial to use a variant of mixout as a regularization technique for the additional output layer; (iii) The proposed mixout $( w _ { \mathrm { p r e } } )$ is helpful to the average dev score and to the finetuning stability in a wider range of its hyperparameter $p$ than dropout.
+
+![](images/7556e26782623269a6d8e06d4274924958e8bafe862607e41a33fa1681ffbf41.jpg)  
+Figure 1: Illustration of mixout $( { \pmb u } )$ . Suppose that $\textbf { \em u }$ and $\pmb { w }$ are a target model parameter and a current model parameter, respectively. (a): We first memorize the parameters of the vanilla network at $\textbf { \em u }$ . (b): In the dropout network, we randomly choose an input neuron to be dropped (a dotted neuron) with a probability of $p$ . That is, all outgoing parameters from the dropped neuron are eliminated (dotted connections). (c): In the mixout $( { \pmb u } )$ network, the eliminated parameters in (b) are replaced by the corresponding parameters in (a). In other words, the mixout $( { \pmb u } )$ network at $\pmb { w }$ is the mixture of the vanilla network at $\textbf { \em u }$ and the dropout network at $\pmb { w }$ with a probability of $p$ .
+
+# 1.1 RELATED WORK
+
+For large-scale pretrained language models (Vaswani et al., 2017; Devlin et al., 2018; Yang et al., 2019), dropout has been used as one of several regularization techniques. The theoretical analysis for dropout as an $L ^ { 2 }$ -regularizer toward 0 was explored by Wan et al. (2013) where 0 is the origin. They provided a sharp characterization of dropout for a simplified setting (generalized linear model). Mianjy & Arora (2019) gave a formal and complete characterization of dropout in deep linear networks with squared loss as a nuclear norm regularization toward 0. However, neither Wan et al. (2013) nor Mianjy $\&$ Arora (2019) gives theoretical analysis for the extension of dropout which uses a point other than 0.
+
+Wiese et al. (2017), Kirkpatrick et al. (2017), and Schwarz et al. (2018) used $L ^ { 2 }$ -penalty toward a pretrained model parameter to improve performance. They focused on preventing catastrophic forgetting to enable their models to learn multiple tasks sequentially. They however do not discuss nor demonstrate the effect of $L ^ { 2 }$ -penalty toward the pretrained model parameter on the stability of finetuning. Barone et al. (2017) introduced tuneout, which is a special case of mixout. They applied various regularization techniques including dropout, tuneout, and $L ^ { 2 }$ -penalty toward a pretrained model parameter to finetune neural machine translation. They however do not demonstrate empirical significance of tuneout compared to other regularization techniques nor its theoretical justification.
+
+# 2 PRELIMINARIES AND NOTATIONS
+
+Norms and Loss Functions Unless explicitly stated, a norm $\| \cdot \|$ refers to $L ^ { 2 }$ -norm. A loss function of a neural network is written as $\begin{array} { r } { \mathcal { L } ( \mathbf { w } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathcal { L } _ { i } ( \ddot { \mathbf { w } } ) } \end{array}$ , where $\pmb { w }$ is a trainable model parameter. $\mathcal { L } _ { i }$ is “a per-example loss function” computed on the $i$ -th data point.
+
+Strong Convexity A differentiable function $f$ is strongly convex if there exists $m > 0$ such that
+
+$$
+f ( pmb { y } ) \geq f ( \pmb { x } ) + \nabla f ( \pmb { x } ) ^ { \top } ( \pmb { y } - \pmb { x } ) + \frac { m } { 2 } \| \pmb { y } - \pmb { x } \| ^ { 2 } ,
+$$
+
+for all $_ { \textbf { \em x } }$ and $\textbf {  { y } }$
+
+Weight Decay We refer as “wdecay $( \pmb { u } , \lambda ) ^ { \prime }$ to minimizing
+
+$$
+\mathcal { L } ( \boldsymbol { \boldsymbol { w } } ) + \frac { \lambda } { 2 } \| \boldsymbol { \boldsymbol { w } } - \boldsymbol { u } \| ^ { 2 } ,
+$$
+
+instead of the original loss function $\mathcal { L } ( w )$ where $\lambda$ is a regularization coefficient. Usual weight decay of $\lambda$ is equivalent to wdecay $( \mathbf { 0 } , \lambda )$ .
+
+Probability for Dropout and Dropconnect Dropout (Srivastava et al., 2014) is a regularization technique selecting a neuron to drop with a probability of $p$ . Dropconnect (Wan et al., 2013) chooses a parameter to drop with a probability of $p$ . To emphasize their hyperparameter $p$ , we write dropout and dropconnect with a drop probability of $p$ as “dropout $( p )$ ” and “dropconnect $( p ) ^ { \dag }$ , respectively. dropout $( p )$ is a special case of dropconnect $( p )$ if we simultaneously drop the parameters outgoing from each dropped neuron.
+
+Inverted Dropout and Dropconnect In the case of dropout $( p )$ , a neuron is retained with a probability of $1 - p$ during training. If we denote the weight parameter of that neuron as $\pmb { w }$ during training, then we use $( 1 - p ) \pmb { w }$ for that weight parameter at test time (Srivastava et al., 2014). This ensures that the expected output of a neuron is the same as the actual output at test time. In this paper, dropout $( p )$ refers to inverted dropout $( p )$ which uses ${ \pmb w } / ( 1 - p )$ instead of $\textbf { \em w }$ during training. By doing so, we do not need to compute the output separately at test time. Similarly, dropconnect $( p )$ refers to inverted dropconnect $( p )$ .
+
+# 3 ANALYSIS OF DROPOUT AND ITS GENERALIZATION
+
+We start our theoretical analysis by investigating dropconnect which is a general form of dropout and then apply the result derived from dropconnect to dropout. The iterative SGD equation for dropconnect $( p )$ with a learning rate of $\eta$ is
+
+$$
+\pmb { w } ^ { ( t + 1 ) } = \pmb { w } ^ { ( t ) } - \eta \pmb { B } ^ { ( t ) } \nabla \mathcal { L } \left( \left( \mathbb { E } B _ { 1 } ^ { ( t ) } \right) ^ { - 1 } \pmb { B } ^ { ( t ) } \pmb { w } ^ { ( t ) } \right) , t = 0 , 1 , 2 , \cdots ,
+$$
+
+where $\pmb { B } ^ { ( t ) } = \mathrm { d i a g } ( \boldsymbol { B } _ { 1 } ^ { ( t ) } , \boldsymbol { B } _ { 2 } ^ { ( t ) }$ , · · · , $B _ { d } ^ { ( t ) } )$ ) and $B _ { i } ^ { ( t ) }$ ’s are mutually independent Bernoulli $( 1 - p )$ random variables with a drop probability of $p$ for all $i$ and $t$ . We regard equation 2 as finding a solution to the minimization problem below:
+
+$$
+\operatorname* { m i n } _ { \pmb { w } } \mathbb { E } \mathcal { L } \left( ( \mathbb { E } B _ { 1 } ) ^ { - 1 } \pmb { B } \pmb { w } \right) ,
+$$
+
+where $B = \operatorname { d i a g } ( B _ { 1 } , ~ B _ { 2 } , ~ \cdot \cdot \cdot , ~ B _ { d } )$ and $B _ { i }$ ’s are mutually independent Bernoulli $( 1 - p )$ random variables with a drop probability of $p$ for all $i$ .
+
+Gaussian dropout (Wang & Manning, 2013) and variational dropout (Kingma et al., 2015) use other random masks to improve dropout rather than Bernoulli random masks. To explain these variants of dropout as well, we set a random mask matrix $M = \mathrm { d i a g } ( M _ { 1 } , M _ { 2 }$ , · · · , $M _ { d } )$ ) to satisfy $\mathbb { E } M _ { i } = \mu$ and $\mathrm { V a r } ( M _ { i } ) = \sigma ^ { 2 }$ for all $i$ . Now we define a random mixture function with respect to $\textbf { \em w }$ from $\textbf { \em u }$ and $M$ as
+
+and a minimization p
+
+$$
+\begin{array} { r l } & { \Phi ( w ; \boldsymbol { u } , M ) = \mu ^ { - 1 } \big ( ( I - M ) \boldsymbol { u } + M \boldsymbol { w } - ( 1 - \mu ) \boldsymbol { u } \big ) , } \\ & { \mathrm { r o b l e m ~ w i t h ~ } ^ { * } \mathrm { m i x c o n n e c t } ( \boldsymbol { u } , \mu , \boldsymbol { \sigma } ^ { 2 } ) ^ { , * } \mathrm { a s } } \\ & { ~ \operatorname* { m i n } _ { \boldsymbol { w } } \mathbb { E } \mathcal { L } \big ( \Phi ( \boldsymbol { w } ; \boldsymbol { u } , M ) \big ) . } \end{array}
+$$
+
+We can view dropconnect $( p )$ equation 3 as a special case of equation 5 where ${ \pmb u } = { \bf 0 }$ and $M = B$ We investigate how mixconnect $( \pmb { u } , \mu , \sigma ^ { 2 } )$ differs from the vanilla minimization problem
+
+$$
+\operatorname* { m i n } _ { \boldsymbol { w } } \mathbb { E } \mathcal { L } ( \boldsymbol { w } ) .
+$$
+
+If the loss function $\mathcal { L }$ is strongly convex, we can derive a lower bound of $\mathbb { E } \mathscr { L } \big ( \Phi ( { \boldsymbol w } ; { \boldsymbol u } , M ) \big )$ as in Theorem 1:
+
+Theorem 1. Assume that the loss function $\mathcal { L }$ is strongly convex. Suppose that a random mixture function with respect to $\pmb { w }$ from $\textbf { \em u }$ and $M$ is given by $\Phi ( { \pmb w } ; { \pmb u } , M )$ in equation $^ { 4 }$ where $M$ is $\mathrm { d i a g } ( M _ { 1 } , ~ M _ { 2 } , ~ \cdot \cdot \cdot , ~ M _ { d } ) $ satisfying $\mathbb { E } M _ { i } = \mu$ and $\mathrm { V a r } ( M _ { i } ) = \sigma ^ { 2 }$ for all $i$ . Then, there exists $m > 0$ such that
+
+$$
+\mathbb { E } \mathcal { L } \big ( \Phi ( { \boldsymbol w } ; { \boldsymbol u } , M ) \big ) \ge \mathcal { L } ( { \boldsymbol w } ) + \frac { m \sigma ^ { 2 } } { 2 \mu ^ { 2 } } \| { \boldsymbol w } - { \boldsymbol u } \| ^ { 2 } ,
+$$
+
+for all $\textbf { \em w }$ (Proof in Supplement $A$ ).
+
+Theorem 1 shows that minimizing the l.h.s. of equation 7 minimizes the r.h.s. of equation 7 when the r.h.s. is a sharp lower limit of the l.h.s. The strong convexity of $\mathcal { L }$ means that $\mathcal { L }$ is bounded from below by a quadratic function, and the inequality of equation 7 comes from the strong convexity. Hence, the equality holds if $\mathcal { L }$ is quadratic, and mixconnect $( \pmb { u } , \mu , \sigma ^ { 2 } )$ is an $L ^ { 2 }$ -regularizer with a regularization coefficient of $m \sigma ^ { 2 } { \dot { / } } \mu ^ { 2 }$ .
+
+# 3.1 MIXCONNECT TO MIXOUT
+
+We propose mixout as a special case of mixconnect, which is motivated by the relationship between dropout and dropconnect. We assume that
+
+$$
+\begin{array} { r } { { \pmb w } = \left( { \pmb w } _ { 1 } ^ { ( N _ { 1 } ) } , \ \cdots , \ { \pmb w } _ { d _ { 1 } } ^ { ( N _ { 1 } ) } , \ w _ { 1 } ^ { ( N _ { 2 } ) } , \ \cdots , \ w _ { d _ { 2 } } ^ { ( N _ { 2 } ) } , \ \cdots \dots , w _ { 1 } ^ { ( N _ { k } ) } , \ \cdots , \ w _ { d _ { k } } ^ { ( N _ { k } ) } \right) , } \end{array}
+$$
+
+where $w _ { j } ^ { ( N _ { i } ) }$ is the $j$ th parameter outgoing from the neuron $N _ { i }$ . We set the corresponding $M$
+
+$$
+M = \mathrm { d i a g } \left( M ^ { ( N _ { 1 } ) } , \ \cdots \ , \ M ^ { ( N _ { 1 } ) } , \ M ^ { ( N _ { 2 } ) } , \ \cdots \ , \ M ^ { ( N _ { 2 } ) } , \ \cdots \ \cdots \ , \ M ^ { ( N _ { k } ) } , \ \cdots \ , \ M ^ { ( N _ { k } ) } \right) ,
+$$
+
+where $\mathbb { E } M ^ { ( N _ { i } ) } = \mu$ and $\mathrm { V a r } ( M ^ { ( N _ { i } ) } ) = \sigma ^ { 2 }$ for all $i$ . In this paper, we set $M ^ { ( N _ { i } ) }$ to Bernoull $( 1 - p )$ for all $i$ and mixout $( { \pmb u } )$ hereafter refers to this correlated version of mixconnect with Bernoulli random masks. We write it as “mixout $( \boldsymbol { u } , \boldsymbol { p } ) ^ { \flat }$ when we emphasize the mix probability $p$ .
+
+Corollary 1.1. Assume that the loss function $\mathcal { L }$ is strongly convex. We denote the random mixture function of mixout $( \pmb { u } , \ p )$ , which is equivalent to that of mixconnect $( \pmb { u } , \ 1 - p , \ p - p ^ { 2 } )$ , as $\Phi ( { \pmb w } ; { \pmb u } , M )$ where $M$ is defined in equation 8. Then, there exists $m > 0$ such that
+
+$$
+\mathbb { E } \mathcal { L } \big ( \Phi ( { \boldsymbol w } ; { \boldsymbol u } , B ) \big ) \ge \mathcal { L } ( { \boldsymbol w } ) + \frac { m p } { 2 ( 1 - p ) } \| { \boldsymbol w } - { \boldsymbol u } \| ^ { 2 } ,
+$$
+
+for all $\textbf { \em w }$
+
+Corollary 1.1 is a straightforward result from Theorem 1. As the mix probability $p$ in equation 9 increases to 1, the $L ^ { 2 }$ -regularization coefficient of $m p / ( 1 - p )$ increases to infinity. It means that $p$ of mixout $( \boldsymbol { \mathscr { u } } , \boldsymbol { p } )$ can adjust the strength of $L ^ { 2 }$ -penalty toward $\textbf { \em u }$ in optimization. mixout $( { \pmb u } )$ differs from wdecay $( { \pmb u } )$ since the regularization coefficient of mixout $( { \pmb u } )$ depends on $m$ determined by the current model parameter $\pmb { w }$ . mixout $( \pmb { u } , \ p )$ indeed regularizes learning to minimize the deviation from $\textbf { \em u }$ . We validate this by performing least squares regression in Supplement D.
+
+We often apply dropout to specific layers. For instance, Simonyan & Zisserman (2014) applied dropout to fully connected layers only. We generalize Theorem 1 to the case in which mixout is only applied to specific layers, and it can be done by constructing $M$ in a particular way. We demonstrate this approach in Supplement B and show that mixout for specific layers adaptively $L ^ { 2 }$ -penalizes their parameters.
+
+# 3.2 MIXOUT FOR PRETRAINED MODELS
+
+Hoffer et al. (2017) have empirically shown that
+
+$$
+\left\| \mathbfcal { w } _ { t } - \mathbfcal { w } _ { 0 } \right\| \sim \log t ,
+$$
+
+where ${ \pmb w } _ { t }$ is a model parameter after the $t$ -th SGD step. When training from scratch, we usually sample an initial model parameter $\pmb { w } _ { 0 }$ from a normal/uniform distribution with mean 0 and small variance. Since ${ \pmb w } _ { 0 }$ is close to the origin, ${ \pmb w } _ { t }$ is away from the origin only with a large $t$ by equation 10. When finetuning, we initialize our model parameter from a pretrained model parameter ${ \pmb w } _ { \mathrm { p r e } }$ . Since we usually obtain ${ \pmb w } _ { \mathrm { p r e } }$ by training from scratch on a large pretraining dataset, ${ \pmb w } _ { \mathrm { p r e } }$ is often far away from the origin. By Corollary 1.1, dropout $L ^ { 2 }$ -penalizes the model parameter for deviating away from the origin rather than ${ \pmb w } _ { \mathrm { p r e } }$ . To explicitly prevent the deviation from ${ \pmb w } _ { \mathrm { p r e } }$ , we instead propose to use mixout $( w _ { \mathrm { p r e } } )$ .
+
+# 4 VERIFICATION OF THEORETICAL RESULTS FOR MIXOUT ON MNIST
+
+Wiese et al. (2017) have highlighted that wdecay $( w _ { \mathrm { p r e } } )$ is an effective regularization technique to avoid catastrophic forgetting during finetuning. Because mixout $( w _ { \mathrm { p r e } } )$ keeps the finetuned model to stay in the vicinity of the pretrained model similarly to wdecay $( \dot { \boldsymbol { w } } _ { \mathrm { p r e } } )$ , we suspect that the proposed mixout $( w _ { \mathrm { p r e } } )$ has a similar effect of alleviating the issue of catastrophic forgetting. To empirically verify this claim, we pretrain a 784-300-100-10 fully-connected network on EMNIST Digits (Cohen et al., 2017), and finetune it on MNIST. For more detailed description of the model architecture and datasets, see Supplement C.1.
+
+In the pretraining stage, we run five random experiments with a batch size of 32 for $\{ 1 , 2 , \cdots , 2 0 \}$ training epochs. We use Adam (Kingma & Ba, 2014) with a learning rate of $1 0 ^ { - 4 }$ , $\beta _ { 1 } ~ = ~ 0 . 9$ , $\beta _ { 2 } ~ = ~ 0 . 9 9 9$ , wdecay $\mathbf { ( 0 , \eta 0 . 0 1 ) }$ , learning rate warm-up over the first $10 \%$ steps of the total steps, and linear decay of the learning rate after the warm-up. We use dropout(0.1) for all layers except the input and output layers. We select ${ \pmb w } _ { \mathrm { p r e } }$ whose validation accuracy on EMNIST Digits is best (0.992) in all experiments.
+
+For finetuning, most of the model hyperparameters are kept same as in pretraining, with the exception of the learning rate, number of training epochs, and regularization techniques. We train with a learning rate of $\zeta \times 1 0 ^ { - 5 }$ for 5 training epochs. We replace dropout $( p )$ with mixout $( \mathbf { \boldsymbol { w } } _ { \mathrm { p r e } } , \mathbf { \boldsymbol { p } } )$ . We do not use any other regularization technique such as wdecay(0) and wdecay $( { \pmb w } _ { \mathrm { p r e } } )$ . We monitor $\lvert | \boldsymbol { w } _ { \mathrm { f t } } - \boldsymbol { w } _ { \mathrm { p r e } } \rvert | ^ { 2 }$ ,1 validation accuracy on MNIST, and validation accuracy on EMNIST Digits to compare mixout $( { \pmb w } _ { \mathrm { p r e } } , \ p )$ to dropout $( p )$ across 10 random restarts.2
+
+![](images/b145e4dfc5e4e4032b2fd0fe02ef963e334cb8b2f25a73b7cb57eafd6a285159.jpg)  
+Figure 2: We present $\lvert | \boldsymbol { w } _ { \mathrm { f t } } - \boldsymbol { w } _ { \mathrm { p r e } } \rvert | ^ { 2 }$ , validation accuracy on MNIST (target task), and validation accuracy on EMNIST Digits (source task), as the function of the probability $p$ where ${ \pmb w } _ { \mathrm { f t } }$ and ${ \pmb w } _ { \mathrm { p r e } }$ are the model parameter after finetuning and the pretrained model parameter, respectively. We report mean (curve) $\pm$ std. (shaded area) across 10 random restarts. (a): mixout $( { \pmb w } _ { \mathrm { p r e } } , \ p ) \ \bar { L } ^ { 2 }$ -penalizes the deviation from ${ \pmb w } _ { \mathrm { p r e } }$ , and this penalty becomes strong as $p$ increases. However, with dropout $( p )$ , ${ \pmb w } _ { \mathrm { f t } }$ becomes away from ${ \pmb w } _ { \mathrm { p r e } }$ as $p$ increases. (b): After finetuning on MNIST, both mixout $( \pmb { w } _ { \mathrm { p r e } } , p )$ and dropout $( p )$ result in high validation accuracy on MNIST for $p \in \{ 0 . 1 , 0 . 2 , 0 . 3 \}$ . (c): Validation accuracy of dropout $( p )$ on EMNIST Digits drops more than that of mixout $( \mathbf { \boldsymbol { w } } _ { \mathrm { p r e } } , \mathbf { \boldsymbol { p } } )$ for all $p$ . mixout $( \mathbf { \boldsymbol { w } } _ { \mathrm { p r e } } , \mathbf { \boldsymbol { p } } )$ minimizes the deviation from ${ \pmb w } _ { \mathrm { p r e } }$ and memorizes the source task better than dropout $( p )$ for all $p$ .
+
+As shown in Figure 2 (a), after finetuning with mixout $( \mathbf { \boldsymbol { w } } _ { \mathrm { p r e } } , \mathbf { \boldsymbol { p } } )$ , the deviation from ${ \pmb w } _ { \mathrm { p r e } }$ is minimized in the $L ^ { 2 }$ -sense. This result verifies Corollary 1.1. We demonstrate that the validation accuracy of mixout $( \mathbf { \boldsymbol { w } } _ { \mathrm { p r e } } , \mathbf { \boldsymbol { p } } )$ has greater robustness to the choice of $p$ than that of dropout $( p )$ . In Figure 2 (b), both dropout $( p )$ and mixout $( \mathbf { \boldsymbol { w } } _ { \mathrm { p r e } } , \mathbf { \boldsymbol { p } } )$ result in high validation accuracy on the target task (MNIST) for $p \in \{ 0 . 1 , \ 0 . 2 , \ 0 . 3 \}$ , although mixout $( \mathbf { \boldsymbol { w } } _ { \mathrm { p r e } } , \mathbf { \boldsymbol { p } } )$ is much more robust with respect to the choice of the mix probability $p$ . In Figure 2 (c), the validation accuracy of mixout $( { \pmb w } _ { \mathrm { p r e } } , \ p )$ on the source task (EMNIST Digits) drops from the validation accuracy of the model at ${ \pmb w } _ { \mathrm { p r e } }$ (0.992) to approximately 0.723 regardless of $p$ . On the other hand, the validation accuracy of dropout $( p )$ on the source task respectively drops by 0.041, 0.074 and 0.105 which are more than those of mixout $( \pmb { w } _ { \mathrm { p r e } } , p )$ for $p \in \{ \bar { 0 } . 1 , 0 . \bar { 2 } , 0 . 3 \bar  \}$ .
+
+# 5 FINETUNING A PRETRAINED LANGUAGE MODEL WITH MIXOUT
+
+In order to experimentally validate the effectiveness of mixout, we finetune BERTLARGE on a subset of GLUE (Wang et al., 2018) tasks (RTE, MRPC, CoLA, and STS-B) with mixout $( w _ { \mathrm { p r e } } )$ . We choose them because Phang et al. (2018) have observed that it was unstable to finetune BERTLARGE on these four tasks. We use the publicly available pretrained model released by Devlin et al. (2018), ported into PyTorch by HuggingFace.3 We use the learning setup and hyperparameters recommended by Devlin et al. (2018). We use Adam with a learning rate of $\overset { \cdot } { 2 } \times \overset { \cdot } { 1 } 0 ^ { - 5 }$ , $\beta _ { 1 } ~ = ~ 0 . 9$ , $\beta _ { 2 } ~ = ~ 0 . 9 \dot { 9 } 9$ , learning rate warmup over the first $10 \%$ steps of the total steps, and linear decay of the learning rate after the warmup finishes. We train with a batch size of 32 for 3 training epochs. Since the pretrained BERTLARGE is the sentence encoder, we have to create an additional output layer, which is not pretrained. We initialize each parameter of it with $\mathcal { N } ( 0 , 0 . 0 2 ^ { 2 } )$ . We describe our experimental setup further in Supplement C.2.
+
+The original regularization strategy used in Devlin et al. (2018) for finetuning BERTLARGE is using both dropout(0.1) and wdecay $\mathbf { ( 0 , \eta 0 . 0 1 ) }$ for all layers except layer normalization and intermediate layers activated by GELU (Hendrycks & Gimpel, 2016). We however cannot use mixout $( { \pmb w } _ { \mathrm { p r e } } )$ nor wdecay $( { \pmb w } _ { \mathrm { p r e } } )$ for the additional output layer which was not pretrained and therefore does not have ${ \pmb w } _ { \mathrm { p r e } }$ . We do not use any regularization for the additional output layer when finetuning BERTLARGE with mixout $( { \boldsymbol { w } } _ { \mathrm { p r e } } )$ and wdecay $( w _ { \mathrm { p r e } } )$ . For the other layers, we replace dropout(0.1) and wdecay $\mathbf { ( 0 , \ 0 . 0 1 ) }$ with mixout $( { \boldsymbol { w } } _ { \mathrm { p r e } } )$ and wdecay $( w _ { \mathrm { p r e } } )$ , respectively.
+
+Phang et al. (2018) have reported that large pretrained models (e.g., $\mathrm { B E R T _ { L A R G E } } \backslash$ ) are prone to degenerate performance when finetuned on a task with a small number of training examples, and that multiple random restarts4 are required to obtain a usable model better than random prediction. To compare finetuning stability of the regularization techniques, we need to demonstrate the distribution of model performance. We therefore train $\mathrm { B E R T _ { L A R G E } }$ with each regularization strategy on each task with 20 random restarts. We validate each random restart on the dev set to observe the behaviour of the proposed mixout and finally evaluate it on the test set for generalization. We present the test score of our proposed regularization strategy on each task in Supplement C.3.
+
+We finetune $\mathrm { B E R T _ { L A R G E } }$ with mixout $( w _ { \mathrm { p r e } }$ , $\{ 0 . 7 , 0 . 8 , 0 . 9 \} \mathrm { , }$ ) on RTE, MRPC, CoLA, and STSB. For the baselines, we finetune $\mathrm { B E R T _ { L A R G E } }$ with both dropout(0.1) and wdecay $\mathbf { ( 0 , \ 0 . 0 1 ) }$ as well as with wdecay $\mathbf { \Delta } _ { w _ { \mathrm { p r e } } }$ , $\{ 0 . 0 1 , 0 . 0 4 , 0 . 0 7 , 0 . 1 0 \} )$ . These choices are made based on the experiments in Section 6.3 and Supplement F. In Section 6.3, we observe that finetuning BERTLARGE with mixout $( \mathbf { \boldsymbol { w } } _ { \mathrm { p r e } } , \mathbf { \boldsymbol { p } } )$ on RTE is significantly more stable with $p \in \{ 0 . 7 , \ 0 . 8 , \ 0 . 9 \}$ while finetuning with $\operatorname { d r o p o u t } ( p )$ becomes unstable as $p$ increases. In Supplement F, we demonstrate that dropout(0.1) is almost optimal for all the tasks in terms of mean dev score although Devlin et al. (2018) selected it to improve the maximum dev score.
+
+In Figure 3, we plot the distributions of the dev scores from 20 random restarts when finetuning BERTLARGE with various regularization strategies on each task. For conciseness, we only show four regularization strategies; Devlin et al. (2018)’s: both dropout(0.1) and wdecay $\mathbf { ( 0 , \theta 0 . 0 1 ) }$ , Wiese et al. (2017)’s: wdecay $( w _ { \mathrm { p r e } } , \ 0 . 0 1 )$ , ours: mixout $( w _ { \mathrm { p r e } } , \ 0 . 7 )$ , and ours $^ +$ Wiese et al. (2017)’s: both mixout $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 7 )$ and wdecay $( w _ { \mathrm { p r e } } , \ 0 . 0 1 )$ . As shown in Figure 3 (a–c), we observe many finetuning runs that fail with the chance-level accuracy when we finetune BERTLARGE with both dropout(0.1) and wdecay $\mathbf { ( 0 , \ 0 . 0 1 ) }$ on RTE, MRPC, and CoLA. We also have a bunch of degenerate model configurations when we use wdecay $( w _ { \mathrm { p r e } } , \ 0 . 0 1 )$ without mixout $( w _ { \mathrm { p r e } } , \ 0 . 7 )$ .
+
+Unlike existing regularization strategies, when we use mixout $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 7 )$ as a regularization technique with or without wdecay $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 0 1 )$ for finetuning $\mathrm { B E R T _ { L A R G E } }$ , the number of degenerate model configurations that fail with a chance-level accuracy significantly decreases. For example, in Figure 3 (c), we have only one degenerate model configuration when finetuning BERTLARGE with mixout $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 7 )$ on CoLA while we observe respectively seven and six degenerate models with Devlin et al. (2018)’s and Wiese et al. (2017)’s regularization strategies.
+
+![](images/726fa0799751348c0c9f773300d3093db9cacd73df1fae0295a6c9fc608e38f3.jpg)  
+Figure 3: Distribution of dev scores on each task from 20 random restarts when finetuning BERTLARGE with Devlin et al. (2018)’s: both dropout(0.1) and wdecay(0, 0.01), Wiese et al. (2017)’s: wdecay $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 0 1 )$ , ours: mixout $( w _ { \mathrm { p r e } } , \ 0 . 7 )$ , and ours $+$ Wiese et al. (2017)’s: both mixout $( w _ { \mathrm { p r e } } , \ 0 . 7 )$ and wdecay( $( w _ { \mathrm { p r e } } , \ 0 . 0 1 )$ . We write them as Devlin (blue), Wiese (orange), Our (green), and $\mathrm { O u r + W }$ (red), respectively. We use the same set of 20 random initializations across all the regularization setups. Error intervals show mean±std. For all the tasks, the number of finetuning runs that fail with the chance-level accuracy is significantly reduced when we use our regularization mixout $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 7 )$ regardless of using wdecay $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 0 1 )$ .
+
+In Figure 3 (a), we further improve the stability of finetuning BERTLARGE by using both mixout $( w _ { \mathrm { p r e } } , 0 . 7 )$ and wdecay( $\mathbf { \Delta } _ { w _ { \mathrm { p r e } } }$ , 0.01). Figure 3 (d) shows respectively two and one degenerate model configurations with Devlin et al. (2018)’s and Wiese et al. (2017)’s, but we do not have any degenerate resulting model with ours and ours $+$ Wiese et al. (2017)’s. In Figure 3 (b, c), we observe that the number of degenerate model configurations increases when we use wdecay $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 0 1 )$ ) additionally to mixout $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 7 )$ . In short, applying our proposed mixout significantly stabilizes the finetuning results of BERTLARGE on small training sets regardless of whether we use wdecay $( \mathbf { \mathscr { w } } _ { \mathrm { p r e } } , \ 0 . 0 1 )$ .
+
+In Table 1, we report the average and the best dev scores across 20 random restarts for each task with various regularization strategies. The average dev scores with mixout( $\mathbf { \Delta } w _ { \mathrm { p r e } }$ , $\{ 0 . 7 , \ 0 . 8 , \ 0 . 9 \} ,$ ) increase for all the tasks. For instance, the mean dev score of finetuning with mixout $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 8 )$ on CoLA is 57.9 which is $4 9 . 2 \%$ increase over 38.8 obtained by finetuning with both dropout $( p )$ and wdecay $\mathbf { ( 0 , \ 0 . 0 1 ) }$ . We observe that using wdecay( $\mathbf { \Delta } w _ { \mathrm { p r e } }$ , $\{ \bar { 0 . 0 1 } , 0 . 0 4 , \mathsf { \bar { 0 } . 0 7 } , 0 . 1 0 \} \}$ ) also improves the average dev scores for most tasks compared to using both dropout $( p )$ and wdecay $\mathbf { ( 0 , \ 0 . 0 1 ) }$ . We however observe that finetuning with mixout $: ( w _ { \mathrm { p r e } } , ~ \{ 0 . 7 , 0 . 8 , 0 . 9 \} )$ outperforms that with wdecay $\mathbf { \Delta } _ { w _ { \mathrm { p r e } } }$ $_ \mathrm { e } , \ \{ 0 . 0 1 , 0 . 0 4 , 0 . 0 7 , 0 . 1 0 \}$ ) on average. This confirms that $\mathtt { m i x o u t } ( w _ { \mathrm { p r e } } )$ has a different effect for finetuning BERTLARGE compared to wdecay $( { \pmb w } _ { \mathrm { p r e } } )$ since mixout $( w _ { \mathrm { p r e } } )$ is an adaptive $L ^ { 2 }$ -regularizer along the optimization trajectory.
+
+Since finetuning a large pretrained language model such as $\mathrm { B E R T _ { L A R G E } }$ on a small training set frequently fails, the final model performance has often been reported as the maximum dev score (Devlin et al., 2018; Phang et al., 2018) among a few random restarts. We thus report the best dev score for each setting in Table 1. According to the best dev scores as well, mixout( $\mathbf { \Delta } _ { w _ { \mathrm { p r e } } }$ , $\{ 0 . 7 , ~ 0 . 8 , ~ 0 . 9 \} )$ improves performance for all the tasks compared to using both dropout $( p )$ and wdecay $\mathbf { ( 0 , \ 0 . 0 1 ) }$ . For instance, using $\mathrm { m i x o u t } ( w _ { \mathrm { p r e } } , \ 0 . 9 )$ improves the maximum dev score by 0.9 compared to using both dropout $( p )$ and wdecay(0, 0.01) on MRPC. Unlike the average dev scores, the best dev scores achieved by using wdecay $( w _ { \mathrm { p r e } }$ , $\{ 0 . 0 1 , 0 . 0 4 , 0 . 0 7 , 0 . 1 0 \}$ ) are better than those achieved by using mixout( $\mathbf { \Delta } _ { w _ { \mathrm { p r e } } }$ , $\{ 0 . 7 , 0 . 8 , 0 . 9 \}$ ) except RTE on which it was better to use mixout $\mathbf { \Delta } _ { w _ { \mathrm { p r e } } }$ , $\left\{ 0 . 7 , 0 . 8 , 0 . 9 \right\}$ ) than wdecay( ${ \pmb w } _ { \mathrm { p r e } }$ , $\{ 0 . 0 \bar { 1 } , 0 . 0 4 , 0 . 0 7 , 0 . 1 0 \}$ ).
+
+Table 1: Mean (max) dev scores across 20 random restarts when finetuning BERTLARGE with various regularization strategies on each task. We show the following baseline results on the first and second cells: Devlin et al. (2018)’s regularization strategy (both dropout $( p )$ and wdecay $\mathbf { ( 0 , \ 0 . 0 1 ) } )$ ) and Wiese et al. (2017)’s regularization strategy (wdecay( $( w _ { \mathrm { p r e } }$ , $\{ 0 . 0 1 , 0 . 0 4 , 0 . 0 7 , 0 . 1 0 \} )$ ). In the third cell, we demonstrate finetuning results with only mixout $( w _ { \mathrm { p r e } }$ , $\{ 0 . 7 , \ 0 . 8 , \ 0 . 9 \} $ ). The results with both mixout $\mathbf { \Delta } _ { w _ { \mathrm { p r e } } }$ , $\{ 0 . 7 , 0 . 8 , \mathbf { \bar { 0 } } . 9 \} )$ and wdecay( $\mathbf { \Delta } w _ { \mathrm { p r e } }$ , 0.01) are also presented in the fourth cell. Bold marks the best of each statistics within each column. The mean dev scores greatly increase for all the tasks when we use mixout( $\mathbf { \Delta } _ { w _ { \mathrm { p r e } } }$ , $\{ 0 . 7 , 0 . 8 , 0 . 9 \}$ ).
+
+<table><tr><td>TECHNIQUE 1</td><td>TECHNIQUE 2</td><td>RTE</td><td>MRPC</td><td>CoLA</td><td>STS-B</td></tr><tr><td>dropout(0.1)</td><td>wdecay(0, 0.01)</td><td>56.5 (73.6)</td><td>83.4 (90.4)</td><td>38.8 (63.3)</td><td>82.4 (90.3)</td></tr><tr><td>-</td><td>wdecay( (wpre, 0.01)</td><td>56.3 (71.5)</td><td>86.2 (91.6)</td><td>41.9 (65.6)</td><td>85.4 (90.5)</td></tr><tr><td></td><td>wdecay( (wpre, 0.04)</td><td>51.5 (70.8)</td><td>85.8 (91.5)</td><td>35.4 (64.7)</td><td>80.7 (90.6)</td></tr><tr><td></td><td>wdecay( (wpre, 0.07</td><td>57.0 (70.4)</td><td>85.8 (91.0)</td><td>48.1 (63.9)</td><td>89.6 (90.3)</td></tr><tr><td>-</td><td>wdecay( (wpre, 0.10)</td><td>54.6 (71.1)</td><td>84.2 (91.8)</td><td>45.6 (63.8)</td><td>84.3 (90.1)</td></tr><tr><td>mixout(wpre, 0.7)</td><td></td><td>61.6 (74.0)</td><td>87.1 (91.1)</td><td>57.4 (62.1)</td><td>89.6 (90.3)</td></tr><tr><td>mixout( (Wpre, 0.8</td><td></td><td>64.0 (74.0)</td><td>89.0 (90.7)</td><td>57.9 (63.8)</td><td>89.4 (90.3)</td></tr><tr><td>mixout( (Wpre, 0.9)</td><td></td><td>64.3 (73.3)</td><td>88.2 (91.4)</td><td>55.2 (63.4)</td><td>89.4 (90.0)</td></tr><tr><td>mixout( 0.7)</td><td>wdecay 0.01)</td><td>65.3 (74.4)</td><td>87.8 (91.8)</td><td>51.9 (64.0)</td><td></td></tr><tr><td>(wpre, mixout( 0.8</td><td>(wpre, wdecay( 0.01)</td><td>62.8 (74.0)</td><td>86.3 (90.9)</td><td>58.3 (65.1)</td><td>89.6 (90.6) 89.7 (90.3)</td></tr><tr><td>(Wpre,</td><td>(wpre,</td><td></td><td></td><td></td><td></td></tr><tr><td>mixout(wpre, 0.9</td><td>wdecay( (wpre, 0.01)</td><td>65.0 (75.5)</td><td>88.6 (91.3)</td><td>58.1 (65.1)</td><td>89.5 (90.0)</td></tr></table>
+
+We investigate the effect of combining both mixout $( { \pmb w } _ { \mathrm { p r e } } )$ and wdecay $( { \pmb w } _ { \mathrm { p r e } } )$ to see whether they are complementary. We finetune BERTLARGE with both mixout( $\mathbf { \Delta } _ { w _ { \mathrm { p r e } } }$ , {0.7, 0.8, 0.9}) and wdecay $( \mathbf { { w } } _ { \mathrm { { p r e } } } , \ 0 . 0 1 )$ . This leads not only to the improvement in the average dev scores but also in the best dev scores compared to using wdecay $( w _ { \mathrm { p r e } }$ , $\{ 0 . 0 1 , 0 . 0 4 , 0 . 0 7 , 0 . 1 0 \} )$ ) and using both $\operatorname { d r o p o u t } ( p )$ and wdecay(0, 0.01). The experiments in this section confirm that using mixout $( w _ { \mathrm { p r e } } )$ as one of several regularization techniques prevents finetuning instability and yields gains in dev scores.
+
+# 6 ABLATION STUDY
+
+In this section, we perform ablation experiments to better understand mixout $( w _ { \mathrm { p r e } } )$ . Unless explicitly stated, all experimental setups are the same as in Section 5.
+
+# 6.1 MIXOUT WITH A SUFFICIENT NUMBER OF TRAINING EXAMPLES
+
+We showed the effectiveness of the proposed mixout finetuning with only a few training examples in Section 5. In this section, we investigate the effectiveness of the proposed mixout in the case of a larger finetuning set. Since it has been stable to finetune $\mathrm { B E R T _ { L A R G E } }$ on a sufficient number of training examples (Devlin et al., 2018; Phang et al., 2018), we expect to see the change in the behaviour of mixout $( { \pmb w } _ { \mathrm { p r e } } )$ when we use it to finetune BERTLARGE on a larger training set.
+
+We train BERTLARGE by using both mixout $( w _ { \mathrm { p r e } } , \ 0 . 7 )$ and wdecay( ${ \pmb w } _ { \mathrm { p r e } }$ , 0.01) with 20 random restarts on SST-2.5 We also train $\mathrm { B E R T _ { L A R G E } }$ by using both dropout $( p )$ and wdecay(0, 0.01) with 20 random restarts on SST-2 as the baseline. In Table 2, we report the mean and maximum of
+
+SST-2 dev scores across 20 random restarts with each regularization strategy. We observe that there is little difference between their mean and maximum dev scores on a larger training set, although using both mixout $( w _ { \mathrm { p r e } } , \ 0 . 7 )$ and wdecay $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 0 1 )$ outperformed using both dropout $( p )$ and wdecay(0, 0.01) on the small training sets in Section 5.
+
+Table 2: Mean (max) SST-2 dev scores across 20 random restarts when finetuning BERTLARGE with each regularization strategy. Bold marks the best of each statistics within each column. For a large training set, both mean and maximum dev scores are similar to each other.   
+
+<table><tr><td>TECHNIQUE 1</td><td>TECHNIQUE 2</td><td>SST-2</td></tr><tr><td>dropout(0.1)</td><td>wdecay(0, ( 0.01)</td><td>93.4 (94.0)</td></tr><tr><td>mixout(wpre, 0.7)</td><td>wdecay(wpre, 0.01)</td><td>93.5 (94.3)</td></tr></table>
+
+# 6.2 EFFECT OF A REGULARIZATION TECHNIQUE FOR AN ADDITIONAL OUTPUT LAYER
+
+In this section, we explore the effect of a regularization technique for an additional output layer. There are two regularization techniques available for the additional output layer: dropout $( p )$ and mixout $( \boldsymbol { w } _ { 0 } , \boldsymbol { p } )$ where $\pmb { w } _ { 0 }$ is its randomly initialized parameter. Either of these strategies differs from the earlier experiments in Section 5 where we did not put any regularization for the additional output layer.
+
+Table 3: We present mean (max) dev scores across 20 random restarts with various regularization techniques for the additional output layers (ADDITIONAL) when finetuning BERTLARGE on each task. For all cases, we apply mixout $( w _ { \mathrm { p r e } } , \ 0 . 7 )$ to the pretrained layers (PRETRAINED). The first row corresponds to the setup in Section 5. In the second row, we apply mixout $( w _ { 0 } , \ 0 . 7 )$ to the additional output layer where $\pmb { w } _ { 0 }$ is its randomly initialized parameter. The third row shows the results obtained by applying dropout(0.7) to the additional output layer. In the fourth row, we demonstrate the best of each result from all the regularization strategies shown in Table 1. Bold marks the best of each statistics within each column. We obtain additional gains in dev scores by varying the regularization technique for the additional output layer.
+
+<table><tr><td>PRETRAINED</td><td>ADDITIONAL</td><td>RTE</td><td>MRPC</td><td>CoLA</td><td>STS-B</td></tr><tr><td>mixout(wpre, 0.7)</td><td>1</td><td>61.6 (74.0)</td><td>87.1 (91.1)</td><td>57.4 (62.1)</td><td>89.6 (90.3)</td></tr><tr><td>mixout( (wpre, 0.7)</td><td>mixout(wo, 0.7)</td><td>66.5 (75.5)</td><td>88.1 (92.4)</td><td>58.7 (65.6)</td><td>89.7 (90.6)</td></tr><tr><td>mixout(wpre, 0.7)</td><td>dropout(0.7)</td><td>57.2 (70.8)</td><td>85.9 (92.5)</td><td>48.9 (64.3)</td><td>89.2 (89.8)</td></tr><tr><td colspan="2">The best of each result from Table 1</td><td>65.3 (75.5)</td><td>89.0 (91.8)</td><td>58.3 (65.6)</td><td>89.7 (90.6)</td></tr></table>
+
+We report the average and best dev scores across 20 random restarts when finetuning BERTLARGE with mixout $( w _ { \mathrm { p r e } } , 0 . 7 )$ while varying the regularization technique for the additional output layer in Table 3.6 We observe that using mixout $( w _ { 0 } , 0 . 7 )$ for the additional output layer improves both the average and best dev score on RTE, CoLA, and STS-B. In the case of MRPC, we have the highest best-dev score by using dropout(0.7) for the additional output layer while the highest mean dev score is obtained by using mixout $( w _ { 0 } , 0 . 7 )$ for it. In Section 3.2, we discussed how mixout $( \pmb { w } _ { 0 } )$ does not differ from dropout when the layer is randomly initialized, since we sample ${ \pmb w } _ { 0 }$ from w whose mean and variance are 0 and small, respectively. Although the additional output layer is randomly initialized, we observe the significant difference between dropout and mixout $( \pmb { w } _ { 0 } )$ in this layer. We conjecture that $\lVert \mathbf { \boldsymbol { w } } _ { 0 } - \mathbf { 0 } \rVert$ is not sufficiently small because $\mathbb { E } \lVert \mathbf { w } - \mathbf { 0 } \rVert$ is proportional to the dimensionality of the layer (2,048). We therefore expect mixout $\mathbf { \Pi } ( \pmb { w } _ { 0 } )$ to behave differently from dropout even for the case of training from scratch.
+
+In the last row of Table 3, we present the best of the corresponding result from Table 1. We have the highest mean and best dev scores when we respectively use mixout $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 7 )$ and mixout $( w _ { 0 } , 0 . 7 )$ for the pretrained layers and the additional output layer on RTE, CoLA, and STSB. The highest mean dev score on MRPC is obtained by using mixout $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 8 )$ for the pretrained layers which is one of the results in Table 1. We have the highest best dev score on MRPC when we use mixout $( w _ { \mathrm { p r e } } , \ 0 . 7 )$ and dropout(0.7) for the pretrained layers and the additional output layer, respectively. The experiments in this section reveal that using mixout $\mathbf { \Pi } ( \pmb { w } _ { 0 } )$ for a randomly initialized layer of a pretrained model is one of the regularization schemes to improve the average dev score and the best dev score.
+
+# 6.3 EFFECT OF MIX PROBABILITY FOR MIXOUT AND DROPOUT
+
+We explore the effect of the hyperparameter $p$ when finetuning BERTLARGE with mixout $( \boldsymbol { \mathbf { \mathit { w } } } _ { \mathrm { p r e } } , \boldsymbol { \mathbf { \mathit { p } } } )$ and dropout $( p )$ . We train $\mathrm { B E R T _ { L A R G E } }$ with mixout $\mathbf { \Delta } ( w _ { \mathrm { p r e } }$ , $\{ 0 . 0 , 0 . 1 , \ \cdot \cdot \ , \ 0 . 9 \} )$ on RTE with 20 random restarts. We also train $\mathrm { B E R T _ { L A R G E } }$ after replacing mixout $( { \pmb w } _ { \mathrm { p r e } } , \ p )$ by dropout $( p )$ with 20 random restarts. We do not use any regularization technique for the additional output layer. Because we use neither wdecay(0) nor wdecay $( { \pmb w } _ { \mathrm { p r e } } )$ in this section, dropout(0.0) and mixout $( { \pmb w } _ { \mathrm { p r e } } , \ 0 . 0 )$ are equivalent to finetuning without regularization.
+
+![](images/ee5e9d873c45ac81b0c4507d9ea83232a40e78746c5ed1944d2a845ec4026a75.jpg)  
+Figure 4: Distribution of RTE dev scores (Accuracy) from 20 random restarts when finetuning BERTLARGE with dropout $( p )$ (orange) or mixout $( \pmb { w } _ { \mathrm { p r e } } , \ p )$ (blue). Error intervals show mean $\pm$ std. We do not use wdecay(0) nor wdecay $( { \boldsymbol { w } } _ { \mathrm { p r e } } )$ . In the case of mixout $( \mathbf { \boldsymbol { w } } _ { \mathrm { p r e } } , \mathbf { \boldsymbol { p } } )$ , the number of usable models after finetuning with mixout $\omega _ { \mathrm { p r e } }$ , $\{ 0 . 7 , \ 0 . 8 , \ 0 . 9 \} ,$ is significantly more than the number of usable models after finetuning with dropout $( p )$ for all $p$ .
+
+It is not helpful to vary $p$ for dropout $( p )$ while mixout $( \pmb { w } _ { \mathrm { p r e } } , p )$ helps significantly in a wide range of $p$ . Figure 4 shows distributions of RTE dev scores across 20 random restarts when finetuning BERTLARGE with dropout $( p )$ and mixout $( \mathbf { \boldsymbol { w } } _ { \mathrm { p r e } } , \mathbf { \boldsymbol { p } } )$ for $p \in \{ 0 . 0 , 0 . 1 , \ \cdot \cdot \cdot , 0 . 9 \}$ . The mean dev score of finetuning BERTLARGE with mixout $( \boldsymbol { \mathbf { \mathit { w } } } _ { \mathrm { p r e } } , \boldsymbol { \mathbf { \mathit { p } } } )$ increases as $p$ increases. On the other hand, the mean dev score of finetuning $\mathrm { B E R T _ { L A R G E } }$ with dropout $( p )$ decreases as $p$ increases. If $p$ is less than 0.4, finetuning with mixout $( \boldsymbol { \mathbf { \mathit { w } } } _ { \mathrm { p r e } } , \boldsymbol { \mathbf { \mathit { p } } } )$ does not improve the finetuning results of using dropout $\left( \{ 0 . 0 , 0 . 1 , 0 . 2 \} \right)$ . We however observe that mixout( $\mathbf { \Delta } _ { w _ { \mathrm { p r e } } }$ , $\{ 0 . 7 , 0 . 8 , 0 . { \bar { 9 } } \} )$ yields better average dev scores than dropout $( p )$ for all $p$ , and significantly reduces the number of finetuning runs that fail with the chance-level accuracy.
+
+We notice that the proposed mixout spends more time than dropout from the experiments in this section. It takes longer to finetune a model with the proposed mixout than with the original dropout, although this increase is not significant especially considering the waste of time from failed finetuning runs using dropout. In Supplement E, we describe more in detail the difference between mixout and dropout in terms of wall-clock time.
+
+# 7 CONCLUSION
+
+The special case of our approach, mixout $( w _ { \mathrm { p r e } } )$ , is one of several regularization techniques modifying a finetuning procedure to prevent catastrophic forgetting. Unlike wdecay $( w _ { \mathrm { p r e } } )$ proposed earlier by Wiese et al. (2017), mixout $( { \pmb w } _ { \mathrm { p r e } } )$ is an adaptive $L ^ { 2 }$ -regularizer toward ${ \pmb w } _ { \mathrm { p r e } }$ in the sense that its regularization coefficient adapts along the optimization path. Due to this difference, the proposed mixout improves the stability of finetuning a big, pretrained language model even with only a few training examples of a target task. Furthermore, our experiments have revealed the proposed approach improves finetuning results in terms of the average accuracy and the best accuracy over multiple runs. We emphasize that our approach can be used with any pretrained language models such as RoBERTa (Liu et al., 2019) and XLNet (Yang et al., 2019), since mixout does not depend on model architectures, and leave it as future work.
+
+# ACKNOWLEDGMENTS
+
+The first and third authors’ work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea government (MOE, MSIT) (NRF-2017R1A2B4011546, NRF2019R1A5A1028324). The second author thanks support by AdeptMind, eBay, TenCent, NVIDIA and CIFAR and was partly supported by Samsung Electronics (Improving Deep Learning using Latent Structure).
+
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+
+SUPPLEMENTARY MATERIAL
+
+A PROOFS FOR THEOREM 1
+
+Theorem 1. Assume that the loss function $\mathcal { L }$ is strongly convex. Suppose that a random mixture function with respect to $\textbf { \em w }$ from $\textbf { \em u }$ and $M$ is given by
+
+$$
+\Phi ( { \pmb w } ; { \pmb u } , M ) = \mu ^ { - 1 } \big ( ( { \pmb I } - M ) { \pmb u } + M { \pmb w } - ( 1 - \mu ) { \pmb u } \big ) ,
+$$
+
+where $M$ is $\mathrm { d i a g } ( M _ { 1 } , ~ M _ { 2 } , ~ \cdot \cdot \cdot , ~ M _ { d } ) $ satisfying $\mathbb { E } M _ { i } = \mu$ and $\mathrm { V a r } ( M _ { i } ) = \sigma ^ { 2 }$ for all $i$ . Then, there exists $m > 0$ such that
+
+$$
+\mathbb { E } \mathcal { L } \big ( \Phi ( { \boldsymbol w } ; { \boldsymbol u } , M ) \big ) \ge \mathcal { L } ( { \boldsymbol w } ) + \frac { m \sigma ^ { 2 } } { 2 \mu ^ { 2 } } \| { \boldsymbol w } - { \boldsymbol u } \| ^ { 2 } ,
+$$
+
+for all $\textbf { \em w }$
+
+Proof. Since $\mathcal { L }$ is strongly convex, there exist $m > 0$ such that
+
+$$
+\begin{array} { r l } & { \mathbb { E } \mathcal { L } \big ( \Phi ( w ; u , M ) \big ) = \mathbb { E } \mathcal { L } \Big ( w + \big ( \Phi ( w ; u , M ) - w \big ) \Big ) } \\ & { \qquad \quad \geq \mathcal { L } ( w ) + \nabla \mathcal { L } ( w ) ^ { \top } \mathbb { E } [ \Phi ( w ; u , M ) - w ] + \frac { m } { 2 } \mathbb { E } \| \Phi ( w ; u , M ) - w \| ^ { 2 } , } \end{array}
+$$
+
+for all $\pmb { w }$ by equation 1. Recall that $\mathbb { E } M _ { i } = \mu$ and $\mathrm { V a r } ( M _ { i } ) = \sigma ^ { 2 }$ for all $i$ . Then, we have
+
+$$
+\mathbb { E } [ \Phi ( { \pmb w } ; { \pmb u } , M ) - { \pmb w } ] = { \bf 0 } ,
+$$
+
+and
+
+$$
+\begin{array} { l } { \displaystyle \mathbb { E } \| \Phi ( \boldsymbol { w } ; ~ \boldsymbol { u } , \boldsymbol { M } ) - \boldsymbol { w } \| ^ { 2 } = \mathbb { E } \left\| \frac { 1 } { \mu } ( \boldsymbol { w } - \boldsymbol { u } ) ( \boldsymbol { M } - \mu \boldsymbol { I } ) \right\| ^ { 2 } } \\ { \displaystyle \qquad = \frac { 1 } { \mu ^ { 2 } } \sum _ { i = 1 } ^ { d } ( w _ { i } - u _ { i } ) ^ { 2 } \mathbb { E } ( M _ { i } - \mu ) ^ { 2 } } \\ { \displaystyle \qquad = \frac { \sigma ^ { 2 } } { \mu ^ { 2 } } \| \boldsymbol { w } - \boldsymbol { u } \| ^ { 2 } . } \end{array}
+$$
+
+By using equation 13 and equation 14, we can rewrite equation 12 as
+
+$$
+\mathbb { E } \mathcal { L } \big ( \Phi ( { \boldsymbol w } ; { \boldsymbol u } , M ) \big ) \ge \mathcal { L } ( { \boldsymbol w } ) + \frac { m \sigma ^ { 2 } } { 2 \mu ^ { 2 } } \| { \boldsymbol w } - { \boldsymbol u } \| ^ { 2 } .
+$$
+
+# B APPLYING TO SPECIFIC LAYERS
+
+We often apply dropout to specific layers. For instance, Simonyan & Zisserman (2014) applied dropout to fully connected layers only. We generalize Theorem 1 to the case in which mixconnect is only applied to specific layers, and it can be done by constructing $M$ in a particular way. To better characterize mixconnect applied to specific layers, we define the index set $\mathbb { I }$ as $\mathbb { I } = \{ i : \ \boldsymbol { M } _ { i } = 1 \}$ . Furthermore, we use $\tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } ^ { v }$ and $\tilde { \mathbf { \pmb { u } } }$ to denote $( w _ { i } ) _ { i \notin \mathbb { I } }$ and $( u _ { i } ) _ { i \notin \mathbb { I } }$ , respectively. Then, we generalize equation 7 to
+
+$$
+\mathbb { E } \mathcal { L } \big ( \Phi ( { \boldsymbol w } ; { \boldsymbol u } , M ) \big ) \ge \mathcal { L } ( { \boldsymbol w } ) + \frac { m \sigma ^ { 2 } } { 2 \mu ^ { 2 } } \| \tilde { { \boldsymbol w } } - \tilde { { \boldsymbol u } } \| ^ { 2 } .
+$$
+
+From equation 15, applying mixconnec $\scriptstyle ; ( u , \mu , \sigma ^ { 2 } )$ is to use adaptive wdecay $( \tilde { u } )$ on the weight parameter of the specific layers $\tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } ^ { \tilde { \mathbf { \Gamma } } } \tilde { \mathbf { \Gamma } } ^ { \tilde { \mathbf { \Gamma } } } \tilde { \mathbf { \Gamma } } ^ { \tilde { \mathbf { \Gamma } } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde { \mathbf { \Gamma } } \tilde \mathrm { \Gamma }$ . Similarly, we can regard applying mixout $( \boldsymbol { \mathscr { u } } , \boldsymbol { \mathscr { p } } )$ to specific layers as adaptive wdecay $( \tilde { u } )$ .
+
+# C EXPERIMENTAL DETAILS
+
+# C.1 FROM EMNIST DIGITS TO MNIST
+
+Model Architecture The model architecture in Section 4 is a 784-300-100-10 fully connected network with a softmax output layer. For each hidden layer, we add layer normalization (Ba et al., 2016) right after the ReLU (Nair & Hinton, 2010) nonlinearity. We initialize each parameter with $\mathcal { N } ( 0 , 0 . \bar { 0 } 2 ^ { 2 } )$ and each bias with 0.
+
+Regularization In the pretraining stage, we use dropout(0.1) and wdecay $\mathbf { ( 0 , \ 0 . 0 1 ) }$ . We apply dropout(0.1) to all hidden layers. That is, we do not drop neurons of the input and output layers. wdecay $\mathbf { ( 0 , \eta 0 . 0 1 ) }$ does not penalize the parameters for bias and layer normalization. When we finetune our model on MNIST, we replace dropout $( p )$ with mixout $( \mathbf { \boldsymbol { w } } _ { \mathrm { p r e } } , \mathbf { \boldsymbol { p } } )$ . We use neither wdecay(0) nor wdecay $( { \pmb w } _ { \mathrm { p r e } } )$ for finetuning.
+
+Dataset For pretraining, we train our model on EMNIST Digits. This dataset has 280,000 characters into 10 balanced classes. These characters are compatible with MNIST characters. EMNIST Digits provides 240,000 characters for training and 40,000 characters for test. We use 240,000 characters provide for training and split these into the training set (216,000 characters) and validation set (24,000 characters). For finetuning, we train our model on MNIST. This has 70,000 characters into 10 balance classes. MNIST provide 60,000 characters for training and 10,000 characters for test. We use 60,000 characters given for training and split these into the training set (54,000 characters) and validation set (6,000 characters).
+
+Data Preprocessing We only use normalization after scaling pixel values into [0, 1]. We do not use any data augmentation.
+
+# C.2 FINETUNING BERT ON PARTIAL GLUE TASKS
+
+Model Architecture Because the model architecture of $\mathrm { B E R T _ { L A R G E } }$ is identical to the original (Devlin et al., 2018), we omit its exhaustive description. Briefly, BERTLARGE has 24 layers, 1024 hidden size, and 16 self-attention heads (total 340M parameters). We use the publicly available pretrained model released by Devlin et al. (2018), ported into PyTorch by HuggingFace.7 We initialize each weight parameter and bias for an additional output layer with $\mathcal { N } ( 0 , 0 . 0 2 ^ { 2 } )$ and 0, respectively.
+
+Regularization In the finetuning stage, Devlin et al. (2018) used wdecay $\mathbf { ( 0 , \ 0 . 0 1 ) }$ for all parameters except bias and layer normalization. They apply dropout(0.1) to all layers except each hidden layer activated by GELU (Hendrycks & Gimpel, 2016) and layer normalization. We substitute wdecay $( { \boldsymbol { w } } _ { \mathrm { p r e } } )$ and mixout $( { \pmb w } _ { \mathrm { p r e } } )$ for wdecay $\mathbf { ( 0 , \ 0 . 0 1 ) }$ and dropout(0.1), respectively.
+
+Dataset We use a subset of GLUE (Wang et al., 2018) tasks. The brief description for each dataset is as the following:
+
+• RTE (2,500 training examples): Binary entailment task (Dagan et al., 2006) • MRPC (3,700 training examples): Semantic similarity (Dolan & Brockett, 2005) • CoLA (8,500 training examples): Acceptability classification (Warstadt et al., 2018) STS-B (7,000 training examples): Semantic textual similarity (Cer et al., 2017) • SST-2 (67,000 training examples): Binary sentiment classification (Socher et al., 2013)
+
+In this paper, we reported F1 accuracy scores for MRPC, Mattew’s correlation scores for CoLA, Spearman correlation scores for STS-B, and accuracy scores for the other tasks.
+
+Data Preprocessing We use the publicly available implementation of BERTLARGE by HuggingFace.8
+
+# C.3 TEST RESULTS ON GLUE TASKS
+
+We expect that using mixout stabilizes finetuning results of BERTLARGE on a small training set. To show this, we demonstrated distributions of dev scores from 20 random restarts on RTE, MRPC, CoLA, and STS-B in Figure 3. We further obtained the highest average/best dev score on each task in Table 3. To confirm the generalization of the our best model on the dev set, we demonstrate the test results scored by the evaluation server9 in Table 4.
+
+Table 4: We present the test score when finetuning BERTLARGE with each regularization strategy on each task. The first row shows the test scores obtained by using both dropout $( p )$ and wdecay $\mathbf { ( 0 , \ 0 . 0 1 ) }$ . These results in the first row are reported by Devlin et al. (2018). They used the learning rate of $\lbrace 2 \times 1 0 ^ { - 5 }$ , $3 \times 1 0 ^ { - 5 }$ , $4 \times 1 0 ^ { - 5 }$ , $5 \times 1 0 ^ { - 5 } \}$ and a batch size of 32 for 3 epochs with multiple random restarts. They selected the best model on each dev set. In the second row, we demonstrate the test scores obtained by using the proposed mixout in Section 6.2: using mixout $( w _ { \mathrm { p r e } } , \ 0 . 7 )$ for the pretrained layers and mixout $( w _ { 0 } , \ 0 . 7 )$ for the additional output layer where $\pmb { w } _ { 0 }$ is its randomly initialized weight parameter. We used the learning rate of $2 \times 1 0 ^ { - 5 }$ and a batch size of 32 for 3 epochs with 20 random restarts. We submitted the best model on each dev set. The third row shows that the test scores obtained by using both dropout $( p )$ and wdecay $\mathbf { ( 0 , \ 0 . 0 1 ) }$ with same experimental setups of the second row. Bold marks the best within each column. The proposed mixout improves the test scores except MRPC compared to the original regularization strategy proposed by Devlin et al. (2018).
+
+<table><tr><td>STRATEGY</td><td>RTE</td><td>MRPC</td><td>CoLA</td><td>STS-B</td></tr><tr><td>Devlin et al. (2018)</td><td>70.1</td><td>89.3</td><td>60.5</td><td>86.5</td></tr><tr><td>mixout(wpre, 0.7)&amp; mixout(wo，0.7)</td><td>70.2</td><td>89.1</td><td>62.1</td><td>87.3</td></tr><tr><td>dropout(p) + wdecay(0, 0.01)</td><td>68.2</td><td>88.3</td><td>59.6</td><td>86.0</td></tr></table>
+
+For all the tasks except MRPC, the test scores obtained by the proposed mixout10 are better than those reported by Devlin et al. (2018). We explored the behaviour of finetuning $\mathrm { B E R T _ { L A R G E } }$ with mixout by using the learning rate of $2 \times 1 0 ^ { - 5 }$ while Devlin et al. (2018) obtained their results by using the learning rate of $\lbrace 2 \times 1 0 ^ { - 5 }$ , $3 \times 1 0 ^ { - 5 }$ , $4 \times 1 0 ^ { - 5 }$ , $5 \times 1 0 ^ { - 5 } \}$ . We thus present the test scores obtained by the regularization strategy of Devlin et al. (2018) when the learning rate is $2 \times 1 0 ^ { - 5 }$ . The results in this section show that the best model on the dev set generalizes well, and all the experiments based on dev scores in this paper are proper to validate the effectiveness of the proposed mixout. For the remaining GLUE tasks such as SST-2 with a sufficient number of training instances, we observed that using mixout does not differs from using dropout in Section 6.1. We therefore omit the test results on the other tasks in GLUE.
+
+# D VERIFICATION OF COROLLARY 1.1 WITH LEAST SQUARES REGRESSION
+
+Corollary 1.1 shows that mixout $( \pmb { u } , p )$ regularizes learning to minimize the deviation from the target model parameter $\textbf { \em u }$ , and the strength of regularization increases as $p$ increases when the loss function is strongly convex. In order to validate this, we explore the behavior of least squares regression with mixout $( \pmb { u } , p )$ on a synthetic dataset. For randomly given $w _ { 1 } ^ { * }$ and $w _ { 2 } ^ { * }$ , we generated an observation $y$ satisfying $y = w _ { 1 } ^ { * } x + w _ { 2 } ^ { * } + \epsilon$ where $\epsilon$ is Gaussian noise. We set the model to $\hat { y } = w _ { 1 } x + w _ { 2 }$ . That is, the model parameter $\pmb { w }$ is given by $( w _ { 1 } , w _ { 2 } )$ . We randomly pick $\textbf { \em u }$ as a target model parameter for $\mathtt { m i x o u t } ( { \boldsymbol { u } } , { \boldsymbol { p } } )$ and perform least squares regression with $\tilde { \mathrm { m i x o u t } } ( u , \ \{ 0 . 0 \dot { , } \ 0 . 3 , \ 0 . 6 , \ 0 . 9 \} )$ . As shown in Figure 5, $\pmb { w }$ converges to the target model parameter $\textbf { \em u }$ rather than the true model parameter $\boldsymbol { w ^ { * } } = ( w _ { 1 } ^ { * } , \ w _ { 2 } ^ { * } )$ as the mix probability $p$ increases.
+
+![](images/b5202dfa5191309276bc694b4c84cda4c5cc55216ff19dfd0485fd47322ff44b.jpg)  
+Figure 5: Behavior of mixout $( \pmb { u } , \pmb { p } )$ for a strongly convex loss function. We plot the line obtained by least squares regression with mixout $( { \pmb u } , \ \{ 0 . \bar { 0 } , \ 0 . 3 , \ 0 . 6 , \ 0 . 9 \} )$ (each green line) on a synthetic dataset (blue dots) generated by the true line (each blue dotted line). As $p$ increases, the regression line (each green line) converges to the target line generated by the target model parameter $\textbf { \em u }$ (each orange dotted line) rather than the true line (each blue dotted line).
+
+# E TIME USAGE OF MIXOUT COMPARED TO DROPOUT
+
+We recorded the training time of the experiment in Section 6.3 to compare the time usage of mixout and that of dropout. It took about 843 seconds to finetune $\mathrm { B E R T _ { L A R G E } }$ with mixout $( { \pmb w } _ { \mathrm { p r e } } )$ . On the other hand, it took about 636 seconds to finetune BERTLARGE with dropout. mixout $( w _ { \mathrm { p r e } } )$ spends $3 2 . 5 \%$ more time than dropout since mixout $( { \boldsymbol { w } } _ { \mathrm { p r e } } )$ needs an additional computation with the pretrained model parameter ${ \pmb w } _ { \mathrm { p r e } }$ . However, as shown in Figure 4, at least 15 finetuning runs among 20 random restarts fail with the chance-level accuracy on RTE with dropout $( p )$ for all $p$ while only 4 finetuning runs out of 20 random restarts are unusable with mixout( $\mathbf { \Delta } _ { w _ { \mathrm { p r e } } }$ , 0.8). From this result, it is reasonable to finetune with the proposed mixout although this requires additional time usage compared to dropout.
+
+# F EXTENSIVE HYPERPARAMETER SEARCH FOR DROPOUT
+
+Devlin et al. (2018) finetuned BERTLARGE with dropout(0.1) on all GLUE (Wang et al., 2018) tasks. They chose it to improve the maximum dev score on each downstream task, but we have reported not only the maximum dev score but also the mean dev score to quantitatively compare various regularization techniques in our paper. In this section, we explore the effect of the hyperparameter $p$ when finetuning BERTLARGE with dropout $( p )$ on RTE, MRPC, CoLA, and STS-B to show dropout(0.1) is optimal in terms of mean dev score. All experimental setups for these experiments are the same as Section 6.3.
+
+![](images/183aea2a13ece2aebace5571f08ab17f72285847b93c248480ebc91c33404e1b.jpg)  
+Figure 6: Distribution of dev scores on each task from 20 random restarts when finetuning BERTLARGE with dropout $( \{ 0 . 0 , 0 . 1 , \ \cdot \cdot \ , 0 . 5 \} )$ . Error intervals show mean±std. When we use dropout(0.1), we have the highest average dev scores on MRPC and STS-B and the second-highest average dev scores on RTE and CoLA. These results show that dropout(0.1) is almost optimal for all tasks in terms of mean dev score.
+
+As shown in Figure 6, we have the highest average dev score on MRPC with dropout(0.1) as well as on STS-B. We obtain the highest average dev scores with dropout(0.0) on RTE and CoLA, but we get the second-highest average dev scores with dropout(0.1) on them. These experiments confirm that the drop probability 0.1 is almost optimal for the highest average dev score on each task.

parse/train/N6JECD-PI5w/N6JECD-PI5w.md

@@ -0,0 +1,288 @@
+# FAIRFIL: CONTRASTIVE NEURAL DEBIASING METHOD FOR PRETRAINED TEXT ENCODERS
+
+Pengyu Cheng∗ , Weituo Hao∗, Siyang Yuan , Shijing Si , Lawrence Carin Department of Electrical and Computer Engineering, Duke University {pengyu.cheng,weituo.hao,siyang.yuan,shijing.si,lcarin}@duke.edu
+
+# ABSTRACT
+
+Pretrained text encoders, such as BERT, have been applied increasingly in various natural language processing (NLP) tasks, and have recently demonstrated significant performance gains. However, recent studies have demonstrated the existence of social bias in these pretrained NLP models. Although prior works have made progress on word-level debiasing, improved sentence-level fairness of pretrained encoders still lacks exploration. In this paper, we proposed the first neural debiasing method for a pretrained sentence encoder, which transforms the pretrained encoder outputs into debiased representations via a fair filter (FairFil) network. To learn the FairFil, we introduce a contrastive learning framework that not only minimizes the correlation between filtered embeddings and bias words but also preserves rich semantic information of the original sentences. On real-world datasets, our FairFil effectively reduces the bias degree of pretrained text encoders, while continuously showing desirable performance on downstream tasks. Moreover, our post hoc method does not require any retraining of the text encoders, further enlarging FairFil’s application space.
+
+# 1 INTRODUCTION
+
+Text encoders, which map raw-text data into low-dimensional embeddings, have become one of the fundamental tools for extensive tasks in natural language processing (Kiros et al., 2015; Lin et al., 2017; Shen et al., 2019; Cheng et al., 2020b). With the development of deep learning, largescale neural sentence encoders pretrained on massive text corpora, such as Infersent (Conneau et al., 2017), ELMo (Peters et al., 2018), BERT (Devlin et al., 2019), and GPT (Radford et al., 2018), have become the mainstream to extract the sentence-level text representations, and have shown desirable performance on many NLP downstream tasks (MacAvaney et al., 2019; Sun et al., 2019; Zhang et al., 2019). Although these pretrained models have been studied comprehensively from many perspectives, such as performance (Joshi et al., 2020), efficiency (Sanh et al., 2019), and robustness (Liu et al., 2019), the fairness of pretrained text encoders has not received significant research attention.
+
+The fairness issue is also broadly recognized as social bias, which denotes the unbalanced model behaviors with respect to some socially sensitive topics, such as gender, race, and religion (Liang et al., 2020). For data-driven NLP models, social bias is an intrinsic problem mainly caused by the unbalanced data of text corpora (Bolukbasi et al., 2016). To quantitatively measure the bias degree of models, prior work proposed several statistical tests (Caliskan et al., 2017; Chaloner & Maldonado, 2019; Brunet et al., 2019), mostly focusing on word-level embedding models. To evaluate the sentence-level bias in the embedding space, May et al. (2019) extended the Word Embedding Association Test (WEAT) (Caliskan et al., 2017) into a Sentence Encoder Association Test (SEAT). Based on the SEAT test, May et al. (2019) claimed the existence of social bias in the pretrained sentence encoders.
+
+Although related works have discussed the measurement of social bias in sentence embeddings, debiasing pretrained sentence encoders remains a challenge. Previous word embedding debiasing methods (Bolukbasi et al., 2016; Kaneko & Bollegala, 2019; Manzini et al., 2019) have limited assistance to sentence-level debiasing, because even if the social bias is eliminated at the word level, the sentence-level bias can still be caused by the unbalanced combination of words in the training text. Besides, retraining a state-of-the-art sentence encoder for debiasing requires a massive amount of computational resources, especially for large-scale deep models like BERT (Devlin et al., 2019) and GPT (Radford et al., 2018). To the best of our knowledge, Liang et al. (2020) proposed the only sentence-level debiasing method (Sent-Debias) for pretrained text encoders, in which the embeddings are revised by subtracting the latent biased direction vectors learned by Principal Component Analysis (PCA) (Wold et al., 1987). However, Sent-Debias makes a strong assumption on the linearity of the bias in the sentence embedding space. Further, the calculation of bias directions depends highly on the embeddings extracted from the training data and the number of principal components, preventing the method from adequate generalization.
+
+In this paper, we proposed the first neural debiasing method for pretrained sentence encoders. For a given pretrained encoder, our method learns a fair filter (FairFil) network, whose inputs are the original embeddings of the encoder, and outputs are the debiased embeddings. Inspired by the multi-view contrastive learning (Chen et al., 2020), for each training sentence, we first generate an augmentation that has the same semantic meaning but in a different potential bias direction. We contrastively train our FairFil by maximizing the mutual information between the debiased embeddings of the original sentences and corresponding augmentations. To further eliminate bias from sensitive words in sentences, we introduce a debiasing regularizer, which minimizes the mutual information between debiased embeddings and the sensitive words’ embeddings. In the experiments, our FairFil outperforms Sent-Debias (Liang et al., 2020) in terms of the fairness and the representativeness of debiased embeddings, indicating our FairFil not only effectively reduces the social bias in the sentence embeddings, but also successfully preserves the rich semantic meaning of input text.
+
+# 2 PRELIMINARIES
+
+Mutual Information (MI) is a measure of the “amount of information” between two variables (Kullback, 1997). The mathematical definition of MI is
+
+$$
+\mathcal { T } ( \pmb { x } ; \pmb { y } ) : = \mathbb { E } _ { p ( \pmb { x } , \pmb { y } ) } \Big [ \log \frac { p ( \pmb { x } , \pmb { y } ) } { p ( \pmb { x } ) p ( \pmb { y } ) } \Big ] ,
+$$
+
+where $p ( { \pmb x } , { \pmb y } )$ is the joint distribution of two variables $( { \pmb x } , { \pmb y } )$ , and $p ( { \pmb x } ) , p ( { \pmb y } )$ are respectively the marginal distributions of $\mathbf { \nabla } _ { \mathbf { x } , \mathbf { y } }$ . Recently, mutual information has achieved considerable success when applied as a learning criterion in diverse deep learning tasks, such as conditional generation (Chen et al., 2016), domain adaptation (Gholami et al., 2020), representation learning (Chen et al., 2020), and fairness (Song et al., 2019). However, the calculation of exact MI in (1) is well-recognized as a challenge, because the expectation w.r.t $p ( { \pmb x } , { \pmb y } )$ is always intractable, especially when only samples from $p ( { \pmb x } , { \pmb y } )$ are provided. To this end, several upper and lower bounds have been introduced to estimate the MI with samples. For MI maximization tasks (Hjelm et al., 2018; Chen et al., 2020), Oord et al. (2018) derived a powerful MI estimator, InfoNCE, based on noise contrastive estimation (NCE) (Gutmann $\&$ Hyvarinen, 2010). Given a batch of sample pairs ¨ $\{ ( \pmb { x } _ { i } , \pmb { y } _ { i } ) \} _ { i = 1 } ^ { N }$ , the InfoNCE estimator is defined with a learnable score function $f ( { \pmb x } , { \pmb y } )$ :
+
+$$
+\mathcal { T } _ { \mathrm { N C E } } : = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \frac { \exp ( f (  { \boldsymbol { { x } } } _ { i } ,  { \boldsymbol { { y } } } _ { i } ) ) } { \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \exp ( f (  { \boldsymbol { { x } } } _ { i } ,  { \boldsymbol { { y } } } _ { j } ) ) } .
+$$
+
+For MI minimization tasks (Alemi et al., 2017; Song et al., 2019), Cheng et al. (2020a) introduced a contrastive log-ratio upper bound (CLUB) based on a variational approximation $q _ { \boldsymbol { \theta } } ( \mathbf { \boldsymbol { y } } | \mathbf { \boldsymbol { x } } )$ of conditional distribution $p ( \pmb { y } | \pmb { x } )$ :
+
+$$
+\mathcal { T } _ { \mathrm { C L U B } } : = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \bigg [ \log q _ { \theta } ( \pmb { y } _ { i } | \pmb { x } _ { i } ) - \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \log q _ { \theta } ( \pmb { y } _ { j } | \pmb { x } _ { i } ) \bigg ] .
+$$
+
+In the following, we use the above two MI estimators to induce the sentence encoder, eliminating the biased information and preserving the semantic information from the original raw text.
+
+# 3 METHOD
+
+Suppose $E ( \cdot )$ is a pretrained sentence encoder, which can encode a sentence $_ { \textbf { \em x } }$ into low-dimensional embedding $\begin{array} { r } { \dot { \boldsymbol { z } } = \boldsymbol { E } ( \boldsymbol { x } ) } \end{array}$ . Each sentence $\pmb { x } = ( w ^ { 1 } , w ^ { 2 } , \dots , w ^ { L } )$ is a sequence of words. The embedding space of $_ z$ has been recognized to have social bias in a series of studies (May et al., 2019; Kurita et al., 2019; Liang et al., 2020). To eliminate the social bias in the embedding space, we aim to learn a fair filter network $f ( \cdot )$ on top of the sentence encoder $E ( \cdot )$ , such that the output embedding of our fair filter $\begin{array} { r } { d = f ( z ) } \end{array}$ can be debiased. To train the fair filter, we design a multi-view contrastive learning framework, which consists of three steps. First, for each input sentence $_ { \textbf { \em x } }$ , we generate an augmented sentence $\mathbf { x } ^ { \prime }$ that has the same semantic meaning as $_ { \textbf { \em x } }$ but in a different potential bias direction. Then, we maximize the mutual information between the original embedding $z = f ( { \pmb x } )$ and the augmented embedding $z ^ { \prime } = f ( x ^ { \prime } )$ with the InfoNCE (Oord et al., 2018) contrastive loss. Further, we design a debiasing regularizer to minimize the mutual information between $^ d$ and sensitive attribute words in $_ { \textbf { \em x } }$ . In the following, we discuss these three steps in detail.
+
+Table 1: Examples of generating an augmentation sentence under the sensitive topic “gender”.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Bias direction</td><td rowspan=1 colspan=1>Sensitive Attribute words</td><td rowspan=1 colspan=2>Text content</td></tr><tr><td rowspan=1 colspan=1>Original</td><td rowspan=1 colspan=1>male</td><td rowspan=1 colspan=1>he,his</td><td rowspan=1 colspan=2>{He} is good at playing {his} basketball.</td></tr><tr><td rowspan=1 colspan=1>Augmentation</td><td rowspan=1 colspan=1>female</td><td rowspan=1 colspan=1>she,her</td><td rowspan=1 colspan=1>{She</td><td rowspan=1 colspan=1>{She} is good at playing {her} basketball.</td></tr></table>
+
+# 3.1 DATA AUGMENTATIONS WITH SENSITIVE ATTRIBUTES
+
+We first describe the sentence data augmentation process for our FairFil contrastive learning. Denote a social sensitive topic as $\mathcal { T } = \{ \mathcal { D } _ { 1 } , \mathcal { D } _ { 2 } , \ldots , \mathcal { D } _ { K } \}$ , where $\mathcal { D } _ { k }$ $( k = 1 , \ldots , K )$ is one of the potential bias directions under the topic. For example, if $\tau$ represents the sensitive topic “gender”, then $\tau$ consists two potential bias directions $\{ \mathcal { D } _ { 1 } , \mathcal { D } _ { 2 } \} \stackrel { - } { = } \{ { } ^ {  } m a l e ^ { , \prime } , \ { } ^ {  } f e m a l e ^ { , \prime } \}$ . Similarly, if $\tau$ is set as the major “religions” of the world, then $\tau$ could contain $\left\{ \mathcal { D } _ { 1 } , \mathcal { D } _ { 2 } , \mathcal { D } _ { 3 } , \mathcal { D } _ { 4 } \right\} \ =$ {“Christianity”, “Islam”, “Judaism”, “Buddhism”} as four components.
+
+For a given social sensitive topic $\mathcal { T } = \{ { D } _ { 1 } , . . . { D } _ { K } \}$ , if a word $w$ is related to one of the potential bias direction $\mathcal { D } _ { k }$ (denote as $w \in \mathcal { D } _ { k } ,$ ), we call $w$ a sensitive attribute word of $\mathcal { D } _ { k }$ (also called bias attribute word in Liang et al. (2020)). For a sensitive attribute word $w \in \mathcal { D } _ { k }$ , suppose we can always find another sensitive attribute word $u \in { \mathcal { D } } _ { j }$ , such that $w$ and $u$ has the equivalent semantic meaning but in a different bias direction. Then we call $u$ as a replaceable word of $w$ in direction $\mathcal { D } _ { j }$ , and denote as $u = r _ { j } ( w )$ . For the topic “gende $\therefore \prime = \{ \stackrel {  } { \cdot } m a l e ^ { \prime \prime } , \stackrel {  } { \cdot } f e m a l e ^ { \prime \prime } \}$ , the word $w =$ “boy” is in the potential bias direction $\mathcal { D } _ { 1 } = \ ^ { \ast } m a l e ^ { \prime \prime }$ ; a replaceable word of “boy” in “female” direction is $r _ { 2 } ( \bar { w } ) = \cdots \mathrm { g i r l } ^ { * } \in \mathcal { D } _ { 2 }$ .
+
+With the above definitions, for each sentence $_ { \textbf { \em x } }$ , we generate an augmented sentence $\mathbf { x } ^ { \prime }$ such that $\mathbf { x } ^ { \prime }$ has the same semantic meaning as $_ { \textbf { \em x } }$ but in a different potential bias direction. More specifically, for a sentence $\pmb { x } = ( w ^ { 1 } , w ^ { 2 } , \dots , \breve { w } ^ { L } )$ , we first find the sensitive word positions as an index set $\mathcal { P }$ , such that each $w ^ { p }$ ${ \bf \Phi } _ { p } \in { \mathcal { P } } ,$ ) is a sensitive attribute words in direction $\mathcal { D } _ { k }$ . We further make a reasonable assumption that the embedding bias of direction $\mathcal { D } _ { k }$ is only caused by the sensitive words $\{ w ^ { p } \} _ { p \in { \mathcal { P } } }$ in $_ { \textbf { \em x } }$ . To sample an augmentation to $_ { \textbf { \em x } }$ , we first select another potential bias direction $\mathcal { D } _ { j }$ , and then replace all sensitive attribute words by their replaceable words in the direction $\mathcal { D } _ { j }$ . That is, $\pmb { x } ^ { \prime } = \{ v ^ { 1 } , v ^ { 2 } , \bot \bot , v ^ { L } \}$ , where $v ^ { l } = w ^ { l }$ if $l \notin \mathcal { P }$ , and $v ^ { l } = r _ { j } ( w ^ { l } )$ if $l \in \mathcal { P }$ . In Table 1, we provide an example for sentence augmentation under the “gender” topic.
+
+# 3.2 CONTRASTIVE LEARNING FRAMEWORK
+
+After obtaining the sentence pair $( { \pmb x } , { \pmb x } ^ { \prime } )$ with the augmentation strategy from Section 3.1, we construct a contrastive learning framework to learn our debiasing fair filter $f ( \cdot )$ . As shown in the Figure 1(a), our framework consists of the following two steps:
+
+(1) We encode sentences $( { \pmb x } , { \pmb x } ^ { \prime } )$ into embeddings $( z , z ^ { \prime } )$ with the pretrained encoder $E ( \cdot )$ . Since $_ { \textbf { \em x } }$ and $\mathbf { x } ^ { \prime }$ have the same meaning but different potential bias directions, the embeddings $( z , z ^ { \prime } )$ will have different bias directions, which are caused by the sensitive attributed words in $_ { \textbf { \em x } }$ and $\mathbf { x } ^ { \prime }$ .
+
+(2) We then feed the sentence embeddings $( z , z ^ { \prime } )$ through our fair filter $f ( \cdot )$ to obtain the debiased embedding outputs $( d , d ^ { \prime } )$ . Ideally, $^ d$ and $\mathbf { \Delta } d ^ { \prime }$ should represent the same semantic meaning without social bias. Inspired by SimCLR (Chen et al., 2020), we encourage the overlapped semantic information between $^ d$ and $\mathbf { \Delta } d ^ { \prime }$ by maximizing their mutual information $\bar { \mathcal { T } } ( d ; d ^ { \prime } )$ .
+
+However, the calculation of $\mathcal { T } ( d ; d ^ { \prime } )$ is practically difficult because only embedding samples of $^ d$ and $\pmb { d } ^ { \prime }$ are available. Therefore, we use the InfoNCE mutual information estimator (Oord et al., 2018) to minimize the lower bound of $\mathcal { T } ( d ; d ^ { \prime } )$ instead. Based on a learnable score function $g ( \cdot , \cdot )$ , the contrastive InfoNCE estimator is calculated within a batch of samples $\{ ( d _ { i } , d _ { i } ^ { \prime } ) \} _ { i = 1 } ^ { N }$
+
+![](images/8c1f7a0b55452733b6dca530e9279647bdf58ae19e81215c4ef29b41a8378f5b.jpg)  
+Figure 1: (a) Contrastive learning framework of FairFil: Sentence $_ { \textbf { \em x } }$ and its augmentation $\mathbf { x } ^ { \prime }$ are encoded into embeddings $^ d$ and $\pmb { d } ^ { \prime }$ , respectively. $\pmb { w } ^ { p }$ is the embedding of a sensitive attribute word selected from $_ { \textbf { \em x } }$ . $\mathcal { T } _ { \mathrm { N C E } }$ maximizes the mutual information between $^ d$ and $\pmb { d } ^ { \prime }$ ; $\scriptstyle { \mathcal { T } } _ { \mathrm { C L U B } }$ eliminates the bias information of $\pmb { w } ^ { p }$ from $\mathbf { \delta } _ { d }$ . (b) Illustration of information in $^ d$ and $\pmb { d } ^ { \prime }$ : The blue and red circles represent the information in $^ d$ and $\mathbf { { \mathbf { { \mathit { d } } } } ^ { \prime } } \mathbf { \Sigma }$ , respectively. The intersection is the mutual information between $^ d$ and $\pmb { d } ^ { \prime }$ . The shadow area represents the bias information of both embeddings.
+
+$$
+\mathcal { T } _ { \mathrm { N C E } } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \frac { \exp ( g ( d _ { i } , d _ { i } ^ { \prime } ) ) } { \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \exp ( g ( d _ { i } , d _ { j } ^ { \prime } ) ) } .
+$$
+
+By maximize $\mathcal { T } _ { \mathrm { N C E } }$ , we encourage the difference between the positive pair score $g ( d _ { i } , d _ { i } ^ { \prime } )$ and the negative pair score $g ( d _ { i } , d _ { j } ^ { \prime } )$ , so that $\mathbf { \ b { d } } _ { i }$ can share more semantic information with $\mathbf { \Delta } d _ { i } ^ { \prime }$ than other embeddings d0j6=i.
+
+# 3.3 DEBIASING REGULARIZER
+
+Practically, the contrastive learning framework in Section 3.2 can already show encouraging debiasing performance (as shown in the Experiments). However, the embedding $^ d$ can contain extra biased information from $_ { z }$ , that only maximizing $\mathcal { T } ( d ; d ^ { \prime } )$ fails to eliminate. To encourage no extra bias in $^ d$ , we introduce a debiasing regularizer which minimizes the mutual information between embedding $^ d$ and the potential bias from embedding $_ z$ . As discussed in Section 3.1, in our framework the potential bias of $_ z$ is assumed to come from the sensitive attribute words in $_ { \textbf { \em x } }$ . Therefore, we should reduce the bias word information from the debiased representation $^ d$ . Let $\pmb { w } ^ { p }$ be the embedding of a sensitive attribute word $w ^ { p }$ in sentence $_ { \textbf { \em x } }$ . The word embedding $\pmb { w } ^ { p }$ can always be obtained from the pretrained text encoders (Bordia $\&$ Bowman, 2019). We then minimize the mutual information $\mathcal { T } ( w ^ { p } ; d )$ , using the CLUB mutual information upper bound (Cheng et al., 2020a) to estimate $\mathcal { T } ( w ^ { p } ; d )$ with embedding samples. Given a batch of embedding pairs $\bar { \{ ( d _ { i } , \boldsymbol { w } ^ { p } ) \} } _ { i = 1 } ^ { N }$ , we can calculate the debiasing regularizer as:
+
+$$
+\mathcal { T } _ { \mathrm { C L U B } } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \Big [ \log q _ { \theta } ( { \pmb w } _ { i } ^ { p } | \pmb d _ { i } ) - \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \log q _ { \theta } ( { \pmb w } _ { j } ^ { p } | \pmb d _ { i } ) \Big ] ,
+$$
+
+where $q _ { \theta }$ is a variational approximation to ground-truth conditional distribution $p ( \pmb { w } | \pmb { d } )$ . We parameterize $q _ { \theta }$ with another neural network. As proved in Cheng et al. (2020a), the better $q _ { \theta } ( { \pmb w } | { \pmb d } )$ approximates $p ( \pmb { w } | \pmb { d } )$ , the more accurate $\scriptstyle { \mathcal { T } } _ { \mathrm { C L U B } }$ serves as the mutual information upper bound. Therefore, besides the loss in (5), we also maximize the log-likelihood of $q _ { \theta } ( { \pmb w } | { \pmb d } )$ with samples $\{ ( d _ { i } , { \boldsymbol { w } } _ { i } ^ { p } ) \} _ { i = 1 } ^ { N }$ .
+
+Based on the above sections, the overall learning scheme of our fair filter (FairFil) is described in Algorithm 1. Also, we provide an intuitive explanation to the two loss terms in our framework. In Figure 1(b), the blue and red circles represent $^ d$ and $\pmb { d } ^ { \prime }$ , respectively, in the embedding space. The intersection $\mathcal { T } ( d ; d ^ { \prime } )$ is the common semantic information extracted from sentences $_ { \textbf { \em x } }$ and $\mathbf { x } ^ { \prime }$ , while the two shadow parts are the extra bias. Note that the perfect debiased embeddings lead to coincident circles. By maximizing $\mathcal { T } _ { \mathrm { N C E } }$ term, we enlarge the overlapped area of $^ d$ and $\pmb { d } ^ { \prime }$ ; by minimizing $\scriptstyle { \mathcal { T } } _ { \mathrm { C L U B } }$ , we shrink the biased shadow parts.
+
+Algorithm 1 Updating the FairFil with a sample batch   
+
+<table><tr><td>Begin with the pretrained text encoder E()and a batch of sentences {x1. Find the sensitive attribute words {wP} and corresponding embeddings {wP}.</td></tr><tr><td></td></tr><tr><td>Generate augmentation x&#x27; from xi,by replacing {wP} with {rj(wp)}.</td></tr><tr><td>Encode (xi,x) into embeddings di=f(E(xi),d𝑖= f(E(x&#x27;)).</td></tr><tr><td>Calculate INcE with {(di,di)}=1 and score function g. if adding debiasing regularizer then</td></tr><tr><td>Update the variational approximation qe(w|d) by maximizing log-likelihood with {(di,w )}</td></tr><tr><td>Calculate IcLuB with qe(wld) and {(di,w)}1</td></tr><tr><td>Learning loss L= -INcE + βIcLUB· else</td></tr><tr><td>Learning loss L = -INcE·</td></tr><tr><td>end if</td></tr><tr><td></td></tr><tr><td>Update FairFil f and score function g by gradient descent with respect to L.</td></tr><tr><td></td></tr></table>
+
+# 4 RELATED WORK
+
+# 4.1 BIAS IN NATURAL LANGUAGE PROCESSING
+
+Social bias has recently been recognized as an important issue in natural language processing (NLP) systems. The studies on bias in NLP are mainly delineated into two categories: bias in the embedding spaces, and bias in downstream tasks (Blodgett et al., 2020). For bias in downstream tasks, the analyses cover comprehensive topics, including machine translation (Stanovsky et al., 2019), language modeling (Bordia & Bowman, 2019), sentiment analysis (Kiritchenko & Mohammad, 2018) and toxicity detection (Dixon et al., 2018). The social bias in embedding spaces has been studied from two important perspectives: bias measurements and and debiasing methods. To measure the bias in an embedding space, Caliskan et al. (2017) proposed a Word Embedding Association Test (WEAT), which compares the similarity between two sets of target words and two sets of attribute words. May et al. (2019) further extended the WEAT to a Sentence Encoder Association Test (SEAT), which replaces the word embeddings by sentence embeddings encoded from pre-defined biased sentence templates. For debiasing methods, most of the prior works focus on word-level representations (Bolukbasi et al., 2016; Bordia & Bowman, 2019). The only sentence-level debiasing method is proposed by Liang et al. (2020), which learns bias directions by PCA and subtracts them in the embedding space.
+
+# 4.2 CONTRASTIVE LEARNING
+
+Contrastive learning is a broad class of training strategies that learns meaningful representations by making positive and negative embedding pairs more distinguishable. Usually, contrastive learning requires a pairwise embedding critic as a similarity/distance of data pairs. Then the learning objective is constructed by maximizing the margin between the critic values of positive data pairs and negative data pairs. Previously contrastive learning has shown encouraging performance in many tasks, including metric learning (Weinberger et al., 2006; Davis et al., 2007), word representation learning (Mikolov et al., 2013), graph learning (Tang et al., 2015; Grover & Leskovec, 2016), etc. Recently, contrastive learning has been applied to the unsupervised visual representation learning task, and significantly reduced the performance gap between supervised and unsupervised learning (He et al., 2020; Chen et al., 2020; Qian et al., 2020). Among these unsupervised methods, Chen et al. (2020) proposed a simple multi-view contrastive learning framework (SimCLR). For each image data, SimCLR generates two augmented images, and then the mutual information of the two augmentation embeddings is maximized within a batch of training data.
+
+# 5 EXPERIMENTS
+
+We first describe the experimental setup in detail, including the pretrained encoders, the training of FairFil, and the downstream tasks. The results of our FairFil are reported and analyzed, along with the previous Sent-Debias method. In general, we evaluate our neural debiasing method from two perspectives: (1) fairness: we compare the bias degree of the original and debiased sentence embeddings for debiasing performance; and (2) representativeness: we apply the debiased embeddings into downstream tasks, and compare the performance with original embeddings.
+
+# 5.1 BIAS EVALUATION METRIC
+
+To evaluate the bias in sentence embeddings, we use the Sentence Encoder Association Test (SEAT) (May et al., 2019), which is an extension of the Word Embedding Association Test (WEAT) (Caliskan et al., 2017). The WEAT test measures the bias in word embeddings by comparing the distances of two sets of target words to two sets of attribute words. More specifically, denote $\mathcal { X }$ and $\mathcal { V }$ as two sets of target word embeddings (e.g., $\mathcal { X }$ includes “male” words such as “boy” and “man”; $\mathcal { V }$ contains “female” words like “girl” and “woman”). The attribute sets $\mathcal { A }$ and $\boldsymbol { B }$ are selected from some social concepts that should be “equal” to $\mathcal { X }$ and $\mathcal { V }$ (e.g., career or personality words). Then the bias degree w.r.t attributes $( A , B )$ of each word embedding $\pmb { t }$ is defined as:
+
+$$
+s ( t , \mathcal { A } , \mathcal { B } ) = \mathrm { m e a n } _ { a \in \mathcal { A } } \cos ( t , a ) - \mathrm { m e a n } _ { b \in \mathcal { B } } \cos ( t , b ) ,
+$$
+
+where $\cos ( \cdot , \cdot )$ is the cosine similarity. Based on (6), the normalized WEAT effect size is:
+
+$$
+d _ { \mathrm { W E A T } } = \frac { \mathrm { m e a n } _ { x \in \mathcal { X } } s ( x , \mathcal { A } , \mathcal { B } ) - \mathrm { m e a n } _ { y \in \mathcal { Y } } s ( y , \mathcal { A } , \mathcal { B } ) } { \mathrm { s t d } _ { t \in \mathcal { X } \cup \mathcal { Y } } s ( t , \mathcal { A } , \mathcal { B } ) } .
+$$
+
+The SEAT test extends WEAT by replacing the word embeddings with sentence embeddings. Both target words and attribute words are converted into sentences with several semantically bleached sentence templates (e.g., “This is ${ < } \mathrm { w o r d } { > } ^ { \mathrm { w } } ,$ ). Then the SEAT statistic is similarly calculated with (7) based on the embeddings of converted sentences. The closer the effect size is to zero, the more fair the embeddings are. Therefore, we report the absolute effect size as the bias measure.
+
+# 5.2 PRETRAINED ENCODERS
+
+We test our neural debiasing method on BERT (Devlin et al., 2019). Since the pretrained BERT requires the additional fine-tuning process for downstream tasks, we report the performance of our FairFil under two scenarios: (1) pretrained BERT: we directly learn our FairFil network based on pretrained BERT without any additional fine-tuning; and (2) BERT post tasks: we fix the parameters of the FairFil network learned on pretrained BERT, and then fine-tune the BERT $^ +$ FairFil together on task-specific data. Note that when fine-tuning, our FairFil will no longer update, which satisfies a fair comparison to Sent-Debias (Liang et al., 2020).
+
+For the downstream tasks of BERT, we follow the setup from Sent-Debias (Liang et al., 2020) and conduct experiments on the following three downstream tasks: (1) SST-2: A sentiment classification task on the Stanford Sentiment Treebank (SST-2) dataset (Socher et al., 2013), on which sentence embeddings are used to predict the corresponding sentiment labels; (2) CoLA: Another sentiment classification task on the Corpus of Linguistic Acceptability (CoLA) grammatical acceptability judgment (Warstadt et al., 2019); and (3) QNLI: A binary question answering task on the Question Natural Language Inference (QNLI) dataset (Wang et al., 2018).
+
+# 5.3 TRAINING OF FAIRFIL
+
+We parameterize the fair filter network with one-layer fully-connected neural networks with the ReLU activation function. The score function $g$ in the InfoNCE estimator is set to a two-layer fully-connected network with one-dimensional output. The variational approximation $q _ { \theta }$ in CLUB estimator is parameterized by a multi-variate Gaussian distribution $q _ { \theta } ( w | d ) = N ( \mu ( d ) , \sigma ^ { 2 } ( d ) )$ , where $\mu ( \cdot )$ and $\sigma ( \cdot )$ are also two-layer fully-connected neural nets. The batch size is set to 128. The learning rate is $1 \times 1 0 ^ { - 5 }$ . We train the fair filter for 10 epochs.
+
+For an appropriate comparison, we follow the setup of Sent-Debias (Liang et al., 2020) and select the same training data for the training of FairFil. The training corpora consist 183,060 sentences from the following five datasets: WikiText-2 (Merity et al., 201y), Stanford Sentiment Treebank (Socher et al., 2013), Reddit (V”olske et al., 2017), MELD (Poria et al., 2019) and POM (Park et al., 2014). Following Liang et al. (2020), we mainly select “gender” as the sensitive topic $\tau$ , and use the same pre-defined word sets of sensitive attribute words and their replaceable words as Sent-Debias did. The word embeddings for training the debiasing regularizer is selected from the token embedding of the pretrained BERT.
+
+# 5.4 DEBIASING RESULTS
+
+In Tables 2 and 3 we report the evaluation results of debiased embeddings on both the absolute SEAT effect size and the downstream classification accuracy. For the SEAT test, we follow the setup in Liang et al. (2020), and test the sentence templates of Terms/Names under different domains designed by Caliskan et al. (2017). The column name Origin refers to the original BERT results, and Sent-D is short for Sent-Debias (Liang et al., 2020). FairFil− and FairFil (as $\mathrm { F a i r F ^ { - } }$ and FairF in the tables) are our method without/with the debiasing regularizer in Section 3.3. The best results of effect size (the lower the better) and classification accuracy (the higher the better) are bold among Sent-D, FairFil−, and FairFil. Since the pretrained BERT does not correspond to any downstream task, the classification accuracy is not reported for it.
+
+Table 2: Performance of debiased embeddings on Pretrained BERT and BERT post SST-2.   
+
+<table><tr><td></td><td colspan="4">Pretrained BERT</td><td colspan="4">BERT post SST-2</td></tr><tr><td></td><td>Origin</td><td>Sent-D</td><td>FairF-</td><td>FairF</td><td>Origin</td><td>Sent-D</td><td>FairF</td><td>FairF</td></tr><tr><td>Names,Career/Family</td><td>0.477</td><td>0.096</td><td>0.218</td><td>0.182</td><td>0.036</td><td>0.109</td><td>0.237</td><td>0.218</td></tr><tr><td>Terms,Career/Family</td><td>0.108</td><td>0.437</td><td>0.086</td><td>0.076</td><td>0.010</td><td>0.057</td><td>0.376</td><td>0.377</td></tr><tr><td>Terms,Math/Arts</td><td>0.253</td><td>0.194</td><td>0.133</td><td>0.124</td><td>0.219</td><td>0.221</td><td>0.301</td><td>0.263</td></tr><tr><td>Names,Math/Arts</td><td>0.254</td><td>0.194</td><td>0.101</td><td>0.082</td><td>1.153</td><td>0.755</td><td>0.084</td><td>0.099</td></tr><tr><td>Terms, Science/Arts</td><td>0.399</td><td>0.075</td><td>0.218</td><td>0.204</td><td>0.103</td><td>0.081</td><td>0.133</td><td>0.127</td></tr><tr><td>Names, Science/Arts</td><td>0.636</td><td>0.540</td><td>0.320</td><td>0.235</td><td>0.222</td><td>0.047</td><td>0.017</td><td>0.005</td></tr><tr><td>Avg. Abs. Effect Size</td><td>0.354</td><td>0.256</td><td>0.179</td><td>0.150</td><td>0.291</td><td>0.212</td><td>0.191</td><td>0.182</td></tr><tr><td>Classification Acc.</td><td>1</td><td>-</td><td>1</td><td>1</td><td>92.7</td><td>89.1</td><td>91.7</td><td>91.6</td></tr></table>
+
+Table 3: Performance of debiased embeddings on BERT post CoLA and BERT post QNLI.   
+
+<table><tr><td></td><td colspan="4">BERT post CoLA</td><td colspan="4">BERT post QNLI</td></tr><tr><td></td><td>Origin</td><td>Sent-D</td><td>FairF-</td><td>FairF</td><td>Origin</td><td>Sent-D</td><td>FairF-</td><td>FairF</td></tr><tr><td>Names, Career/Family</td><td>0.009</td><td>0.149</td><td>0.273</td><td>0.034</td><td>0.261</td><td>0.054</td><td>0.196</td><td>0.103</td></tr><tr><td>Terms, Career/Family</td><td>0.199</td><td>0.186</td><td>0.156</td><td>0.119</td><td>0.155</td><td>0.004</td><td>0.050</td><td>0.206</td></tr><tr><td>Terms,Math/Arts</td><td>0.268</td><td>0.311</td><td>0.008</td><td>0.092</td><td>0.584</td><td>0.083</td><td>0.306</td><td>0.323</td></tr><tr><td>Names,Math/Arts</td><td>0.150</td><td>0.308</td><td>0.060</td><td>0.101</td><td>0.581</td><td>0.629</td><td>0.168</td><td>0.288</td></tr><tr><td>Terms, Science/Arts</td><td>0.425</td><td>0.163</td><td>0.245</td><td>0.249</td><td>0.087</td><td>0.716</td><td>0.500</td><td>0.245</td></tr><tr><td>Names,Science/Arts</td><td>0.032</td><td>0.192</td><td>0.102</td><td>0.127</td><td>0.521</td><td>0.443</td><td>0.378</td><td>0.167</td></tr><tr><td>Avg. Abs.Effect Size</td><td>0.181</td><td>0.217</td><td>0.141</td><td>0.120</td><td>0.365</td><td>0.321</td><td>0.266</td><td>0.222</td></tr><tr><td>Classification Acc.</td><td>57.6</td><td>55.4</td><td>56.5</td><td>56.5</td><td>91.3</td><td>90.6</td><td>91.0</td><td>90.8</td></tr></table>
+
+From the SEAT test results, our contrastive learning framework effectively reduces the gender bias for both pretrained BERT and fine-tuned BERT under most test scenarios. Comparing with Sent-Debias, our FairFil reaches a lower bias degree on the majority of the individual SEAT tests. Considering the average of absolute effect size, our FairFil is distinguished by a significant margin to Sent-Debias. Moreover, our FairFil achieves higher downstream classification accuracy than Sent-Debias, which indicates learning neural filter networks can preserve more semantic meaning than subtracting bias directions learned from PCA.
+
+Table 4: Comparison of average debiasing performance on pretrained BERT   
+
+<table><tr><td>Method</td><td>Bias Degree</td></tr><tr><td>BERT origin (Devlin et al.,2019) FastText (Bojanowski etal., 2017)</td><td>0.354</td></tr><tr><td></td><td>0.565</td></tr><tr><td>BERT word (Bolukbasi et al., 2016)</td><td>0.861</td></tr><tr><td>BERT simple (May et al., 2019)</td><td>0.298</td></tr><tr><td>Sent-Debias (Liang et al.,2020)</td><td>0.256</td></tr><tr><td>FairFil- (Ours)</td><td>0.179</td></tr><tr><td>FairFil (Ours)</td><td>0.150</td></tr></table>
+
+For the ablation study, we also report the results of FairFil without the debiasing regularizer, as in FairF−. Only with the contrastive learning framework, $\mathrm { F a i r F ^ { - } }$ already reduces the bias effectively and even achieves better effect size than the FairF on some of the SEAT tests. With the debiasing regularizer, FairF has better average SEAT effect sizes but slightly loses in terms of the downstream performance. However, the overall performance of FairF and FairF− shows a trade-off between fairness and representativeness of the filter network.
+
+We also compare the debiasing performance on a broader class of baselines, including word-level debiasing methods, and report the average absolute SEAT effect size on the pretrained BERT encoder. Both FairF− and FairF achieve a lower bias degree than other baselines. The word-level debiasing methods (FastText (Bojanowski et al., 2017) and BERT word (Bolukbasi et al., 2016))
+
+![](images/b1af18f81a0530ed7cdc54ea708ebd8933b60de57ffa2ef5a690d953331c31cd.jpg)  
+Figure 2: Influence of the training data proportion to debias degree of BERT.
+
+![](images/73e47c5a70cdb751cd0be57c568fba5cb6b2d7062571501831b7227b38e7b83d.jpg)  
+Figure 3: T-SNE plots of sentence embedding mean of each words contextualized in templates. The left-hand side is from the original pretrained BERT; the right-hand side is from our FairFil.
+
+have the worst debiasing performance, which validates our observation that the word-level debiasing methods cannot reduce sentence-level social bias in NLP models.
+
+# 5.5 ANALYSIS
+
+To test the influence of data proportion on the model’s debiasing performance, we select WikiText-2 with 13,750 sentences as the training corpora following the setup in Liang et al. (2020). Then we randomly divide the training data into 5 equal-sized partitions. We evaluate the bias degree of the sentence debiasing methods on different combinations of the partitions, specifically with training data proportions $20 \%$ , $40 \%$ , $60 \%$ , $80 \%$ , $100 \%$ ). Under each data proportion, we repeat the training 5 times to obtain the mean and variance of the absolute SEAT effect size. In Figure 2, we plot the bias degree of BERT post tasks with different training data proportions. In general, both Sent-Debias and FairFil achieve better performance and smaller variance when the proportion of training data is larger. Under a $20 \%$ training proportion, our FairFil can better remove bias in text encoder, which shows FairFil has better data efficiency with the contrastive learning framework.
+
+To further study output debiased sentence embedding, we visualize the relative distances of attributes and targets of SEAT before/after our debiasing process. We choose the target words as “he” and “she.” Attributes are selected from different social domains. We first contextualize the selected words into sentence templates as described in Section 5.1. We then average the original/debiased embeddings of these sentence template and plot the t-SNE (Maaten & Hinton, 2008) in Figure 3. From the t-SNE, the debiased encoder provides more balanced distances from gender targets “he/she” to the attribute concepts.
+
+# 6 CONCLUSIONS
+
+This paper has developed a novel debiasing method for large-scale pretrained text encoder neural networks. We proposed a fair filter (FairFil) network, which takes the original sentence embeddings as input and outputs the debiased sentence embeddings. To train the fair filter, we constructed a multi-view contrast learning framework, which maximizes the mutual information between each sentence and its augmentation. The augmented sentence is generated by replacing sensitive words in the original sentence with words in a similar semantic but different bias directions. Further, we designed a debiasing regularizer that minimizes the mutual information between the debiased embeddings and the corresponding sensitive words in sentences. Experimental results demonstrate the proposed FairFil not only reduces the bias in sentence embedding space, but also maintains the semantic meaning of the embeddings. This post hoc method does not require access to the training corpora, or any retraining process of the pretrained text encoder, which enhances its applicability.
+
+# ACKNOWLEDGEMENTS
+
+This research was supported in part by the DOE, NSF and ONR.
+
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+        "type": "text",
+        "text": "Pengyu Cheng∗ , Weituo Hao∗, Siyang Yuan , Shijing Si , Lawrence Carin Department of Electrical and Computer Engineering, Duke University {pengyu.cheng,weituo.hao,siyang.yuan,shijing.si,lcarin}@duke.edu ",
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+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "Pretrained text encoders, such as BERT, have been applied increasingly in various natural language processing (NLP) tasks, and have recently demonstrated significant performance gains. However, recent studies have demonstrated the existence of social bias in these pretrained NLP models. Although prior works have made progress on word-level debiasing, improved sentence-level fairness of pretrained encoders still lacks exploration. In this paper, we proposed the first neural debiasing method for a pretrained sentence encoder, which transforms the pretrained encoder outputs into debiased representations via a fair filter (FairFil) network. To learn the FairFil, we introduce a contrastive learning framework that not only minimizes the correlation between filtered embeddings and bias words but also preserves rich semantic information of the original sentences. On real-world datasets, our FairFil effectively reduces the bias degree of pretrained text encoders, while continuously showing desirable performance on downstream tasks. Moreover, our post hoc method does not require any retraining of the text encoders, further enlarging FairFil’s application space. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Text encoders, which map raw-text data into low-dimensional embeddings, have become one of the fundamental tools for extensive tasks in natural language processing (Kiros et al., 2015; Lin et al., 2017; Shen et al., 2019; Cheng et al., 2020b). With the development of deep learning, largescale neural sentence encoders pretrained on massive text corpora, such as Infersent (Conneau et al., 2017), ELMo (Peters et al., 2018), BERT (Devlin et al., 2019), and GPT (Radford et al., 2018), have become the mainstream to extract the sentence-level text representations, and have shown desirable performance on many NLP downstream tasks (MacAvaney et al., 2019; Sun et al., 2019; Zhang et al., 2019). Although these pretrained models have been studied comprehensively from many perspectives, such as performance (Joshi et al., 2020), efficiency (Sanh et al., 2019), and robustness (Liu et al., 2019), the fairness of pretrained text encoders has not received significant research attention. ",
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+        "text": "The fairness issue is also broadly recognized as social bias, which denotes the unbalanced model behaviors with respect to some socially sensitive topics, such as gender, race, and religion (Liang et al., 2020). For data-driven NLP models, social bias is an intrinsic problem mainly caused by the unbalanced data of text corpora (Bolukbasi et al., 2016). To quantitatively measure the bias degree of models, prior work proposed several statistical tests (Caliskan et al., 2017; Chaloner & Maldonado, 2019; Brunet et al., 2019), mostly focusing on word-level embedding models. To evaluate the sentence-level bias in the embedding space, May et al. (2019) extended the Word Embedding Association Test (WEAT) (Caliskan et al., 2017) into a Sentence Encoder Association Test (SEAT). Based on the SEAT test, May et al. (2019) claimed the existence of social bias in the pretrained sentence encoders. ",
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+        "text": "Although related works have discussed the measurement of social bias in sentence embeddings, debiasing pretrained sentence encoders remains a challenge. Previous word embedding debiasing methods (Bolukbasi et al., 2016; Kaneko & Bollegala, 2019; Manzini et al., 2019) have limited assistance to sentence-level debiasing, because even if the social bias is eliminated at the word level, the sentence-level bias can still be caused by the unbalanced combination of words in the training text. Besides, retraining a state-of-the-art sentence encoder for debiasing requires a massive amount of computational resources, especially for large-scale deep models like BERT (Devlin et al., 2019) and GPT (Radford et al., 2018). To the best of our knowledge, Liang et al. (2020) proposed the only sentence-level debiasing method (Sent-Debias) for pretrained text encoders, in which the embeddings are revised by subtracting the latent biased direction vectors learned by Principal Component Analysis (PCA) (Wold et al., 1987). However, Sent-Debias makes a strong assumption on the linearity of the bias in the sentence embedding space. Further, the calculation of bias directions depends highly on the embeddings extracted from the training data and the number of principal components, preventing the method from adequate generalization. ",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "In this paper, we proposed the first neural debiasing method for pretrained sentence encoders. For a given pretrained encoder, our method learns a fair filter (FairFil) network, whose inputs are the original embeddings of the encoder, and outputs are the debiased embeddings. Inspired by the multi-view contrastive learning (Chen et al., 2020), for each training sentence, we first generate an augmentation that has the same semantic meaning but in a different potential bias direction. We contrastively train our FairFil by maximizing the mutual information between the debiased embeddings of the original sentences and corresponding augmentations. To further eliminate bias from sensitive words in sentences, we introduce a debiasing regularizer, which minimizes the mutual information between debiased embeddings and the sensitive words’ embeddings. In the experiments, our FairFil outperforms Sent-Debias (Liang et al., 2020) in terms of the fairness and the representativeness of debiased embeddings, indicating our FairFil not only effectively reduces the social bias in the sentence embeddings, but also successfully preserves the rich semantic meaning of input text. ",
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+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
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+        "text": "Mutual Information (MI) is a measure of the “amount of information” between two variables (Kullback, 1997). The mathematical definition of MI is ",
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+        "type": "equation",
+        "img_path": "images/0cfd62ec28d0d26275b94cfcdfb84f2e15e838caa664f0f44b3619ef87293c3b.jpg",
+        "text": "$$\n\\mathcal { T } ( \\pmb { x } ; \\pmb { y } ) : = \\mathbb { E } _ { p ( \\pmb { x } , \\pmb { y } ) } \\Big [ \\log \\frac { p ( \\pmb { x } , \\pmb { y } ) } { p ( \\pmb { x } ) p ( \\pmb { y } ) } \\Big ] ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $p ( { \\pmb x } , { \\pmb y } )$ is the joint distribution of two variables $( { \\pmb x } , { \\pmb y } )$ , and $p ( { \\pmb x } ) , p ( { \\pmb y } )$ are respectively the marginal distributions of $\\mathbf { \\nabla } _ { \\mathbf { x } , \\mathbf { y } }$ . Recently, mutual information has achieved considerable success when applied as a learning criterion in diverse deep learning tasks, such as conditional generation (Chen et al., 2016), domain adaptation (Gholami et al., 2020), representation learning (Chen et al., 2020), and fairness (Song et al., 2019). However, the calculation of exact MI in (1) is well-recognized as a challenge, because the expectation w.r.t $p ( { \\pmb x } , { \\pmb y } )$ is always intractable, especially when only samples from $p ( { \\pmb x } , { \\pmb y } )$ are provided. To this end, several upper and lower bounds have been introduced to estimate the MI with samples. For MI maximization tasks (Hjelm et al., 2018; Chen et al., 2020), Oord et al. (2018) derived a powerful MI estimator, InfoNCE, based on noise contrastive estimation (NCE) (Gutmann $\\&$ Hyvarinen, 2010). Given a batch of sample pairs ¨ $\\{ ( \\pmb { x } _ { i } , \\pmb { y } _ { i } ) \\} _ { i = 1 } ^ { N }$ , the InfoNCE estimator is defined with a learnable score function $f ( { \\pmb x } , { \\pmb y } )$ : ",
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+        "type": "equation",
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+        "text": "$$\n\\mathcal { T } _ { \\mathrm { N C E } } : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\frac { \\exp ( f (  { \\boldsymbol { { x } } } _ { i } ,  { \\boldsymbol { { y } } } _ { i } ) ) } { \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\exp ( f (  { \\boldsymbol { { x } } } _ { i } ,  { \\boldsymbol { { y } } } _ { j } ) ) } .\n$$",
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+        "type": "text",
+        "text": "For MI minimization tasks (Alemi et al., 2017; Song et al., 2019), Cheng et al. (2020a) introduced a contrastive log-ratio upper bound (CLUB) based on a variational approximation $q _ { \\boldsymbol { \\theta } } ( \\mathbf { \\boldsymbol { y } } | \\mathbf { \\boldsymbol { x } } )$ of conditional distribution $p ( \\pmb { y } | \\pmb { x } )$ : ",
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+    {
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+        "text": "$$\n\\mathcal { T } _ { \\mathrm { C L U B } } : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\bigg [ \\log q _ { \\theta } ( \\pmb { y } _ { i } | \\pmb { x } _ { i } ) - \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\log q _ { \\theta } ( \\pmb { y } _ { j } | \\pmb { x } _ { i } ) \\bigg ] .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "In the following, we use the above two MI estimators to induce the sentence encoder, eliminating the biased information and preserving the semantic information from the original raw text. ",
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+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Suppose $E ( \\cdot )$ is a pretrained sentence encoder, which can encode a sentence $_ { \\textbf { \\em x } }$ into low-dimensional embedding $\\begin{array} { r } { \\dot { \\boldsymbol { z } } = \\boldsymbol { E } ( \\boldsymbol { x } ) } \\end{array}$ . Each sentence $\\pmb { x } = ( w ^ { 1 } , w ^ { 2 } , \\dots , w ^ { L } )$ is a sequence of words. The embedding space of $_ z$ has been recognized to have social bias in a series of studies (May et al., 2019; Kurita et al., 2019; Liang et al., 2020). To eliminate the social bias in the embedding space, we aim to learn a fair filter network $f ( \\cdot )$ on top of the sentence encoder $E ( \\cdot )$ , such that the output embedding of our fair filter $\\begin{array} { r } { d = f ( z ) } \\end{array}$ can be debiased. To train the fair filter, we design a multi-view contrastive learning framework, which consists of three steps. First, for each input sentence $_ { \\textbf { \\em x } }$ , we generate an augmented sentence $\\mathbf { x } ^ { \\prime }$ that has the same semantic meaning as $_ { \\textbf { \\em x } }$ but in a different potential bias direction. Then, we maximize the mutual information between the original embedding $z = f ( { \\pmb x } )$ and the augmented embedding $z ^ { \\prime } = f ( x ^ { \\prime } )$ with the InfoNCE (Oord et al., 2018) contrastive loss. Further, we design a debiasing regularizer to minimize the mutual information between $^ d$ and sensitive attribute words in $_ { \\textbf { \\em x } }$ . In the following, we discuss these three steps in detail. ",
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+        "type": "table",
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+        "table_caption": [
+            "Table 1: Examples of generating an augmentation sentence under the sensitive topic “gender”. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Bias direction</td><td rowspan=1 colspan=1>Sensitive Attribute words</td><td rowspan=1 colspan=2>Text content</td></tr><tr><td rowspan=1 colspan=1>Original</td><td rowspan=1 colspan=1>male</td><td rowspan=1 colspan=1>he,his</td><td rowspan=1 colspan=2>{He} is good at playing {his} basketball.</td></tr><tr><td rowspan=1 colspan=1>Augmentation</td><td rowspan=1 colspan=1>female</td><td rowspan=1 colspan=1>she,her</td><td rowspan=1 colspan=1>{She</td><td rowspan=1 colspan=1>{She} is good at playing {her} basketball.</td></tr></table>",
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+        "text": "3.1 DATA AUGMENTATIONS WITH SENSITIVE ATTRIBUTES ",
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+        "text": "We first describe the sentence data augmentation process for our FairFil contrastive learning. Denote a social sensitive topic as $\\mathcal { T } = \\{ \\mathcal { D } _ { 1 } , \\mathcal { D } _ { 2 } , \\ldots , \\mathcal { D } _ { K } \\}$ , where $\\mathcal { D } _ { k }$ $( k = 1 , \\ldots , K )$ is one of the potential bias directions under the topic. For example, if $\\tau$ represents the sensitive topic “gender”, then $\\tau$ consists two potential bias directions $\\{ \\mathcal { D } _ { 1 } , \\mathcal { D } _ { 2 } \\} \\stackrel { - } { = } \\{ { } ^ {  } m a l e ^ { , \\prime } , \\ { } ^ {  } f e m a l e ^ { , \\prime } \\}$ . Similarly, if $\\tau$ is set as the major “religions” of the world, then $\\tau$ could contain $\\left\\{ \\mathcal { D } _ { 1 } , \\mathcal { D } _ { 2 } , \\mathcal { D } _ { 3 } , \\mathcal { D } _ { 4 } \\right\\} \\ =$ {“Christianity”, “Islam”, “Judaism”, “Buddhism”} as four components. ",
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+        "type": "text",
+        "text": "For a given social sensitive topic $\\mathcal { T } = \\{ { D } _ { 1 } , . . . { D } _ { K } \\}$ , if a word $w$ is related to one of the potential bias direction $\\mathcal { D } _ { k }$ (denote as $w \\in \\mathcal { D } _ { k } ,$ ), we call $w$ a sensitive attribute word of $\\mathcal { D } _ { k }$ (also called bias attribute word in Liang et al. (2020)). For a sensitive attribute word $w \\in \\mathcal { D } _ { k }$ , suppose we can always find another sensitive attribute word $u \\in { \\mathcal { D } } _ { j }$ , such that $w$ and $u$ has the equivalent semantic meaning but in a different bias direction. Then we call $u$ as a replaceable word of $w$ in direction $\\mathcal { D } _ { j }$ , and denote as $u = r _ { j } ( w )$ . For the topic “gende $\\therefore \\prime = \\{ \\stackrel {  } { \\cdot } m a l e ^ { \\prime \\prime } , \\stackrel {  } { \\cdot } f e m a l e ^ { \\prime \\prime } \\}$ , the word $w =$ “boy” is in the potential bias direction $\\mathcal { D } _ { 1 } = \\ ^ { \\ast } m a l e ^ { \\prime \\prime }$ ; a replaceable word of “boy” in “female” direction is $r _ { 2 } ( \\bar { w } ) = \\cdots \\mathrm { g i r l } ^ { * } \\in \\mathcal { D } _ { 2 }$ . ",
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+        "type": "text",
+        "text": "With the above definitions, for each sentence $_ { \\textbf { \\em x } }$ , we generate an augmented sentence $\\mathbf { x } ^ { \\prime }$ such that $\\mathbf { x } ^ { \\prime }$ has the same semantic meaning as $_ { \\textbf { \\em x } }$ but in a different potential bias direction. More specifically, for a sentence $\\pmb { x } = ( w ^ { 1 } , w ^ { 2 } , \\dots , \\breve { w } ^ { L } )$ , we first find the sensitive word positions as an index set $\\mathcal { P }$ , such that each $w ^ { p }$ ${ \\bf \\Phi } _ { p } \\in { \\mathcal { P } } ,$ ) is a sensitive attribute words in direction $\\mathcal { D } _ { k }$ . We further make a reasonable assumption that the embedding bias of direction $\\mathcal { D } _ { k }$ is only caused by the sensitive words $\\{ w ^ { p } \\} _ { p \\in { \\mathcal { P } } }$ in $_ { \\textbf { \\em x } }$ . To sample an augmentation to $_ { \\textbf { \\em x } }$ , we first select another potential bias direction $\\mathcal { D } _ { j }$ , and then replace all sensitive attribute words by their replaceable words in the direction $\\mathcal { D } _ { j }$ . That is, $\\pmb { x } ^ { \\prime } = \\{ v ^ { 1 } , v ^ { 2 } , \\bot \\bot , v ^ { L } \\}$ , where $v ^ { l } = w ^ { l }$ if $l \\notin \\mathcal { P }$ , and $v ^ { l } = r _ { j } ( w ^ { l } )$ if $l \\in \\mathcal { P }$ . In Table 1, we provide an example for sentence augmentation under the “gender” topic. ",
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+        "text": "3.2 CONTRASTIVE LEARNING FRAMEWORK ",
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+        "text": "After obtaining the sentence pair $( { \\pmb x } , { \\pmb x } ^ { \\prime } )$ with the augmentation strategy from Section 3.1, we construct a contrastive learning framework to learn our debiasing fair filter $f ( \\cdot )$ . As shown in the Figure 1(a), our framework consists of the following two steps: ",
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+        "text": "(1) We encode sentences $( { \\pmb x } , { \\pmb x } ^ { \\prime } )$ into embeddings $( z , z ^ { \\prime } )$ with the pretrained encoder $E ( \\cdot )$ . Since $_ { \\textbf { \\em x } }$ and $\\mathbf { x } ^ { \\prime }$ have the same meaning but different potential bias directions, the embeddings $( z , z ^ { \\prime } )$ will have different bias directions, which are caused by the sensitive attributed words in $_ { \\textbf { \\em x } }$ and $\\mathbf { x } ^ { \\prime }$ . ",
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+        "text": "(2) We then feed the sentence embeddings $( z , z ^ { \\prime } )$ through our fair filter $f ( \\cdot )$ to obtain the debiased embedding outputs $( d , d ^ { \\prime } )$ . Ideally, $^ d$ and $\\mathbf { \\Delta } d ^ { \\prime }$ should represent the same semantic meaning without social bias. Inspired by SimCLR (Chen et al., 2020), we encourage the overlapped semantic information between $^ d$ and $\\mathbf { \\Delta } d ^ { \\prime }$ by maximizing their mutual information $\\bar { \\mathcal { T } } ( d ; d ^ { \\prime } )$ . ",
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+        "type": "text",
+        "text": "However, the calculation of $\\mathcal { T } ( d ; d ^ { \\prime } )$ is practically difficult because only embedding samples of $^ d$ and $\\pmb { d } ^ { \\prime }$ are available. Therefore, we use the InfoNCE mutual information estimator (Oord et al., 2018) to minimize the lower bound of $\\mathcal { T } ( d ; d ^ { \\prime } )$ instead. Based on a learnable score function $g ( \\cdot , \\cdot )$ , the contrastive InfoNCE estimator is calculated within a batch of samples $\\{ ( d _ { i } , d _ { i } ^ { \\prime } ) \\} _ { i = 1 } ^ { N }$ ",
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+    {
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+        "img_path": "images/8c1f7a0b55452733b6dca530e9279647bdf58ae19e81215c4ef29b41a8378f5b.jpg",
+        "image_caption": [
+            "Figure 1: (a) Contrastive learning framework of FairFil: Sentence $_ { \\textbf { \\em x } }$ and its augmentation $\\mathbf { x } ^ { \\prime }$ are encoded into embeddings $^ d$ and $\\pmb { d } ^ { \\prime }$ , respectively. $\\pmb { w } ^ { p }$ is the embedding of a sensitive attribute word selected from $_ { \\textbf { \\em x } }$ . $\\mathcal { T } _ { \\mathrm { N C E } }$ maximizes the mutual information between $^ d$ and $\\pmb { d } ^ { \\prime }$ ; $\\scriptstyle { \\mathcal { T } } _ { \\mathrm { C L U B } }$ eliminates the bias information of $\\pmb { w } ^ { p }$ from $\\mathbf { \\delta } _ { d }$ . (b) Illustration of information in $^ d$ and $\\pmb { d } ^ { \\prime }$ : The blue and red circles represent the information in $^ d$ and $\\mathbf { { \\mathbf { { \\mathit { d } } } } ^ { \\prime } } \\mathbf { \\Sigma }$ , respectively. The intersection is the mutual information between $^ d$ and $\\pmb { d } ^ { \\prime }$ . The shadow area represents the bias information of both embeddings. "
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+        "text": "$$\n\\mathcal { T } _ { \\mathrm { N C E } } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\frac { \\exp ( g ( d _ { i } , d _ { i } ^ { \\prime } ) ) } { \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\exp ( g ( d _ { i } , d _ { j } ^ { \\prime } ) ) } .\n$$",
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+        "text": "By maximize $\\mathcal { T } _ { \\mathrm { N C E } }$ , we encourage the difference between the positive pair score $g ( d _ { i } , d _ { i } ^ { \\prime } )$ and the negative pair score $g ( d _ { i } , d _ { j } ^ { \\prime } )$ , so that $\\mathbf { \\ b { d } } _ { i }$ can share more semantic information with $\\mathbf { \\Delta } d _ { i } ^ { \\prime }$ than other embeddings d0j6=i. ",
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+        "text": "3.3 DEBIASING REGULARIZER ",
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+        "text": "Practically, the contrastive learning framework in Section 3.2 can already show encouraging debiasing performance (as shown in the Experiments). However, the embedding $^ d$ can contain extra biased information from $_ { z }$ , that only maximizing $\\mathcal { T } ( d ; d ^ { \\prime } )$ fails to eliminate. To encourage no extra bias in $^ d$ , we introduce a debiasing regularizer which minimizes the mutual information between embedding $^ d$ and the potential bias from embedding $_ z$ . As discussed in Section 3.1, in our framework the potential bias of $_ z$ is assumed to come from the sensitive attribute words in $_ { \\textbf { \\em x } }$ . Therefore, we should reduce the bias word information from the debiased representation $^ d$ . Let $\\pmb { w } ^ { p }$ be the embedding of a sensitive attribute word $w ^ { p }$ in sentence $_ { \\textbf { \\em x } }$ . The word embedding $\\pmb { w } ^ { p }$ can always be obtained from the pretrained text encoders (Bordia $\\&$ Bowman, 2019). We then minimize the mutual information $\\mathcal { T } ( w ^ { p } ; d )$ , using the CLUB mutual information upper bound (Cheng et al., 2020a) to estimate $\\mathcal { T } ( w ^ { p } ; d )$ with embedding samples. Given a batch of embedding pairs $\\bar { \\{ ( d _ { i } , \\boldsymbol { w } ^ { p } ) \\} } _ { i = 1 } ^ { N }$ , we can calculate the debiasing regularizer as: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a63c40840b1faa97447066e2f554e7dfffa8be0097875d90080e8ba1df426f47.jpg",
+        "text": "$$\n\\mathcal { T } _ { \\mathrm { C L U B } } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\Big [ \\log q _ { \\theta } ( { \\pmb w } _ { i } ^ { p } | \\pmb d _ { i } ) - \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\log q _ { \\theta } ( { \\pmb w } _ { j } ^ { p } | \\pmb d _ { i } ) \\Big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $q _ { \\theta }$ is a variational approximation to ground-truth conditional distribution $p ( \\pmb { w } | \\pmb { d } )$ . We parameterize $q _ { \\theta }$ with another neural network. As proved in Cheng et al. (2020a), the better $q _ { \\theta } ( { \\pmb w } | { \\pmb d } )$ approximates $p ( \\pmb { w } | \\pmb { d } )$ , the more accurate $\\scriptstyle { \\mathcal { T } } _ { \\mathrm { C L U B } }$ serves as the mutual information upper bound. Therefore, besides the loss in (5), we also maximize the log-likelihood of $q _ { \\theta } ( { \\pmb w } | { \\pmb d } )$ with samples $\\{ ( d _ { i } , { \\boldsymbol { w } } _ { i } ^ { p } ) \\} _ { i = 1 } ^ { N }$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "Based on the above sections, the overall learning scheme of our fair filter (FairFil) is described in Algorithm 1. Also, we provide an intuitive explanation to the two loss terms in our framework. In Figure 1(b), the blue and red circles represent $^ d$ and $\\pmb { d } ^ { \\prime }$ , respectively, in the embedding space. The intersection $\\mathcal { T } ( d ; d ^ { \\prime } )$ is the common semantic information extracted from sentences $_ { \\textbf { \\em x } }$ and $\\mathbf { x } ^ { \\prime }$ , while the two shadow parts are the extra bias. Note that the perfect debiased embeddings lead to coincident circles. By maximizing $\\mathcal { T } _ { \\mathrm { N C E } }$ term, we enlarge the overlapped area of $^ d$ and $\\pmb { d } ^ { \\prime }$ ; by minimizing $\\scriptstyle { \\mathcal { T } } _ { \\mathrm { C L U B } }$ , we shrink the biased shadow parts. ",
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+    {
+        "type": "table",
+        "img_path": "images/03a1adeb3cc52d169b4f8ecb0c39e58cde5e3c402f96562d90cae020d0fa9fb9.jpg",
+        "table_caption": [
+            "Algorithm 1 Updating the FairFil with a sample batch "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Begin with the pretrained text encoder E()and a batch of sentences {x1. Find the sensitive attribute words {wP} and corresponding embeddings {wP}.</td></tr><tr><td></td></tr><tr><td>Generate augmentation x&#x27; from xi,by replacing {wP} with {rj(wp)}.</td></tr><tr><td>Encode (xi,x) into embeddings di=f(E(xi),d𝑖= f(E(x&#x27;)).</td></tr><tr><td>Calculate INcE with {(di,di)}=1 and score function g. if adding debiasing regularizer then</td></tr><tr><td>Update the variational approximation qe(w|d) by maximizing log-likelihood with {(di,w )}</td></tr><tr><td>Calculate IcLuB with qe(wld) and {(di,w)}1</td></tr><tr><td>Learning loss L= -INcE + βIcLUB· else</td></tr><tr><td>Learning loss L = -INcE·</td></tr><tr><td>end if</td></tr><tr><td></td></tr><tr><td>Update FairFil f and score function g by gradient descent with respect to L.</td></tr><tr><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 BIAS IN NATURAL LANGUAGE PROCESSING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Social bias has recently been recognized as an important issue in natural language processing (NLP) systems. The studies on bias in NLP are mainly delineated into two categories: bias in the embedding spaces, and bias in downstream tasks (Blodgett et al., 2020). For bias in downstream tasks, the analyses cover comprehensive topics, including machine translation (Stanovsky et al., 2019), language modeling (Bordia & Bowman, 2019), sentiment analysis (Kiritchenko & Mohammad, 2018) and toxicity detection (Dixon et al., 2018). The social bias in embedding spaces has been studied from two important perspectives: bias measurements and and debiasing methods. To measure the bias in an embedding space, Caliskan et al. (2017) proposed a Word Embedding Association Test (WEAT), which compares the similarity between two sets of target words and two sets of attribute words. May et al. (2019) further extended the WEAT to a Sentence Encoder Association Test (SEAT), which replaces the word embeddings by sentence embeddings encoded from pre-defined biased sentence templates. For debiasing methods, most of the prior works focus on word-level representations (Bolukbasi et al., 2016; Bordia & Bowman, 2019). The only sentence-level debiasing method is proposed by Liang et al. (2020), which learns bias directions by PCA and subtracts them in the embedding space. ",
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+    {
+        "type": "text",
+        "text": "4.2 CONTRASTIVE LEARNING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Contrastive learning is a broad class of training strategies that learns meaningful representations by making positive and negative embedding pairs more distinguishable. Usually, contrastive learning requires a pairwise embedding critic as a similarity/distance of data pairs. Then the learning objective is constructed by maximizing the margin between the critic values of positive data pairs and negative data pairs. Previously contrastive learning has shown encouraging performance in many tasks, including metric learning (Weinberger et al., 2006; Davis et al., 2007), word representation learning (Mikolov et al., 2013), graph learning (Tang et al., 2015; Grover & Leskovec, 2016), etc. Recently, contrastive learning has been applied to the unsupervised visual representation learning task, and significantly reduced the performance gap between supervised and unsupervised learning (He et al., 2020; Chen et al., 2020; Qian et al., 2020). Among these unsupervised methods, Chen et al. (2020) proposed a simple multi-view contrastive learning framework (SimCLR). For each image data, SimCLR generates two augmented images, and then the mutual information of the two augmentation embeddings is maximized within a batch of training data. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first describe the experimental setup in detail, including the pretrained encoders, the training of FairFil, and the downstream tasks. The results of our FairFil are reported and analyzed, along with the previous Sent-Debias method. In general, we evaluate our neural debiasing method from two perspectives: (1) fairness: we compare the bias degree of the original and debiased sentence embeddings for debiasing performance; and (2) representativeness: we apply the debiased embeddings into downstream tasks, and compare the performance with original embeddings. ",
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+    {
+        "type": "text",
+        "text": "5.1 BIAS EVALUATION METRIC ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To evaluate the bias in sentence embeddings, we use the Sentence Encoder Association Test (SEAT) (May et al., 2019), which is an extension of the Word Embedding Association Test (WEAT) (Caliskan et al., 2017). The WEAT test measures the bias in word embeddings by comparing the distances of two sets of target words to two sets of attribute words. More specifically, denote $\\mathcal { X }$ and $\\mathcal { V }$ as two sets of target word embeddings (e.g., $\\mathcal { X }$ includes “male” words such as “boy” and “man”; $\\mathcal { V }$ contains “female” words like “girl” and “woman”). The attribute sets $\\mathcal { A }$ and $\\boldsymbol { B }$ are selected from some social concepts that should be “equal” to $\\mathcal { X }$ and $\\mathcal { V }$ (e.g., career or personality words). Then the bias degree w.r.t attributes $( A , B )$ of each word embedding $\\pmb { t }$ is defined as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/e34930275b800071a949db344a09873492ef9a369806dcd04f9678b87e84d1b2.jpg",
+        "text": "$$\ns ( t , \\mathcal { A } , \\mathcal { B } ) = \\mathrm { m e a n } _ { a \\in \\mathcal { A } } \\cos ( t , a ) - \\mathrm { m e a n } _ { b \\in \\mathcal { B } } \\cos ( t , b ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\cos ( \\cdot , \\cdot )$ is the cosine similarity. Based on (6), the normalized WEAT effect size is: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/681a0f99acde358feb5acbd3dcc422754a639d34602fcd02bf4ec03d0d0ff340.jpg",
+        "text": "$$\nd _ { \\mathrm { W E A T } } = \\frac { \\mathrm { m e a n } _ { x \\in \\mathcal { X } } s ( x , \\mathcal { A } , \\mathcal { B } ) - \\mathrm { m e a n } _ { y \\in \\mathcal { Y } } s ( y , \\mathcal { A } , \\mathcal { B } ) } { \\mathrm { s t d } _ { t \\in \\mathcal { X } \\cup \\mathcal { Y } } s ( t , \\mathcal { A } , \\mathcal { B } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The SEAT test extends WEAT by replacing the word embeddings with sentence embeddings. Both target words and attribute words are converted into sentences with several semantically bleached sentence templates (e.g., “This is ${ < } \\mathrm { w o r d } { > } ^ { \\mathrm { w } } ,$ ). Then the SEAT statistic is similarly calculated with (7) based on the embeddings of converted sentences. The closer the effect size is to zero, the more fair the embeddings are. Therefore, we report the absolute effect size as the bias measure. ",
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+    {
+        "type": "text",
+        "text": "5.2 PRETRAINED ENCODERS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We test our neural debiasing method on BERT (Devlin et al., 2019). Since the pretrained BERT requires the additional fine-tuning process for downstream tasks, we report the performance of our FairFil under two scenarios: (1) pretrained BERT: we directly learn our FairFil network based on pretrained BERT without any additional fine-tuning; and (2) BERT post tasks: we fix the parameters of the FairFil network learned on pretrained BERT, and then fine-tune the BERT $^ +$ FairFil together on task-specific data. Note that when fine-tuning, our FairFil will no longer update, which satisfies a fair comparison to Sent-Debias (Liang et al., 2020). ",
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+        "type": "text",
+        "text": "For the downstream tasks of BERT, we follow the setup from Sent-Debias (Liang et al., 2020) and conduct experiments on the following three downstream tasks: (1) SST-2: A sentiment classification task on the Stanford Sentiment Treebank (SST-2) dataset (Socher et al., 2013), on which sentence embeddings are used to predict the corresponding sentiment labels; (2) CoLA: Another sentiment classification task on the Corpus of Linguistic Acceptability (CoLA) grammatical acceptability judgment (Warstadt et al., 2019); and (3) QNLI: A binary question answering task on the Question Natural Language Inference (QNLI) dataset (Wang et al., 2018). ",
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+    {
+        "type": "text",
+        "text": "5.3 TRAINING OF FAIRFIL ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We parameterize the fair filter network with one-layer fully-connected neural networks with the ReLU activation function. The score function $g$ in the InfoNCE estimator is set to a two-layer fully-connected network with one-dimensional output. The variational approximation $q _ { \\theta }$ in CLUB estimator is parameterized by a multi-variate Gaussian distribution $q _ { \\theta } ( w | d ) = N ( \\mu ( d ) , \\sigma ^ { 2 } ( d ) )$ , where $\\mu ( \\cdot )$ and $\\sigma ( \\cdot )$ are also two-layer fully-connected neural nets. The batch size is set to 128. The learning rate is $1 \\times 1 0 ^ { - 5 }$ . We train the fair filter for 10 epochs. ",
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+        "type": "text",
+        "text": "For an appropriate comparison, we follow the setup of Sent-Debias (Liang et al., 2020) and select the same training data for the training of FairFil. The training corpora consist 183,060 sentences from the following five datasets: WikiText-2 (Merity et al., 201y), Stanford Sentiment Treebank (Socher et al., 2013), Reddit (V”olske et al., 2017), MELD (Poria et al., 2019) and POM (Park et al., 2014). Following Liang et al. (2020), we mainly select “gender” as the sensitive topic $\\tau$ , and use the same pre-defined word sets of sensitive attribute words and their replaceable words as Sent-Debias did. The word embeddings for training the debiasing regularizer is selected from the token embedding of the pretrained BERT. ",
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+    {
+        "type": "text",
+        "text": "5.4 DEBIASING RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In Tables 2 and 3 we report the evaluation results of debiased embeddings on both the absolute SEAT effect size and the downstream classification accuracy. For the SEAT test, we follow the setup in Liang et al. (2020), and test the sentence templates of Terms/Names under different domains designed by Caliskan et al. (2017). The column name Origin refers to the original BERT results, and Sent-D is short for Sent-Debias (Liang et al., 2020). FairFil− and FairFil (as $\\mathrm { F a i r F ^ { - } }$ and FairF in the tables) are our method without/with the debiasing regularizer in Section 3.3. The best results of effect size (the lower the better) and classification accuracy (the higher the better) are bold among Sent-D, FairFil−, and FairFil. Since the pretrained BERT does not correspond to any downstream task, the classification accuracy is not reported for it. ",
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+    {
+        "type": "table",
+        "img_path": "images/34e9e64ff39ac335347ce8e54b7dbdc9e8b6fdf1638f6ecd2440b494110833cc.jpg",
+        "table_caption": [
+            "Table 2: Performance of debiased embeddings on Pretrained BERT and BERT post SST-2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">Pretrained BERT</td><td colspan=\"4\">BERT post SST-2</td></tr><tr><td></td><td>Origin</td><td>Sent-D</td><td>FairF-</td><td>FairF</td><td>Origin</td><td>Sent-D</td><td>FairF</td><td>FairF</td></tr><tr><td>Names,Career/Family</td><td>0.477</td><td>0.096</td><td>0.218</td><td>0.182</td><td>0.036</td><td>0.109</td><td>0.237</td><td>0.218</td></tr><tr><td>Terms,Career/Family</td><td>0.108</td><td>0.437</td><td>0.086</td><td>0.076</td><td>0.010</td><td>0.057</td><td>0.376</td><td>0.377</td></tr><tr><td>Terms,Math/Arts</td><td>0.253</td><td>0.194</td><td>0.133</td><td>0.124</td><td>0.219</td><td>0.221</td><td>0.301</td><td>0.263</td></tr><tr><td>Names,Math/Arts</td><td>0.254</td><td>0.194</td><td>0.101</td><td>0.082</td><td>1.153</td><td>0.755</td><td>0.084</td><td>0.099</td></tr><tr><td>Terms, Science/Arts</td><td>0.399</td><td>0.075</td><td>0.218</td><td>0.204</td><td>0.103</td><td>0.081</td><td>0.133</td><td>0.127</td></tr><tr><td>Names, Science/Arts</td><td>0.636</td><td>0.540</td><td>0.320</td><td>0.235</td><td>0.222</td><td>0.047</td><td>0.017</td><td>0.005</td></tr><tr><td>Avg. Abs. Effect Size</td><td>0.354</td><td>0.256</td><td>0.179</td><td>0.150</td><td>0.291</td><td>0.212</td><td>0.191</td><td>0.182</td></tr><tr><td>Classification Acc.</td><td>1</td><td>-</td><td>1</td><td>1</td><td>92.7</td><td>89.1</td><td>91.7</td><td>91.6</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f5d8fc9a49cc5afb531787f31ea81ae3b037ec4af7301043842334f2f54e715a.jpg",
+        "table_caption": [
+            "Table 3: Performance of debiased embeddings on BERT post CoLA and BERT post QNLI. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">BERT post CoLA</td><td colspan=\"4\">BERT post QNLI</td></tr><tr><td></td><td>Origin</td><td>Sent-D</td><td>FairF-</td><td>FairF</td><td>Origin</td><td>Sent-D</td><td>FairF-</td><td>FairF</td></tr><tr><td>Names, Career/Family</td><td>0.009</td><td>0.149</td><td>0.273</td><td>0.034</td><td>0.261</td><td>0.054</td><td>0.196</td><td>0.103</td></tr><tr><td>Terms, Career/Family</td><td>0.199</td><td>0.186</td><td>0.156</td><td>0.119</td><td>0.155</td><td>0.004</td><td>0.050</td><td>0.206</td></tr><tr><td>Terms,Math/Arts</td><td>0.268</td><td>0.311</td><td>0.008</td><td>0.092</td><td>0.584</td><td>0.083</td><td>0.306</td><td>0.323</td></tr><tr><td>Names,Math/Arts</td><td>0.150</td><td>0.308</td><td>0.060</td><td>0.101</td><td>0.581</td><td>0.629</td><td>0.168</td><td>0.288</td></tr><tr><td>Terms, Science/Arts</td><td>0.425</td><td>0.163</td><td>0.245</td><td>0.249</td><td>0.087</td><td>0.716</td><td>0.500</td><td>0.245</td></tr><tr><td>Names,Science/Arts</td><td>0.032</td><td>0.192</td><td>0.102</td><td>0.127</td><td>0.521</td><td>0.443</td><td>0.378</td><td>0.167</td></tr><tr><td>Avg. Abs.Effect Size</td><td>0.181</td><td>0.217</td><td>0.141</td><td>0.120</td><td>0.365</td><td>0.321</td><td>0.266</td><td>0.222</td></tr><tr><td>Classification Acc.</td><td>57.6</td><td>55.4</td><td>56.5</td><td>56.5</td><td>91.3</td><td>90.6</td><td>91.0</td><td>90.8</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "From the SEAT test results, our contrastive learning framework effectively reduces the gender bias for both pretrained BERT and fine-tuned BERT under most test scenarios. Comparing with Sent-Debias, our FairFil reaches a lower bias degree on the majority of the individual SEAT tests. Considering the average of absolute effect size, our FairFil is distinguished by a significant margin to Sent-Debias. Moreover, our FairFil achieves higher downstream classification accuracy than Sent-Debias, which indicates learning neural filter networks can preserve more semantic meaning than subtracting bias directions learned from PCA. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/d5cec6c5fa8e102545c3d64716e74071e9781ac802d7f69b45a869964a36d06d.jpg",
+        "table_caption": [
+            "Table 4: Comparison of average debiasing performance on pretrained BERT "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Bias Degree</td></tr><tr><td>BERT origin (Devlin et al.,2019) FastText (Bojanowski etal., 2017)</td><td>0.354</td></tr><tr><td></td><td>0.565</td></tr><tr><td>BERT word (Bolukbasi et al., 2016)</td><td>0.861</td></tr><tr><td>BERT simple (May et al., 2019)</td><td>0.298</td></tr><tr><td>Sent-Debias (Liang et al.,2020)</td><td>0.256</td></tr><tr><td>FairFil- (Ours)</td><td>0.179</td></tr><tr><td>FairFil (Ours)</td><td>0.150</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "For the ablation study, we also report the results of FairFil without the debiasing regularizer, as in FairF−. Only with the contrastive learning framework, $\\mathrm { F a i r F ^ { - } }$ already reduces the bias effectively and even achieves better effect size than the FairF on some of the SEAT tests. With the debiasing regularizer, FairF has better average SEAT effect sizes but slightly loses in terms of the downstream performance. However, the overall performance of FairF and FairF− shows a trade-off between fairness and representativeness of the filter network. ",
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+        "type": "text",
+        "text": "We also compare the debiasing performance on a broader class of baselines, including word-level debiasing methods, and report the average absolute SEAT effect size on the pretrained BERT encoder. Both FairF− and FairF achieve a lower bias degree than other baselines. The word-level debiasing methods (FastText (Bojanowski et al., 2017) and BERT word (Bolukbasi et al., 2016)) ",
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+        "img_path": "images/b1af18f81a0530ed7cdc54ea708ebd8933b60de57ffa2ef5a690d953331c31cd.jpg",
+        "image_caption": [
+            "Figure 2: Influence of the training data proportion to debias degree of BERT. "
+        ],
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+        "img_path": "images/73e47c5a70cdb751cd0be57c568fba5cb6b2d7062571501831b7227b38e7b83d.jpg",
+        "image_caption": [
+            "Figure 3: T-SNE plots of sentence embedding mean of each words contextualized in templates. The left-hand side is from the original pretrained BERT; the right-hand side is from our FairFil. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "have the worst debiasing performance, which validates our observation that the word-level debiasing methods cannot reduce sentence-level social bias in NLP models. ",
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+    {
+        "type": "text",
+        "text": "5.5 ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To test the influence of data proportion on the model’s debiasing performance, we select WikiText-2 with 13,750 sentences as the training corpora following the setup in Liang et al. (2020). Then we randomly divide the training data into 5 equal-sized partitions. We evaluate the bias degree of the sentence debiasing methods on different combinations of the partitions, specifically with training data proportions $20 \\%$ , $40 \\%$ , $60 \\%$ , $80 \\%$ , $100 \\%$ ). Under each data proportion, we repeat the training 5 times to obtain the mean and variance of the absolute SEAT effect size. In Figure 2, we plot the bias degree of BERT post tasks with different training data proportions. In general, both Sent-Debias and FairFil achieve better performance and smaller variance when the proportion of training data is larger. Under a $20 \\%$ training proportion, our FairFil can better remove bias in text encoder, which shows FairFil has better data efficiency with the contrastive learning framework. ",
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+    {
+        "type": "text",
+        "text": "To further study output debiased sentence embedding, we visualize the relative distances of attributes and targets of SEAT before/after our debiasing process. We choose the target words as “he” and “she.” Attributes are selected from different social domains. We first contextualize the selected words into sentence templates as described in Section 5.1. We then average the original/debiased embeddings of these sentence template and plot the t-SNE (Maaten & Hinton, 2008) in Figure 3. From the t-SNE, the debiased encoder provides more balanced distances from gender targets “he/she” to the attribute concepts. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This paper has developed a novel debiasing method for large-scale pretrained text encoder neural networks. We proposed a fair filter (FairFil) network, which takes the original sentence embeddings as input and outputs the debiased sentence embeddings. To train the fair filter, we constructed a multi-view contrast learning framework, which maximizes the mutual information between each sentence and its augmentation. The augmented sentence is generated by replacing sensitive words in the original sentence with words in a similar semantic but different bias directions. Further, we designed a debiasing regularizer that minimizes the mutual information between the debiased embeddings and the corresponding sensitive words in sentences. Experimental results demonstrate the proposed FairFil not only reduces the bias in sentence embedding space, but also maintains the semantic meaning of the embeddings. This post hoc method does not require access to the training corpora, or any retraining process of the pretrained text encoder, which enhances its applicability. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
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+        "type": "text",
+        "text": "This research was supported in part by the DOE, NSF and ONR. ",
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+# Prototypical Cross-Attention Networks for Multiple Object Tracking and Segmentation
+
+Lei $\mathbf { K e } ^ { 1 , 2 }$ Xia Li1 Martin Danelljan1 Yu-Wing Tai3 Chi-Keung Tang2 Fisher ${ \bf { Y } } { \bf { u } } ^ { 1 }$ 1ETH Zürich 2HKUST 3Kuaishou Technology {lkeab,cktang}@cse.ust.hk, {xia.li,martin.danelljan}@vision.ee.ethz.ch yuwing@gmail.com, i@yf.io
+
+# Abstract
+
+Multiple object tracking and segmentation requires detecting, tracking, and segmenting objects belonging to a set of given classes. Most approaches only exploit the temporal dimension to address the association problem, while relying on single frame predictions for the segmentation mask itself. We propose Prototypical Cross-Attention Network (PCAN), capable of leveraging rich spatio-temporal information for online multiple object tracking and segmentation. PCAN first distills a space-time memory into a set of prototypes and then employs cross-attention to retrieve rich information from the past frames. To segment each object, PCAN adopts a prototypical appearance module to learn a set of contrastive foreground and background prototypes, which are then propagated over time. Extensive experiments demonstrate that PCAN outperforms current video instance tracking and segmentation competition winners on both Youtube-VIS and BDD100K datasets, and shows efficacy to both one-stage and two-stage segmentation frameworks. Code and video resources are available at http://vis.xyz/pub/pcan.
+
+# 1 Introduction
+
+Multiple object tracking and segmentation (MOTS), also known as Video Instance Segmentation (VIS), is an important problem with many real-world applications, including autonomous driving [10, 26] and video analysis [4, 46]. The task involves tracking and segmenting all objects within a video from a given set of semantic classes. We are witnessing rapidly growing research interest on MOTS thanks to the introduction of large scale benchmarks [46, 50, 37]. State-of-the-art methods [46, 5, 37, 29] for MOTS mainly follow the tracking-by-detection paradigm, where objects are first detected and segmented in individual frames and then associated over time.
+
+Although methods based on the popular tracking-by-detection philosophy have shown promising results, temporal modeling is limited to the object association phase [46, 5, 22] and only between two adjacent frames [37, 18]. On the other hand, the temporal dimension carries rich information about the scene. The information encoded in multiple temporal views of an object has the potential of improving the quality of predicted segmentation, localization, and categories. However, effectively and efficiently leveraging the rich temporal information remains a challenge. While sequential modeling has been applied for video processing [40, 41, 9, 28, 12], these methods generally operate directly on the high-resolution deep features, requiring large computational and memory consumption, which greatly limits their use.
+
+We propose a Prototypical Cross-Attention Module, termed PCAM, to leverage temporal information for multiple object tracking and segmentation. As illustrated in Figure 1, the module first distills spatiotemporal information into condensed prototypes using clustering based on Expectation Maximization. The resulting prototypes, composed of Gaussian Components, yield a rich and generalizable yet compact representation of the past visual features. Given a deep feature embedding of the current frame, PCAM then employs prototypical cross-attention to read relevant information from prior frames.
+
+![](images/f9dada9f50f662cbde9d5b8cc24676fef8cba7e325d838b30e8e3626365616f6.jpg)  
+Figure 1: We propose Prototypical Cross-Attention Network for MOTS, which first condenses the space-time memory and high-resolution frame embeddings into frame-level and instance-level prototypes. These are then employed to retrieve rich temporal information from past frames by our efficient prototypical cross-attention operation.
+
+Based on the noise-reduced clustered video features information, we further develop a Prototypical Cross-Attention Network (PCAN) for MOTS, that integrates the general PCAM at two stages in the network: on the frame-level and instance-level. The former reconstructs and aligns temporal past frame features with current frame, while the instance level integrates specific information about each object in the video. For robustness to object appearance change, PCAN represents each object instance by learning sets of contrastive foreground and background prototypes, which are propagated in an online manner. With a limited number of prototypes for each instance or frame, PCAN efficiently performs long-range feature aggregation and propagation in a video with linear complexity. Consequently, our PCAN outperforms standard non-local attention [40] and video transformer [41] on both the large-scale Youtube-VIS and BDD100K MOTS benchmarks.
+
+Our main contributions are summarized as follows: (i) We introduce the PCAN module for efficiently utilizing long-term spatio-temporal video information. (ii) We develop a MOTS approach that employs PCAN on frame and instance-level. (iii) We further represent the appearance of each video tracklet with contrastive foreground and background prototypes, which are propagated over time. (iv) We extensively analyze our approach. Our PCAN outperforms previous approaches on the challenging self-driving dataset BDD100K [50] and the semantically diverse YouTube-VIS dataset [46].
+
+# 2 Related work
+
+Video instance segmentation (VIS) Existing VIS methods [46, 2, 21] widely adapt the twostage paradigm of Mask R-CNN [11] and its variants [13, 15] by adding an additional tracking branch. Thus, their typical pipelines first detect regions of interest (RoIs) and then use the instance features after RoIAlign to regress object mask and associate cross-frame instances. More recent works [5, 18, 22, 48] employ a one-stage instance segmentation method, e.g. the anchor-free FCOS detector [34], which predicts a linear combination of mask bases [3] as its final segmentation. The aforementioned approaches make very limited use of temporal information to enhance the quality of the segmentation, instead relying on single image-based mask prediction, or only model short-term temporal correlation between two consecutive frames [18, 30]. In the context of long-term temporal association, the offline method VisTr [41] adapts vision transformer [6] for VIS, but suffers from a huge computational burden and memory consumption due to the dense pixel-level attention operations over long sequences. Compared to these methods, our PCAN temporally aggregates and propagates the prototypical features with both the long-term benefit and linear complexity.
+
+Multiple Object Tracking and Segmentation (MOTS) Similar to VIS, MOTS methods [37, 27, 29] mainly follow the tracking-by-detection paradigm. Objects are first detected and segmented, followed by association between frames. Track R-CNN [37] integrates temporal context feature from two neighboring frames using 3D convolutions. TrackFormer [25] performs joint object detection and tracking by recurrently using Transformers, while Stem-Seg [1] adopts a short 3D convolutional spatio-temporal volume to learn pixel embedding by treating segmentation as a bottom-up grouping. In contrast, our approach clusters appearance features in a long spatio-temporal volume with explicit foreground and background prototypes that are updates online. Besides, the mixture Gaussian components in instance appearance module equips PCAN a stronger modeling ability compared to instance-level average pooling [33, 49] or single Gaussian model [51, 14].
+
+Temporal attention models Video understanding usually requires long-range sequential modeling of relations between spatio-temporal locations. Recently, attention-based approaches, such as non-local attention [40, 39, 28, 12] and transformers [8, 35, 16], have been successfully adopted in video classification and action recognition. These tasks [23, 32, 43] involve dense pixel-level attention, leading to quadratic complexity in the sequence length, thus making them excessively expensive for long sequences. Improved temporal attention models mainly include double attention mechanism [7] on image recognition with global-local decomposition, and clustered attention Transformer [38] for language sequence modeling. Besides, recent prototypical methods [19, 45] use the EM algorithm for single-image semantic segmentation or few-shot learning [33]. Unlike these methods, our PCAN uses compact prototypical representation both for temporal feature aggregation and compact instance appearance feature propagation.
+
+# 3 Method
+
+We propose an approach for Multiple Object Tracking and Segmentation. Given a video sequence, the goal is to detect, track, and segment objects from a predefined set of object categories. Specifically, we consider the online setting, where the predictions only depend on current and past frames.
+
+# 3.1 Traditional Cross-Attention
+
+To utilize the rich temporal information to improve the segmentation prediction, recent approaches [28, 12] have employed cross-attention. We consider past spatio-temporal information encoded in a memory M, consisting of deep features of size $H \times W \times T \times C$ . The memory encapsulates valuable information about the past appearances and predictions of objects and background in a scene. To attend to the memory, the information is first separately embedded into key $\mathbf { k } ^ { M }$ and value $\mathbf { v } ^ { M }$ feature vectors. The keys are used to address relevant memories whose corresponding values are returned. The standard memory reading process is a non-local operation computed as the weighted sum,
+
+$$
+y _ { i } = \frac { 1 } { Z _ { i } } \sum _ { j = 1 } ^ { H \times W \times T } \exp ( \mathbf { k } _ { i } ^ { Q } \cdot \mathbf { k } _ { j } ^ { M } ) \mathbf { v } _ { j } ^ { M } ,
+$$
+
+where $\mathbf { k } ^ { Q }$ denotes query key map, which is predicted from the current frame. Further, $i$ and $j$ are the index of each query and the memory location, and $\begin{array} { r } { Z _ { i } = \sum _ { j } \exp ( \mathbf { k } _ { i } ^ { Q } \cdot \mathbf { k } _ { j } ^ { M } ) } \end{array}$ is the normalizing factor.
+
+Although proven effective, the standard attention operation (1) is known to suffer from poor computational and memory scaling properties [20]. In particular, since all queries are matched to all keys, it experiences a quadratic scaling $\mathcal { O } ( ( H W ) ^ { 2 } )$ of computations in the spatial size $H W$ of the feature map. This is particularly problematic for segmentation tasks, where fine-grained high-resolution information is desired to improve the quality of the predictions.
+
+# 3.2 Prototypical Cross-Attention
+
+To address the aforementioned limitations of the standard cross-attention, we introduce the prototypical cross-attention to first condense sets of high-resolution feature vectors in the past frames. Our approach is based on a clustered memory $\mathbf { M } _ { c }$ . We call these clusters prototypes, since they correspond to representative items in the memory. While clustering effectively reduces the number of items in the memory, it also serves to deprecate noisy information, leading to a more generalizable and robust representation of the memory.
+
+To employ an attention mechanism, similar to (1), we require a clustering of the memory that generates a principled continuous and differentiable clustering assignment function. We therefore cluster the keys in the memory by fitting a Gaussian Mixture Model (GMM),
+
+![](images/b16e350d7758f3867593885cc547299195adefa0c2ec495c97b222a2328de4e2.jpg)  
+Figure 2: Overview of our frame-level prototypical cross-attention. For a frame $\hat { t }$ in the memory we first perform GMM-based clustering to achieve the key $\mathbf { k } _ { \hat { t } j } ^ { \mu }$ and value $\mathbf { v } _ { \hat { t } j } ^ { \mu }$ prototypes. Given the key encoding $\mathbf { k } _ { t }$ of the current frame, we attend to the prototypes to generate the reconstructed feature $\mathbf { y } _ { \hat { t } }$ , which are then aggregated temporally and fused with the current value encoding $\mathbf { v } _ { t }$ .
+
+$$
+p ( \mathbf { k } ) = \frac { 1 } { N } \sum _ { j = 1 } ^ { N } p ( \mathbf { k } | z = j ) , \qquad p ( \mathbf { k } | z = j ) = \frac { 1 } { ( 2 \pi \sigma ^ { 2 } ) ^ { \frac { D } { 2 } } } \exp \left( - \frac { 1 } { 2 \sigma ^ { 2 } } \| \mathbf { k } - \mathbf { k } _ { j } ^ { \mu } \| ^ { 2 } \right)
+$$
+
+Here, $N$ denotes the number of Gaussian mixtures, $D$ is the feature dimension of the keys. We use a constant variance parameter $\sigma ^ { 2 }$ and uniform cluster priors $\begin{array} { r } { p ( z = j ) = \frac { 1 } { N } } \end{array}$ , where $z$ denotes the latent cluster assignment variable. The component means $\mathbf { k } ^ { \mu }$ represent the prototype keys in the memory. We generate the clustering (2) using the standard Expectation-Maximization algorithm.
+
+The GMM allows us to compute a soft cluster assignment by evaluating the posterior probability of the latent assignment variable $z$ . Using Bayes rule, the probability of a key value $\mathbf { k }$ to be assigned to the $j$ th prototype is derived as,
+
+$$
+p ( z = j | \mathbf { k } ) = \frac { p ( \mathbf { k } | z = j ) p ( z = j ) } { \sum _ { l = 1 } ^ { N } p ( \mathbf { k } | z = l ) p ( z = l ) } = \frac { \exp \left( - \frac { 1 } { 2 \sigma ^ { 2 } } \| \mathbf { k } - \mathbf { k } _ { j } ^ { \mu } \| ^ { 2 } \right) } { \sum _ { l = 1 } ^ { N } \exp \left( - \frac { 1 } { 2 \sigma ^ { 2 } } \| \mathbf { k } - \mathbf { k } _ { l } ^ { \mu } \| ^ { 2 } \right) } .
+$$
+
+The resulting cluster assignment can thus be written as a SoftMax operation, where the corresponding logits are provided by the negative cluster distance $\| \mathbf { k } - \mathbf { k } _ { j } ^ { \mu } \| ^ { 2 }$ scaled with a temperature of $2 \sigma ^ { 2 }$ .
+
+Since the clustering is performed in the key space of the memory, we next retrieve the corresponding value prototypes. To this end, we employ the key cluster assignment probabilities in (3) to compute the values for each memory prototype,
+
+$$
+\mathbf { v } _ { j } ^ { \mu } = \sum _ { l = 1 } ^ { H \times W } p ( z = j | \mathbf { k } _ { l } ^ { M } ) \mathbf { v } _ { l } ^ { M } .
+$$
+
+For attending to our clustered memory, we first predict the key encodings We then read from the clustered memory by computing the average over t $\mathbf { k } _ { i } ^ { Q }$ of the query imvalue prototypes $\mathbf { v } _ { j } ^ { \mu }$ weighted with the cluster assignment probabilities,
+
+$$
+\mathbf { y } _ { i } = \sum _ { j = 1 } ^ { N } p ( z = j | \mathbf { k } _ { i } ^ { Q } ) \mathbf { v } _ { j } ^ { \mu } = \frac { 1 } { Z _ { i } } \sum _ { j = 1 } ^ { N } \exp \left( - \frac { 1 } { 2 \sigma ^ { 2 } } \| \mathbf { k } _ { i } ^ { Q } - \mathbf { k } _ { j } ^ { \mu } \| ^ { 2 } \right) \mathbf { v } _ { j } ^ { \mu } .
+$$
+
+The final attention operation has much similarity with the original dot-product cross attention (1). Note that the key-query similarity in our approach is measured by Euclidian distance instead of a dot-product. Importantly, our formulation (5) attends to a reduced set of $N$ prototypes, while the original attention (1) requires attending to the full spatio-temporal memory of size $H \times W \times T$ .
+
+# 3.3 Prototypical Cross-Attention Network
+
+Here, we propose the Prototypical Cross-Attention Network (PCAN) for MOTS by integrating our prototypical cross-attention module into both the frame-level and instance-level. The former aims to align and aggregate temporal frame features stored in memory, while the latter is for propagating the instance appearance features over time and produce instance cross-attention maps to help segmentation. Besides, we also design a prototypical instance appearance module to represent each video tracklet with contrastive mixture foreground and background prototypes.
+
+# 3.3.1 Frame-level Prototypical Cross-Attention
+
+In Figure 2, prototypical cross-attention first produces prototypes by fitting a Gaussian mixtures model (2) to the feature in the memory. To provide further flexibility when dynamically updating the memory compute the $\mathbf { M }$ , we first perforkey prototypes wise clustering for each reference frame feature at , and retrieve the corresponding value embeddings $\hat { t }$ $N$ $\{ \mathbf { k } _ { \hat { t } i } ^ { \mu } \} _ { j = 1 } ^ { N }$ $\{ \mathbf { v } _ { \hat { t } j } ^ { \mu } \} _ { j = 1 } ^ { N }$ using (4) for each memory frame $\hat { t }$ independently. The key and value features are predicted using two parallel convolutional layers.
+
+Frame-wise prototypical memory attention Given the query key encoding $\mathbf { k } _ { t i } ^ { Q }$ of the current frame $t$ , we perform prototypical cross-attention to each memory frame $\hat { t }$ independently using our formulation (3) as,
+
+$$
+{ \bf y } _ { \hat { t } i } = \frac { 1 } { Z _ { \hat { t } \hat { t } } } \sum _ { j = 1 } ^ { N } \exp \left( - \frac { 1 } { 2 \sigma ^ { 2 } } \| { \bf k } _ { t i } - { \bf k } _ { \hat { t } j } ^ { \mu } \| ^ { 2 } \right) { \bf v } _ { \hat { t } j } ^ { \mu } , \qquad Z _ { \hat { t } i } = \sum _ { l = 1 } ^ { N } \exp \left( - \frac { 1 } { 2 \sigma ^ { 2 } } \| { \bf k } _ { t i } ^ { Q } - { \bf k } _ { \hat { t } \hat { t } } ^ { \mu } \| ^ { 2 } \right) .
+$$
+
+Note that the index $i$ refers to a spatial coordinate in the current frame. The resulting feature map $\mathbf { y } _ { \hat { t } }$ can intuitively be seen as a projection of features from frame $\hat { t }$ to the current frame. This projection essentially aligns the condensed feature information in frame $\hat { t }$ with the current frame.
+
+Temporal feature aggregation Since frame-wise attention does not fuse temporal information, we perform a temporal aggregation. The temporal information $\mathbf { y } _ { \hat { t } }$ in (6) from different frames $\hat { t }$ are fused as a linear combination, weighted by the feature similarity with the current frame. Specifically, the temporally aggregated representation is obtained as
+
+$$
+\bar { \mathbf { y } } _ { t i } = \sum _ { \hat { t } = 1 } ^ { t } w _ { \hat { t } i } \mathbf { y } _ { \hat { t } i } , \qquad w _ { \hat { t } i } = \frac { \exp ( \mathbf { y } _ { t i } \cdot \mathbf { y } _ { \hat { t } i } ) } { \sum _ { s = 1 } ^ { t } \exp ( \mathbf { y } _ { t i } \cdot \mathbf { y } _ { s i } ) } .
+$$
+
+Note that ${ \hat { t } } = t$ in the sum refers to the value embedding $\mathbf { y } _ { t i } = \mathbf { v } _ { t i } ^ { Q }$ extracted from the current frame. The contribution of each frame $\hat { t }$ is thus weighted by the similarity to this current frame prediction using the attention weights $w _ { \hat { t } i }$ . This strategy ensures that incorrect or dissimilar regions are suppressed when computing the final aggregated feature embedding $\bar { \mathbf { y } } _ { t }$ . To handle object with large-scale variation and produce more fine-grained instance mask prediction, we further extend temporal aggregation to multi-level using different levels of the extracted FPN features, as detailed in the supplementary material.
+
+# 3.3.2 Instance-level Prototypical Cross-Attention
+
+Contrastive foreground and background representation In additional to the condensed frame-level representation, for more accurate segmentation results, we further encode each tracked object with compact and robust appearance prototypes. To further empower our proposed attention mechanism, we utilize the initially detected object mask to identify each foreground instance. We then separately model the extracted foreground and background features using a GMM (2). We denote the resulting foreground prototypes as $\mathbf { k } _ { t j . } ^ { + }$ and background prototypes as $\mathbf { k } _ { t j } ^ { - }$ . The former thus focuses on the appearance of the specific object, creating a rich and dynamic appearance model. When employed in our prototypical cross-attention framework (Section 3.2), it provides fine-grained attention from localized prototypes that naturally learn to focus specific parts of views of the object, as visualized in Fig. 3. Furthermore, the background prototypes $\mathbf { k } _ { t j } ^ { - }$ capture valuable information about the background appearance, which can greatly alleviate the segmentation process. For each object instance we attend to the foreground and background prototypes separately using (3). The results are concatenated together with the initial mask detection to the Temporal Segmentation Head (TSM) for final prediction, as illustrated in Figure 3.
+
+![](images/0d0831e8e286b63817527cb523139e218cd86186d20191f7a86218e7cec18a22.jpg)  
+Figure 3: Our instance-level prototypical attention with foreground and background prototypes and temporal propagation. The foreground/background attention maps from (bottom) demonstrate the localized and discriminative appearance representation. Temporal Segmentation Module (TSM) takes the current frame, initial mask, and instance attention maps as input and generates the final mask.
+
+Tracklet feature propagation and updating To effectively model the object appearance change and preserve the most relevant information, we design a recurrent instance appearance updating scheme. From the first video frame where object appears, the accumulated prototypes $\bar { \mathbf { k } } _ { t j } ^ { + }$ , $\bar { \mathbf { k } } _ { t j } ^ { - }$ for the instance are propagated to the subsequent frames and updated with new appearance prototypes $\mathbf { k } _ { t j } ^ { + }$ , $\mathbf { k } _ { t j } ^ { - }$ using an update rate $\lambda$ as,
+
+$$
+\bar { \mathbf { k } } _ { t j } ^ { + } = ( 1 - \lambda ) \bar { \mathbf { k } } _ { t - 1 , j } ^ { + } + \lambda \mathbf { k } _ { t j } ^ { + } , \qquad \bar { \mathbf { k } } _ { t j } ^ { - } = ( 1 - \lambda ) \bar { \mathbf { k } } _ { t - 1 , j } ^ { - } + \lambda \mathbf { k } _ { t j } ^ { - } .
+$$
+
+Figure 3 also reveals the consistency of the attended region of a specific prototype $j$
+
+# 4 Experiments
+
+Here, we present comprehensive evaluation and analysis of our approach. Experiments are performed on two large scale datasets, namely YouTube-VIS [46] and BDD100K [50].
+
+# 4.1 Experiment setup
+
+Youtube-VIS YouTube-VIS-2019 [46] dataset contains 2,883 high quality videos with 131k annotated object instances belonging to 40 diverse categories. The task is to simultaneously classifying, segment and track object instances belonging to these categories. The evaluation metrics for this task are an adaptation of the Average Precision (AP) and Average Recall (AR) of image instance segmentation.
+
+BDD100K We also evaluate on the large-scale tracking and segmentation dataset of BDD100K [50], which is a challenging self-driving dataset with 154 videos (30,817 images) for training, 32 videos (6,475 images) for validation, and 37 videos (7,484 images) for testing. The dataset provides 8 annotated categories for evaluation, where the images in the tracking set are annotated per 5 FPS with 30 FPS frame rate. We adopt the well-established MOTS metrics [37] to our task.
+
+Implementation details We implement PCAN based on two different existing MOTS approaches. For Youtube-VIS, we adopt ResNet with FPN pre-trained on COCO as the backbone, and build our segmentation tracker on the one-stage segmentation model [5]. Both the instance and frame cross-attention is built on the extracted FPN features. Our model is trained with initial learning rate 0.0025 on 4 GPUs using SGD, and executes with a speed of 15.0 FPS on ResNet-50. Similar to [46, 22, 18], we use the input size $3 6 0 \times 6 4 0$ for training. On BDD100K, we build PCAN by extending the two-stage MOT method [29] with our temporal segmentation modules. We follow the same training strategy of QDTrack-mots [29]. More details can be found in supplemental material.
+
+Table 1: Comparison with state-of-the-art on the YouTube-VIS validation set. Results are reported in terms of mask accuracy (AP) and recall (AR). Asterisks ∗ denote concurrent works on arXiv.   
+
+<table><tr><td>Method</td><td>Backbone</td><td>Type</td><td>Online</td><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td>VisTr*[41]</td><td>ResNet-50</td><td>Transformer</td><td>×</td><td>35.6</td><td>56.8</td><td>37.0</td><td>35.2</td><td>40.2</td></tr><tr><td>OSMN [47]</td><td>ResNet-50</td><td>Two-stage</td><td>√</td><td>23.4</td><td>36.5</td><td>25.7</td><td>28.9</td><td>31.1</td></tr><tr><td>FEELVOS [36]</td><td>ResNet-50</td><td>Two-stage</td><td>√</td><td>26.9</td><td>42.0</td><td>29.7</td><td>29.9</td><td>33.4</td></tr><tr><td>DeepSORT[42]</td><td>ResNet-50</td><td>Two-stage</td><td>√</td><td>26.1</td><td>42.9</td><td>26.1</td><td>27.8</td><td>31.3</td></tr><tr><td>MaskTrack R-CNN [46]</td><td>ResNet-50</td><td>Two-stage</td><td>√</td><td>30.3</td><td>51.1</td><td>32.6</td><td>31.0</td><td>35.5</td></tr><tr><td>STEm-Seg[1]</td><td>ResNet-50</td><td> One-stage</td><td></td><td>30.6</td><td>50.7</td><td>33.5</td><td>31.6</td><td>37.1</td></tr><tr><td>SipMask [5]</td><td>ResNet-50</td><td>One-stage</td><td></td><td>32.5</td><td>53.0</td><td>33.3</td><td>33.5</td><td>38.9</td></tr><tr><td>STMask*[18]</td><td>ResNet-50</td><td>One-stage</td><td>x&lt;&gt;</td><td>33.5</td><td>52.1</td><td>36.9</td><td>31.1</td><td>39.2</td></tr><tr><td>SG-Net* [22]</td><td>ResNet-50</td><td>One-stage</td><td>√</td><td>34.8</td><td>56.1</td><td>36.8</td><td>35.8</td><td>40.8</td></tr><tr><td>PCAN (Ours)</td><td>ResNet-50</td><td>One-stage</td><td>√</td><td>36.1</td><td>54.9</td><td>39.4</td><td>36.3</td><td>41.6</td></tr><tr><td>STMask*[18]</td><td>ResNet-101</td><td>One-stage</td><td>√</td><td>36.3</td><td>55.2</td><td>39.9</td><td>33.7</td><td>42.0</td></tr><tr><td>SG-Net* [22]</td><td>ResNet-101</td><td>One-stage</td><td>√</td><td>36.3</td><td>57.1</td><td>39.6</td><td>35.9</td><td>43.0</td></tr><tr><td>PCAN (Ours)</td><td>ResNet-101</td><td>One-stage</td><td>√</td><td>37.6</td><td>57.2</td><td>41.3</td><td>37.2</td><td>43.9</td></tr></table>
+
+Table 2: State-of-the-art comparison on the BDD100K segmentation tracking validation set. I: ImageNet. C: COCO. S: Cityscapes. B: BDD100K. "-fix" means adopting the pretrained model from the BDD100K tracking set, fixing the existing parts, and only training the added mask head.   
+
+<table><tr><td>Method</td><td>Pretrained</td><td>Online</td><td>mMOTSA↑</td><td>mMOTSP↑</td><td>mIDF个</td><td>ID sw.↓</td><td>mAP↑</td></tr><tr><td>SortIoU</td><td>I, C, S</td><td>√</td><td>10.3</td><td>59.9</td><td>21.8</td><td>15951</td><td>22.2</td></tr><tr><td>MaskTrackRCNN [36]</td><td>I, C, S</td><td>√</td><td>12.3</td><td>59.9</td><td>26.2</td><td>9116</td><td>22.0</td></tr><tr><td>STEm-Seg [1]</td><td>1,C, s</td><td>×</td><td>12.2</td><td>58.2</td><td>25.4</td><td>8732</td><td>21.8</td></tr><tr><td>QDTrack-mots [29]</td><td>1, C,S</td><td>√</td><td>22.5</td><td>59.6</td><td>40.8</td><td>1340</td><td>22.4</td></tr><tr><td>QDTrack-mots-fix [29]</td><td>I, B</td><td>√</td><td>23.5</td><td>66.3</td><td>44.5</td><td>973</td><td>25.5</td></tr><tr><td>PCAN (Ours)</td><td>I,B</td><td>√</td><td>27.4</td><td>66.7</td><td>45.1</td><td>876</td><td>26.6</td></tr></table>
+
+# 4.2 State-of-the-Art Comparison
+
+We compare our approach with the state-of-the-art methods on the aforementioned large-scale MOTS/VIS benchmarks Youtube-VIS and BDD100K, where PCAN outperforms all existing methods without bells and whistles, and shows efficacy to both one-stage and two-stage segmentation frameworks. We follow the official metrics of each benchmark to evaluate our model.
+
+Youtube-VIS The results of Youtube-VIS benchmark is in Table 1, where PCAN achieves the best mask AP of $3 6 . 1 \%$ using ResNet-50 and $3 7 . 6 \%$ using ResNet-101 respectively, while being an online method. Our approach consistently surpasses most recent SOTA methods, including STMask [18] and SG-Net [22] by a significant margin. These methods only conduct temporal modeling between two adjacent frames for feature correlation. Compared to our baseline SipMask [5], a single-image based segmentation with object centerness association, PCAN improves the mask AP from $3 2 . 5 \%$ to $3 6 . 1 \%$ , which shows the effectiveness of long-term temporal modeling in helping object tracking and segmentation.
+
+BDD100K Table 2 shows our results on BDD100K tracking and segmentation benchmark, where PCAN outperforms the strong baseline methods MaskTrackRCNN [46] and QDTrack-mots [29]. Our approach achieves a large advantage in mMOTSA, with over 3 points gain and around $10 \%$ ID switches decrease. MOTSA measures segmentation as well as tracking quality, while ID Switches can measure the performance of identity consistency. The significant advancements demonstrate that our method with prototypical cross-attention enables more accurate pixel-wise object tracking by effectively exploiting temporal information.
+
+# 4.3 Ablation study and analysis
+
+We conduct detailed ablation studies on Youtube-VIS validation set, where we investigate the effect of our proposed prototypical cross-attention components for MOTS during training and testing.
+
+Effect of frame-level prototypical cross-attention module To study the importance of temporal information amount, we conduct an ablation study on models with different input temporal window lengths in Table 3. A temporal length of 1 thus means that no prior temporal information guidance is used during video instance segmentation. By varying the frame length from 1 to 32, the mask AP increases from $3 2 . 5 \%$ to $3 5 . 4 \%$ , which reveals that richer temporal information with multiple views of a segmented object indeed brings more gain to model performance. For the number of frame-level prototypes, we used 64 during training and testing. The results on YouTube-VIS in Table 8 show that the precision saturates for larger numbers of prototypes.
+
+Table 3: Results of varying temporal memory length in our PCAN on YouTube-VIS.   
+
+<table><tr><td>Length</td><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td>1</td><td>32.5</td><td>53.0</td><td>33.3</td><td>33.5</td><td>38.9</td></tr><tr><td>2</td><td>33.7</td><td>53.8</td><td>35.3</td><td>33.9</td><td>39.5</td></tr><tr><td>4</td><td>33.9</td><td>54.0</td><td>36.8</td><td>34.1</td><td>40.0</td></tr><tr><td>8</td><td>34.2</td><td>53.7</td><td>37.6</td><td>34.4</td><td>40.3</td></tr><tr><td>16</td><td>34.6</td><td>53.7</td><td>38.3</td><td>35.4</td><td>40.5</td></tr><tr><td>32</td><td>35.4</td><td>53.8</td><td>39.1</td><td>35.9</td><td>41.0</td></tr></table>
+
+Table 4: Effect of multi-layer prototypical feature fusion with tube length 4 on YouTube-VIS.   
+
+<table><tr><td>FPN Layer</td><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td>P3</td><td>30.8</td><td>51.7</td><td>32.0</td><td>32.6</td><td>37.0</td></tr><tr><td>P4</td><td>32.0</td><td>51.5</td><td>34.1</td><td>32.6</td><td>37.2</td></tr><tr><td>P5</td><td>32.9</td><td>52.1</td><td>35.9</td><td>33.2</td><td>38.6</td></tr><tr><td>P3-P4</td><td>33.1</td><td>52.3</td><td>35.6</td><td>33.6</td><td>38.5</td></tr><tr><td>P3-P5</td><td>33.9</td><td>54.0</td><td>36.8</td><td>34.1</td><td>40.0</td></tr></table>
+
+Table 5: Comparison with non-local attention [39] and transformer [6, 41] on YouTube-VIS.   
+
+<table><tr><td rowspan="2">Length</td><td colspan="3">Prototypical Cross-Attention</td><td colspan="3">Non-local Attention</td><td colspan="3">Transformer (Multi-Head Self-Attention)</td></tr><tr><td>AP</td><td>FLOPs(B)</td><td>Memory(M)</td><td>AP</td><td>FLOPs(B)</td><td>Memory(M)</td><td>AP</td><td>FLOPs(B)</td><td>Memory(M)</td></tr><tr><td>2</td><td>33.7</td><td>5.8</td><td>323</td><td>33.2</td><td>24.3</td><td>2497</td><td>24.6</td><td>103.8</td><td>5321</td></tr><tr><td>4</td><td>33.9</td><td>12.0</td><td>652</td><td>33.3</td><td>49.1</td><td>4763</td><td>25.8</td><td>387.2</td><td>9844</td></tr><tr><td>8</td><td>34.2</td><td>23.7</td><td>1419</td><td>33.6</td><td>99.6</td><td>9631</td><td>28.3</td><td>1413.3</td><td>18762</td></tr></table>
+
+![](images/c76ee7c6980393304488d4699b6a5c3971b1bb8d922650744576ad064f7e6827.jpg)  
+Figure 4: Qualitative impact of our PCAM on YouTube-VIS. Mask colors encode object identity. Our frame-level PCAM (second row) helps provide consistent detections and preserve identities compared to the baseline (first row). The instance-level PCAM (fourth row) provides more accurate masks, while further improving identity consistency compared to not employing our module (third row).
+
+Effect of multi-layer temporal aggregation Since we perform temporal feature aggregation on the extracted FPN features, to help deal with objects with partial occlusion and large-scale variation, we also study the effect of using different levels of the extracted FPN features. In Table 4, we select the FPN feature map from P3-P5 layers for (excluding P6 and P7 due to impractical computation cost), and perform prototypical temporal aggregation on each FPN layer. We find that multi-layer information is also important to final model performance.
+
+Computation and memory efficiency In Table 5 we analyze different attention mechanisms. Compared to standard space-time memory reading using non-local attention [39, 28] or recent popular transformer [41, 6] with multi-head self-attention layer, the prototypical cross-attention with condensed prototypes not only enjoys high accuracy advantage, but also largely reduces the memory consumption and computation amount. For input tube length 8, the prototypical memory consumption is less than $10 \%$ of the transformer with negligible FLOPs computation due to the small number of representative prototypes in (5).
+
+Effect of instance-level prototypical appearance module We analyze the instance-level prototypical cross-attention module, which represents each video tracklet using the contrastive prototypes. In
+
+Table 6: Ablation study on number of instancelevel prototypes on YouTube-VIS.   
+
+<table><tr><td rowspan=1 colspan=1>Pos.Proto.Number丨Neg.Proto.Number</td><td rowspan=1 colspan=1>Pos.Proto.Number丨Neg.Proto.Number</td><td rowspan=1 colspan=1>AP AP50</td></tr><tr><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>32.5 53.0</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>32.4 52.3</td></tr><tr><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>32.1 52.4</td></tr><tr><td rowspan=2 colspan=1>15</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>32.7 52.8</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>33.1 53.6</td></tr><tr><td rowspan=1 colspan=1>30</td><td rowspan=1 colspan=1>30</td><td rowspan=1 colspan=1>33.9 54.1</td></tr><tr><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>33.6 53.8</td></tr></table>
+
+Table 7: Ablation on instance-level EM feature propagation and updating on YouTube-VIS.   
+
+<table><tr><td>version</td><td>AP</td><td>AP50</td></tr><tr><td>No instance prototype propagation</td><td>33.5</td><td>53.2</td></tr><tr><td>Using initial instance prototype</td><td>33.0</td><td>52.8</td></tr><tr><td>Update momentum = 0.2</td><td>34.3</td><td>53.8</td></tr><tr><td>Update momentum = 0.5</td><td>34.0</td><td>53.6</td></tr></table>
+
+Table 8: Ablation on number of framelevel prototypes on YouTube-VIS.   
+
+<table><tr><td>Proto.Number</td><td>AP</td><td>AP50</td></tr><tr><td>8</td><td>32.6</td><td>52.8</td></tr><tr><td>16</td><td>33.1</td><td>53.3</td></tr><tr><td>32</td><td>33.9</td><td>53.5</td></tr><tr><td>64</td><td>34.2</td><td>53.7</td></tr><tr><td>128</td><td>34.1</td><td>53.8</td></tr></table>
+
+Table 9: Results of varying EM iterations for our PCAN on YouTube-VIS.   
+
+<table><tr><td>Iteration number</td><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td>1</td><td>33.3</td><td>53.4</td><td>35.8</td><td>33.2</td><td>38.8</td></tr><tr><td>2</td><td>33.7</td><td>53.9</td><td>36.4</td><td>33.6</td><td>39.3</td></tr><tr><td>4</td><td>33.7</td><td>54.1</td><td>36.5</td><td>33.9</td><td>39.5</td></tr><tr><td>6</td><td>33.9</td><td>54.0</td><td>36.8</td><td>34.1</td><td>40.0</td></tr><tr><td>8</td><td>33.6</td><td>53.6</td><td>36.1</td><td>33.7</td><td>39.3</td></tr></table>
+
+Table 6, we study the influence of instance prototype number and the effect of foreground-background contrasting. Using both positive and negative prototypes improves AP from $3 2 . 5 \%$ to $3 3 . 9 \%$ . Compared to the single prototype representation, the GMM demonstrate a stronger appearance modeling ability. We further find that the performance saturates when the number is larger than 60. In the Figure 6 and supplementary file, we provide additional instance cross-attention maps visualization to highlight the various attended regions.
+
+In Table 7, we investigate the effectiveness of instance prototype (including the both positive and negative ones) propagation in an online manner, and compared it with using the instance prototype in the initial frame or current frame. We find that updating object prototypes recurrently with a momentum of 0.2 improves video segmentation AP of $1 . 3 \%$ .
+
+Influence of EM iteration number We study the influence of EM iteration number $T$ during condensing prototypes and the results are shown in Table 9. Using temporal memory length 4, we find that the accuracy gains of PCAN increase with more iterations from 1 to 6, and the improvement starts to saturate when $T \geqslant 6$ . We use the same iteration number during training and test.
+
+Ablation study on KITTI-MOTS We also train PCAN on the KITTI-MOTS [37] training set and conduct ablations on the instance and frame PCAMs. In Table 10, PCAN with window size 8 on val set also shows significant improvements compared to the TrackR-CNN [37] (a two-stage tracker based on Mask R-CNN) on the benchmark. Note that many published methods on KITTI-MOTS, such as Vip-DeepLab [31], EagerMOT [17] and MOTSFusion [24], use 3D bounding boxes, LIDAR point clouds, or optical flow (PointTrack [44]). In contrast, our method only relies on RGB images.
+
+Qualitative analysis In Figure 4, we showcase qualitative ablation results of PCAN on Youtube-VIS. Compared to the baseline, we see that our model results in more consistent segmentation and better tracking using prototypical cross-attention module. We also provide visual results on BDD100K in Figure 5, where PCAN produces robust tracking and segmentation results even under large object appearance change (first row) or low illumination (second row). In the 3rd row, PCAN has limitations in handling missing detections (the person in the first frame) with limited appearance information under extreme lighting, and produce tracking errors in the second frame when visible parts of the same car is totally different across frame and with low appearance similarity.
+
+Cross-Attention Visualization In Figure 6, we visualize instance-level prototypical cross-attention of the interested car for both the corresponding foreground and background regions on three continuous frames on BDD100K, where the attended region of each object prototype reveals the implicit unsupervised temporal consistency. More visualization cases on instance and frame cross-attention maps and relevant analysis are in the supplementary file.
+
+Societal impact PCAN has high potential impact in important applications, such as transportation, sports analysis, and self-driving vehicles. However, this powerful technology can be deployed in human monitoring and surveillance as well which raise ethical and privacy issues. Potential negative impact can be avoided by enforcing a strict and secure data privacy regulation such as the GDPR,
+
+Table 10: Ablation study of PCAN on KITTI-MOTS [37] validation set.   
+
+<table><tr><td>Method</td><td>|Car-MOTSA</td><td>Ped-MOTSA</td><td>Car-MOTSP</td><td>Ped-MOTSP</td></tr><tr><td>TrackR-CNN [37]</td><td>87.8</td><td>65.1</td><td>87.2</td><td>75.7</td></tr><tr><td rowspan="3">PCAN w/o frame PCAM PCAN w/o instance PCAM</td><td>87.3</td><td>65.3</td><td>86.9</td><td>75.0</td></tr><tr><td>87.8</td><td>65.8</td><td>87.1</td><td>75.5</td></tr><tr><td>89.6</td><td>66.4</td><td>88.3</td><td>76.1</td></tr></table>
+
+![](images/43f2b11ee05e974f625ef7fa9c6ca494b65750d3481407d6beccb7f126aa5d06.jpg)  
+Figure 5: Qualitative results of our method on BDD100K. PCAN produces robust tracking and segmentation results under large motion and appearance changes (1st row) and heavy traffic in low-light conditions (2nd row). In the 3rd row, PCAN misses a detection (the person to the left in 1st frame), and produces tracking errors (2nd frame) when it covers totally different regions of the car with low appearance similarity. Zoom for better view. Video results are in the suppl. file.
+
+![](images/4685f107e82d0332bd9ca6ab4b5b4295cba1dbb068fc8f484d749cefb0c32c5f.jpg)  
+Figure 6: Instance cross-attention maps visualization for the car specified by the red dotted bounding box on BDD100K. We select the first four foreground/background prototypes as example, where each one focuses on specific car sub-regions with implicit unsupervised temporal consistency over time. proper technology management education, and having an open dialogue among various stakeholders on how such technology should be deployed and regulated.
+
+# 5 Conclusion
+
+We present PCAN, a new online method for MOTS. PCAN first distills the space-time memory into a set of frame-level and instance-level prototypes, followed by cross-attention to retrieve rich information from the past frames. In contrast to most previous MOTS methods with limited temporal consideration, PCAN efficiently performs long-term temporal propagation and aggregation, and achieves large performance gain on the two largest MOTS benchmarks with low computation and memory cost. We validate the efficacy of PCAN on both the existing one-stage and two-stage trackers. We believe PCAN will significantly benefit more video understanding tasks in the future.
+
+# Acknowledgments and Disclosure of Funding
+
+This research is supported in part by the Research Grant Council of the Hong Kong SAR under grant no. 16201818 and Kuaishou Technology.
+
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parse/train/OkFPq7ZtsQ/OkFPq7ZtsQ_content_list.json

@@ -0,0 +1,1209 @@
+[
+    {
+        "type": "text",
+        "text": "Prototypical Cross-Attention Networks for Multiple Object Tracking and Segmentation ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Lei $\\mathbf { K e } ^ { 1 , 2 }$ Xia Li1 Martin Danelljan1 Yu-Wing Tai3 Chi-Keung Tang2 Fisher ${ \\bf { Y } } { \\bf { u } } ^ { 1 }$ 1ETH Zürich 2HKUST 3Kuaishou Technology {lkeab,cktang}@cse.ust.hk, {xia.li,martin.danelljan}@vision.ee.ethz.ch yuwing@gmail.com, i@yf.io ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            318,
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+            334
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Multiple object tracking and segmentation requires detecting, tracking, and segmenting objects belonging to a set of given classes. Most approaches only exploit the temporal dimension to address the association problem, while relying on single frame predictions for the segmentation mask itself. We propose Prototypical Cross-Attention Network (PCAN), capable of leveraging rich spatio-temporal information for online multiple object tracking and segmentation. PCAN first distills a space-time memory into a set of prototypes and then employs cross-attention to retrieve rich information from the past frames. To segment each object, PCAN adopts a prototypical appearance module to learn a set of contrastive foreground and background prototypes, which are then propagated over time. Extensive experiments demonstrate that PCAN outperforms current video instance tracking and segmentation competition winners on both Youtube-VIS and BDD100K datasets, and shows efficacy to both one-stage and two-stage segmentation frameworks. Code and video resources are available at http://vis.xyz/pub/pcan. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Multiple object tracking and segmentation (MOTS), also known as Video Instance Segmentation (VIS), is an important problem with many real-world applications, including autonomous driving [10, 26] and video analysis [4, 46]. The task involves tracking and segmenting all objects within a video from a given set of semantic classes. We are witnessing rapidly growing research interest on MOTS thanks to the introduction of large scale benchmarks [46, 50, 37]. State-of-the-art methods [46, 5, 37, 29] for MOTS mainly follow the tracking-by-detection paradigm, where objects are first detected and segmented in individual frames and then associated over time. ",
+        "bbox": [
+            174,
+            598,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Although methods based on the popular tracking-by-detection philosophy have shown promising results, temporal modeling is limited to the object association phase [46, 5, 22] and only between two adjacent frames [37, 18]. On the other hand, the temporal dimension carries rich information about the scene. The information encoded in multiple temporal views of an object has the potential of improving the quality of predicted segmentation, localization, and categories. However, effectively and efficiently leveraging the rich temporal information remains a challenge. While sequential modeling has been applied for video processing [40, 41, 9, 28, 12], these methods generally operate directly on the high-resolution deep features, requiring large computational and memory consumption, which greatly limits their use. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a Prototypical Cross-Attention Module, termed PCAM, to leverage temporal information for multiple object tracking and segmentation. As illustrated in Figure 1, the module first distills spatiotemporal information into condensed prototypes using clustering based on Expectation Maximization. The resulting prototypes, composed of Gaussian Components, yield a rich and generalizable yet compact representation of the past visual features. Given a deep feature embedding of the current frame, PCAM then employs prototypical cross-attention to read relevant information from prior frames. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/f9dada9f50f662cbde9d5b8cc24676fef8cba7e325d838b30e8e3626365616f6.jpg",
+        "image_caption": [
+            "Figure 1: We propose Prototypical Cross-Attention Network for MOTS, which first condenses the space-time memory and high-resolution frame embeddings into frame-level and instance-level prototypes. These are then employed to retrieve rich temporal information from past frames by our efficient prototypical cross-attention operation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Based on the noise-reduced clustered video features information, we further develop a Prototypical Cross-Attention Network (PCAN) for MOTS, that integrates the general PCAM at two stages in the network: on the frame-level and instance-level. The former reconstructs and aligns temporal past frame features with current frame, while the instance level integrates specific information about each object in the video. For robustness to object appearance change, PCAN represents each object instance by learning sets of contrastive foreground and background prototypes, which are propagated in an online manner. With a limited number of prototypes for each instance or frame, PCAN efficiently performs long-range feature aggregation and propagation in a video with linear complexity. Consequently, our PCAN outperforms standard non-local attention [40] and video transformer [41] on both the large-scale Youtube-VIS and BDD100K MOTS benchmarks. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our main contributions are summarized as follows: (i) We introduce the PCAN module for efficiently utilizing long-term spatio-temporal video information. (ii) We develop a MOTS approach that employs PCAN on frame and instance-level. (iii) We further represent the appearance of each video tracklet with contrastive foreground and background prototypes, which are propagated over time. (iv) We extensively analyze our approach. Our PCAN outperforms previous approaches on the challenging self-driving dataset BDD100K [50] and the semantically diverse YouTube-VIS dataset [46]. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            665,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Video instance segmentation (VIS) Existing VIS methods [46, 2, 21] widely adapt the twostage paradigm of Mask R-CNN [11] and its variants [13, 15] by adding an additional tracking branch. Thus, their typical pipelines first detect regions of interest (RoIs) and then use the instance features after RoIAlign to regress object mask and associate cross-frame instances. More recent works [5, 18, 22, 48] employ a one-stage instance segmentation method, e.g. the anchor-free FCOS detector [34], which predicts a linear combination of mask bases [3] as its final segmentation. The aforementioned approaches make very limited use of temporal information to enhance the quality of the segmentation, instead relying on single image-based mask prediction, or only model short-term temporal correlation between two consecutive frames [18, 30]. In the context of long-term temporal association, the offline method VisTr [41] adapts vision transformer [6] for VIS, but suffers from a huge computational burden and memory consumption due to the dense pixel-level attention operations over long sequences. Compared to these methods, our PCAN temporally aggregates and propagates the prototypical features with both the long-term benefit and linear complexity. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Multiple Object Tracking and Segmentation (MOTS) Similar to VIS, MOTS methods [37, 27, 29] mainly follow the tracking-by-detection paradigm. Objects are first detected and segmented, followed by association between frames. Track R-CNN [37] integrates temporal context feature from two neighboring frames using 3D convolutions. TrackFormer [25] performs joint object detection and tracking by recurrently using Transformers, while Stem-Seg [1] adopts a short 3D convolutional spatio-temporal volume to learn pixel embedding by treating segmentation as a bottom-up grouping. In contrast, our approach clusters appearance features in a long spatio-temporal volume with explicit foreground and background prototypes that are updates online. Besides, the mixture Gaussian components in instance appearance module equips PCAN a stronger modeling ability compared to instance-level average pooling [33, 49] or single Gaussian model [51, 14]. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Temporal attention models Video understanding usually requires long-range sequential modeling of relations between spatio-temporal locations. Recently, attention-based approaches, such as non-local attention [40, 39, 28, 12] and transformers [8, 35, 16], have been successfully adopted in video classification and action recognition. These tasks [23, 32, 43] involve dense pixel-level attention, leading to quadratic complexity in the sequence length, thus making them excessively expensive for long sequences. Improved temporal attention models mainly include double attention mechanism [7] on image recognition with global-local decomposition, and clustered attention Transformer [38] for language sequence modeling. Besides, recent prototypical methods [19, 45] use the EM algorithm for single-image semantic segmentation or few-shot learning [33]. Unlike these methods, our PCAN uses compact prototypical representation both for temporal feature aggregation and compact instance appearance feature propagation. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Method ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose an approach for Multiple Object Tracking and Segmentation. Given a video sequence, the goal is to detect, track, and segment objects from a predefined set of object categories. Specifically, we consider the online setting, where the predictions only depend on current and past frames. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Traditional Cross-Attention ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To utilize the rich temporal information to improve the segmentation prediction, recent approaches [28, 12] have employed cross-attention. We consider past spatio-temporal information encoded in a memory M, consisting of deep features of size $H \\times W \\times T \\times C$ . The memory encapsulates valuable information about the past appearances and predictions of objects and background in a scene. To attend to the memory, the information is first separately embedded into key $\\mathbf { k } ^ { M }$ and value $\\mathbf { v } ^ { M }$ feature vectors. The keys are used to address relevant memories whose corresponding values are returned. The standard memory reading process is a non-local operation computed as the weighted sum, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fb353697a8e0eab2e7444438043fd302d2da09a647e6bbe216e08adc1b8988a8.jpg",
+        "text": "$$\ny _ { i } = \\frac { 1 } { Z _ { i } } \\sum _ { j = 1 } ^ { H \\times W \\times T } \\exp ( \\mathbf { k } _ { i } ^ { Q } \\cdot \\mathbf { k } _ { j } ^ { M } ) \\mathbf { v } _ { j } ^ { M } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            594,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { k } ^ { Q }$ denotes query key map, which is predicted from the current frame. Further, $i$ and $j$ are the index of each query and the memory location, and $\\begin{array} { r } { Z _ { i } = \\sum _ { j } \\exp ( \\mathbf { k } _ { i } ^ { Q } \\cdot \\mathbf { k } _ { j } ^ { M } ) } \\end{array}$ is the normalizing factor. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Although proven effective, the standard attention operation (1) is known to suffer from poor computational and memory scaling properties [20]. In particular, since all queries are matched to all keys, it experiences a quadratic scaling $\\mathcal { O } ( ( H W ) ^ { 2 } )$ of computations in the spatial size $H W$ of the feature map. This is particularly problematic for segmentation tasks, where fine-grained high-resolution information is desired to improve the quality of the predictions. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 Prototypical Cross-Attention ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            767,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To address the aforementioned limitations of the standard cross-attention, we introduce the prototypical cross-attention to first condense sets of high-resolution feature vectors in the past frames. Our approach is based on a clustered memory $\\mathbf { M } _ { c }$ . We call these clusters prototypes, since they correspond to representative items in the memory. While clustering effectively reduces the number of items in the memory, it also serves to deprecate noisy information, leading to a more generalizable and robust representation of the memory. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To employ an attention mechanism, similar to (1), we require a clustering of the memory that generates a principled continuous and differentiable clustering assignment function. We therefore cluster the keys in the memory by fitting a Gaussian Mixture Model (GMM), ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/b16e350d7758f3867593885cc547299195adefa0c2ec495c97b222a2328de4e2.jpg",
+        "image_caption": [
+            "Figure 2: Overview of our frame-level prototypical cross-attention. For a frame $\\hat { t }$ in the memory we first perform GMM-based clustering to achieve the key $\\mathbf { k } _ { \\hat { t } j } ^ { \\mu }$ and value $\\mathbf { v } _ { \\hat { t } j } ^ { \\mu }$ prototypes. Given the key encoding $\\mathbf { k } _ { t }$ of the current frame, we attend to the prototypes to generate the reconstructed feature $\\mathbf { y } _ { \\hat { t } }$ , which are then aggregated temporally and fused with the current value encoding $\\mathbf { v } _ { t }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            90,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            406,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/17661c3cb86cc2fb9007908dc47fec9a18fa15456690cfeccf122038c357c9f9.jpg",
+        "text": "$$\np ( \\mathbf { k } ) = \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } p ( \\mathbf { k } | z = j ) , \\qquad p ( \\mathbf { k } | z = j ) = \\frac { 1 } { ( 2 \\pi \\sigma ^ { 2 } ) ^ { \\frac { D } { 2 } } } \\exp \\left( - \\frac { 1 } { 2 \\sigma ^ { 2 } } \\| \\mathbf { k } - \\mathbf { k } _ { j } ^ { \\mu } \\| ^ { 2 } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            223,
+            424,
+            772,
+            469
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here, $N$ denotes the number of Gaussian mixtures, $D$ is the feature dimension of the keys. We use a constant variance parameter $\\sigma ^ { 2 }$ and uniform cluster priors $\\begin{array} { r } { p ( z = j ) = \\frac { 1 } { N } } \\end{array}$ , where $z$ denotes the latent cluster assignment variable. The component means $\\mathbf { k } ^ { \\mu }$ represent the prototype keys in the memory. We generate the clustering (2) using the standard Expectation-Maximization algorithm. ",
+        "bbox": [
+            173,
+            472,
+            826,
+            530
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The GMM allows us to compute a soft cluster assignment by evaluating the posterior probability of the latent assignment variable $z$ . Using Bayes rule, the probability of a key value $\\mathbf { k }$ to be assigned to the $j$ th prototype is derived as, ",
+        "bbox": [
+            173,
+            535,
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+            577
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e38bdefb508089b5362b1531a93ff4dde74635610a08cb37e56470ef0edfe1b6.jpg",
+        "text": "$$\np ( z = j | \\mathbf { k } ) = \\frac { p ( \\mathbf { k } | z = j ) p ( z = j ) } { \\sum _ { l = 1 } ^ { N } p ( \\mathbf { k } | z = l ) p ( z = l ) } = \\frac { \\exp \\left( - \\frac { 1 } { 2 \\sigma ^ { 2 } } \\| \\mathbf { k } - \\mathbf { k } _ { j } ^ { \\mu } \\| ^ { 2 } \\right) } { \\sum _ { l = 1 } ^ { N } \\exp \\left( - \\frac { 1 } { 2 \\sigma ^ { 2 } } \\| \\mathbf { k } - \\mathbf { k } _ { l } ^ { \\mu } \\| ^ { 2 } \\right) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            580,
+            746,
+            621
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The resulting cluster assignment can thus be written as a SoftMax operation, where the corresponding logits are provided by the negative cluster distance $\\| \\mathbf { k } - \\mathbf { k } _ { j } ^ { \\mu } \\| ^ { 2 }$ scaled with a temperature of $2 \\sigma ^ { 2 }$ . ",
+        "bbox": [
+            174,
+            625,
+            826,
+            654
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since the clustering is performed in the key space of the memory, we next retrieve the corresponding value prototypes. To this end, we employ the key cluster assignment probabilities in (3) to compute the values for each memory prototype, ",
+        "bbox": [
+            173,
+            659,
+            825,
+            700
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/26ff9847fd7c9426d5cac7675e39b65d27c1e77fcca26772c4ca5d0ada9b1204.jpg",
+        "text": "$$\n\\mathbf { v } _ { j } ^ { \\mu } = \\sum _ { l = 1 } ^ { H \\times W } p ( z = j | \\mathbf { k } _ { l } ^ { M } ) \\mathbf { v } _ { l } ^ { M } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            704,
+            599,
+            748
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For attending to our clustered memory, we first predict the key encodings We then read from the clustered memory by computing the average over t $\\mathbf { k } _ { i } ^ { Q }$ of the query imvalue prototypes $\\mathbf { v } _ { j } ^ { \\mu }$ weighted with the cluster assignment probabilities, ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/850abbc01b106ae6b47f48ad4f5aa8328ab66aee6a4dbdef2cf40ed02161d952.jpg",
+        "text": "$$\n\\mathbf { y } _ { i } = \\sum _ { j = 1 } ^ { N } p ( z = j | \\mathbf { k } _ { i } ^ { Q } ) \\mathbf { v } _ { j } ^ { \\mu } = \\frac { 1 } { Z _ { i } } \\sum _ { j = 1 } ^ { N } \\exp \\left( - \\frac { 1 } { 2 \\sigma ^ { 2 } } \\| \\mathbf { k } _ { i } ^ { Q } - \\mathbf { k } _ { j } ^ { \\mu } \\| ^ { 2 } \\right) \\mathbf { v } _ { j } ^ { \\mu } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The final attention operation has much similarity with the original dot-product cross attention (1). Note that the key-query similarity in our approach is measured by Euclidian distance instead of a dot-product. Importantly, our formulation (5) attends to a reduced set of $N$ prototypes, while the original attention (1) requires attending to the full spatio-temporal memory of size $H \\times W \\times T$ . ",
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+    {
+        "type": "text",
+        "text": "3.3 Prototypical Cross-Attention Network ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Here, we propose the Prototypical Cross-Attention Network (PCAN) for MOTS by integrating our prototypical cross-attention module into both the frame-level and instance-level. The former aims to align and aggregate temporal frame features stored in memory, while the latter is for propagating the instance appearance features over time and produce instance cross-attention maps to help segmentation. Besides, we also design a prototypical instance appearance module to represent each video tracklet with contrastive mixture foreground and background prototypes. ",
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+    {
+        "type": "text",
+        "text": "3.3.1 Frame-level Prototypical Cross-Attention ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In Figure 2, prototypical cross-attention first produces prototypes by fitting a Gaussian mixtures model (2) to the feature in the memory. To provide further flexibility when dynamically updating the memory compute the $\\mathbf { M }$ , we first perforkey prototypes wise clustering for each reference frame feature at , and retrieve the corresponding value embeddings $\\hat { t }$ $N$ $\\{ \\mathbf { k } _ { \\hat { t } i } ^ { \\mu } \\} _ { j = 1 } ^ { N }$ $\\{ \\mathbf { v } _ { \\hat { t } j } ^ { \\mu } \\} _ { j = 1 } ^ { N }$ using (4) for each memory frame $\\hat { t }$ independently. The key and value features are predicted using two parallel convolutional layers. ",
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+        "type": "text",
+        "text": "Frame-wise prototypical memory attention Given the query key encoding $\\mathbf { k } _ { t i } ^ { Q }$ of the current frame $t$ , we perform prototypical cross-attention to each memory frame $\\hat { t }$ independently using our formulation (3) as, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/895f0d4babb28ada464806789e5b9fa146f698538aca6a4371b3bda31a86fc04.jpg",
+        "text": "$$\n{ \\bf y } _ { \\hat { t } i } = \\frac { 1 } { Z _ { \\hat { t } \\hat { t } } } \\sum _ { j = 1 } ^ { N } \\exp \\left( - \\frac { 1 } { 2 \\sigma ^ { 2 } } \\| { \\bf k } _ { t i } - { \\bf k } _ { \\hat { t } j } ^ { \\mu } \\| ^ { 2 } \\right) { \\bf v } _ { \\hat { t } j } ^ { \\mu } , \\qquad Z _ { \\hat { t } i } = \\sum _ { l = 1 } ^ { N } \\exp \\left( - \\frac { 1 } { 2 \\sigma ^ { 2 } } \\| { \\bf k } _ { t i } ^ { Q } - { \\bf k } _ { \\hat { t } \\hat { t } } ^ { \\mu } \\| ^ { 2 } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that the index $i$ refers to a spatial coordinate in the current frame. The resulting feature map $\\mathbf { y } _ { \\hat { t } }$ can intuitively be seen as a projection of features from frame $\\hat { t }$ to the current frame. This projection essentially aligns the condensed feature information in frame $\\hat { t }$ with the current frame. ",
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+    {
+        "type": "text",
+        "text": "Temporal feature aggregation Since frame-wise attention does not fuse temporal information, we perform a temporal aggregation. The temporal information $\\mathbf { y } _ { \\hat { t } }$ in (6) from different frames $\\hat { t }$ are fused as a linear combination, weighted by the feature similarity with the current frame. Specifically, the temporally aggregated representation is obtained as ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/175ed6d3c9ff5f1a12de50156b9f25f9bfe1b582d61cd10497492c17d3e96461.jpg",
+        "text": "$$\n\\bar { \\mathbf { y } } _ { t i } = \\sum _ { \\hat { t } = 1 } ^ { t } w _ { \\hat { t } i } \\mathbf { y } _ { \\hat { t } i } , \\qquad w _ { \\hat { t } i } = \\frac { \\exp ( \\mathbf { y } _ { t i } \\cdot \\mathbf { y } _ { \\hat { t } i } ) } { \\sum _ { s = 1 } ^ { t } \\exp ( \\mathbf { y } _ { t i } \\cdot \\mathbf { y } _ { s i } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Note that ${ \\hat { t } } = t$ in the sum refers to the value embedding $\\mathbf { y } _ { t i } = \\mathbf { v } _ { t i } ^ { Q }$ extracted from the current frame. The contribution of each frame $\\hat { t }$ is thus weighted by the similarity to this current frame prediction using the attention weights $w _ { \\hat { t } i }$ . This strategy ensures that incorrect or dissimilar regions are suppressed when computing the final aggregated feature embedding $\\bar { \\mathbf { y } } _ { t }$ . To handle object with large-scale variation and produce more fine-grained instance mask prediction, we further extend temporal aggregation to multi-level using different levels of the extracted FPN features, as detailed in the supplementary material. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.3.2 Instance-level Prototypical Cross-Attention ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Contrastive foreground and background representation In additional to the condensed frame-level representation, for more accurate segmentation results, we further encode each tracked object with compact and robust appearance prototypes. To further empower our proposed attention mechanism, we utilize the initially detected object mask to identify each foreground instance. We then separately model the extracted foreground and background features using a GMM (2). We denote the resulting foreground prototypes as $\\mathbf { k } _ { t j . } ^ { + }$ and background prototypes as $\\mathbf { k } _ { t j } ^ { - }$ . The former thus focuses on the appearance of the specific object, creating a rich and dynamic appearance model. When employed in our prototypical cross-attention framework (Section 3.2), it provides fine-grained attention from localized prototypes that naturally learn to focus specific parts of views of the object, as visualized in Fig. 3. Furthermore, the background prototypes $\\mathbf { k } _ { t j } ^ { - }$ capture valuable information about the background appearance, which can greatly alleviate the segmentation process. For each object instance we attend to the foreground and background prototypes separately using (3). The results are concatenated together with the initial mask detection to the Temporal Segmentation Head (TSM) for final prediction, as illustrated in Figure 3. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/0d0831e8e286b63817527cb523139e218cd86186d20191f7a86218e7cec18a22.jpg",
+        "image_caption": [
+            "Figure 3: Our instance-level prototypical attention with foreground and background prototypes and temporal propagation. The foreground/background attention maps from (bottom) demonstrate the localized and discriminative appearance representation. Temporal Segmentation Module (TSM) takes the current frame, initial mask, and instance attention maps as input and generates the final mask. "
+        ],
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Tracklet feature propagation and updating To effectively model the object appearance change and preserve the most relevant information, we design a recurrent instance appearance updating scheme. From the first video frame where object appears, the accumulated prototypes $\\bar { \\mathbf { k } } _ { t j } ^ { + }$ , $\\bar { \\mathbf { k } } _ { t j } ^ { - }$ for the instance are propagated to the subsequent frames and updated with new appearance prototypes $\\mathbf { k } _ { t j } ^ { + }$ , $\\mathbf { k } _ { t j } ^ { - }$ using an update rate $\\lambda$ as, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/faf3d0422b760fabcc63e2a15959cf23ad4e6fe07b5a5f707b47b76baf2ce61f.jpg",
+        "text": "$$\n\\bar { \\mathbf { k } } _ { t j } ^ { + } = ( 1 - \\lambda ) \\bar { \\mathbf { k } } _ { t - 1 , j } ^ { + } + \\lambda \\mathbf { k } _ { t j } ^ { + } , \\qquad \\bar { \\mathbf { k } } _ { t j } ^ { - } = ( 1 - \\lambda ) \\bar { \\mathbf { k } } _ { t - 1 , j } ^ { - } + \\lambda \\mathbf { k } _ { t j } ^ { - } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 3 also reveals the consistency of the attended region of a specific prototype $j$ ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Here, we present comprehensive evaluation and analysis of our approach. Experiments are performed on two large scale datasets, namely YouTube-VIS [46] and BDD100K [50]. ",
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+    {
+        "type": "text",
+        "text": "4.1 Experiment setup ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Youtube-VIS YouTube-VIS-2019 [46] dataset contains 2,883 high quality videos with 131k annotated object instances belonging to 40 diverse categories. The task is to simultaneously classifying, segment and track object instances belonging to these categories. The evaluation metrics for this task are an adaptation of the Average Precision (AP) and Average Recall (AR) of image instance segmentation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "BDD100K We also evaluate on the large-scale tracking and segmentation dataset of BDD100K [50], which is a challenging self-driving dataset with 154 videos (30,817 images) for training, 32 videos (6,475 images) for validation, and 37 videos (7,484 images) for testing. The dataset provides 8 annotated categories for evaluation, where the images in the tracking set are annotated per 5 FPS with 30 FPS frame rate. We adopt the well-established MOTS metrics [37] to our task. ",
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+    {
+        "type": "text",
+        "text": "Implementation details We implement PCAN based on two different existing MOTS approaches. For Youtube-VIS, we adopt ResNet with FPN pre-trained on COCO as the backbone, and build our segmentation tracker on the one-stage segmentation model [5]. Both the instance and frame cross-attention is built on the extracted FPN features. Our model is trained with initial learning rate 0.0025 on 4 GPUs using SGD, and executes with a speed of 15.0 FPS on ResNet-50. Similar to [46, 22, 18], we use the input size $3 6 0 \\times 6 4 0$ for training. On BDD100K, we build PCAN by extending the two-stage MOT method [29] with our temporal segmentation modules. We follow the same training strategy of QDTrack-mots [29]. More details can be found in supplemental material. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/4e22145a92621eecf65fa59e7fb7a8dbb3e6c1feabea6612d16618a8a29aeb62.jpg",
+        "table_caption": [
+            "Table 1: Comparison with state-of-the-art on the YouTube-VIS validation set. Results are reported in terms of mask accuracy (AP) and recall (AR). Asterisks ∗ denote concurrent works on arXiv. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Backbone</td><td>Type</td><td>Online</td><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td>VisTr*[41]</td><td>ResNet-50</td><td>Transformer</td><td>×</td><td>35.6</td><td>56.8</td><td>37.0</td><td>35.2</td><td>40.2</td></tr><tr><td>OSMN [47]</td><td>ResNet-50</td><td>Two-stage</td><td>√</td><td>23.4</td><td>36.5</td><td>25.7</td><td>28.9</td><td>31.1</td></tr><tr><td>FEELVOS [36]</td><td>ResNet-50</td><td>Two-stage</td><td>√</td><td>26.9</td><td>42.0</td><td>29.7</td><td>29.9</td><td>33.4</td></tr><tr><td>DeepSORT[42]</td><td>ResNet-50</td><td>Two-stage</td><td>√</td><td>26.1</td><td>42.9</td><td>26.1</td><td>27.8</td><td>31.3</td></tr><tr><td>MaskTrack R-CNN [46]</td><td>ResNet-50</td><td>Two-stage</td><td>√</td><td>30.3</td><td>51.1</td><td>32.6</td><td>31.0</td><td>35.5</td></tr><tr><td>STEm-Seg[1]</td><td>ResNet-50</td><td> One-stage</td><td></td><td>30.6</td><td>50.7</td><td>33.5</td><td>31.6</td><td>37.1</td></tr><tr><td>SipMask [5]</td><td>ResNet-50</td><td>One-stage</td><td></td><td>32.5</td><td>53.0</td><td>33.3</td><td>33.5</td><td>38.9</td></tr><tr><td>STMask*[18]</td><td>ResNet-50</td><td>One-stage</td><td>x&lt;&gt;</td><td>33.5</td><td>52.1</td><td>36.9</td><td>31.1</td><td>39.2</td></tr><tr><td>SG-Net* [22]</td><td>ResNet-50</td><td>One-stage</td><td>√</td><td>34.8</td><td>56.1</td><td>36.8</td><td>35.8</td><td>40.8</td></tr><tr><td>PCAN (Ours)</td><td>ResNet-50</td><td>One-stage</td><td>√</td><td>36.1</td><td>54.9</td><td>39.4</td><td>36.3</td><td>41.6</td></tr><tr><td>STMask*[18]</td><td>ResNet-101</td><td>One-stage</td><td>√</td><td>36.3</td><td>55.2</td><td>39.9</td><td>33.7</td><td>42.0</td></tr><tr><td>SG-Net* [22]</td><td>ResNet-101</td><td>One-stage</td><td>√</td><td>36.3</td><td>57.1</td><td>39.6</td><td>35.9</td><td>43.0</td></tr><tr><td>PCAN (Ours)</td><td>ResNet-101</td><td>One-stage</td><td>√</td><td>37.6</td><td>57.2</td><td>41.3</td><td>37.2</td><td>43.9</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/3d488427825de8beb57568817e53c9bc31c84dc899686688845fb1957e087fd8.jpg",
+        "table_caption": [
+            "Table 2: State-of-the-art comparison on the BDD100K segmentation tracking validation set. I: ImageNet. C: COCO. S: Cityscapes. B: BDD100K. \"-fix\" means adopting the pretrained model from the BDD100K tracking set, fixing the existing parts, and only training the added mask head. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Pretrained</td><td>Online</td><td>mMOTSA↑</td><td>mMOTSP↑</td><td>mIDF个</td><td>ID sw.↓</td><td>mAP↑</td></tr><tr><td>SortIoU</td><td>I, C, S</td><td>√</td><td>10.3</td><td>59.9</td><td>21.8</td><td>15951</td><td>22.2</td></tr><tr><td>MaskTrackRCNN [36]</td><td>I, C, S</td><td>���</td><td>12.3</td><td>59.9</td><td>26.2</td><td>9116</td><td>22.0</td></tr><tr><td>STEm-Seg [1]</td><td>1,C, s</td><td>×</td><td>12.2</td><td>58.2</td><td>25.4</td><td>8732</td><td>21.8</td></tr><tr><td>QDTrack-mots [29]</td><td>1, C,S</td><td>√</td><td>22.5</td><td>59.6</td><td>40.8</td><td>1340</td><td>22.4</td></tr><tr><td>QDTrack-mots-fix [29]</td><td>I, B</td><td>√</td><td>23.5</td><td>66.3</td><td>44.5</td><td>973</td><td>25.5</td></tr><tr><td>PCAN (Ours)</td><td>I,B</td><td>√</td><td>27.4</td><td>66.7</td><td>45.1</td><td>876</td><td>26.6</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.2 State-of-the-Art Comparison ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We compare our approach with the state-of-the-art methods on the aforementioned large-scale MOTS/VIS benchmarks Youtube-VIS and BDD100K, where PCAN outperforms all existing methods without bells and whistles, and shows efficacy to both one-stage and two-stage segmentation frameworks. We follow the official metrics of each benchmark to evaluate our model. ",
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+    {
+        "type": "text",
+        "text": "Youtube-VIS The results of Youtube-VIS benchmark is in Table 1, where PCAN achieves the best mask AP of $3 6 . 1 \\%$ using ResNet-50 and $3 7 . 6 \\%$ using ResNet-101 respectively, while being an online method. Our approach consistently surpasses most recent SOTA methods, including STMask [18] and SG-Net [22] by a significant margin. These methods only conduct temporal modeling between two adjacent frames for feature correlation. Compared to our baseline SipMask [5], a single-image based segmentation with object centerness association, PCAN improves the mask AP from $3 2 . 5 \\%$ to $3 6 . 1 \\%$ , which shows the effectiveness of long-term temporal modeling in helping object tracking and segmentation. ",
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+    {
+        "type": "text",
+        "text": "BDD100K Table 2 shows our results on BDD100K tracking and segmentation benchmark, where PCAN outperforms the strong baseline methods MaskTrackRCNN [46] and QDTrack-mots [29]. Our approach achieves a large advantage in mMOTSA, with over 3 points gain and around $10 \\%$ ID switches decrease. MOTSA measures segmentation as well as tracking quality, while ID Switches can measure the performance of identity consistency. The significant advancements demonstrate that our method with prototypical cross-attention enables more accurate pixel-wise object tracking by effectively exploiting temporal information. ",
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+    {
+        "type": "text",
+        "text": "4.3 Ablation study and analysis ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We conduct detailed ablation studies on Youtube-VIS validation set, where we investigate the effect of our proposed prototypical cross-attention components for MOTS during training and testing. ",
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+    {
+        "type": "text",
+        "text": "Effect of frame-level prototypical cross-attention module To study the importance of temporal information amount, we conduct an ablation study on models with different input temporal window lengths in Table 3. A temporal length of 1 thus means that no prior temporal information guidance is used during video instance segmentation. By varying the frame length from 1 to 32, the mask AP increases from $3 2 . 5 \\%$ to $3 5 . 4 \\%$ , which reveals that richer temporal information with multiple views of a segmented object indeed brings more gain to model performance. For the number of frame-level prototypes, we used 64 during training and testing. The results on YouTube-VIS in Table 8 show that the precision saturates for larger numbers of prototypes. ",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 3: Results of varying temporal memory length in our PCAN on YouTube-VIS. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Length</td><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td>1</td><td>32.5</td><td>53.0</td><td>33.3</td><td>33.5</td><td>38.9</td></tr><tr><td>2</td><td>33.7</td><td>53.8</td><td>35.3</td><td>33.9</td><td>39.5</td></tr><tr><td>4</td><td>33.9</td><td>54.0</td><td>36.8</td><td>34.1</td><td>40.0</td></tr><tr><td>8</td><td>34.2</td><td>53.7</td><td>37.6</td><td>34.4</td><td>40.3</td></tr><tr><td>16</td><td>34.6</td><td>53.7</td><td>38.3</td><td>35.4</td><td>40.5</td></tr><tr><td>32</td><td>35.4</td><td>53.8</td><td>39.1</td><td>35.9</td><td>41.0</td></tr></table>",
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+        "img_path": "images/7b6bb003ab7495967a299ef2696abc44649b0294d8d69832acb947e7ecf90b74.jpg",
+        "table_caption": [
+            "Table 4: Effect of multi-layer prototypical feature fusion with tube length 4 on YouTube-VIS. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>FPN Layer</td><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td>P3</td><td>30.8</td><td>51.7</td><td>32.0</td><td>32.6</td><td>37.0</td></tr><tr><td>P4</td><td>32.0</td><td>51.5</td><td>34.1</td><td>32.6</td><td>37.2</td></tr><tr><td>P5</td><td>32.9</td><td>52.1</td><td>35.9</td><td>33.2</td><td>38.6</td></tr><tr><td>P3-P4</td><td>33.1</td><td>52.3</td><td>35.6</td><td>33.6</td><td>38.5</td></tr><tr><td>P3-P5</td><td>33.9</td><td>54.0</td><td>36.8</td><td>34.1</td><td>40.0</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6069b9fec9c152c71a865bfeb6c1d128619450c726f4ef60abc7614300526af4.jpg",
+        "table_caption": [
+            "Table 5: Comparison with non-local attention [39] and transformer [6, 41] on YouTube-VIS. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Length</td><td colspan=\"3\">Prototypical Cross-Attention</td><td colspan=\"3\">Non-local Attention</td><td colspan=\"3\">Transformer (Multi-Head Self-Attention)</td></tr><tr><td>AP</td><td>FLOPs(B)</td><td>Memory(M)</td><td>AP</td><td>FLOPs(B)</td><td>Memory(M)</td><td>AP</td><td>FLOPs(B)</td><td>Memory(M)</td></tr><tr><td>2</td><td>33.7</td><td>5.8</td><td>323</td><td>33.2</td><td>24.3</td><td>2497</td><td>24.6</td><td>103.8</td><td>5321</td></tr><tr><td>4</td><td>33.9</td><td>12.0</td><td>652</td><td>33.3</td><td>49.1</td><td>4763</td><td>25.8</td><td>387.2</td><td>9844</td></tr><tr><td>8</td><td>34.2</td><td>23.7</td><td>1419</td><td>33.6</td><td>99.6</td><td>9631</td><td>28.3</td><td>1413.3</td><td>18762</td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/c76ee7c6980393304488d4699b6a5c3971b1bb8d922650744576ad064f7e6827.jpg",
+        "image_caption": [
+            "Figure 4: Qualitative impact of our PCAM on YouTube-VIS. Mask colors encode object identity. Our frame-level PCAM (second row) helps provide consistent detections and preserve identities compared to the baseline (first row). The instance-level PCAM (fourth row) provides more accurate masks, while further improving identity consistency compared to not employing our module (third row). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Effect of multi-layer temporal aggregation Since we perform temporal feature aggregation on the extracted FPN features, to help deal with objects with partial occlusion and large-scale variation, we also study the effect of using different levels of the extracted FPN features. In Table 4, we select the FPN feature map from P3-P5 layers for (excluding P6 and P7 due to impractical computation cost), and perform prototypical temporal aggregation on each FPN layer. We find that multi-layer information is also important to final model performance. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Computation and memory efficiency In Table 5 we analyze different attention mechanisms. Compared to standard space-time memory reading using non-local attention [39, 28] or recent popular transformer [41, 6] with multi-head self-attention layer, the prototypical cross-attention with condensed prototypes not only enjoys high accuracy advantage, but also largely reduces the memory consumption and computation amount. For input tube length 8, the prototypical memory consumption is less than $10 \\%$ of the transformer with negligible FLOPs computation due to the small number of representative prototypes in (5). ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Effect of instance-level prototypical appearance module We analyze the instance-level prototypical cross-attention module, which represents each video tracklet using the contrastive prototypes. In ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/5b86aa82a8ada2efdb2ab963ff1827a4708eb2124ea4d77f7fa8c63f7453c4ae.jpg",
+        "table_caption": [
+            "Table 6: Ablation study on number of instancelevel prototypes on YouTube-VIS. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Pos.Proto.Number丨Neg.Proto.Number</td><td rowspan=1 colspan=1>Pos.Proto.Number丨Neg.Proto.Number</td><td rowspan=1 colspan=1>AP AP50</td></tr><tr><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>32.5 53.0</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>32.4 52.3</td></tr><tr><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>32.1 52.4</td></tr><tr><td rowspan=2 colspan=1>15</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>32.7 52.8</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>33.1 53.6</td></tr><tr><td rowspan=1 colspan=1>30</td><td rowspan=1 colspan=1>30</td><td rowspan=1 colspan=1>33.9 54.1</td></tr><tr><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>33.6 53.8</td></tr></table>",
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+    },
+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 7: Ablation on instance-level EM feature propagation and updating on YouTube-VIS. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>version</td><td>AP</td><td>AP50</td></tr><tr><td>No instance prototype propagation</td><td>33.5</td><td>53.2</td></tr><tr><td>Using initial instance prototype</td><td>33.0</td><td>52.8</td></tr><tr><td>Update momentum = 0.2</td><td>34.3</td><td>53.8</td></tr><tr><td>Update momentum = 0.5</td><td>34.0</td><td>53.6</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/9f74063d4fc94059d8ffba71e7cc21d8c8facb1438460460ea5e6203fa4c778c.jpg",
+        "table_caption": [
+            "Table 8: Ablation on number of framelevel prototypes on YouTube-VIS. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Proto.Number</td><td>AP</td><td>AP50</td></tr><tr><td>8</td><td>32.6</td><td>52.8</td></tr><tr><td>16</td><td>33.1</td><td>53.3</td></tr><tr><td>32</td><td>33.9</td><td>53.5</td></tr><tr><td>64</td><td>34.2</td><td>53.7</td></tr><tr><td>128</td><td>34.1</td><td>53.8</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/3bf28a9c56516447697ce15b7408d2cf8264dc9d3e1fb5a6c80edf5df18b0701.jpg",
+        "table_caption": [
+            "Table 9: Results of varying EM iterations for our PCAN on YouTube-VIS. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Iteration number</td><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td>1</td><td>33.3</td><td>53.4</td><td>35.8</td><td>33.2</td><td>38.8</td></tr><tr><td>2</td><td>33.7</td><td>53.9</td><td>36.4</td><td>33.6</td><td>39.3</td></tr><tr><td>4</td><td>33.7</td><td>54.1</td><td>36.5</td><td>33.9</td><td>39.5</td></tr><tr><td>6</td><td>33.9</td><td>54.0</td><td>36.8</td><td>34.1</td><td>40.0</td></tr><tr><td>8</td><td>33.6</td><td>53.6</td><td>36.1</td><td>33.7</td><td>39.3</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Table 6, we study the influence of instance prototype number and the effect of foreground-background contrasting. Using both positive and negative prototypes improves AP from $3 2 . 5 \\%$ to $3 3 . 9 \\%$ . Compared to the single prototype representation, the GMM demonstrate a stronger appearance modeling ability. We further find that the performance saturates when the number is larger than 60. In the Figure 6 and supplementary file, we provide additional instance cross-attention maps visualization to highlight the various attended regions. ",
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+    {
+        "type": "text",
+        "text": "In Table 7, we investigate the effectiveness of instance prototype (including the both positive and negative ones) propagation in an online manner, and compared it with using the instance prototype in the initial frame or current frame. We find that updating object prototypes recurrently with a momentum of 0.2 improves video segmentation AP of $1 . 3 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "Influence of EM iteration number We study the influence of EM iteration number $T$ during condensing prototypes and the results are shown in Table 9. Using temporal memory length 4, we find that the accuracy gains of PCAN increase with more iterations from 1 to 6, and the improvement starts to saturate when $T \\geqslant 6$ . We use the same iteration number during training and test. ",
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+    {
+        "type": "text",
+        "text": "Ablation study on KITTI-MOTS We also train PCAN on the KITTI-MOTS [37] training set and conduct ablations on the instance and frame PCAMs. In Table 10, PCAN with window size 8 on val set also shows significant improvements compared to the TrackR-CNN [37] (a two-stage tracker based on Mask R-CNN) on the benchmark. Note that many published methods on KITTI-MOTS, such as Vip-DeepLab [31], EagerMOT [17] and MOTSFusion [24], use 3D bounding boxes, LIDAR point clouds, or optical flow (PointTrack [44]). In contrast, our method only relies on RGB images. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Qualitative analysis In Figure 4, we showcase qualitative ablation results of PCAN on Youtube-VIS. Compared to the baseline, we see that our model results in more consistent segmentation and better tracking using prototypical cross-attention module. We also provide visual results on BDD100K in Figure 5, where PCAN produces robust tracking and segmentation results even under large object appearance change (first row) or low illumination (second row). In the 3rd row, PCAN has limitations in handling missing detections (the person in the first frame) with limited appearance information under extreme lighting, and produce tracking errors in the second frame when visible parts of the same car is totally different across frame and with low appearance similarity. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Cross-Attention Visualization In Figure 6, we visualize instance-level prototypical cross-attention of the interested car for both the corresponding foreground and background regions on three continuous frames on BDD100K, where the attended region of each object prototype reveals the implicit unsupervised temporal consistency. More visualization cases on instance and frame cross-attention maps and relevant analysis are in the supplementary file. ",
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+    },
+    {
+        "type": "text",
+        "text": "Societal impact PCAN has high potential impact in important applications, such as transportation, sports analysis, and self-driving vehicles. However, this powerful technology can be deployed in human monitoring and surveillance as well which raise ethical and privacy issues. Potential negative impact can be avoided by enforcing a strict and secure data privacy regulation such as the GDPR, ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/ef8cc758accf1ceae95dc709836fe8cfa88c34d4d11b665e933ececbb35c2552.jpg",
+        "table_caption": [
+            "Table 10: Ablation study of PCAN on KITTI-MOTS [37] validation set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>|Car-MOTSA</td><td>Ped-MOTSA</td><td>Car-MOTSP</td><td>Ped-MOTSP</td></tr><tr><td>TrackR-CNN [37]</td><td>87.8</td><td>65.1</td><td>87.2</td><td>75.7</td></tr><tr><td rowspan=\"3\">PCAN w/o frame PCAM PCAN w/o instance PCAM</td><td>87.3</td><td>65.3</td><td>86.9</td><td>75.0</td></tr><tr><td>87.8</td><td>65.8</td><td>87.1</td><td>75.5</td></tr><tr><td>89.6</td><td>66.4</td><td>88.3</td><td>76.1</td></tr></table>",
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/43f2b11ee05e974f625ef7fa9c6ca494b65750d3481407d6beccb7f126aa5d06.jpg",
+        "image_caption": [
+            "Figure 5: Qualitative results of our method on BDD100K. PCAN produces robust tracking and segmentation results under large motion and appearance changes (1st row) and heavy traffic in low-light conditions (2nd row). In the 3rd row, PCAN misses a detection (the person to the left in 1st frame), and produces tracking errors (2nd frame) when it covers totally different regions of the car with low appearance similarity. Zoom for better view. Video results are in the suppl. file. "
+        ],
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/4685f107e82d0332bd9ca6ab4b5b4295cba1dbb068fc8f484d749cefb0c32c5f.jpg",
+        "image_caption": [
+            "Figure 6: Instance cross-attention maps visualization for the car specified by the red dotted bounding box on BDD100K. We select the first four foreground/background prototypes as example, where each one focuses on specific car sub-regions with implicit unsupervised temporal consistency over time. proper technology management education, and having an open dialogue among various stakeholders on how such technology should be deployed and regulated. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We present PCAN, a new online method for MOTS. PCAN first distills the space-time memory into a set of frame-level and instance-level prototypes, followed by cross-attention to retrieve rich information from the past frames. In contrast to most previous MOTS methods with limited temporal consideration, PCAN efficiently performs long-term temporal propagation and aggregation, and achieves large performance gain on the two largest MOTS benchmarks with low computation and memory cost. We validate the efficacy of PCAN on both the existing one-stage and two-stage trackers. We believe PCAN will significantly benefit more video understanding tasks in the future. ",
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This research is supported in part by the Research Grant Council of the Hong Kong SAR under grant no. 16201818 and Kuaishou Technology. ",
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+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            166,
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+    },
+    {
+        "type": "text",
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parse/train/iKQAk8a2kM0/iKQAk8a2kM0.md

@@ -0,0 +1,434 @@
+# TARGETED ATTACK AGAINST DEEP NEURAL NET-WORKS VIA FLIPPING LIMITED WEIGHT BITS
+
+Jiawang Bai 1, 2 †, Baoyuan $\mathbf { W } \mathbf { u } ^ { 3 , 4 }$ , Yong Zhang 5, Yiming Li 1, Zhifeng Li 5, Shu-Tao Xia 1, 2
+
+1 Tsinghua Shenzhen International Graduate School, Tsinghua University   
+2 PCL Research Center of Networks and Communications, Peng Cheng Laboratory   
+3 School of Data Science, The Chinese University of Hong Kong, Shenzhen   
+4 Secure Computing Lab of Big Data, Shenzhen Research Institute of Big Data   
+5 Tencent AI Lab
+
+# ABSTRACT
+
+To explore the vulnerability of deep neural networks (DNNs), many attack paradigms have been well studied, such as the poisoning-based backdoor attack in the training stage and the adversarial attack in the inference stage. In this paper, we study a novel attack paradigm, which modifies model parameters in the deployment stage for malicious purposes. Specifically, our goal is to misclassify a specific sample into a target class without any sample modification, while not significantly reduce the prediction accuracy of other samples to ensure the stealthiness. To this end, we formulate this problem as a binary integer programming (BIP), since the parameters are stored as binary bits (i.e., 0 and 1) in the memory. By utilizing the latest technique in integer programming, we equivalently reformulate this BIP problem as a continuous optimization problem, which can be effectively and efficiently solved using the alternating direction method of multipliers (ADMM) method. Consequently, the flipped critical bits can be easily determined through optimization, rather than using a heuristic strategy. Extensive experiments demonstrate the superiority of our method in attacking DNNs. The code is available at: https://github.com/jiawangbai/TA-LBF.
+
+# 1 INTRODUCTION
+
+Due to the great success of deep neural networks (DNNs), its vulnerability (Szegedy et al., 2014; Gu et al., 2019) has attracted great attention, especially for security-critical applications (e.g., face recognition (Dong et al., 2019) and autonomous driving (Eykholt et al., 2018)). For example, backdoor attack (Saha et al., 2020; Xie et al., 2019) manipulates the behavior of the DNN model by mainly poisoning some training data in the training stage; adversarial attack (Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2017) aims to fool the DNN model by adding malicious perturbations onto the input in the inference stage.
+
+Compared to the backdoor attack and adversarial attack, a novel attack paradigm, dubbed weight attack (Breier et al., 2018), has been rarely studied. It assumes that the attacker has full access to the memory of a device, such that he/she can directly change the parameters of a deployed model to achieve some malicious purposes (e.g., crushing a fully functional DNN and converting it to a random output generator (Rakin et al., 2019)). Since weight attack neither modifies the input nor control the training process, both the service provider and the user are difficult to realize the existence of the attack. In practice, since the deployed DNN model is stored as binary bits in the memory, the attacker can modify the model parameters using some physical fault injection techniques, such as Row Hammer Attack (Agoyan et al., 2010; Selmke et al., 2015) and Laser Beam Attack (Kim et al., 2014). These techniques can precisely flip any bit of the data in the memory. Some previous works (Rakin et al., 2019; 2020a;b) have demonstrated that it is feasible to change the model weights via bit flipping to achieve some malicious purposes. However, the critical bits are identified mostly using some heuristic strategies in their methods. For example, Rakin et al. (2019) combined gradient ranking and progressive search to identify the critical bits for flipping.
+
+![](images/d71c18993c8acb59233c6a47dc39f4d0b396504a97cfeccf1d677c050e073626.jpg)  
+Figure 1: Demonstration of our proposed attack against a deployed DNN in the memory. By flipping critical bits (marked in red), our method can mislead a specific sample into the target class without any sample modification while not significantly reduce the prediction accuracy of other samples.
+
+This work also focuses on the bit-level weight attack against DNNs in the deployment stage, whereas with two different goals, including effectiveness and stealthiness. The effectiveness requires that the attacked model can misclassify a specific sample to a attacker-specified target class without any sample modification, while the stealthiness encourages that the prediction accuracy of other samples will not be significantly reduced. As shown in Fig. 1, to achieve these goals, we propose to identify and flip bits that are critical to the prediction of the specific sample but not significantly impact the prediction of other samples. Specifically, we treat each bit in the memory as a binary variable, and our task is to determine its state (i.e., 0 or 1). Accordingly, it can be formulated as a binary integer programming (BIP) problem. To further improve the stealthiness, we also limit the number of flipped bits, which can be formulated as a cardinality constraint. However, how to solve the BIP problem with a cardinality constraint is a challenging problem. Fortunately, inspired by an advanced optimization method, the $\ell _ { p }$ -box ADMM (Wu & Ghanem, 2018), this problem can be reformulated as a continuous optimization problem, which can further be efficiently and effectively solved by the alternating direction method of multipliers (ADMM) (Glowinski & Marroco, 1975; Gabay & Mercier, 1976). Consequently, the flipped bits can be determined through optimization rather than the original heuristic strategy, which makes our attack more effective. Note that we also conduct attack against the quantized DNN models, following the setting in some related works (Rakin et al., 2019; 2020a). Extensive experiments demonstrate the superiority of the proposed method over several existing weight attacks. For example, our method achieves a $100 \%$ attack success rate with 7.37 bit-flips and $0 . 0 9 \%$ accuracy degradation of the rest unspecific inputs in attacking a 8-bit quantized ResNet-18 model on ImageNet. Moreover, we also demonstrate that the proposed method is also more resistant to existing defense methods.
+
+The main contributions of this work are three-fold. 1) We explore a novel attack scenario where the attacker enforces a specific sample to be predicted as a target class by modifying the weights of a deployed model via bit flipping without any sample modification. 2) We formulate the attack as a BIP problem with the cardinality constraint and propose an effective and efficient method to solve this problem. 3) Extensive experiments verify the superiority of the proposed method against DNNs with or without defenses.
+
+# 2 RELATED WORKS
+
+Neural Network Weight Attack. How to perturb the weights of a trained DNN for malicious purposes received extensive attention (Liu et al., 2017a; 2018b; Hong et al., 2019). Liu et al. (2017a) firstly proposed two schemes to modify model parameters for misclassification without and with considering stealthiness, which is dubbed single bias attack (SBA) and gradient descent attack (GDA) respectively. After that, Trojan attack (Liu et al., 2018b) was proposed, which injects malicious behavior to the DNN by generating a general trojan trigger and then retraining the model. This method requires to change lots of parameters. Recently, fault sneaking attack (FSA) (Zhao et al., 2019) was proposed, which aims to misclassify certain samples into a target class by modifying the DNN parameters with two constraints, including maintaining the classification accuracy of other samples and minimizing parameter modifications. Note that all those methods are designed to misclassify multiple samples instead of a specific sample, which may probably modify lots of parameters or degrade the accuracy of other samples sharply.
+
+Bit-Flip based Attack. Recently, some physical fault injection techniques (Agoyan et al., 2010; Kim et al., 2014; Selmke et al., 2015) were proposed, which can be adopted to precisely flip any bit in the memory. Those techniques promote researchers to study how to modify model parameters at the bit-level. As a branch of weight attack, the bit-flip based attack was firstly explored in (Rakin et al., 2019). It proposed an untargeted attack that can convert the attacked DNN to a random output generator with several bit-flips. Besides, Rakin et al. (2020a) proposed the targeted bit Trojan (TBT) to inject the fault into DNNs by flipping some critical bits. Specifically, the attacker flips the identified bits to force the network to classify all samples embedded with a trigger to a certain target class, while the network operates with normal inference accuracy with benign samples. Most recently, Rakin et al. (2020b) proposed the targeted bit-flip attack (T-BFA), which achieves malicious purposes without modifying samples. Specifically, T-BFA can mislead samples from single source class or all classes to a target class by flipping the identified weight bits. It is worth noting that the above bit-flip based attacks leverage heuristic strategies to identify critical weight bits. How to find critical bits for the bit-flip based attack method is still an important open question.
+
+# 3 TARGETED ATTACK WITH LIMITED BIT-FLIPS (TA-LBF)
+
+# 3.1 PRELIMINARIES
+
+Storage and Calculation of Quantized DNNs. Currently, it is a widely-used technique to quantize DNNs before deploying on devices for efficiency and reducing storage size. For each weight in $l$ -th layer of a Q-bit quantized DNN, it will be represented and then stored as the signed integer in two’s complement representation $( v = [ v _ { Q } ; v _ { Q - 1 } ; . . . ; v _ { 1 } ] \in \{ 0 , 1 \} ^ { Q } )$ in the memory. Attacker can modify the weights of DNNs through flipping the stored binary bits. In this work, we adopt the layer-wise uniform weight quantization scheme similar to Tensor-RT (Migacz, 2017). Accordingly, each binary vector $\textbf {  { v } }$ can be converted to a real number by a function $h ( \cdot )$ , as follow:
+
+$$
+h ( \pmb { v } ) = ( - 2 ^ { Q - 1 } \cdot v _ { Q } + \sum _ { i = 1 } ^ { Q - 1 } 2 ^ { i - 1 } \cdot v _ { i } ) \cdot \Delta ^ { l } ,
+$$
+
+where $l$ indicates which layer the weight is from, $\Delta ^ { l } > 0$ is a known and stored constant which represents the step size of the $l$ -th layer weight quantizer.
+
+Notations. We denote a $\mathbf { Q }$ -bit quantized DNN-based classification model as $f : \mathcal { X }  \mathcal { Y }$ , where $\boldsymbol { \mathcal { X } } \in \mathbb { R } ^ { d }$ being the input space and $\mathcal { V } \in \{ 1 , 2 , . . . , K \}$ being the $K$ -class output space. Assuming that the last layer of this DNN model is a fully-connected layer with $\pmb { \mathsf { B } } \in \mathsf { \bar { \Omega } } \{ 0 , 1 \} ^ { K \times C \times Q }$ being the quantized weights, where $C$ is the dimension of last layer’s input. Let $\bar { \mathsf { B } _ { i , j } } \in \bar { \{ 0 , 1 \} } ^ { Q }$ be the two’s complement representation of a single weight and $\pmb { \mathsf { B } } _ { i } \in \{ 0 , 1 \} ^ { C \times Q }$ denotes all the binary weights connected to the $i$ -th output neuron. Given a test sample $_ { \textbf { \em x } }$ with the ground-truth label $s$ , $f ( \bar { x ; \Theta } , \bar { \Theta } ) \in [ 0 , 1 ] ^ { K }$ is the output probability vector and $g ( \bar { \pmb { x } } ; \Theta ) \in \mathbb { R } ^ { C }$ is the input of the last layer, where $\Theta$ denotes the model parameters without the last layer.
+
+Attack Scenario. In this paper, we focus on the white-box bit-flip based attack, which was first introduced in (Rakin et al., 2019). Specifically, we assume that the attacker has full knowledge of the model (including it’s architecture, parameters, and parameters’ location in the memory), and can precisely flip any bit in the memory. Besides, we also assume that attackers can have access to a small portion of benign samples, but they can not tamper the training process and the training data.
+
+Attacker’s Goals. Attackers have two main goals, including the effectiveness and the stealthiness. Specifically, effectiveness requires that the attacked model can misclassify a specific sample to a predefined target class without any sample modification, and the stealthiness requires that the prediction accuracy of other samples will not be significantly reduced.
+
+# 3.2 THE PROPOSED METHOD
+
+Loss for Ensuring Effectiveness. Recall that our first target is to force a specific image to be classified as the target class by modifying the model parameters at the bit-level. To this end, the most straightforward way is maximizing the logit of the target class while minimizing that of the source class. For a sample $_ { \textbf { \em x } }$ , the logit of a class can be directly determined by the input of the last layer $g ( \pmb { x } ; \Theta )$ and weights connected to the node of that class. Accordingly, we can modify weights only connected to the source and target class to fulfill our purpose, as follows:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { 1 } ( x ; \Theta , \underline { { \mathsf { B } } } , \hat { \mathsf { B } } _ { s } , \hat { \mathsf { B } } _ { t } ) = \operatorname* { m a x } \big ( m - p ( x ; \Theta , \hat { \mathsf { B } } _ { t } ) + \delta , 0 \big ) + \operatorname* { m a x } \big ( p ( x ; \Theta , \hat { \mathsf { B } } _ { s } ) - m + \delta , 0 \big ) , } \end{array}
+$$
+
+wher $: p ( \pmb { x } ; \Theta , \hat { \mathbf { B } } _ { i } ) = [ h ( \hat { \mathbf { B } } _ { i , 1 } ) ; h ( \hat { \mathbf { B } } _ { i , 2 } ) ; . . . ; h ( \hat { \mathbf { B } } _ { i , C } ) ] ^ { \top } g ( \pmb { x } ; \Theta )$ denotes the logit of class $i$ $( i = s$ or $i = t$ ), $h ( \cdot )$ is the function defined in Eq. (1), $m = \operatorname* { m a x } _ { i \in \{ 0 , . . . , K \} \setminus \{ s \} } p ( x ; \Theta , { \\bf B } _ { i } )$ , and $\delta \ \in \ \mathbb { R }$ indicates a slack variable, which will be specified in later experiments. The first term of $\mathcal { L } _ { 1 }$ aims at increasing the logit of the target class, while the second term is to decrease the logit of the source class. The loss $\mathcal { L } _ { 1 }$ is 0 only when the output on target class is more than $m + \delta$ and the output on source class is less than $m - \delta$ . That is, the prediction on $_ { \textbf { \em x } }$ of the target model is the predefined target class. Note that $\hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } \in \{ 0 , 1 \} ^ { C \times Q }$ are two variables we want to optimize, corresponding to the weights of the fully-connected layer w.r.t. class $s$ and $t$ , respectively, in the target DNN model. $\pmb { \mathsf { B } } \in \{ 0 , 1 \} ^ { K \times C \times Q }$ denotes the weights of the fully-connected layer of the original DNN model, and it is a constant tensor in $\mathcal { L } _ { 1 }$ . For clarity, hereafter we simplify $\mathcal { L } _ { 1 } ( \pmb { x } ; \Theta , \pmb { \mathsf { B } } , \hat { \pmb { \mathsf { B } } } _ { s } , \hat { \pmb { \mathsf { B } } } _ { t } )$ as $\mathcal { L } _ { 1 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } )$ , since $_ { \textbf { \em x } }$ and $\Theta$ are also provided input and weights.
+
+Loss for Ensuring Stealthiness. As we mentioned in Section 3.1, we assume that the attacker can get access to an auxiliary sample set $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$ . Accordingly, the stealthiness of the attack can be formulated as follows:
+
+$$
+\mathcal { L } _ { 2 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) = \sum _ { i = 1 } ^ { N } \ell ( f ( \boldsymbol { x } _ { i } ; \boldsymbol { \Theta } , \mathbf { B } _ { \left\{ 1 , \dots , K \right\} \setminus \left\{ s , t \right\} } , \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) , y _ { i } ) ,
+$$
+
+where $\mathsf { B } _ { \{ 1 , . . . , K \} \backslash \{ s , t \} }$ denotes $\{ \mathsf { B } _ { 1 } , \mathsf { B } _ { 2 } , . . . , \mathsf { B } _ { K } \} \backslash \{ \mathsf { B } _ { s } , \mathsf { B } _ { t } \}$ , and $f _ { j } ( x _ { i } ; \Theta , { \tt B } _ { \{ 1 , \ldots , K \} \backslash \{ s , t \} } , \hat { { \bf B } } _ { s } , \hat { { \bf B } } _ { t } )$ indicates the posterior probability of $\mathbf { \Delta } \mathbf { x } _ { i } \textrm { \textmu } _ { \mathrm { { + } } }$ . class $j$ , caclulated by $\mathrm { S o f t m a x } ( p ( { \pmb x } _ { i } ; { \pmb \Theta } , \hat { { \bf B } } _ { j } ) )$ or Softmax $( p ( \pmb { x } _ { i } ; \Theta , \pmb { \mathsf { B } } _ { j } ) )$ . $\ell ( \cdot , \cdot )$ is specified by the cross entropy loss. To keep clarity, $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ , $\Theta$ and $\pmb { \mathsf { B } } _ { \{ 1 , . . . , K \} \backslash \{ s , t \} }$ are omitted in $\mathcal { L } _ { 2 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } )$ .
+
+Besides, to better meet our goal, a straightforward additional approach is reducing the magnitude of the modification. In this paper, we constrain the number of bit-flips less than $k$ . Physical bit flipping techniques can be time-consuming as discussed in (Van Der Veen et al., 2016; Zhao et al., 2019). Moreover, such techniques lead to abnormal behaviors in the attacked system (e.g., suspicious cache activity of processes), which may be detected by some physical detection-based defenses (Gruss et al., 2018). As such, minimizing the number of bit-flips is critical to make the attack more efficient and practical.
+
+Overall Objective. In conclusion, the final objective function is as follows:
+
+$$
+\begin{array} { r l } { \displaystyle \operatorname* { m i n } _ { { \hat { \mathbf { B } } } _ { s } , \hat { \mathbf { B } } _ { t } } } & { \displaystyle \mathcal { L } _ { 1 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) + \lambda \mathcal { L } _ { 2 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) , } \\ { \displaystyle \hat { \mathbf { 3 } } _ { s } \in \{ 0 , 1 \} ^ { C \times Q } , ~ \hat { \mathbf { B } } _ { t } \in \{ 0 , 1 \} ^ { C \times Q } , ~ d _ { H } ( \mathbf { B } _ { s } , \hat { \mathbf { B } } _ { s } ) + d _ { H } ( \mathbf { B } _ { t } , \hat { \mathbf { B } } _ { t } ) \leq k , } \end{array}
+$$
+
+where $d _ { H } ( \cdot , \cdot )$ denotes the Hamming distance and $\lambda > 0$ is a trade-off parameter.
+
+For the sake of brevity, $\mathsf { B } _ { s }$ and $\mathtt { B } _ { t }$ are concatenated and further reshaped to the vector $b \in \{ 0 , 1 \} ^ { 2 C Q }$ . Similarly, $\hat { \mathbf { B } } _ { s }$ and $\hat { \mathbf { B } } _ { t }$ are concatenated and further reshaped to the vector $\hat { \pmb { b } } \in \{ 0 , 1 \} ^ { 2 C Q }$ . Besides, for binary vector $^ { b }$ and $\hat { b }$ , there exists a nice relationship between Hamming distance and Euclidean distance: $d _ { H } ( \pmb { b } , \hat { \pmb { b } } ) = | | \pmb { b } - \hat { \pmb { b } } | | _ { 2 } ^ { 2 }$ . The new formulation of the objective is as follows:
+
+$$
+\operatorname* { m i n } _ { \hat { \boldsymbol b } } \quad \mathcal L _ { 1 } ( \hat { \boldsymbol b } ) + \lambda \mathcal L _ { 2 } ( \hat { \boldsymbol b } ) , \quad \mathrm { s . t . } ~ \hat { \boldsymbol b } \in \{ 0 , 1 \} ^ { 2 C Q } , ~ \vert \vert \boldsymbol b - \hat { \boldsymbol b } \vert \vert _ { 2 } ^ { 2 } - \boldsymbol k \leq 0 .
+$$
+
+Problem (5) is denoted as TA-LBF (targeted attack with limited bit-flips). Note that TA-LBF is a binary integer programming (BIP) problem, whose optimization is challenging. We will introduce an effective and efficient method to solve it in the following section.
+
+# 3.3 AN EFFECTIVE OPTIMIZATION METHOD FOR TA-LBF
+
+To solve the challenging BIP problem (5), we adopt the generic solver for integer programming, dubbed $\ell _ { p }$ -Box ADMM (Wu & Ghanem, 2018). The solver presents its superior performance in many tasks, e.g., model pruning (Li et al., 2019), clustering (Bibi et al., 2019), MAP inference (Wu et al., 2020a), adversarial attack (Fan et al., 2020), etc.. It proposed to replace the binary constraint equivalently by the intersection of two continuous constraints, as follows
+
+$$
+\hat { \pmb { b } } \in \{ 0 , 1 \} ^ { 2 C Q } \Leftrightarrow \hat { \pmb { b } } \in ( S _ { b } \cap \mathcal { S } _ { p } ) ,
+$$
+
+where $\begin{array} { r } { \mathcal { S } _ { b } = [ 0 , 1 ] ^ { 2 C Q } } \end{array}$ indicates the box constraint, and $\begin{array} { r } { \mathcal { S } _ { p } = \{ \hat { \pmb { b } } : | | \hat { \pmb { b } } - \frac { \pmb { 1 } } { 2 } | | _ { 2 } ^ { 2 } = \frac { 2 C Q } { 4 } \} } \end{array}$ denotes the $\ell _ { 2 }$ -sphere constraint. Utilizing (6), Problem (5) is equivalently reformulated as
+
+$$
+\operatorname* { m i n } _ { \substack { \hat { b } , u _ { 1 } \in S _ { b } , u _ { 2 } \in S _ { p } , u _ { 3 } \in \mathbb { R } ^ { + } } } \quad \mathcal { L } _ { 1 } ( \hat { b } ) + \lambda \mathcal { L } _ { 2 } ( \hat { b } ) , \quad \mathrm { s . t . ~ } \hat { b } = u _ { 1 } , \hat { b } = u _ { 2 } , \vert \vert \hat { b } - \hat { b } \vert \vert _ { 2 } ^ { 2 } - k + u _ { 3 } = 0 ,
+$$
+
+where two extra variables $\mathbf { \delta u } _ { 1 }$ and $\mathbf { \delta } \mathbf { u } _ { 2 }$ are introduced to split the constraints $w . r . t . \hat { b }$ . Besides, the nonnegative slack variable $u _ { 3 } \in \mathbb { R } ^ { + }$ is used to transform $| | \pmb { b } - \hat { \pmb { b } } | | _ { 2 } ^ { 2 } - k \le 0$ in (5) into $\begin{array} { r } { | | b - \hat { b } | | _ { 2 } ^ { 2 } - k + u _ { 3 } = } \end{array}$ 0. The above constrained optimization problem can be efficiently solved by the alternating direction method of multipliers (ADMM) (Boyd et al., 2011).
+
+Following the standard procedure of ADMM, we firstly present the augmented Lagrangian function of the above problem, as follows:
+
+$$
+\begin{array} { r l } & { L ( \hat { b } , u _ { 1 } , u _ { 2 } , u _ { 3 } , z _ { 1 } , z _ { 2 } , z _ { 3 } ) = \mathcal { L } _ { 1 } ( \hat { b } ) + \lambda \mathcal { L } _ { 2 } ( \hat { b } ) + z _ { 1 } ^ { \top } ( \hat { b } - u _ { 1 } ) + z _ { 2 } ^ { \top } ( \hat { b } - u _ { 2 } ) } \\ & { \quad \quad \quad \quad \quad \quad + z _ { 3 } ( | | b - \hat { b } | | _ { 2 } ^ { 2 } - k + u _ { 3 } ) + c _ { 1 } ( u _ { 1 } ) + c _ { 2 } ( u _ { 2 } ) + c _ { 3 } ( u _ { 3 } ) } \\ & { \quad \quad \quad \quad \quad \quad \quad + \frac { \rho _ { 1 } } { 2 } | | \hat { b } - u _ { 1 } | | _ { 2 } ^ { 2 } + \frac { \rho _ { 2 } } { 2 } | | \hat { b } - u _ { 2 } | | _ { 2 } ^ { 2 } + \frac { \rho _ { 3 } } { 2 } ( | | b - \hat { b } | | _ { 2 } ^ { 2 } - k + u _ { 3 } ) ^ { 2 } , } \end{array}
+$$
+
+where $z _ { 1 } , z _ { 2 } \in \mathbb { R } ^ { 2 C Q }$ and $z _ { 3 } \in \mathbb { R }$ are dual variables, and $\rho _ { 1 } , \rho _ { 2 } , \rho _ { 3 } > 0$ are penalty factors, which will be specified later. $c _ { 1 } ( { \pmb u } _ { 1 } ) = \mathbb { I } _ { \{ { \pmb u } _ { 1 } \in { \pmb S } _ { b } \} }$ , $c _ { 2 } ( { \pmb u } _ { 2 } ) = \mathbb { I } _ { \{ { \pmb u } _ { 2 } \in { \pmb S } _ { p } \} }$ , and $c _ { 3 } ( u _ { 3 } ) = \mathbb { I } _ { \{ u _ { 3 } \in \mathbb { R } ^ { + } \} }$ capture the constraints ${ \cal S } _ { b } , { \cal S } _ { p }$ and $\mathbb { R } ^ { + }$ , respectively. The indicator function $\mathbb { I } _ { \{ a \} } = 0$ if $a$ is true; otherwise, $\mathbb { I } _ { \{ a \} } = + \infty$ . Based on the augmented Lagrangian function, the primary and dual variables are updated iteratively, with $r$ indicating the iteration index.
+
+Given $( \hat { b } ^ { r } , z _ { 1 } ^ { r } , z _ { 2 } ^ { r } , z _ { 3 } ^ { r } )$ , update $( { \pmb u } _ { 1 } ^ { r + 1 } , { \pmb u } _ { 2 } ^ { r + 1 } , { \pmb u } _ { 3 } ^ { r + 1 } )$ . Given $( \hat { b } ^ { r } , z _ { 1 } ^ { r } , z _ { 2 } ^ { r } , z _ { 3 } ^ { r } )$ , $( { \pmb u } _ { 1 } , { \pmb u } _ { 2 } , { \pmb u } _ { 3 } )$ are independent, and they can be optimized in parallel, as follows
+
+$$
+\left\{ \begin{array} { l l } { u _ { 1 } ^ { r + 1 } = \underset { u _ { 1 } \in S _ { b } } { \mathrm { a r g } \mathrm { m i n } } \ ( z _ { 1 } ^ { r } ) ^ { \top } ( \hat { b } ^ { r } - u _ { 1 } ) + \frac { \rho _ { 1 } } { 2 } | | \hat { b } ^ { r } - u _ { 1 } | | _ { 2 } ^ { 2 } = \mathcal { P } _ { S _ { b } } ( \hat { b } ^ { r } + \frac { z _ { 1 } ^ { r } } { \rho _ { 1 } } ) , } \\ { u _ { 2 } ^ { r + 1 } = \underset { u _ { 2 } \in S _ { p } } { \mathrm { a r g } \mathrm { m i n } } \ ( z _ { 2 } ^ { r } ) ^ { \top } ( \hat { b } ^ { r } - u _ { 2 } ) + \frac { \rho _ { 2 } } { 2 } | | \hat { b } ^ { r } - u _ { 2 } | | _ { 2 } ^ { 2 } = \mathcal { P } _ { S _ { p } } ( \hat { b } ^ { r } + \frac { z _ { 2 } ^ { r } } { \rho _ { 2 } } ) , } \\ { u _ { 3 } ^ { r + 1 } = \underset { u _ { 3 } \in \mathbb { R } ^ { + } } { \mathrm { a r g } \mathrm { m i n } } z _ { 3 } ^ { r } ( | | b - \hat { b } ^ { r } | | _ { 2 } ^ { 2 } - k + u _ { 3 } ) + \frac { \rho _ { 3 } } { 2 } ( | | b - \hat { b } ^ { r } | | _ { 2 } ^ { 2 } - k + u _ { 3 } ) ^ { 2 } } \\ { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad = \mathcal { P } _ { \mathbb { R } ^ { + } } ( - | | b - \hat { b } ^ { r } | | _ { 2 } ^ { 2 } + k - \frac { z _ { 3 } ^ { r } } { \rho _ { 3 } } ) , } \end{array} \right.
+$$
+
+where $\mathcal { P } _ { S _ { b } } ( \pmb { a } ) = \operatorname* { m i n } ( ( \mathbf { 1 } , \operatorname* { m a x } ( \mathbf { 0 } , \pmb { a } ) )$ with $\mathbf { \pmb { a } } \in \mathbb { R } ^ { n }$ is the projection onto the box constraint $\boldsymbol { S } _ { b }$ ; $\begin{array} { r } { \mathcal { P } _ { S _ { p } } ( \pmb { a } ) = \frac { \sqrt { n } } { 2 } \frac { \bar { \pmb { a } } } { | | \pmb { a } | | } + \frac { 1 } { 2 } } \end{array}$ with $\begin{array} { r } { \bar { \mathbf { a } } = \mathbf { a } - \frac { \mathbf { 1 } } { 2 } } \end{array}$ indicates the projection onto the $\ell _ { 2 }$ -sphere constraint $S _ { p }$ (Wu & Ghanem, 2018); ${ \mathcal { P } } _ { \mathbb { R } ^ { + } } ( a ) = \operatorname* { m a x } ( 0 , a )$ with $a \in \mathbb { R }$ indicates the projection onto $\mathbb { R } ^ { + }$ .
+
+Given $( { \pmb u } _ { 1 } ^ { r + 1 } , { \pmb u } _ { 2 } ^ { r + 1 } , { \pmb u } _ { 3 } ^ { r + 1 } , z _ { 1 } ^ { r } , z _ { 2 } ^ { r } , z _ { 3 } ^ { r } )$ , update $\hat { b } ^ { r + 1 }$ . Although there is no closed-form solution to $\hat { b } ^ { r + 1 }$ , it can be easily updated by the gradient descent method, as both $\mathcal { L } _ { 1 } ( \hat { b } )$ and $\mathcal { L } _ { 2 } ( \hat { b } )$ are differentiable w.r.t. $\hat { b }$ , as follows
+
+$$
+\hat { \boldsymbol { b } } ^ { r + 1 } \gets \hat { \boldsymbol { b } } ^ { r } - \eta \cdot \frac { \partial L ( \hat { \boldsymbol { b } } , \boldsymbol { u } _ { 1 } ^ { r + 1 } , \boldsymbol { u } _ { 2 } ^ { r + 1 } , \boldsymbol { u } _ { 3 } ^ { r + 1 } , z _ { 1 } ^ { r } , z _ { 2 } ^ { r } , z _ { 3 } ^ { r } ) } { \partial \hat { \boldsymbol { b } } } \Big | _ { \hat { \boldsymbol { b } } = \hat { \boldsymbol { b } } ^ { r } } ,
+$$
+
+where $\eta > 0$ denotes the step size. Note that we can run multiple steps of gradient descent in the above update. Both the number of steps and $\eta$ will be specified in later experiments. Besides, due to the space limit, the detailed derivation of $\partial L / \partial \hat { \boldsymbol { b } }$ will be presented in Appendix A.
+
+Given $( \hat { b } ^ { r + 1 } , { \pmb u } _ { 1 } ^ { r + 1 } , { \pmb u } _ { 2 } ^ { r + 1 } , u _ { 3 } ^ { r + 1 } )$ , update $( z _ { 1 } ^ { r + 1 } , z _ { 2 } ^ { r + 1 } , z _ { 3 } ^ { r + 1 } )$ . The dual variables are updated by the gradient ascent method, as follows
+
+$$
+\left\{ \begin{array} { l } { z _ { 1 } ^ { r + 1 } = z _ { 1 } ^ { r } + \rho _ { 1 } ( \hat { b } ^ { r + 1 } - { \pmb u } _ { 1 } ^ { r + 1 } ) , } \\ { z _ { 2 } ^ { r + 1 } = z _ { 2 } ^ { r } + \rho _ { 2 } ( \hat { b } ^ { r + 1 } - { \pmb u } _ { 2 } ^ { r + 1 } ) , } \\ { z _ { 3 } ^ { r + 1 } = z _ { 3 } ^ { r } + \rho _ { 3 } ( | | { \pmb b } - \hat { { \pmb b } } ^ { r + 1 } | | _ { 2 } ^ { 2 } - k + u _ { 3 } ^ { r + 1 } ) . } \end{array} \right.
+$$
+
+Remarks. 1) Note that since $( { \pmb u } _ { 1 } ^ { r + 1 } , { \pmb u } _ { 2 } ^ { r + 1 } , { \pmb u } _ { 3 } ^ { r + 1 } )$ are updated in parallel, their updates belong to the same block. Thus, the above algorithm is a two-block ADMM algorithm. We provide the algorithm outline in Appendix B. 2) Except for the update of $\hat { b } ^ { r + 1 }$ , all other updates are very simple and efficient. The computational cost of the whole algorithm will be analyzed in Appendix C. 3) Due to the inexact solution to $\hat { b } ^ { r + 1 }$ using gradient descent, the theoretical convergence of the whole ADMM algorithm cannot be guaranteed. However, as demonstrated in many previous works (Gol’shtein & Tret’yakov, 1979; Eckstein & Bertsekas, 1992; Boyd et al., 2011), the inexact two-block ADMM often shows good practical convergence, which is also the case in our later experiments. Besides, the numerical convergence analysis is presented in Appendix D. 4) The proper adjustment of $( \rho _ { 1 } , \rho _ { 2 } , \rho _ { 3 } )$ could accelerate the practical convergence, which will be specified later .
+
+# 4 EXPERIMENTS
+
+# 4.1 EVALUATION SETUP
+
+Settings. We compare our method (TA-LBF) with GDA (Liu et al., 2017a), FSA (Zhao et al., 2019), T-BFA (Rakin et al., 2020b), and TBT (Rakin et al., 2020a). All those methods can be adopted to misclassify a specific image into a target class. We also take the fine-tuning (FT) of the last fully-connected layer as a baseline method. We conduct experiments on CIFAR-10 (Krizhevsky et al., 2009) and ImageNet (Russakovsky et al., 2015). We randomly select 1,000 images from each dataset as the evaluation set for all methods. Specifically, for each of the 10 classes in CIFAR-10, we perform attacks on the 100 randomly selected validation images from the other 9 classes. For ImageNet, we randomly choose 50 target classes. For each target class, we perform attacks on 20 images randomly selected from the rest classes in the validation set. Besides, for all methods except GDA which does not employ auxiliary samples, we provide 128 and 512 auxiliary samples on CIFAR-10 and ImageNet, respectively. Following the setting in (Rakin et al., 2020a;b), we adopt the quantized ResNet (He et al., 2016) and VGG (Simonyan & Zisserman, 2015) as the target models. For our TA-LBF, the trade-off parameter $\lambda$ and the constraint parameter $k$ affect the attack stealthiness and the attack success rate. We adopt a strategy for jointly searching $\lambda$ and $k$ , which is specified in Appendix E.3. More descriptions of our settings are provided in Appendix E.
+
+Evaluation Metrics. We adopt three metrics to evaluate the attack performance, i.e., the post attack accuracy (PA-ACC), the attack success rate (ASR), and the number of bit-flips $( \mathrm { N _ { f l i p } } )$ . PA-ACC denotes the post attack accuracy on the validation set except for the specific attacked sample and the auxiliary samples. ASR is defined as the ratio of attacked samples that are successfully attacked into the target class among all 1,000 attacked samples. $\mathrm { { N _ { f l i p } } }$ is the number of bit-flips required for an attack. A better attack performance corresponds to a higher PA-ACC and ASR, while a lower $\mathrm { { N _ { f l i p } } }$ . Besides, we also show the accuracy of the original model, denoted as ACC.
+
+# 4.2 MAIN RESULTS
+
+Results on CIFAR-10. The results of all methods on CIFAR-10 are shown in Table 1. Our method achieves a $100 \%$ ASR with the fewest $\mathrm { { N _ { f l i p } } }$ for all the bit-widths and architectures. FT modifies the maximum number of bits among all methods since there is no limitation of parameter modifications. Due to the absence of the training data, the PA-ACC of FT is also poor. These results indicate that fine-tuning the trained DNN as an attack method is infeasible. Although T-BFA flips the secondfewest bits under three cases, it fails to achieve a higher ASR than GDA and FSA. In terms of PA-ACC, TA-LBF is comparable to other methods. Note that the PA-ACC of TA-LBF significantly outperforms that of GDA, which is the most competitive w.r.t. ASR and $\mathrm { { N _ { f l i p } } }$ among all the baseline methods. The PA-ACC of GDA is relatively poor, because it does not employ auxiliary samples. Achieving the highest ASR, the lowest $\mathrm { { N _ { f l i p } } }$ , and the comparable PA-ACC demonstrates that our optimization-based method is more superior than other heuristic methods (TBT, T-BFA and GDA).
+
+Table 1: Results of all attack methods across different bit-widths and architectures on CIFAR-10 and ImageNet (bold: the best; underline: the second best). The mean and standard deviation of PA-ACC and $\mathrm { { N _ { f l i p } } }$ are calculated by attacking the 1,000 images. Our method is denoted as TA-LBF.   
+
+<table><tr><td>Dataset</td><td>Method</td><td>Target Model</td><td>PA-ACC (%) 85.01±2.90</td><td>ASR (%)</td><td>Nfip</td><td>Target Model</td><td>PA-ACC (%</td><td>ASR (%)</td><td>Nfip</td></tr><tr><td colspan="16" rowspan="19" rowspan="19">CIPAIII1</td><td>84.31±3.10 VGG</td><td>98.7</td><td></td><td>11298.74±830.36</td></tr><tr><td>100.0 97.3</td><td>1507.51±86.54 246.70±8.19</td><td>77.79±23.35</td><td>51.6</td><td>599.40±19.53</td></tr><tr><td>ResNet 8-bit</td><td>88.07±0.84 87.56±2.22 98.7</td><td></td><td>9.91±2.33</td><td>8-bit 89.83±3.92</td><td>96.7</td><td>14.53±3.74</td></tr><tr><td>T-BFA FSA ACC:</td><td>88.38±2.28</td><td>98.9</td><td>185.51±54.93</td><td>88.80±2.86 ACC:</td><td>96.8</td><td>253.92±122.06</td></tr><tr><td>GDA 92.16%</td><td>86.73±3.50</td><td>99.8</td><td>26.83±12.50</td><td>85.51±2.88</td><td>100.0</td><td></td></tr><tr><td>TA-LBF</td><td>88.20±2.64</td><td>100.0</td><td>5.57±1.58</td><td>93.20% 86.06±3.17</td><td>100.0</td><td>21.54±6.79 7.40±2.72</td></tr><tr><td>FT</td><td>84.37±2.94</td><td>100.0</td><td>392.48±47.26</td><td>VGG</td><td>83.31±3.76 94.5</td><td>2270.52±324.69</td></tr><tr><td>TBT</td><td>ResNet 4-bit</td><td>87.79±1.86 96.0</td><td>118.20±15.03</td><td>4-bit</td><td>83.90±2.63</td><td>62.4</td></tr><tr><td>T-BFA</td><td></td><td>86.46±2.80 97.9</td><td>8.80±2.01</td><td>ACC:</td><td>88.74±4.52 96.2</td><td>266.40±18.70 11.23±2.36</td></tr><tr><td>FSA</td><td>87.73±2.36 ACC:</td><td>98.4</td><td>76.83±25.27</td><td>87.58±3.06</td><td>97.5</td><td>75.03±29.75</td></tr><tr><td>GDA</td><td>86.25±3.59</td><td>99.8</td><td>14.08±7.94</td><td>85.08±2.82</td><td>100.0</td><td>10.31±3.77</td></tr><tr><td>TA-LBF</td><td>91.90% 87.82±2.60</td><td>100.0</td><td>5.25±1.09</td><td>92.61%</td><td>85.91±3.29 100.0</td><td>6.26±2.37</td></tr><tr><td rowspan="12">aeegee</td><td></td><td>59.33±0.93</td><td>100.0</td><td>277424.29±12136.34</td><td>VGG</td><td>62.08±2.33</td><td>100.0 1729685.22±137539.54</td></tr><tr><td>FT ResNet TBT 8-bit</td><td>69.18±0.03</td><td>99.9</td><td>577.40±19.42</td><td>8-bit</td><td>72.99±0.02 99.2</td><td>4115.26±191.25</td></tr><tr><td>T-BFA</td><td>68.71±0.36</td><td>79.3</td><td>24.57±20.03</td><td>73.09±0.12</td><td>84.5</td><td>363.78±153.28</td></tr><tr><td>FSA ACC:</td><td>69.27±0.15</td><td>99.7</td><td>441.21±119.45</td><td>73.28±0.03</td><td>100.0</td><td>1030.03±260.30</td></tr><tr><td>GDA 69.50%</td><td>69.26±0.22</td><td>100.0</td><td>18.54±6.14</td><td>ACC: 73.29±0.02</td><td>100.0</td><td></td></tr><tr><td>TA-LBF</td><td>69.41±0.08</td><td>100.0</td><td>7.37±2.18</td><td>73.31% 73.28±0.03</td><td>100.0</td><td>197.05±49.85</td></tr><tr><td>FT</td><td></td><td>15.65±4.52 100.0</td><td>135854.50±21399.94</td><td>VGG</td><td>17.76±1.71</td><td>100.0</td><td>69.89±18.42 1900751.70±37329.44</td></tr><tr><td>TBT</td><td>ResNet</td><td>99.8</td><td>271.24±15.98</td><td>4-bit</td><td>71.18±0.03</td><td>100.0</td><td>3231.00±345.68</td></tr><tr><td>T-BFA</td><td>4-bit</td><td>66.36±0.07</td><td></td><td></td><td>71.49±0.15</td><td></td><td></td></tr><tr><td>FSA</td><td></td><td>65.86±0.42</td><td>80.4</td><td>24.79±19.02</td><td>71.69±0.09</td><td>84.3</td><td>350.33±158.57</td></tr><tr><td></td><td>ACC:</td><td>66.44±0.21</td><td>99.9</td><td>157.53±33.66</td><td>ACC:</td><td>100.0</td><td>441.32±111.26</td></tr><tr><td>GDA</td><td>66.77%</td><td>66.54±0.22</td><td>100.0</td><td>11.45±3.82</td><td>71.76%</td><td>71.73±0.03</td><td>100.0</td><td>107.18±28.70</td></tr><tr><td>TA-LBF</td><td></td><td>66.69±0.07</td><td>100.0</td><td>7.96±2.50</td><td></td><td>71.73±0.03</td><td>100.0</td><td>69.72±18.84</td></tr></table>
+
+Results on ImageNet. The results on ImageNet are shown in Table 1. It can be observed that GDA shows very competitive performance compared to other methods. However, our method obtains the highest PA-ACC, the fewest bit-flips (less than 8), and a $100 \%$ ASR in attacking ResNet. For VGG, our method also achieves a $100 \%$ ASR with the fewest $\mathrm { { N _ { f l i p } } }$ for both bit-widths. The $\mathrm { { N _ { f l i p } } }$ results of our method are mainly attributed to the cardinality constraint on the number of bit-flips. Moreover, for our method, the average PA-ACC degradation over four cases on ImageNet is only $0 . 0 6 \%$ , which demonstrates the stealthiness of our attack. When comparing the results of ResNet and VGG, an interesting observation is that all methods require significantly more bit-flips for VGG. One reason is that VGG is much wider than ResNet. Similar to the claim in (He et al., 2020), increasing the network width contributes to the robustness against the bit-flip based attack.
+
+# 4.3 RESISTANCE TO DEFENSE METHODS
+
+Resistance to Piece-wise Clustering. He et al. (2020) proposed a novel training technique, called piece-wise clustering, to enhance the network robustness against the bit-flip based attack. Such a training technique introduces an additional weight penalty to the inference loss, which has the effect of eliminating close-to-zero weights (He et al., 2020). We test the resistance of all attack methods to the piece-wise clustering. We conduct experiments with the 8-bit quantized ResNet on CIFAR-10 and ImageNet. Following the ideal configuration in (He et al., 2020), the clustering coefficient, which is a hyper-parameter of piece-wise clustering, is set to 0.001 in our evaluation. For our method, the initial $k$ is set to 50 on ImageNet and the rest settings are the same as those in Section 4.1. Besides the three metrics in Section 4.1, we also present the number of increased $\mathrm { { N _ { f l i p } } }$ compared to the model without defense (i.e., results in Table 1), denoted as $\Delta \mathrm { N _ { f l i p } }$ .
+
+The results of the resistance to the piece-wise clustering of all attack methods are shown in Table 2. It shows that the model trained with piece-wise clustering can improve the number of required bit-flips for all attack methods. However, our method still achieves a $100 \%$ ASR with the least number of bit-flips on both two datasets. Although TBT achieves a smaller $\Delta \mathrm { N _ { f l i p } }$ than ours on CIFAR-10, its ASR is only $5 2 . 3 \%$ , which also verifies the defense effectiveness of the piece-wise clustering. Compared with other methods, TA-LBF achieves the fewest $\Delta \mathrm { N _ { f l i p } }$ on ImageNet and the best PA-ACC on both datasets. These results demonstrate the superiority of our method over other methods when attacking models trained with piece-wise clustering.
+
+Table 2: Results of all attack methods against the models with defense on CIFAR-10 and ImageNet (bold: the best; underline: the second best). The mean and standard deviation of PA-ACC and $\mathrm { { N _ { f l i p } } }$ are calculated by attacking the 1,000 images. Our method is denoted as TA-LBF. $\Delta \mathrm { N _ { f l i p } }$ denotes the increased $\mathrm { { N _ { f l i p } } }$ compared to the corresponding result in Table 1.   
+
+<table><tr><td rowspan=1 colspan=1>Defense</td><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>ACC(%</td><td rowspan=1 colspan=1>PA-ACC    ASR         Nflip           △Nfip(%      (%</td></tr><tr><td rowspan=6 colspan=1>Prreerreseeceeecs</td><td rowspan=4 colspan=1>CIFAIIIO</td><td rowspan=4 colspan=1>FTTBTT-BFAFSAGDATA-LBF</td><td rowspan=4 colspan=1>91.01</td><td rowspan=1 colspan=1>84.06±3.56    99.5      1893.55±68.98       386.04</td></tr><tr><td rowspan=1 colspan=1>87.05±1.69    52.3       254.20±10.22        7.50</td></tr><tr><td rowspan=1 colspan=1>85.82±1.89    98.6       45.51±9.47         35.6086.61±2.51    98.6      246.11±75.36        60.6084.12±4.77   100.0      52.76±16.29         25.93</td></tr><tr><td rowspan=1 colspan=1>87.30±2.74    100.0       18.93±7.11         13.36</td></tr><tr><td rowspan=2 colspan=1>1aeeeee</td><td rowspan=2 colspan=1>FTTBTT-BFAFSAGDATA-LBF</td><td rowspan=2 colspan=1>63.62</td><td rowspan=1 colspan=1>43.44±2.07    92.2    762267.56±52179.46   484843.2763.07±0.04    81.8      1184.14±30.30       606.74</td></tr><tr><td rowspan=1 colspan=1>62.82±0.27    90.1      273.56±191.29       248.99</td></tr><tr><td rowspan=10 colspan=1>ereereereter</td><td rowspan=4 colspan=1>CI-AIIIT</td><td rowspan=4 colspan=1>FTTBTT-BFAFSAGDATA-LBF</td><td rowspan=4 colspan=1>94.29</td><td rowspan=1 colspan=1>86.46±2.84    100.0     2753.43±188.27      1245.92</td></tr><tr><td rowspan=1 colspan=1>89.72±2.99    89.5      366.90±12.09        120.20</td></tr><tr><td rowspan=1 colspan=1>91.16±1.42    98.7       17.91±4.64         8.0090.70±2.37    98.5      271.27±65.18        85.76</td></tr><tr><td rowspan=1 colspan=1>89.83±3.02   100.0      48.96±21.03         22.1390.96±2.63    100.0       8.79±2.44          3.22</td></tr><tr><td rowspan=6 colspan=1>Jeagee</td><td rowspan=6 colspan=1>FTTBTT-BFAFSAGDATA-LBF</td><td rowspan=6 colspan=1>71.35</td><td rowspan=1 colspan=1>63.51±1.29    100.0   507456.61±34517.04    230032.32</td></tr><tr><td rowspan=1 colspan=1>71.12±0.04    99.9      1138.34±44.23       560.94</td></tr><tr><td rowspan=1 colspan=1>70.84±0.30    88.9       40.23±27.29         15.66</td></tr><tr><td rowspan=1 colspan=1>71.30±0.04   100.0      449.70±106.42        8.49</td></tr><tr><td rowspan=1 colspan=1>71.30±0.05   100.0       20.01±6.04          1.47</td></tr><tr><td rowspan=1 colspan=1>71.30±0.04    100.0       8.48±2.52          1.11</td></tr></table>
+
+![](images/7850006755107c70048d1572f16f755a12adc7c14126cb92e99c44cc3ff87ea1.jpg)  
+Figure 2: Results of TA-LBF with different parameters $\lambda , k$ , and the number of auxiliary samples $N$ on CIFAR-10. Regions in shadow indicate the standard deviation of attacking the 1,000 images.
+
+Resistance to Larger Model Capacity. Previous studies (He et al., 2020; Rakin et al., 2020b) observed that increasing the network capacity can improve the robustness against the bit-flip based attack. Accordingly, we evaluate all attack methods against the models with a larger capacity using the 8-bit quantized ResNet on both datasets. Similar to the strategy in (He et al., 2020), we increase the model capacity by varying the network width (i.e., $2 \times$ width in our experiments). All settings of our method are the same as those used in Section 4.1.
+
+The results are presented in Table 2. We observe that all methods require more bit-flips to attack the model with the $2 \times$ width. To some extent, it demonstrates that the wider network with the same architecture is more robust against the bit-flip based attack. However, our method still achieves a $100 \%$ ASR with the fewest $\mathrm { { N _ { f l i p } } }$ and $\Delta \mathrm { N _ { f l i p } }$ . Moreover, when comparing the two defense methods, we find that piece-wise clustering performs better than the model with a larger capacity in terms of $\Delta \mathrm { N _ { f l i p } }$ . However, piece-wise clustering training also causes the accuracy decrease of the original model (e.g., from $9 2 . 1 6 \%$ to $9 1 . 0 1 \%$ on CIFAR-10). We provide more results in attacking models with defense under different settings in Appendix $\mathbf { F }$ .
+
+![](images/129dd219c5a4c72c37d6fbb3f19e93c93cb229ae23a639c705a4f769da5f6d17.jpg)  
+Figure 3: Visualization of decision boundaries of the original model and the post attack models. The attacked sample from Class 3 is misclassified into the Class 1 by FSA, GDA, and our method.
+
+# 4.4 ABLATION STUDY
+
+We perform ablation studies on parameters $\lambda$ and $k$ , and the number of auxiliary samples $N$ . We use the 8-bit quantized ResNet on CIFAR-10 as the representative for analysis. We discuss the attack performance of TA-LBF under different values of $\lambda$ while $k$ is fixed at 20, and under different values of $k$ while $\lambda$ is fixed at 10. To analyze the effect of $N$ , we configure $N$ from 25 to 800 and keep other settings the same as those in Section 4.1. The results are presented in Fig. 2. We observe that our method achieves a $100 \%$ ASR when $\lambda$ is less than 20. As expected, the PA-ACC increases while the ASR decreases along with the increase of $\lambda$ . The plot of parameter $k$ presents that $k$ can exactly limit the number of bit-flips, while other attack methods do not involve such constraint. This advantage is critical since it allows the attacker to identify limited bits to perform an attack when the budget is fixed. As shown in the figure, the number of auxiliary samples less than 200 has a marked positive impact on the PA-ACC. It’s intuitive that more auxiliary samples can lead to a better PA-ACC. The observation also indicates that TA-LBF still works well without too many auxiliary samples.
+
+# 4.5 VISUALIZATION OF DECISION BOUNDARY
+
+To further compare FSA and GDA with our method, we visualize the decision boundaries of the original and the post attack models in Fig. 3. We adopt a four-layer Multi-Layer Perceptron trained with the simulated 2-D Blob dataset from 4 classes. The original decision boundary indicates that the original model classifies all data points almost perfectly. The attacked sample is classified into Class 3 by all methods. Visually, GDA modifies the decision boundary drastically, especially for Class 0. However, our method modifies the decision boundary mainly around the attacked sample. Althoug FSA is comparable to ours visually in Fig. 3, it flips $1 0 \times$ bits than GDA and TA-LBF. In terms of the numerical results, TA-LBF achieves the best PA-ACC and the fewest $\mathrm { { N _ { f l i p } } }$ . This finding verifies that our method can achieve a successful attack even only tweaking the original classifier.
+
+# 5 CONCLUSION
+
+In this work, we have presented a novel attack paradigm that the weights of a deployed DNN can be slightly changed via bit flipping in the memory, to give a target prediction for a specific sample, while the predictions on other samples are not significantly influenced. Since the weights are stored as binary bits in the memory, we formulate this attack as a binary integer programming (BIP) problem, which can be effectively and efficiently solved by a continuous algorithm. Since the critical bits are determined through optimization, the proposed method can achieve the attack goals by flipping a few bits, and it shows very good performance under different experimental settings.
+
+# ACKNOWLEDGMENTS
+
+This work is supported in part by the National Key Research and Development Program of China under Grant 2018YFB1800204, the National Natural Science Foundation of China under Grant 61771273, the R&D Program of Shenzhen under Grant JCYJ20180508152204044. Baoyuan Wu is supported by the Natural Science Foundation of China under grant No. 62076213, and the university development fund of the Chinese University of Hong Kong, Shenzhen under grant No. 01001810.
+
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+
+# A UPDATE $\hat { b }$ BY GRADIENT DESCENT
+
+In this section, we derive the gradient of $\textit { L w } . r . t . \textit { \textbf { b } }$ , which is adopted to update $\hat { b } ^ { r + 1 }$ by gradient descent (see Eq. (10)). The derivation consists of the following parts.
+
+Derivation of $\partial \mathcal { L } _ { 1 } ( \hat { b } ) / \partial \hat { b }$ . For clarity, here we firstly repeat some definitions,
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { L } _ { 1 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) = \operatorname* { m a x } \big ( m - p ( x ; \boldsymbol { \Theta } , \hat { \mathbf { B } } _ { t } ) + \delta , 0 \big ) + \operatorname* { m a x } \big ( p ( x ; \boldsymbol { \Theta } , \hat { \mathbf { B } } _ { s } ) - m + \delta , 0 \big ) , } \\ { \displaystyle p ( x ; \boldsymbol { \Theta } , \hat { \mathbf { B } } _ { i } ) = [ h ( \hat { \mathbf { B } } _ { i , 1 } ) ; h ( \hat { \mathbf { B } } _ { i , 2 } ) ; . . . ; h ( \hat { \mathbf { B } } _ { i , C } ) ] ^ { \top } g ( x ; \boldsymbol { \Theta } ) , } \\ { \displaystyle h ( v ) = ( - 2 ^ { Q - 1 } \cdot v _ { Q } + \sum _ { i = 1 } ^ { Q - 1 } 2 ^ { i - 1 } \cdot v _ { i } ) \cdot \Delta ^ { l } . } \end{array}
+$$
+
+Then, we obtain that
+
+$$
+\frac { \partial p ( x ; \boldsymbol { \Theta } , \hat { \mathbf { B } } _ { i } ) } { \partial \hat { \mathbf { B } } _ { i } } = [ g _ { 1 } ( x ; \boldsymbol { \Theta } ) \cdot ( \frac { \nabla h ( \hat { \mathbf { B } } _ { i , 1 } ) } { \partial \hat { \mathbf { B } } _ { i , 1 } } ) ^ { \top } ; . . . ; g _ { C } ( x ; \boldsymbol { \Theta } ) \cdot ( \frac { \nabla h ( \hat { \mathbf { B } } _ { i , C } ) } { \nabla \hat { \mathbf { B } } _ { i , C } } ) ^ { \top } ] ,
+$$
+
+where $\begin{array} { r } { \frac { \nabla h ( \pmb { v } ) } { \nabla \pmb { v } } = [ 2 ^ { 0 } ; 2 ^ { 1 } , \dots , 2 ^ { Q - 2 } ; - 2 ^ { Q - 1 } ] \cdot \Delta ^ { l } } \end{array}$ is a constant, and here $l$ indicates the last layer; $g _ { j } ( { \pmb x } ; { \Theta } )$ ∇v  denotes the $j$ -th entry of the vector $g ( \pmb { x } ; \mathbf { \Theta } \Theta )$ . Utilizing (15), we have
+
+$$
+\begin{array} { r l } & { \frac { \partial \mathcal { L } _ { 1 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) } { \partial \hat { \mathbf { B } } _ { s } } = \left\{ \begin{array} { l l } { \frac { \partial p ( x ; \Theta , \hat { \mathbf { B } } _ { s } ) } { \partial \hat { \mathbf { B } } _ { s } } , \mathrm { ~ i f ~ } p ( x ; \Theta , \mathbf { B } _ { s } ) > m - \delta } \\ { \mathbf { 0 } , \mathrm { ~ o t h e r w i s e } } \end{array} \right. , } \\ & { \frac { \partial \mathcal { L } _ { 1 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) } { \partial \hat { \mathbf { B } } _ { t } } = \left\{ \begin{array} { l l } { - \frac { \partial p ( x ; \Theta , \hat { \mathbf { B } } _ { t } ) } { \partial \hat { \mathbf { B } } _ { t } } , \mathrm { ~ i f ~ } p ( x ; \Theta , \mathbf { B } _ { t } ) < m + \delta } \\ { \mathbf { 0 } , \mathrm { ~ o t h e r w i s e } } \end{array} \right. . } \end{array}
+$$
+
+Thus, we obtain that
+
+$$
+\frac { \partial \mathcal { L } _ { 1 } ( \hat { \boldsymbol { b } } ) } { \partial \hat { \boldsymbol { b } } } = \big [ \mathrm { R e s h a p e } \big ( \frac { \partial \mathcal { L } _ { 1 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) } { \partial \hat { \mathbf { B } } _ { s } } \big ) ; \mathrm { R e s h a p e } \big ( \frac { \partial \mathcal { L } _ { 1 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) } { \partial \hat { \mathbf { B } } _ { t } } \big ) \big ] ,
+$$
+
+where Reshape $( \cdot )$ elongates a matrix to a vector along the column.
+
+Derivation of $\partial \mathcal { L } _ { 2 } ( \hat { b } ) / \partial \hat { b }$ . For clarity, here we firstly repeat the following definition
+
+$$
+\mathcal { L } _ { 2 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) = \sum _ { i = 1 } ^ { N } \ell \big ( f ( \pmb { x } _ { i } ; \boldsymbol { \Theta } , \mathbf { B } _ { \{ 1 , \dots , K \} \setminus \{ s , t \} } , \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) , y _ { i } \big ) ,
+$$
+
+where $f _ { j } ( { \pmb x } _ { i } ; \Theta , { \pmb \mathsf { B } } _ { \{ 1 , \dots , K \} \backslash \{ s , t \} } , { \hat { \pmb \mathsf { B } } } _ { s } , { \hat { \pmb \mathsf { B } } } _ { t } ) = \operatorname { S o f t m a x } ( p ( { \pmb x } _ { i } ; \Theta , { \hat { \pmb \mathsf { B } } } _ { j } ) )$ or $\mathrm { S o f t m a x } ( p ( { \pmb x } _ { i } ; { \pmb \Theta } , { \pmb B } _ { j } ) )$ indicates the posterior probability of $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ w.r.t. class $j$ , and we simply denote $f ( \pmb { x } _ { i } ) \in [ 0 , 1 ] ^ { K }$ as the posterior probability vector of $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ . $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$ denotes the auxiliary sample set. $\ell ( \cdot , \cdot )$ is specified as the cross entropy loss. Then, we have
+
+$$
+\begin{array} { r l } & { \displaystyle \frac { \partial \mathcal { L } _ { 2 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) } { \partial \hat { \mathbf { B } } _ { s } } = \sum _ { i = 1 } ^ { N } \bigg [ \big ( \mathbb { I } ( y _ { i } = s ) - f _ { s } ( \pmb { x } _ { i } ; \boldsymbol { \Theta } , \mathbf { B } _ { \{ 1 , \dots , K \} \setminus \{ s , t \} } , \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) \big ) \cdot \frac { \partial p ( \pmb { x } _ { i } ; \boldsymbol { \Theta } , \hat { \mathbf { B } } _ { s } ) } { \partial \hat { \mathbf { B } } _ { s } } \bigg ] , } \\ & { \displaystyle \frac { \partial \mathcal { L } _ { 2 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) } { \partial \hat { \mathbf { B } } _ { t } } = \sum _ { i = 1 } ^ { N } \bigg [ \big ( \mathbb { I } ( y _ { i } = t ) - f _ { t } ( \pmb { x } _ { i } ; \boldsymbol { \Theta } , \mathbf { B } _ { \{ 1 , \dots , K \} \setminus \{ s , t \} } , \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) \big ) \cdot \frac { \partial p ( \pmb { x } _ { i } ; \boldsymbol { \Theta } , \hat { \mathbf { B } } _ { t } ) } { \partial \hat { \mathbf { B } } _ { t } } \bigg ] , } \end{array}
+$$
+
+where $\mathbb { I } ( a ) = 1$ of $a$ is true, otherwise $\mathbb { I } ( a ) = 0$ . Thus, we obtain
+
+$$
+\frac { \partial \mathcal { L } _ { 2 } ( \hat { \boldsymbol { b } } ) } { \partial \hat { \boldsymbol { b } } } = \bigg [ \mathrm { R e s h a p e } \big ( \frac { \partial \mathcal { L } _ { 2 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) } { \partial \hat { \mathbf { B } } _ { s } } \big ) ; \mathrm { R e s h a p e } \big ( \frac { \partial \mathcal { L } _ { 2 } ( \hat { \mathbf { B } } _ { s } , \hat { \mathbf { B } } _ { t } ) } { \partial \hat { \mathbf { B } } _ { t } } \big ) \bigg ] .
+$$
+
+Derivation of $\partial L ( \hat { b } ) / \partial \hat { b }$ . According to Eq. (8), and utilizing Eqs. (17) and (21), we obtain
+
+$$
+\frac { \partial L ( \hat { b } ) } { \partial \hat { b } } = \frac { \partial \mathcal { L } _ { 1 } ( \hat { b } ) } { \partial \hat { b } } + \frac { \partial \mathcal { L } _ { 2 } ( \hat { b } ) } { \partial \hat { b } } + z _ { 1 } + z _ { 2 } + \rho _ { 1 } ( \hat { b } - u _ { 1 } ) + \rho _ { 2 } ( \hat { b } - u _ { 2 } ) + 2 ( \hat { b } - b ) \cdot \big [ z _ { 3 } + \rho _ { 3 } \lvert \lvert \hat { b } - b \rvert \rvert _ { 2 } ^ { 2 } - k + u _ { 3 } \big ] .
+$$
+
+# B ALGORITHM OUTLINE
+
+# Algorithm 1 Continuous optimization for the BIP problem (5).
+
+Input: The original quantized DNN model $f$ with weights $\Theta , \mathsf { B }$ , attacked sample $_ { \textbf { \em x } }$ with groundtruth label $s$ , target class $t$ , auxiliary sample set $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$ , hyper-parameters $\lambda , k$ , and $\delta$ .
+
+# Output: $\hat { b }$
+
+1: Initial $\pmb { u } _ { 1 } ^ { 0 }$ , ${ \pmb u } _ { 2 } ^ { 0 }$ , $u _ { 3 } ^ { 0 }$ , $z _ { 1 } ^ { 0 }$ , $z _ { 2 } ^ { 0 }$ , $z _ { 3 } ^ { 0 }$ , $\hat { b } ^ { 0 }$ and let $r \gets 0$ ;   
+2: while not converged do   
+3: Update $\pmb { u } _ { 1 } ^ { r + 1 }$ , $u _ { 2 } ^ { r + 1 }$ and $u _ { 3 } ^ { r + 1 }$ as Eq. (9);   
+4: Update $\hat { \pmb { b } } ^ { r + 1 }$ as Eq. (10);   
+5: Update $z _ { 1 } ^ { r + 1 }$ +1 , z r+12 a nd $z _ { 3 } ^ { r + 1 }$ as Eq. (11);   
+6: $r  r + 1$ .
+
+# C COMPLEXITY ANALYSIS
+
+Table 3: Running time (seconds) of attacking one image for different methods. The mean and standard deviation are calculated by 10 attacks.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>FT</td><td rowspan=1 colspan=1>TBT</td><td rowspan=1 colspan=1>T-BFA</td><td rowspan=1 colspan=1>FSA</td><td rowspan=1 colspan=1>GDA</td><td rowspan=1 colspan=1>TA-LBF</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10ImageNet</td><td rowspan=1 colspan=1>15.54±1.64124.32±3.61</td><td rowspan=1 colspan=1>389.12±27.7931425.81±540.60</td><td rowspan=1 colspan=1>35.05±15.7919.16±3.52</td><td rowspan=1 colspan=1>2.71±0.4865.28±2.49</td><td rowspan=1 colspan=1>0.67±0.5461.97±1.59</td><td rowspan=1 colspan=1>113.38±6.54222.95±9.39</td></tr></table>
+
+7: end while
+
+The computational complexity of the proposed algorithm (i.e., Algorithm 1) consists of two parts, the forward and backward pass. In terms of the forward pass, since $\Theta , \mathsf { B } _ { \{ 1 , \dots , K \} \backslash \{ s , t \} }$ are fixed during the optimization, their involved terms, including $g ( \pmb { x } ; \Theta )$ and $p ( \pmb { x } ; \pmb { \Theta } , \pmb { \mathsf { B } } _ { i } ) | _ { i \neq s , t }$ , are calculated only one time. The main cost from $\hat { \mathbf { B } } _ { s }$ and $\hat { \mathbf { B } } _ { t }$ is $O ( 2 ( N + 1 ) C ^ { 2 } Q )$ per iteration, as there are $N + 1$ samples. In terms of the backward pass, the main cost is from the update of $\hat { b } ^ { r + 1 }$ , which is $O ( 2 ( N + 1 ) C Q )$ per iteration in the gradient descent. Since all other updates are very simple, their costs are omitted here. Thus, the overall computational cost is $O \big ( T _ { o u t e r } [ 2 ( \dot { N } + 1 ) \dot { C } Q \cdot ( C + T _ { i n n e r } ) ] \big )$ , with $T _ { o u t e r }$ being the iteration of the overall algorithm and $T _ { i n n e r }$ indicating the number of gradient steps in updating $\hat { b } ^ { r + 1 }$ . As shown in Section D, the proposed method TA-LBF always converges very fast in our experiments, thus $T _ { o u t e r }$ is not very large. As demonstrated in Section E.3, $T _ { i n n e r }$ is set to 5 in our experiments. In short, the proposed method can be optimized very efficiently.
+
+Besides, we also compare the computational complexity of different attacks empirically. Specifically, we compare the running time of attacking one image of different methods against the 8-bit quantized ResNet on CIFAR-10 and ImageNet dataset. As shown in Table 3, TBT is the most timeconsuming method among all attacks. Although the proposed TA-LBF is not superior to T-BFA, FSA, and GDA in running time, this gap can be tolerated when attacking a single image in the deployment stage. Besides, our method performs better in terms of PA-ACC, ASR, and $\mathrm { { N _ { f l i p } } }$ as demonstrated in our experiments.
+
+# D NUMERICAL CONVERGENCE ANALYSIS
+
+We present the numerical convergence of TA-LBF in Fig. 4. Note that $| | \hat { b } - u _ { 1 } | | _ { 2 } ^ { 2 }$ and $| | \hat { b } - u _ { 2 } | | _ { 2 } ^ { 2 }$ characterize the degree of satisfaction of the box and $\ell _ { 2 }$ -sphere constraint, respectively. For the two examples of CIFAR-10 and ImageNet, the values of both indicators first increase, then drop, and finally close to 0. Another interesting observation is that $\mathcal { L } _ { 1 } + \lambda \mathcal { L } _ { 2 }$ first decreases evidently and then increases slightly. Such findings illustrate the optimization process of TA-LBF. In the early iterations, modifying the model parameters tends to achieve the two goals mentioned in Section 3.1; in the late iterations, $\hat { b }$ is encouraged to satisfy the box and $l _ { 2 }$ -sphere constraint. We also observe that both examples stop when meeting $| | \hat { \pmb b } - { \pmb u } _ { 1 } | | _ { 2 } ^ { 2 } \leq 1 0 ^ { - 4 }$ and $| | \hat { \pmb b } - { \pmb u } _ { 2 } | | _ { 2 } ^ { 2 } \leq 1 0 ^ { - 4 }$ and do not exceed the maximum number of iterations (i.e., 2000). The numerical results demonstrate the fast convergence of our method in practice.
+
+![](images/e0ef249e908ba2087f6c6bbc84b7a205cdd2c1723cc9fba8f4631f8f5394f980.jpg)  
+Figure 4: Numerical convergence analysis of TA-LBF $w . r . t .$ . the attacked sample on CIFAR-10 and ImageNet, respectively. We present the values of $| | \hat { \pmb { b } } - \pmb { u } _ { 1 } | | _ { 2 } ^ { 2 } , | | \hat { \pmb { b } } - \pmb { u } _ { 2 } | | _ { 2 } ^ { 2 }$ and $\mathcal { L } _ { 1 } + \lambda \mathcal { L } _ { 2 }$ at different iterations in attacking 8-bit quantized ResNet. Note that $\lambda$ in the left figure is 100 and $\lambda$ in the right figure is $1 0 ^ { 4 }$ .
+
+# E EVALUATION SETUP
+
+# E.1 BASELINE METHODS
+
+Since GDA (Liu et al., 2017a) and FSA (Zhao et al., 2019) are originally designed for attacking the full-precision network, we adapt these two methods to attack the quantized network by applying quantization-aware training (Jacob et al., 2018). We adopt the $\ell _ { 0 }$ -norm for FSA (Liu et al., 2017a) and modification compression for GDA (Zhao et al., 2019) to reduce the number of the modified parameters. Among three types of T-BFA (Rakin et al., 2020b), we compare to the most comparable method: the 1-to-1 stealthy attack scheme. The purpose of this attack scheme is to misclassify samples of a single source class into the target class while maintaining the prediction accuracy of other samples. Besides, we take the fine-tuning (FT) of the last fully-connected layer as a basic attack and present its results. We perform attack once for each selected image except TBT (Rakin et al., 2020a) and totally 1,000 attacks on each dataset. The attack objective of TBT is that the attacked DNN model misclassifies all inputs with a trigger to a certain target class. Due to such objective, the number of attacks for TBT is equal to the number of target classes (i.e., 10 attacks on CIFAR-10 and 50 attacks on ImageNet).
+
+# E.2 TARGET MODELS
+
+According to the setting in (Rakin et al., 2020a;b), we adopt two popular network architectures: ResNet (He et al., 2016) and VGG (Simonyan & Zisserman, 2015) for evaluation. On CIFAR-10, we perform experiments on ResNet-20 and VGG-16. On ImageNet, we use the pre-trained ResNet$1 8 ^ { * }$ and VGG- $\bar { 1 } 6 ^ { \dagger }$ network. We quantize all networks to the 4-bit and 8-bit quantization level using the layer-wise uniform weight quantization scheme, which is similar to the one involved in the Tensor-RT solution (Migacz, 2017).
+
+# E.3 PARAMETER SETTINGS OF TA-LBF
+
+For each attack, we adopt a strategy for jointly searching $\lambda$ and $k$ . Specifically, for an initially given $k$ , we search $\lambda$ from a relatively large initial value and divide it by 2 if the attack does not succeed. The maximum search times of $\lambda$ for a fixed $k$ is set to 8. If it exceeds the maximum search times, we double $k$ and search $\lambda$ from the relatively large initial value. The maximum search times of $k$ is set to 4. On CIFAR-10, the initial $k$ and $\lambda$ are set to 5 and 100. On ImageNet, $\lambda$ is initialized as $1 0 ^ { 4 }$ ; $k$ is initialized as 5 and 50 for ResNet and VGG, respectively. On CIFAR-10, the $\delta$ in $\mathcal { L } _ { 1 }$ is set to 10. On ImageNet, the $\delta$ is set to 3 and increased to 10 if the attack fails. $\mathbf { \delta u } _ { 1 }$ and $\mathbf { \delta } \mathbf { u } _ { 2 }$ are initialized as $^ { b }$ and $u _ { 3 }$ is initialized as 0. $z _ { 1 }$ and $z _ { 2 }$ are initialized as 0 and $z _ { 3 }$ is initialized as 0. $\hat { b }$ is initialized as $^ { b }$ . During each iteration, the number of gradient steps for updating $\hat { b }$ is 5 and the step size is set to 0.01 on both datasets. Hyper-parameters $( \rho _ { 1 } , \rho _ { 2 } , \rho _ { 3 } )$ (see Eq. (11)) are initialized as $( 1 0 ^ { - 4 } , 1 0 ^ { - 4 } , 1 0 ^ { - 5 } )$ on both datasets, and increase by $\rho _ { i } \gets \rho _ { i } \times 1 . 0 1$ , $i = { 1 , 2 , 3 }$ after each iteration. The maximum values of $( \rho _ { 1 } , \rho _ { 2 } , \rho _ { 3 } )$ are set to (50, 50, 5) on both datasets. Besides the maximum number of iterations (i.e., 2000), we also set another stopping criterion, i.e., $| | \hat { \pmb { b } } - { \pmb { u } } _ { 1 } | | _ { 2 } ^ { 2 } \leq 1 0 ^ { - 4 }$ and $| | \hat { b } - { \pmb u } _ { 2 } | | _ { 2 } ^ { 2 } \leq 1 0 ^ { - 4 }$ .
+
+Table 4: Results of all attack methods against models trained with piece-wise clustering on CIFAR10 (bold: the best; underline: the second best). We adopt different clustering coefficients, including 0.0005, 0.005, and 0.01. The mean and standard deviation of PA-ACC and $\mathrm { { N _ { f l i p } } }$ are calculated by attacking the 1,000 images. Our method is denoted as TA-LBF. $\Delta \mathrm { N _ { f l i p } }$ denotes the increased $\mathrm { { N _ { f l i p } } }$ compared to the corresponding result in Table 1.   
+
+<table><tr><td>Clustering Coefficient</td><td>Method</td><td>ACC (%)</td><td>PA-ACC (%） 84.28±3.49</td><td>ASR (%）</td><td>Nflip</td><td>△Nfip</td></tr><tr><td>0.0005</td><td>FT TBT T-BFA FSA GDA TA-LBF</td><td>91.42</td><td>87.97±1.75 86.20±1.96 87.17±2.44 85.28±4.16 87.92±2.54</td><td>100.0 66.1 98.5 98.5 100.0 100.0</td><td>1868.26±72.48 250.30±10.97 30.95±6.50 222.70±56.52 41.33±12.84 13.47±5.34</td><td>360.75 3.60 21.04 37.19 14.50 7.90</td></tr><tr><td>0.005</td><td>FT TBT T-BFA FSA GDA TA-LBF</td><td>88.03</td><td>81.08±3.61 82.96±2.18 80.80±2.64 83.10±2.75 79.23±6.25 83.63±3.47</td><td>97.9 12.7 98.1 98.4 99.9 100.0</td><td>1774.69±51.47 246.80±16.06 61.72±12.17 231.66±89.21 64.87±22.78 25.52±11.59</td><td>267.18 0.10 51.81 46.15 38.04 19.95</td></tr><tr><td>0.01</td><td>FT TBT T-BFA FSA GDA TA-LBF</td><td>85.65</td><td>78.73±3.54 79.86±2.04 76.67±3.41 80.45±3.14 75.33±7.83 80.51±4.39</td><td>98.3 10.1 98.1 98.0 99.7 100.0</td><td>1748.54±46.19 236.50±10.93 55.49±11.77 220.28±101.01 59.17±23.63 24.60±13.03</td><td>241.03 -10.20 45.58 34.77 32.34 19.03</td></tr></table>
+
+# F MORE RESULTS ON RESISTANCE TO DEFENSE METHODS
+
+# F.1 RESISTANCE TO PIECE-WISE CLUSTERING
+
+We conduct experiments using the 8-bit quantized ResNet on CIFAR-10 with different clustering coefficients. We set the maximum search times of $k$ to 5 for clustering coefficient 0.005 and 0.01 and keep the rest settings the same as those in Section 4.1. The results are presented in Table 4. As shown in the table, all values of $\mathrm { { N _ { f l i p } } }$ are larger than attacking models without defense for all methods, which is similar to Table 2. Our method achieves a $100 \%$ ASR with the fewest $\mathrm { { N _ { f l i p } } }$ under the three clustering coefficients. Although TBT obtains a smaller $\Delta \mathrm { N _ { f l i p } }$ than our method, it fails to achieve a satisfactory ASR. For example, TBT achieves only a $10 . 1 \%$ ASR when the clustering coefficient is set to 0.01. We observe that for all clustering coefficients, piece-wise clustering reduces the original accuracy. Such a phenomenon is more significant as the clustering coefficient increases. The results also show that there is no guarantee that if the clustering coefficient is larger (e.g., 0.01), the model is more robust, which is consistent with the finding in (He et al., 2020).
+
+Table 5: Results of all attack methods against models with a larger capacity on CIFAR-10. We adopt $3 \times$ and $4 \times$ width networks. The mean and standard deviation of PA-ACC and $\mathrm { { N _ { f l i p } } }$ are calculated by attacking the 1,000 images. $\Delta \mathrm { N _ { f l i p } }$ denotes the increased $\mathrm { { N _ { f l i p } } }$ compared to the corresponding result in Table 1.   
+
+<table><tr><td>Model Width</td><td>Method</td><td>ACC (%）</td><td>PA-ACC (%）</td><td>ASR (%）</td><td>Nfip</td><td>△Nfip</td></tr><tr><td>3×</td><td>FT TBT T-BFA FSA GDA TA-LBF</td><td>94.90</td><td>86.96±2.79 90.67±5.23 92.18±1.14 91.40±2.38 90.79±2.91 91.42±2.81</td><td>100.0 74.1 98.9 99.0 100.0 100.0</td><td>4002.52±281.24 504.70±20.44 30.50±7.52 342.20±79.44 67.53±27.45 12.29±4.18</td><td>2495.01 258.00 20.59 156.69 40.70 6.72</td></tr><tr><td>4×</td><td>FT TBT T-BFA FSA GDA TA-LBF</td><td>95.02</td><td>86.94±2.78 85.39±5.08 92.49±1.22 91.60±2.42 90.76±3.00 90.94±3.11</td><td>100.0 90.1 99.4 98.7 100.0 100.0</td><td>4527.68±369.35 625.50±32.38 19.14±5.04 338.93±100.12 66.92±40.32 8.37±2.80</td><td>3020.17 378.80 9.23 153.42 40.09 2.80</td></tr></table>
+
+# F.2 RESISTANCE TO LARGER MODEL CAPACITY
+
+Besides the results of networks with a $2 \times$ width shown in Section 4.3, we also evaluate all methods against models with a $3 \times$ and $4 \times$ width. All settings are the same as those used in Section 4.1. The results are provided in Table 5. Among all attack methods, our method is least affected by increasing the network width. Especially for the network with a $4 \times$ width, our $\Delta \mathrm { N _ { f l i p } }$ is only 2.80. The results demonstrate the superiority of the formulated BIP problem and optimization. Moreover, compared with piece-wise clustering, having a larger model capacity can improve the original accuracy, but increases the model size and the computation complexity.
+
+# G DISCUSSIONS
+
+# G.1 COMPARING BACKDOOR, ADVERSARIAL, AND WEIGHT ATTACK
+
+An attacker can achieve malicious purposes utilizing backdoor, adversarial, and weight attacks. In this section, we emphasize the differences among them.
+
+Backdoor attack happens in the training stage and requires that the attacker can tamper the training data even the training process (Liu et al., 2020b; Li et al., 2020). Through poisoning some training samples with a trigger, the attacker can control the behavior of the attacked DNN in the inference stage. For example, images with reflections are misclassified into a target class, while benign images are classified normally (Liu et al., 2020a). However, such an attack paradigm causes the accuracy degradation on benign samples, which makes it detectable for users. Besides, these methods also require to modify samples in the inference stage, which is sometimes impossible for the attacker. Many defense methods against backdoor attack have been proposed, such as the preprocessingbased defense (Liu et al., 2017b), the model reconstruction-based defense (Liu et al., 2018a), and the trigger synthesis-based defense (Wang et al., 2019).
+
+Adversarial attack modifies samples in the inference stage by adding small perturbations that remain imperceptible to the human vision system (Akhtar & Mian, 2018). Since adversarial attack only modifies inputs while keeping the model unchanged, it has no effect on the benign samples. Besides the basic white-box attack, the black-box attack (Wu et al., 2020b; Chen et al., 2020) and universal attack (Zhang et al., 2020b;a) have attracted wide attention. Inspired by its success in the classification, it also has been extended to other tasks, including image captioning (Xu et al., 2019), retrieval (Bai et al., 2020; Feng et al., 2020), etc.. Similarly, recent studies have demonstrated many defense methods against adversarial attack, including the preprocessing-based defense (Xie et al., 2018), the detection-based defense (Xu et al., 2017), and the adversarial learning-based defense (Carmon et al., 2019; Wu et al., 2020c).
+
+Weight attack modifies model parameters in the deployment stage, which is the studied paradigm in this work. Weight attack generally aims at misleading the DNN model on the selected sample(s), while having a minor effect on other samples (Zhao et al., 2019; Rakin et al., 2020b). Many studies (Yao et al., 2020; Breier et al., 2018; Pan, 2020) have demonstrated that the DNN parameters can be modified in the bit-level in memory using fault injection techniques (Agoyan et al., 2010; Kim et al., 2014; Selmke et al., 2015) in practice. Note that the defense methods against weight attack have been not well studied. Although some defense methods (He et al., 2020) were proposed, they cannot achieve satisfactory performance. For example, our method can still achieve a $100 \%$ attack success rate against two proposed defense methods. Our work would encourage further investigation on the security of the model parameters from both attack and defense sides.
+
+# G.2 COMPARING TA-LBF WITH OTHER WEIGHT ATTACKS
+
+We compare our TA-LBF with other weight attack methods, including TBT (Rakin et al., 2020a), TBFA (Rakin et al., 2020b), GDA (Liu et al., 2017a), and FSA (Zhao et al., 2019) in this section. TBT tampers both the test sample and the model parameters. Specifically, it first locates critical bits and generates a trigger, and then flips these bits to classify all inputs embedded with the trigger to a target class. However, the malicious samples are easily detected by human inspection or many detection methods (Tran et al., 2018; Du et al., 2020). We do not modify the samples to perform TA-LBF, which makes the attack more stealthy. Rakin et al. (2020b) proposed T-BFA which misclassifies all samples (N-to-1 version) or samples from a source class (1-to-1 version) into a target class. Our method aims at misclassifying a specific sample, which meets the attacker’s requirement in some scenarios. For example, the attacker wants to manipulate the behavior of a face recognition engine on a specific input. Since it affects multiple samples, T-BFA maybe not stealthy enough in attacking real-world applications. GDA (Liu et al., 2017a) and FSA (Zhao et al., 2019) modify model parameters at the weight-level rather than bit-level. They are designed for misclassifying multiple samples from arbitrary classes, which makes it infeasible for them to only modify the parameters connected to the source and target class. They modify more parameters than our method as shown in the experiments, it might be due to the reason discussed above. Besides, TBT, T-BFA, and GDA determine the critical weights to modify using heuristic strategies, while our TA-LBF adopts optimization-based methods. Although FSA applies ADMM for solving the optimization problem, it has no explicit constraint to control the number of modified parameters, which makes it intends to modify more parameters than GDA and our TA-LBF.
+
+![](images/390662740639be5c401051769981654152398edcb8bfa25b1eb7d17b4bc7f640.jpg)  
+H TRADE-OFF BETWEEN THREE EVALUATION METRICS   
+Figure 5: Curves of the trade-off between PA-ACC and $\mathrm { { N _ { f l i p } } }$ and the trade-off between PA-ACC and ASR for the proposed TA-LBF on two datasets.
+
+In this section, we investigate the trade-off between three adopted evaluation metrics (i.e., PA-ACC, ASR, and $\mathrm { { N } _ { f l i p . } }$ ) for our attack. All experiments are conducted on CIFAR-10 and ImageNet dataset in attacking the 8-bit quantized ResNet.
+
+We firstly discuss the trade-off between PA-ACC and $\mathrm { { N _ { f l i p } } }$ by fixing the ASR as $100 \%$ using the search strategy in Appendix E.3 and adjusting the initial $\lambda$ and $k$ to obtain different attack results. The two curves on the left show that increasing the $\mathrm { { N _ { f l i p } } }$ can improve the PA-ACC when $\mathrm { { N _ { f l i p } } }$ is relatively small; the PA-ACC decreases with the increase of $\mathrm { { N _ { f l i p } } }$ when $\mathrm { { N _ { f l i p } } }$ is greater than a threshold. This phenomenon demonstrates that constraining the number of bit-flips is essential to ensure the attack stealthiness, as mentioned in Section 3.2. To study the trade-off between PA-ACC and ASR, we fix the parameter $k$ as 10 for approximately 10 bit-flips and adjust the parameter $\lambda$ to obtain different PA-ACC and ASR results. The trade-off curves between PA-ACC and ASR show that increasing ASR can decrease the PA-ACC significantly. Therefore, how to achieve high ASR and PA-ACC simultaneously is still an important open problem.