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+# MOTION FORECASTING WITH UNLIKELIHOOD TRAINING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Motion forecasting is essential for making safe and intelligent decisions in robotic applications such as autonomous driving. State-of-the-art methods formulate it as a sequence-to-sequence prediction problem, which is solved in an encoderdecoder framework with a maximum likelihood estimation objective. In this paper, we show that the likelihood objective itself results in a model assigning too much probability to trajectories that are unlikely given the contextual information such as maps and states of surrounding agents. This is despite the fact that many state-of-the-art models do take contextual information as part of their input. We propose a new objective, unlikelihood training, which forces generated trajectories that conflict with contextual information to be assigned a lower probability by our model. We demonstrate that our method can improve state-of-art models’ performance on challenging real-world trajectory forecasting datasets (nuScenes and Argoverse) by $8 \%$ and reduce the standard deviation by up to $50 \%$ . Code will be made available.
+
+# 1 INTRODUCTION
+
+For robotic applications deployed in the real world, the ability to foresee the future motions of agents in the surrounding environment plays an essential role for safe and intelligent decision making. This is a very challenging task. For example, in the autonomous driving domain, to predict nearby agents’ future trajectories, an agent needs to consider contextual information such as their past trajectories, potential interactions, and maps. State of the art prediction models (Salzmann et al., 2020; Tang & Salakhutdinov, 2019; Rhinehart et al., 2019) directly take contextual information as part of their input and use techniques such as graph neural networks to extract high-level features for prediction. They are typically trained with a maximum likelihood estimation (MLE) objective that maximizes the likelihood of ground truth trajectories in the predicted distribution. Although MLE loss encourages the prediction to be close to the ground truth geometrically, it does not focus on learning a good distribution that is plausible with respect to the contextual information. These models predict trajectories that violate the contextual information (e.g., go to opposite driving direction or out of the driving area) but still closes to ground truth. In contrast, humans can easily notice that these trajectories are unlikely in a specific context. This phenomenon suggests that simply applying MLE loss cannot fully exploit contextual information.
+
+To address the problem, we propose a novel and simple method, unlikelihood training, that injects contextual information into the learning signal. Our loss penalizes the trajectories that violate the contextual information, called negative trajectories, by minimizing their likelihood in the predicted distribution. To generate negative trajectories, we first draw a number of candidate trajectories from our model’s predicted distribution. Then, a context checker is used to cut out the trajectories that violate contextual information as negative trajectories. This context checker does not need to be differentiable. By minimizing the likelihood of negative trajectories, the model is forced to use the contextual information to avoid predictions that violate context. Therefore, the prediction quality is improved.
+
+Existing methods (Casas et al., 2020; Park et al., 2020) using contextual information as learning signals either introduce new learning parameters or using high-variance learning methods such as the REINFORCE algorithm (Casas et al., 2020). In contrast, our method injects rich contextual information into the training objective and keeps the training process simple.
+
+Unlikelihood training (Welleck et al., 2019) has been applied to neural text generation. We are the first to propose unlikelihood training for continuous space of trajectories. For the discrete space of token sequences, repeating tokens or n-grams in the generated sequence are chosen as negative tokens. In contrast, we design a context checker to select negative trajectories sampled from the continuous distribution of model predictions.
+
+Our method can be viewed as a simple add-on to any models that estimate the distribution of future trajectories. It improves their performance by encouraging models to focus more on contextual information without increasing the complexity of its original training process.
+
+Our contributions are summarized as follows:
+
+• We propose a novel and simple method, unlikelihood training for motion forecasting in autonomous driving that encourages models to use contextual information by minimizing the likelihood of trajectories that violate contextual information. Our method can be easily incorporated into state-of-the-art models.   
+• Our experimental results on challenging real-world trajectory forecasting datasets, nuScenes and Argoverse, shows that unlikelihood training can improve prediction performance by $8 \%$ and reduce the standard deviation by up to $50 \%$ .
+
+# 2 RELATED WORK
+
+In this section, we briefly review the two most related topics.
+
+Trajectory Forecasting Trajectory forecasting of dynamic agents, a core problem for robotic applications such as autonomous driving and social robots, has been well studied in the literature. State-of-the-art models solves it as a sequence-to-sequence multi-modal prediction problem (Lee et al., 2017; Cui et al., 2018; Chai et al., 2019; Rhinehart et al., 2019; Kosaraju et al., 2019; Tang & Salakhutdinov, 2019; Ridel et al., 2020; Salzmann et al., 2020; Huang et al., 2019). (Cui et al., 2018; Chai et al., 2019; Ridel et al., 2020) predicts multiple future trajectories without learning low dimensional latent agent behaviors. (Lee et al., 2017; Kosaraju et al., 2019; Rhinehart et al., 2019; Huang et al., 2019) encodes agent behaviors in continuous low dimensional latent space while (Tang & Salakhutdinov, 2019; Salzmann et al., 2020) uses discrete latent variables. Discrete latent variables succinctly capture semantically meaningful modes such as turn left, turn right. (Tang & Salakhutdinov, 2019; Salzmann et al., 2020) learns discrete latent variables without explicit labels. All of them use a maximum likelihood estimation (MLE) objective or its approximations (e.g., VAE). In this paper, we show that MLE loss can ignore contextual information such as maps and states of surrounding agents. As a result, models with such a loss can assign too much probability to unlikely trajectories. We propose an unlikelihood training objective to avoid such cases. All models with the maximum likelihood estimation objective can potentially benefit from our methods.
+
+Contrastive learning and unlikelihood training To date, several studies have investigated the possibilities to benefit from negative data. One of the popular direction is contrastive learning. Contrastive learning has achieved significant success in many fields (Oord et al., 2018; Kipf et al., 2019; Ma & Collins, 2018; Abid & Zou, 2019; Welleck et al., 2019). NCE (Ma & Collins, 2018) CPC (Oord et al., 2018) maximizes the mutual information between data and latent representation by a novel contrastive loss to extract useful representation from data. C-SWMs (Kipf et al., 2019) utilizes contrastive learning to learn a better world model for reinforcement learning tasks. Recently, unlikelihood training (Welleck et al., 2019) proposes a new method to utilize negative data. In addition, to maximize the likelihood of the ground truth token, it minimizes the likelihood of negative tokens for better text generation. Their method is on the discrete space of token sequences. Repeating tokens or n-grams in the generated sequence is chosen as negative tokens. In contrast, our proposed method works in the continuous space of trajectories. We design a novel method, context checker, to select negative trajectories.
+
+# 3 METHOD
+
+# 3.1 PROBLEM FORMULATION
+
+We are targeting at better predicting the future trajectory $\mathbf { Y } _ { i }$ of a vehicle $i$ given input $\mathbf { X } _ { i }$ . $\mathbf { X } _ { i }$ can include any related information like rasterized maps or past positions of vehicle $i$ and surrounding agents, depends on the design of the method. Here we skip the detailed choice of input and denote it as $\mathbf { X } _ { i }$ for conciseness. Due to different driving strategies, driving intents, and the complex traffic environment, there are usually multiple possible future trajectories given an input $x _ { i }$ (although there is only one ground truth future trajectory $y _ { i , g t }$ in a dataset recorded in the real world). To handle this situation, most state of the art methods (Salzmann et al., 2020) model a distribution of possible future trajectories $p _ { \theta } ( \mathbf { Y } _ { i } \mid \mathbf { X } _ { i } )$ to cover all the possibilities given the input $\mathbf { X } _ { i }$ instead of predicting one trajectory. $\theta$ denotes the learning parameters of the model. To train such methods, most stateof-the-art models usually use maximum likelihood estimation (MLE) to maximize the likelihood of ground truth trajectory $\mathbf { Y } _ { i , g t }$ in the predicted distribution. For example, the loss of CVAE-based model Trajectron $^ { + + }$ (Salzmann et al., 2020) is Eq.1. This loss is used to maximize the lower bound of ground truth’s likelihood when the coefficient $k = 1$ .
+
+$$
+\begin{array} { r l } & { L _ { \mathrm { t r a j + + } } = - \mathbb { E } _ { \hat { z } \sim q _ { \theta 3 } ( z | X _ { i } , Y _ { i , g t } ) } \bigl [ \log p _ { \theta 2 } ( Y _ { i , g t } \mid X _ { i } , \hat { z } ) \bigr ] } \\ & { \qquad + k D _ { \mathrm { K L } } \bigl ( q _ { \theta 3 } ( z \mid X _ { i } , Y _ { i , g t } ) \bigr ) \bigl \| p _ { \theta 1 } ( z \mid X _ { i } ) \bigr ) - I _ { q } ( X _ { i } ; z ) } \\ & { \qquad \geq - \log p ( Y _ { i , g t } \mid X _ { i } , Y _ { i , g t } ) - I _ { q } ( X _ { i } ; z ) , \qquad \mathrm { w h e n ~ } k = 1 } \end{array}
+$$
+
+Limitation of MLE on Motion Forecasting MLE encourages the model to predict a distribution that allocates reasonable probability mass to the region where $\mathbf { Y _ { i } }$ is located by minimizing the KL-divergence of predicted distribution and ground truth distribution. Because the domain of trajectory distribution is over the geometric locations, MLE makes these two distributions ”close” to each other geometrically. However, we argue that maintaining the geometrical nearness only is not good enough for motion forecasting task in autonomous driving. In complex traffic scenarios, there can be many potential trajectories close enough to the ground truth geometrically but are very unlikely to happen. For example, if the ground truth trajectory $Y _ { i , g t }$ is on the outermost lane, a trajectory that is close to $Y _ { i , g t }$ but outside the drivable region is unlike to happen in the real world. However, MLE loss will not impose a significant enough penalty on such a case to avoid such a prediction. Fig.1 demonstrates a prediction example from Trajectron $^ { + + }$ (Salzmann et al., 2020) where part of the distribution is outside of the derivable region or on the lane with the wrong direction. The MLE-based loss
+
+![](images/1847de5db5a5b37e5aedcaf818bf8b15925a84c83024d377c47a7cf211c8cb7d.jpg)  
+Figure 1: Examples of predicted distribution from Trajectron $^ { + + }$ (Salzmann et al., 2020). White points denote the ground truth trajectory $\mathbf { Y } _ { i , g t }$ and the color region indicates the predicted distribution. Some of the prediction go outside of the drivable region or go to the lane in opposite direction.
+
+only offers learning signals that contain the geometric location information of the ground truth trajectories. All the other contextual information, like the drivable region and the lane direction, are missing in the learning signals. While the inputs to a model contain rich contextual information, the model cannot use it to avoid the prediction that is geometrically close to ground truth but violates context. In contrast, this is quite a simple task for humans.
+
+# 3.2 UNLIKELIHOOD LOSS
+
+To mitigate this problem, we design a new loss term that encourages the model to consider the contextual information. Inspired by contrastive learning and unlikelihood training, we additionally train our model to minimize the likelihood of trajectories that violate the contextual information given input $X _ { i }$ . We denote them as negative trajectories ${ Y _ { i , n e g } }$ . Let’s first assume that we already have a distribution of negative trajectories $p _ { \mathrm { n e g } } ( \bar { \mathbf { Y } } _ { i } \mid X _ { i } )$ . One intuitive way is to directly minimize the log likelihood of ${ Y _ { i , n e g } }$ in our predicted distribution, similar to MLE but in an opposite manner
+
+$$
+L _ { \mathrm { u n l i k e } } = \mathbb { E } _ { X _ { i } , \sim \mathbb { D } , Y _ { i , n e g } \sim p _ { \mathrm { n e g } } ( Y _ { i } | X _ { i } ) } [ \log p _ { \theta } ( Y _ { i , n e g } \mid X _ { i } ) ]
+$$
+
+However, the gradient of log function tends to infinity when the input tends to 0, which leads to unstable training since the model are optimized to minimize $p _ { \theta } ( \bar { \mathbf { Y } _ { i , n e g } } \mid \mathbf { X } _ { i } )$ and $p _ { \theta } ( \mathbf { Y } _ { i , n e g } \mid$ $\mathbf { X } _ { i } ) \geq 0$ . To avoid the infinity gradient region of log function, we add a small constant $\epsilon$ to the likelihood. The final loss term we propose is
+
+$$
+L _ { \mathrm { u n l i k e } } = \mathbb { E } _ { X _ { i } , \sim \mathbb { D } , Y _ { i , n e g } \sim p _ { \mathrm { n e g } } ( \mathbf { Y } _ { i } \mid \mathbf { X } ) } [ \log ( p _ { \theta } ( { Y } _ { i , n e g } \mid X _ { i } ) + \epsilon ) ) ]
+$$
+
+We call it unlikelihood loss. We use a coefficient $\gamma$ to balance $L _ { \mathrm { u n l i k e } }$ . The final training objective in case we combine our method with Trajectron $^ { + + }$ is
+
+$$
+L = L _ { \mathrm { t r a j + + } } + \gamma L _ { \mathrm { u n l i k e } }
+$$
+
+Eq.4 is also easily adapted to combine with any other models that predict trajectory distribution as output. With the help of $L _ { \mathrm { u n l i k e } }$ , we inject the contextual information into the learning signal, force the model to better extract and use contextual information in $\mathbf { X } _ { i }$ , and generate more reasonable predicted distribution to avoid high $L _ { \mathrm { u n l i k e } }$ .
+
+# 3.3 NEGATIVE TRAJECTORIES
+
+Our proposed loss term is highly dependent on the negative samples ${ Y } _ { i , n e g }$ from the distribution $p _ { \mathrm { n e g } } ( Y _ { i } \mid X _ { i } )$ . However, these are not given in the dataset. To solve this issue, we approximate the samples by directly drawing a set of trajectories from the predicted distribution and select the trajectories that violate the contextual information out by a context checker. Note that this checker does not need to be differentiable and it can be as complex and advanced as necessary. The type of unlike predictions the model learns to avoid by our method depends on the type of unlike trajectories the checker can detect.
+
+Design of Our Checker We implement a map-based checker to judge whether a given trajectory suits the context or not. In detail, the checker examines whether the trajectory goes into the lane in the opposite direction or out of the road. We create a map that stores the lane direction at every location of lanes and the drivable region. Two examples are shown in Fig.2. We first check whether all the locations of a given trajectory are in the drivable region. If so, we further calculate angles between velocity and the lane direction at each time step to see whether they are all inside a 90 degree. The velocities are approximated by differentiating the trajectory. The trajectories that fail to pass the exam are judged as negative trajectories ${ Y _ { i , n e g } }$ . Note that the lane direction information originally comes from the dataset and is usually incomplete or invalid like in the intersection. In this case, we only use the drivable region information. In addition, there are also a small part of ground truth trajectories in the dataset that violates the lane direction or drivable region. In this case, our checker skips the checking and we train it without $L _ { \mathrm { u n l i k e } }$ to allow similar prediction. Note that this checker is not perfect due to the incomplete information and simple checking mechanism. A trajectory that passes the exam of the checker doesn’t mean it is $100 \%$ compatible with the context. However, our method can still work properly, because our method only depends on the negative trajectories that do not pass the exam and has nothing to do with the passing ones. Our checker design offers reasonable negative trajectories to support our approach. But of course, a more advanced checker helps the model to avoid more complex unlikely prediction. We leave it open for future research.
+
+# 3.4 ALGORITHM
+
+The finial algorithm is shown in Alg.1. At each iteration, we first run the forward pass of the model to get the output distribution $p _ { \theta } ( \pmb { Y } _ { i } \mid \pmb { X } _ { i } )$ given the input $X _ { i }$ . Then, $K$ negative candidate trajectories are drawn from this distribution and we select the negative trajectories out ${ Y _ { i , n e g } }$ via our checker. After that, ground truth trajectory and negative trajectories are used to calculate the loss function, update the model, and go to next iteration. Note that if there are no ${ Y _ { i , n e g } }$ in the $K$ negative candidates judged by our checker for data $i$ , we don’t apply $L _ { \mathrm { u n l i k e } }$ on $i$ .
+
+![](images/3cc40cce52f70c1a0954a903aa911a5f9324ffbf776ff889b682efe3fa35338e.jpg)  
+Figure 2: Examples of maps used in the checker in nuScences (Caesar et al., 2019a). Green and blue region together denote the drivable region and the blue means that we have lane direction information here. Random locations are sampled and their lane directions are plotted as red arrows to show the concrete directions. Blue line denotes the trajectories to check and the velocity directions are represented as yellow arrows. (a) shows a negative trajectory that goes out of the road. (b) is a passing case.
+
+# 3.5 GRADIENT ANALYSIS
+
+Let’s assume a single-mode prediction case that the future position $_ { { \mathbf { \mathit { y } } } _ { i , g t , t } }$ at time step $t$ of agent $i$ is modeled by a simple Gaussian distribution $\mathcal { N } ( \boldsymbol { y } _ { i , t } ; \bar { \boldsymbol { \mu } } _ { i , t } , \hat { \sigma } _ { i , t } \bar { \boldsymbol { I } } )$ . $\hat { \pmb { \mu } }$ and $\hat { \sigma } _ { i , t }$ are calculated by the model. With a single negative position ${ \bf { \it { y } } } _ { i , n e g , t }$ we define a simple loss for step $t$ $L _ { t } = - \log \mathcal { N } ( y _ { g t , t } ; \hat { \mu } _ { t } , \hat { \sigma } _ { t } I ) + \log \bar { \mathcal { N } } ( y _ { n e g , t } ; \hat { \mu } _ { t } , \hat { \sigma } _ { t } I )$ and omit the subscript $i$ for brevity. The gradient of $L _ { t }$ with respect to $\hat { \pmb { \mu } } _ { t }$ and $\hat { \sigma } _ { t }$ in this case is (Derivation in Appx.A):
+
+$$
+\frac { \partial { { L } _ { t } } } { \partial { { \hat { \pmb { \mu } } } _ { t } } } = - \frac { 1 } { { { \hat { \sigma } } _ { t } ^ { 2 } } } ( ( \pmb { y } _ { g t , t } - { { \hat { \pmb { \mu } } } _ { t } } ) + ( \pmb { { \hat { \mu } } } _ { t } - \pmb { y } _ { n e g , t } ) )
+$$
+
+$$
+\frac { \partial L } { \partial \hat { \sigma } _ { t } } = - \frac { 1 } { { \hat { \sigma } _ { t } } ^ { 3 } } ( | | y _ { g t , t } - \hat { \mu } _ { t } | | ^ { 2 } - | | y _ { n e g , t } - \hat { \mu } _ { t } | | ^ { 2 } )
+$$
+
+Eq.5 shows that the center of the predicted distribution $\hat { \pmb { \mu } } _ { t }$ is pushed towards ${ \bf \nabla } _ { { \bf { y } } _ { g t , t } }$ and pushed away from yneg,t by this learning objective. In Eq.6, when ygt,t is closer to the center than yneg,t, ∂L∂σˆt is positive and $\hat { \sigma } _ { t }$ is decreased. Note that ${ { y } _ { n e g , t } }$ is selected out from samples of $\mathcal { N } ( \hat { \mu } _ { t } , \hat { \sigma } _ { t } \pmb { I } ) )$ , this means when $\mathcal { N } ( \hat { \mu } _ { t } , \hat { \sigma } _ { t } \pmb { I } ) )$ ) covers context-violated region and this region is farther than ground truth region, $\mathcal { N } ( \hat { \mu } _ { t } , \hat { \sigma } _ { t } \pmb { I } ) )$ will shrink to exclude the negative region and become a better estimation to the true data distribution.
+
+When the prediction is not so accurate (e.g. at the beginning of training), our ground truth $\scriptstyle { \boldsymbol { \mathbf { \mathit { y } } } } _ { \mathit { g t , t } }$ may be farther than the negative location ${ \bf { \mathscr { y } } } _ { n e g , t }$ . In this case, $\mathcal { N } ( \hat { \mu } _ { t } , \hat { \sigma } _ { t } \pmb { I } )$ will expand to better cover the ground truth and make the prediction more uncertain. A simple approach to alleviate this issue is turning off our unlikelihood loss $L _ { \mathrm { u n l i k e } }$ in the first few training epochs. We implement this by making $\gamma$ in Eq.4 as a sigmoid function centered at a specified epoch. By this way, we smoothly turn on $L _ { \mathrm { u n l i k e } }$ during training.
+
+In this section, we present the experimental results of our method to demonstrate our performance. Our method is easily to applied on state of the art models that generate a future trajectory distribution and can further improve their performance. In our experiments, we select Trajectron $^ { + + }$ (Salzmann et al., 2020), one of the state-of-the-art methods on NuScenes dataset (Caesar et al., 2019b) with open-source implementation, as our base model. We extend the implementation to work on Argoverse dataset (Chang et al., 2019). We evaluate our approach on these two motion forecasting datasets.
+
+<table><tr><td>Algorithm 1: Training process (Use Trajec- tron++ as base model)</td></tr><tr><td>Initialize the model parameters 0; Initialize learning rate α and coeifficient γ; while not converge do X,Yi,gt ~ D run forward pass to compute pe(Yi | Xi) draw K trajectotries Yi,k ~ pe(Yi |Xi) select Yi,neg via checker Compute Ltraj++ using Eq.1 Compute Lunlike using Eq.3 L = Ltraj++ + γLunlike 0=0-αVθ(L) end</td></tr></table>
+
+Test Model Trajectron $^ { + + }$ (Salzmann et al.,   
+2020) is a CVAE-based (Sohn et al., 2015) model. Its input $X _ { i }$ contains positions, velocities, heading of the predicted and surrounding vehicles, and a map patch. The output distribution $\begin{array} { r } { p _ { \theta } ( \boldsymbol { Y _ { i } } ~ \{ \textbf {  { X } } _ { i } ) ~ = ~ \sum _ { z } p _ { \theta 1 } ( z ~ \lvert ~ \boldsymbol { X } _ { i } ) p _ { \theta 2 } ( \boldsymbol { Y _ { i } } ~ \{ \textbf {  { X } } _ { i } , { z } \} | } \end{array}$ is a Gaussian mixture model with 25 components and modeled by an encoder net $p _ { \theta 1 } ( z \mid X _ { i } )$ and a decoder net $p _ { \theta 1 } ( Y _ { i } \mid X _ { i } , z )$ . In addition, it has another encoder net ${ q _ { \theta 3 } } ( z \mid X _ { i } , Y _ { i , g t } )$ used only in training. The original learning objective is shown in Eq.1. $I _ { q }$ denotes the mutual information.
+
+Evaluation Metrics we use average $l _ { 2 }$ displacement error (ADE) and final $l _ { 2 }$ displacement error (FDE) to evaluate the prediction performance. Each of them contains some sub-versions. FDE-1 is the FDE calculated using only 1 predicted trajectory. In both original Trajectron $^ { + + }$ and our approach, this single trajectory is drawn from predicted distribution by greedy search step by step. ADE-Full/FDE-Full represents the quality of the whole output distribution. To compute ADEFull/FDE-Full, we randomly sample 200 trajectories and calculate the average performance as the reported scores. In addition, we use our context checker to measure the context-violation rate in these 200 trajectories as a metric to show the context-related performance.
+
+# 4.1 NUSCENES DATASET
+
+nuScenes dataset (Caesar et al., 2019b) contains 1000 city driving scenes from both left-hand (Singapore) and right-hand (Boston) traffic regions. Each scene is 20s long and recorded in $2 \mathrm { H z }$ . It is one of the biggest open-source motion forecasting datasets with detailed semantic maps.
+
+Experiments The batch size is set to 1024. Models are trained for 35 epochs and we test the weights from the best epoch measured by average ADE on validation set. The coefficient $\gamma$ in Alg.1 increases gradually from 0 to 1 as a sigmoid function centered at 24th epoch. Initial learning rate is 3e-3 and it decays exponentially by 0.9995 per iteration. These hyperparameters except $\gamma$ are optimized for Trajectron $^ { + + }$ and lead to better performance than that in original paper. In addition, we rotate the scenes randomly from $1 5 ^ { \circ }$ to $3 4 5 ^ { \circ }$ in the training set for data augmentation following the setting of original Trajectron $^ { + + }$ . For each model, we run 5 experiments and report the mean and standard deviation of the measured metrics. The models are trained to predict 3 seconds into the future. To evaluate on generalization beyond the training horizon, we test on both 3 second and 4 second prediction horizons.
+
+Tab.1 shows the quantitative results 3s and 4s prediction measured by the FDE/ADE-Full metrics and the context-violation rate. Our unlikelihood loss improves Trajectron $^ { + + }$ by about $8 \%$ for both metrics in both 3 and 4 second prediction horizons. Results indicate that our method helps improve the accuracy of the predicted distribution. This is also demonstrated in the qualitative comparison in Fig.3. The predicted distribution from our methods covers the not-drivable region less compared to original Trajectron $^ { + + }$ without our proposed loss. In contrast, original Trajectron $^ { + + }$ tends to violate the contextual information when prediction horizon is long. This shows that our method encourages the model to be more sensitive to the road boundary and the lane direction. In addition, we observe a reduction of the performance variance measured by standard deviation in Tab1, indicates that our
+
+Table 1: nuScenes: Experiment results on nuScenes dataset (Caesar et al., 2019b) with Trajectron $^ { + + }$ (Salzmann et al., 2020). FDE and ADE are averaged over 200 trajectories drawn from the predicted distribution. Our proposed loss improves the performance of Trajectron $^ { + + }$ by about $8 \%$ , and avoid $16 \%$ context-violated prediction compared to Trajectron $^ { + + }$ , which indicates a better predicted distribution. Mean and standard deviation are calculated over 5 runs.   
+
+<table><tr><td>Model</td><td colspan="2">FDE-Full</td><td colspan="2">ADE-Full</td><td colspan="2">Context-Violation-Rate</td></tr><tr><td></td><td>3s</td><td>4s</td><td>3s</td><td>4s</td><td>3s</td><td>4s</td></tr><tr><td>Trajectron++</td><td>1.46±0.07</td><td>2.74±0.10</td><td>0.59±0.04</td><td>1.04±0.05</td><td>7.29%±0.22%</td><td>10.59%±0.54%</td></tr><tr><td>Ours</td><td>1.34±0.04</td><td>2.51±0.06</td><td>0.54±0.02</td><td>0.95±0.03</td><td>6.57 %±0.15%</td><td>8.85% ± 0.32%</td></tr></table>
+
+Table 2: nuScenes: Experimental results on the nuScenes dataset for single prediction. FDE and ADE are computed by only one predicted trajectory. For both Trajectron $^ { + + }$ and our method, this trajectory is sampled by greedy search. Our method helps to improve the predicted accuracy. Mean and standard deviation are calculated over 5 runs.
+
+<table><tr><td>Model</td><td colspan="4">FDE-1</td></tr><tr><td></td><td>1s</td><td>2s</td><td>3s</td><td>4s</td></tr><tr><td>Const. Velocity (Salzmann et al., 2020)</td><td>0.32</td><td>0.89</td><td>1.70</td><td>2.73</td></tr><tr><td>S-LSTM (Alahi et al., 2016)</td><td>0.47</td><td></td><td>1.61</td><td>-</td></tr><tr><td>CSP (Deo &amp; Trivedi,2018)</td><td>0.46</td><td>2.35</td><td>1.50</td><td></td></tr><tr><td>CAR-Net (Sadeghian et al., 2018)</td><td>0.38</td><td>-</td><td>1.35</td><td></td></tr><tr><td>SpAGNN (Casas et al., 2019)</td><td>0.36</td><td>=</td><td>1.23</td><td>-</td></tr><tr><td>Trajectron++ (Salzmann et al., 2020)</td><td>0.07</td><td>0.45</td><td>1.14</td><td>2.20</td></tr><tr><td>Trajectron++ (our hyperparameters)</td><td>0.06±0.01</td><td>0.43±0.01</td><td>1.08±0.04</td><td>2.05±0.08</td></tr><tr><td>Ours</td><td>0.05±0.00</td><td>0.42±0.01</td><td>1.05±0.02</td><td>1.99±0.05</td></tr></table>
+
+method helps to stable the training process. Comparison with other methods is shown in Tab.2. The FDE for single predicted trajectory is also improved by our method.
+
+# 4.2 ARGOVERSE DATASET
+
+Argoverse dataset (Chang et al., 2019) contains 300,000 5-second tracked scenarios in 2 American cities Miami and Pittsburgh. The data is recorded in $1 0 \ \mathrm { H z }$ . The first 2 seconds are used as input to predict the next 3 seconds future. It is also one of the biggest open-source motion forecasting datasets that offer semantic maps.
+
+Experiments We downsample the data from $1 0 \mathrm { H z }$ to $2 \mathrm { H z }$ to make the setting similar to nuScenes following the setting of (Park et al., 2020). Argoverse does not release the ground truth future trajectories for the test dataset. Therefore, we use the original validation set as our test set in this experiments and randomly split the original training set into our training set with $9 5 \%$ data and our validation set with $5 \%$ data. Batch size is 256. Initial learning rate is 3e-3 and it decays exponentially by 0.9995 per iteration. The model is trained for maximal 60 epochs and we select the weights from the best epoch measured by average ADE in validation set. The coeffient $\gamma$ in Alg.1 increases gradually from 0 to 1 as a sigmoid function centered at 18th epoch. Trajectron $^ { + + }$ is not designed for Argoverse. To make the experiments runnable, we made some small modifications that are explained in Appx.D. We execute 6 training instances for both original Trajectron $^ { + + }$ and our method, report the average performance and the standard deviation. The results are listed in Tab.3. Compared to Trajectron $^ { + + }$ without our method, our model improves the accuracy of the prediction by about $8 \%$ with our simple loss and reduces the performance variance by about $50 \%$ measured by standard deviation.
+
+# 4.3 ABLATION STUDY
+
+Prediction horizon for negative candidates Assume we have a predicted trajectory with a length six, and it is on the drivable region and obeys the lane direction. However, the trajectory tends to hit the road boundary in the near future (e.g., in 1 second). Such a trajectory can pass our checker’s exam, but it is still unlikely to happen in the real world. We can easily select out such a trajectory by extending the prediction horizon for the candidate trajectories and examining our checker’s extended version. To verify whether this helps us build a better checker, we extend the prediction horizon for negative candidates from 3 seconds to 4, 5, 6 seconds, respectively, and examine them by our original checker. The selected negative trajectories are truncated back to 3 seconds for computing $L _ { \mathrm { u n l i k e } }$ . We can see in table 4 that the model benefits from an adequately extended prediction horizon. The prediction horizon for ground truth trajectory is 3s. By extending the negative trajectories 1 second more, we improve the prediction accuracy. Numbers are averaged over two training instances.
+
+![](images/10373648cea99d219d79d7cb06ab1e5382c63c59f66797cc160ed58c20678d7d.jpg)  
+Figure 3: Qualitative results of our method and Trajectron $^ { + + }$ in a complex scenario. Some of the predicted distribution of Trajectron $^ { + + }$ are out of the road or cover the lane with wrong direction. Our method helps alleviate this issue. Predicted distribution is plotted as colored region and white points denotes the ground truth trajectories. More results are in Appx.F
+
+Table 3: Argoverse: Experimental results on Argoverse dataset. Compared to our base model Trajectron $^ { + + }$ , our proposed method helps increase the accuracy and stable the performance.   
+
+<table><tr><td>Model</td><td>ADE-1</td><td>FDE-1</td><td>ADE-Full</td><td>FDE-Full</td></tr><tr><td>Trajectron++</td><td>1.15±0.14</td><td>2.73±0.24</td><td>1.43±0.15</td><td>3.40±0.20</td></tr><tr><td>Ours</td><td>1.06±0.08</td><td>2.58±0.14</td><td>1.34±0.06</td><td>3.25±0.10</td></tr></table>
+
+Table 4: nuScenes: Ablation study on prediction horizon of negative candidates on nuScenes dataset.   
+
+<table><tr><td></td><td></td><td colspan="5">FDE Full</td><td colspan="3">FDE ML</td><td colspan="5">B.Violations 3s</td></tr><tr><td>Model</td><td>PHn</td><td>1s</td><td>2s</td><td>3s</td><td>4s</td><td>1s</td><td>2s</td><td>3s</td><td>4s</td><td>1s</td><td>2s</td><td></td><td></td><td>4s</td></tr><tr><td>Trajectron++</td><td>-</td><td>0.107</td><td>0.560</td><td>1.378</td><td>2.621</td><td>0.052</td><td>0.396</td><td>1.010</td><td></td><td>1.950</td><td>9.169%</td><td>9.710%</td><td>13.048%</td><td>21.369%</td></tr><tr><td>Ours</td><td>3s</td><td>0.105</td><td>0.538</td><td>1.320</td><td></td><td>2.494</td><td>0.054</td><td>0.414</td><td>1.029</td><td>1.960</td><td>9.188%</td><td>9.619%</td><td>11.818%</td><td>17.460%</td></tr><tr><td>Ours</td><td>4s</td><td>0.087</td><td>0.498</td><td>1.238</td><td>2.339</td><td></td><td>0.050</td><td>0.387</td><td>0.981</td><td>1.869</td><td>9.183%</td><td>9.620%</td><td>11.872%</td><td>17.551%</td></tr><tr><td>Ours</td><td>5s</td><td>0.087</td><td>0.497</td><td>1.259</td><td>2.406</td><td></td><td>0.056</td><td>0.397</td><td>1.002</td><td>1.889</td><td>9.174%</td><td>9.667%</td><td>12.106%</td><td>17.992%</td></tr><tr><td>Ours</td><td>6s</td><td>0.099</td><td>0.521</td><td>1.297</td><td>2.474</td><td></td><td>0.059</td><td>0.405</td><td>1.004</td><td>1.912</td><td>9.170%</td><td>9.636%</td><td>12.046%</td><td>17.761%</td></tr></table>
+
+# 5 CONCLUSION
+
+We present unlikelihood guided trajectory prediction method, that minimizes the probability of unlikely trajectories. During training, our context checker detects predicted unlikely trajectories and their probabilities are reduced through an unlikelihood loss. Our method can be incorporated into state-of-the-art models with a maximum likelihood estimation objective. Our experimental results demonstrate that our method significantly improves state-of-the-art trajectory prediction models.
+
+We hope that our work may encourage future work on exploring better unlikelihood methods for trajectory prediction and improved context checker models.
+
+# REFERENCES
+
+Abubakar Abid and James Zou. Contrastive variational autoencoder enhances salient features. arXiv preprint arXiv:1902.04601, 2019.
+
+Alexandre Alahi, Kratarth Goel, Vignesh Ramanathan, Alexandre Robicquet, Li Fei-Fei, and Silvio Savarese. Social lstm: Human trajectory prediction in crowded spaces. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 961–971, 2016.
+
+Holger Caesar, Varun Bankiti, Alex H Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom. nuscenes: A multimodal dataset for autonomous driving. arXiv preprint arXiv:1903.11027, 2019a.
+
+Holger Caesar, Varun Bankiti, Alex H. Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom. nuscenes: A multimodal dataset for autonomous driving. arXiv preprint arXiv:1903.11027, 2019b.
+
+Sergio Casas, Cole Gulino, Renjie Liao, and Raquel Urtasun. Spatially-aware graph neural networks for relational behavior forecasting from sensor data. arXiv preprint arXiv:1910.08233, 2019.
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+Sergio Casas, Cole Gulino, Simon Suo, and Raquel Urtasun. The importance of prior knowledge in precise multimodal prediction. arXiv preprint arXiv:2006.02636, 2020.
+
+Yuning Chai, Benjamin Sapp, Mayank Bansal, and Dragomir Anguelov. Multipath: Multiple probabilistic anchor trajectory hypotheses for behavior prediction. arXiv preprint arXiv:1910.05449, 2019.
+
+Ming-Fang Chang, John Lambert, Patsorn Sangkloy, Jagjeet Singh, Slawomir Bak, Andrew Hartnett, De Wang, Peter Carr, Simon Lucey, Deva Ramanan, et al. Argoverse: 3d tracking and forecasting with rich maps. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 8748–8757, 2019.
+
+Henggang Cui, Vladan Radosavljevic, Fang-Chieh Chou, Tsung-Han Lin, Thi Nguyen, TzuKuo Huang, Jeff Schneider, and Nemanja Djuric. Multimodal trajectory predictions for autonomous driving using deep convolutional networks. CoRR, abs/1809.10732, 2018. URL http://arxiv.org/abs/1809.10732.
+
+Nachiket Deo and Mohan M Trivedi. Multi-modal trajectory prediction of surrounding vehicles with maneuver based lstms. In 2018 IEEE Intelligent Vehicles Symposium (IV), pp. 1179–1184. IEEE, 2018.
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+Xin Huang, Stephen G. McGill, Jonathan A. DeCastro, Brian Charles Williams, Luke Fletcher, John J. Leonard, and Guy Rosman. Diversity-aware vehicle motion prediction via latent semantic sampling. ArXiv, abs/1911.12736, 2019.
+
+Thomas Kipf, Elise van der Pol, and Max Welling. Contrastive learning of structured world models. arXiv preprint arXiv:1911.12247, 2019.
+
+Vineet Kosaraju, Amir Sadeghian, Roberto Mart´ın-Mart´ın, Ian Reid, S Hamid Rezatofighi, and Silvio Savarese. Social-bigat: Multimodal trajectory forecasting using bicycle-gan and graph attention networks. arXiv preprint arXiv:1907.03395, 2019.
+
+Namhoon Lee, Wongun Choi, Paul Vernaza, Christopher B. Choy, Philip H. S. Torr, and Manmohan Chandraker. DESIRE: distant future prediction in dynamic scenes with interacting agents. In 2017 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017, Honolulu, HI, USA, July 21-26, 2017, pp. 2165–2174. IEEE Computer Society, 2017. doi: 10.1109/CVPR.2017.233. URL https://doi.org/10.1109/CVPR.2017.233.
+
+Zhuang Ma and Michael Collins. Noise contrastive estimation and negative sampling for conditional models: Consistency and statistical efficiency. arXiv preprint arXiv:1809.01812, 2018.
+
+Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018.
+
+Seong Hyeon Park, Gyubok Lee, Manoj Bhat, Jimin Seo, Minseok Kang, Jonathan Francis, Ashwin R Jadhav, Paul Pu Liang, and Louis-Philippe Morency. Diverse and admissible trajectory forecasting through multimodal context understanding. European Conference on Computer Vision, 2020.
+
+Nicholas Rhinehart, Rowan McAllister, Kris Kitani, and Sergey Levine. Precog: Prediction conditioned on goals in visual multi-agent settings. In The IEEE International Conference on Computer Vision (ICCV), October 2019.
+
+Daniela Ridel, Nachiket Deo, Denis Wolf, and Mohan Trivedi. Scene compliant trajectory forecast with agent-centric spatio-temporal grids. IEEE Robotics and Automation Letters, 5(2):2816– 2823, 2020.
+
+Amir Sadeghian, Ferdinand Legros, Maxime Voisin, Ricky Vesel, Alexandre Alahi, and Silvio Savarese. Car-net: Clairvoyant attentive recurrent network. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 151–167, 2018.
+
+Tim Salzmann, Boris Ivanovic, Punarjay Chakravarty, and Marco Pavone. Trajectron $^ { + + }$ : Multiagent generative trajectory forecasting with heterogeneous data for control. arXiv preprint arXiv:2001.03093, 2020.
+
+Kihyuk Sohn, Honglak Lee, and Xinchen Yan. Learning structured output representation using deep conditional generative models. In Advances in neural information processing systems, pp. 3483–3491, 2015.
+
+Charlie Tang and Russ R Salakhutdinov. Multiple futures prediction. In Advances in Neural Information Processing Systems, pp. 15398–15408, 2019.
+
+Sean Welleck, Ilia Kulikov, Stephen Roller, Emily Dinan, Kyunghyun Cho, and Jason Weston. Neural text generation with unlikelihood training. arXiv preprint arXiv:1908.04319, 2019.
+
+Tianyang Zhao, Yifei Xu, Mathew Monfort, Wongun Choi, Chris Baker, Yibiao Zhao, Yizhou Wang, and Ying Nian Wu. Multi-agent tensor fusion for contextual trajectory prediction. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 12126–12134, 2019.
+
+# A DERIVATION OF GRADIENT
+
+Here we show how to obtain Eq.5 and Eq.6
+
+$$
+\begin{array} { r l } & { L _ { t } = - \log \mathcal { N } ( y _ { g t , t } ; \hat { \mu } _ { t } , \hat { \sigma } _ { t } I ) + \log \mathcal { N } ( y _ { n e g , t } ; \hat { \mu } _ { t } , \hat { \sigma } _ { t } I ) } \\ & { \quad = \displaystyle \frac { 1 } { 2 } ( \log 2 \pi + \log \hat { \sigma } _ { t } ^ { 2 } + \frac { | | y _ { g t , t } - \hat { \mu } _ { t } | | ^ { 2 } } { \hat { \sigma } _ { t } ^ { 2 } } ) - \frac { 1 } { 2 } ( \log 2 \pi + \log \hat { \sigma } _ { t } ^ { 2 } + \frac { | | y _ { n e g , t } - \hat { \mu } _ { t } | | ^ { 2 } } { \hat { \sigma } _ { t } ^ { 2 } } ) } \\ & { \quad = \displaystyle \frac { 1 } { 2 \hat { \sigma } _ { t } ^ { 2 } } ( | | y _ { g t , t } - \hat { \mu } _ { t } | | ^ { 2 } - | | y _ { n e g , t } - \hat { \mu } _ { t } | | ^ { 2 } ) } \end{array}
+$$
+
+$$
+\begin{array} { r l r } {  { \frac { \partial L _ { t } } { \partial \hat { \pmb { \mu } } _ { t } } = \frac { \partial } { \partial \hat { \pmb { \mu } } _ { t } } \frac { 1 } { 2 \hat { \sigma } _ { t } ^ { 2 } } ( | | \pmb { y } _ { g t , t } - \hat { \pmb { \mu } } _ { t } | | ^ { 2 } - | | \pmb { y } _ { n e g , t } - \hat { \pmb { \mu } } _ { t } | | ^ { 2 } ) } } \\ & { } & { = - \frac { 1 } { \hat { \sigma } _ { t } ^ { 2 } } ( ( \pmb { y } _ { g t , t } - \hat { \pmb { \mu } } _ { t } ) + ( \hat { \pmb { \mu } } _ { t } - \pmb { y } _ { n e g , t } ) ) } \end{array}
+$$
+
+$$
+\begin{array} { r l r } & { } & { \displaystyle \frac { \partial L } { \partial \hat { \sigma } _ { t } } = \frac { \partial } { \partial \hat { \sigma } _ { t } } \frac { 1 } { 2 \hat { \sigma } _ { t } ^ { 2 } } ( | | y _ { g t , t } - \hat { \pmb { \mu } } _ { t } | | ^ { 2 } - | | y _ { n e g , t } - \hat { \pmb { \mu } } _ { t } | | ^ { 2 } ) } \\ & { } & { \mathrm { ~ \ ~ \ } = - \frac { 1 } { \hat { \sigma } _ { t } ^ { 3 } } ( | | y _ { g t , t } - \hat { \pmb { \mu } } _ { t } | | ^ { 2 } - | | y _ { n e g , t } - \hat { \pmb { \mu } } _ { t } | | ^ { 2 } ) } \end{array}
+$$
+
+# B GRADIENT ANALYSIS IN GAUSSIAN MIXTURE MODEL
+
+In case we model the output distribution as a Gaussian mixture model $\begin{array} { r l } { p _ { \mathrm { G M M } } ( \pmb { y } _ { t } ) } & { { } = } \end{array}$ $\begin{array} { r l } { \sum _ { i } \phi _ { i } \mathcal { N } ( \pmb { y } _ { t } ; \hat { \pmb { \mu } } _ { i , t } , \hat { \sigma } _ { i , t } \pmb { I } ) } & { { } } \end{array}$ , the gradient of $L _ { t }$ w.r.t. the mean of component $i$ is
+
+$$
+\begin{array} { r l } & { \frac { \partial L _ { t } } { \partial \hat { \mu } _ { i , t } } = \frac { \partial } { \partial \hat { \mu } _ { i , t } } ( - \log \frac { p _ { \mathrm { G M M } } ( y _ { g , t , t } ) } { p _ { \mathrm { G M M } } ( y _ { n e g , t } ) } ) } \\ & { \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 \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 \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 } \\ & { \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 \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 \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 \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 \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 \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 \quad \quad \quad \ \end{array}
+$$
+
+This gradient shows that the center $\hat { \mu } _ { i , t }$ of component $i$ will be pushed towards the ground truth location ${ \bf \nabla } _ { { \bf { y } } _ { g t , t } }$ and way from the negative location ${ \bf { \mathscr { y } } } _ { n e g , t }$ . For $\hat { \sigma } _ { i , t }$ we have
+
+$$
+\begin{array} { r l r } {  { \frac { \partial L } { \partial \hat { \sigma } _ { i , t } } = - \phi _ { i } ( \frac { 1 } { p _ { \mathrm { G M M } } ( y _ { g t , t } ) } \frac { \partial \mathcal { N } ( y _ { g t , t } ; \hat { \mu } _ { i , t } , \hat { \sigma } _ { i , t } I ) } { \partial \hat { \sigma } _ { i , t } } - \frac { 1 } { p _ { \mathrm { G M M } } ( y _ { n e g , t } ) } \frac { \partial \mathcal { N } ( y _ { n e g , t } ; \hat { \mu } _ { i , t } , \hat { \sigma } _ { i , t } I ) } { \partial \hat { \sigma } _ { i , t } } ) } } \\ & { } & { = - \frac { \phi _ { i } } { \sigma _ { i , t } ^ { 3 } } ( \frac { \mathcal { N } ( y _ { g t , t } ; \hat { \mu } _ { i , t } , \hat { \sigma } _ { i , t } I ) } { p _ { \mathrm { G M M } } ( y _ { g t , t } ) } ( \vert \vert y _ { g t , t } - \hat { \mu } _ { i , t } \vert \vert ^ { 2 } - \sigma _ { i , t } ^ { 2 } ) } \\ & { } & { \quad - \frac { \mathcal { N } ( y _ { n e g , t } ; \hat { \mu } _ { i , t } , \hat { \sigma } _ { i , t } I ) } { p _ { \mathrm { G M M } } ( y _ { n e g , t } ) } ( \vert \vert y _ { n e g , t } - \hat { \mu } _ { i , t } \vert \vert ^ { 2 } - \sigma _ { i , t } ^ { 2 } ) ) } \end{array}
+$$
+
+# C THE INFLUENCE OF OUR LOSS WITHOUT MAP INPUT
+
+In this experiment, we remove the map input to the model and demonstrate the influence of our unlikelihood loss in this case. Results are shown in Tab.5 Interestingly, our loss helps improve the performance from 1.52 to 1.42 measured by the FDE-Full in 3 seconds case although the model doesn’t see the context during inference. We think the reason is that compared to the Trajectron $^ { + + }$ without map input, our unlikelihood loss still offers context information to support the training since this loss is calculated using the context. Therefore, our model receives more information during training and performs better. Besides, our method without map input even achieves a comparable result (FDE-FUll 1.42) compared to Trajectron $^ { + + }$ with map input (FDE-Full 1.46). This experiment shows that our loss can inject context information into learning signals. In addition, map input improves FDE-Full of Trajectron $^ { + + }$ in 3s prediction from 1.52 to 1.46, which is about $6 \mathrm { c m }$ . However, when we further add our unlikelihood loss, performance improved from 1.46 to 1.34, which is 12 cm. Our loss triple the contribution of context information, which is significant.
+
+Table 5: The influence of our loss without map input to the model on nuScenes. Compared to Trajectron $^ { + + }$ without map input, our loss performs better, which indicates that our loss can inject context information into learning signals   
+
+<table><tr><td>Model</td><td>FDE-Full 3s</td><td>ADE-Full 3s</td></tr><tr><td>Trajectron++ without map input</td><td>1.52±0.04</td><td>0.61±0.02</td></tr><tr><td>Trajectron++</td><td>1.46±0.07</td><td>0.59±0.04</td></tr><tr><td>Ours without map input</td><td>1.42±0.03</td><td>0.57±0.01</td></tr><tr><td>Ours</td><td>1.34±0.04</td><td>0.54±0.02</td></tr></table>
+
+# D MODIFICATION ON TRAJECTRON $^ { + + }$ FOR ARGOVERSE EVALUATION
+
+The original Trajectron+ $^ { - + }$ (Salzmann et al., 2020) is evaluated on nuScenes dataset (Caesar et al., 2019b) but not on Argoverse (Chang et al., 2019). To make it runnable on Argoverse, we make some modifications on Trajectron $^ { + + }$ . The input states of the full version of Trajectron $^ { + + }$ consist of the positions, velocities, accelerations, heading angles and heading angular velocities. However, Argoverse doesn’t offer the data for heading angles and heading angular velocities. Therefore, we remove them from the input states. In addition, Trajectron $^ { + + }$ has a unicycle physics model to convert the predicted velocity from neural networks to locations. The unicycle physics model requires also the angular velocities information. Therefore, we replace the unicycle model by the single integrator model that requires only velocities and initial positions. The maps offered by nuScens and Argoverse are different, too. We stack the drivable region and region of interest offered by Argoverse and upsample them to the same resolution as nuScenes as the input map. Please refer to Argoverse for the details. Lastly, Trajectron $^ { + + }$ use separate modules to process surrounding vehicles and surrounding pedestrians in nuScenes. However, Argoverse doesn’t offer the labels for the surrouding agents and just simply denote them as ”others”. Therefore, we remove the pedestrian modules in Trajectron $^ { + + }$ and use the vehicle modules to process both surrounding pedestrians and vehicles.
+
+# E COMPARISON WITH OTHER METHODS ON ARGOVERSE
+
+Our experiment setting on Argoverse follows AttGlobal-CAM-Nf (Park et al., 2020). Here we list the detailed comparison with their numbers using their evaluation metrics minADE-12 and minFDE12, which are the minimal ADE and FDE over 12 prediction candidates, in Tab.6. Both Trajectron++ and our method outperform their numbers. In addition, our unlikelihood loss helps improve the performance of Trajectron $^ { + + }$ . Our scores and the standard deviation are calculated over 4 training instances.
+
+Table 6: Experimental results on Argoverse dataset   
+
+<table><tr><td>Model</td><td>minADE-12</td><td>minFDE-12</td></tr><tr><td>CSP (Deo &amp; Trivedi,2018; Park et al.,2020)</td><td>1.39</td><td>2.57</td></tr><tr><td>DESIRE (Lee et al.,2017; Park et al.,2020)</td><td>0.90</td><td>1.45</td></tr><tr><td>MATF-GAN (Zhao et al.,2019;Park et al., 2020)</td><td>1.26</td><td>2.31</td></tr><tr><td>R2P2-MA(Rhinehart et al.,2019;Park etal.,2020)</td><td>1.11</td><td>1.77</td></tr><tr><td>AttGlobal-CAM-Nf (Park et al., 2020)</td><td>0.73</td><td>1.12</td></tr><tr><td>Trajectron++ (Salzmann et al., 2020)</td><td>0.65±0.09</td><td>1.08±0.07</td></tr><tr><td>Ours</td><td>0.63±0.04</td><td>1.08±0.06</td></tr></table>
+
+# F QUALITATIVE RESULTS
+
+Here, we demonstrate our method’s qualitative results compared with Trajectron $^ { + + }$ for 3 seconds prediction. We randomly sample 50 trajectories from the predicted prediction, use kernel density estimation (KDE) to approximate the total output distribution from the samples, and print it out in Fig.4. White points represent the ground truth trajectories. Compared to Trajectron $^ { + + }$ , our method suits the contextual information more and therefore is more accurate and plausible.
+
+![](images/4a3f8749352651ddd592c8b50fb69586771e280e08d9c3215cbe03f98ac3c4c4.jpg)  
+Figure 4: Qualitative results of our method and Trajectron $^ { + + }$ .

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+        "type": "text",
+        "text": "MOTION FORECASTING WITH UNLIKELIHOOD TRAINING ",
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+        "bbox": [
+            176,
+            98,
+            821,
+            145
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
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+        "text": "Motion forecasting is essential for making safe and intelligent decisions in robotic applications such as autonomous driving. State-of-the-art methods formulate it as a sequence-to-sequence prediction problem, which is solved in an encoderdecoder framework with a maximum likelihood estimation objective. In this paper, we show that the likelihood objective itself results in a model assigning too much probability to trajectories that are unlikely given the contextual information such as maps and states of surrounding agents. This is despite the fact that many state-of-the-art models do take contextual information as part of their input. We propose a new objective, unlikelihood training, which forces generated trajectories that conflict with contextual information to be assigned a lower probability by our model. We demonstrate that our method can improve state-of-art models’ performance on challenging real-world trajectory forecasting datasets (nuScenes and Argoverse) by $8 \\%$ and reduce the standard deviation by up to $50 \\%$ . Code will be made available. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "For robotic applications deployed in the real world, the ability to foresee the future motions of agents in the surrounding environment plays an essential role for safe and intelligent decision making. This is a very challenging task. For example, in the autonomous driving domain, to predict nearby agents’ future trajectories, an agent needs to consider contextual information such as their past trajectories, potential interactions, and maps. State of the art prediction models (Salzmann et al., 2020; Tang & Salakhutdinov, 2019; Rhinehart et al., 2019) directly take contextual information as part of their input and use techniques such as graph neural networks to extract high-level features for prediction. They are typically trained with a maximum likelihood estimation (MLE) objective that maximizes the likelihood of ground truth trajectories in the predicted distribution. Although MLE loss encourages the prediction to be close to the ground truth geometrically, it does not focus on learning a good distribution that is plausible with respect to the contextual information. These models predict trajectories that violate the contextual information (e.g., go to opposite driving direction or out of the driving area) but still closes to ground truth. In contrast, humans can easily notice that these trajectories are unlikely in a specific context. This phenomenon suggests that simply applying MLE loss cannot fully exploit contextual information. ",
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+        "text": "To address the problem, we propose a novel and simple method, unlikelihood training, that injects contextual information into the learning signal. Our loss penalizes the trajectories that violate the contextual information, called negative trajectories, by minimizing their likelihood in the predicted distribution. To generate negative trajectories, we first draw a number of candidate trajectories from our model’s predicted distribution. Then, a context checker is used to cut out the trajectories that violate contextual information as negative trajectories. This context checker does not need to be differentiable. By minimizing the likelihood of negative trajectories, the model is forced to use the contextual information to avoid predictions that violate context. Therefore, the prediction quality is improved. ",
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+        "text": "Existing methods (Casas et al., 2020; Park et al., 2020) using contextual information as learning signals either introduce new learning parameters or using high-variance learning methods such as the REINFORCE algorithm (Casas et al., 2020). In contrast, our method injects rich contextual information into the training objective and keeps the training process simple. ",
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+        "text": "Unlikelihood training (Welleck et al., 2019) has been applied to neural text generation. We are the first to propose unlikelihood training for continuous space of trajectories. For the discrete space of token sequences, repeating tokens or n-grams in the generated sequence are chosen as negative tokens. In contrast, we design a context checker to select negative trajectories sampled from the continuous distribution of model predictions. ",
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+    {
+        "type": "text",
+        "text": "Our method can be viewed as a simple add-on to any models that estimate the distribution of future trajectories. It improves their performance by encouraging models to focus more on contextual information without increasing the complexity of its original training process. ",
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+        "text": "Our contributions are summarized as follows: ",
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+        "text": "• We propose a novel and simple method, unlikelihood training for motion forecasting in autonomous driving that encourages models to use contextual information by minimizing the likelihood of trajectories that violate contextual information. Our method can be easily incorporated into state-of-the-art models.   \n• Our experimental results on challenging real-world trajectory forecasting datasets, nuScenes and Argoverse, shows that unlikelihood training can improve prediction performance by $8 \\%$ and reduce the standard deviation by up to $50 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we briefly review the two most related topics. ",
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+    {
+        "type": "text",
+        "text": "Trajectory Forecasting Trajectory forecasting of dynamic agents, a core problem for robotic applications such as autonomous driving and social robots, has been well studied in the literature. State-of-the-art models solves it as a sequence-to-sequence multi-modal prediction problem (Lee et al., 2017; Cui et al., 2018; Chai et al., 2019; Rhinehart et al., 2019; Kosaraju et al., 2019; Tang & Salakhutdinov, 2019; Ridel et al., 2020; Salzmann et al., 2020; Huang et al., 2019). (Cui et al., 2018; Chai et al., 2019; Ridel et al., 2020) predicts multiple future trajectories without learning low dimensional latent agent behaviors. (Lee et al., 2017; Kosaraju et al., 2019; Rhinehart et al., 2019; Huang et al., 2019) encodes agent behaviors in continuous low dimensional latent space while (Tang & Salakhutdinov, 2019; Salzmann et al., 2020) uses discrete latent variables. Discrete latent variables succinctly capture semantically meaningful modes such as turn left, turn right. (Tang & Salakhutdinov, 2019; Salzmann et al., 2020) learns discrete latent variables without explicit labels. All of them use a maximum likelihood estimation (MLE) objective or its approximations (e.g., VAE). In this paper, we show that MLE loss can ignore contextual information such as maps and states of surrounding agents. As a result, models with such a loss can assign too much probability to unlikely trajectories. We propose an unlikelihood training objective to avoid such cases. All models with the maximum likelihood estimation objective can potentially benefit from our methods. ",
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+    {
+        "type": "text",
+        "text": "Contrastive learning and unlikelihood training To date, several studies have investigated the possibilities to benefit from negative data. One of the popular direction is contrastive learning. Contrastive learning has achieved significant success in many fields (Oord et al., 2018; Kipf et al., 2019; Ma & Collins, 2018; Abid & Zou, 2019; Welleck et al., 2019). NCE (Ma & Collins, 2018) CPC (Oord et al., 2018) maximizes the mutual information between data and latent representation by a novel contrastive loss to extract useful representation from data. C-SWMs (Kipf et al., 2019) utilizes contrastive learning to learn a better world model for reinforcement learning tasks. Recently, unlikelihood training (Welleck et al., 2019) proposes a new method to utilize negative data. In addition, to maximize the likelihood of the ground truth token, it minimizes the likelihood of negative tokens for better text generation. Their method is on the discrete space of token sequences. Repeating tokens or n-grams in the generated sequence is chosen as negative tokens. In contrast, our proposed method works in the continuous space of trajectories. We design a novel method, context checker, to select negative trajectories. ",
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+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 PROBLEM FORMULATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We are targeting at better predicting the future trajectory $\\mathbf { Y } _ { i }$ of a vehicle $i$ given input $\\mathbf { X } _ { i }$ . $\\mathbf { X } _ { i }$ can include any related information like rasterized maps or past positions of vehicle $i$ and surrounding agents, depends on the design of the method. Here we skip the detailed choice of input and denote it as $\\mathbf { X } _ { i }$ for conciseness. Due to different driving strategies, driving intents, and the complex traffic environment, there are usually multiple possible future trajectories given an input $x _ { i }$ (although there is only one ground truth future trajectory $y _ { i , g t }$ in a dataset recorded in the real world). To handle this situation, most state of the art methods (Salzmann et al., 2020) model a distribution of possible future trajectories $p _ { \\theta } ( \\mathbf { Y } _ { i } \\mid \\mathbf { X } _ { i } )$ to cover all the possibilities given the input $\\mathbf { X } _ { i }$ instead of predicting one trajectory. $\\theta$ denotes the learning parameters of the model. To train such methods, most stateof-the-art models usually use maximum likelihood estimation (MLE) to maximize the likelihood of ground truth trajectory $\\mathbf { Y } _ { i , g t }$ in the predicted distribution. For example, the loss of CVAE-based model Trajectron $^ { + + }$ (Salzmann et al., 2020) is Eq.1. This loss is used to maximize the lower bound of ground truth’s likelihood when the coefficient $k = 1$ . ",
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+    {
+        "type": "equation",
+        "img_path": "images/781105400a1111a0e3fa91d530f94715f23464a8c12287377eb4499ebc1d78ae.jpg",
+        "text": "$$\n\\begin{array} { r l } & { L _ { \\mathrm { t r a j + + } } = - \\mathbb { E } _ { \\hat { z } \\sim q _ { \\theta 3 } ( z | X _ { i } , Y _ { i , g t } ) } \\bigl [ \\log p _ { \\theta 2 } ( Y _ { i , g t } \\mid X _ { i } , \\hat { z } ) \\bigr ] } \\\\ & { \\qquad + k D _ { \\mathrm { K L } } \\bigl ( q _ { \\theta 3 } ( z \\mid X _ { i } , Y _ { i , g t } ) \\bigr ) \\bigl \\| p _ { \\theta 1 } ( z \\mid X _ { i } ) \\bigr ) - I _ { q } ( X _ { i } ; z ) } \\\\ & { \\qquad \\geq - \\log p ( Y _ { i , g t } \\mid X _ { i } , Y _ { i , g t } ) - I _ { q } ( X _ { i } ; z ) , \\qquad \\mathrm { w h e n ~ } k = 1 } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Limitation of MLE on Motion Forecasting MLE encourages the model to predict a distribution that allocates reasonable probability mass to the region where $\\mathbf { Y _ { i } }$ is located by minimizing the KL-divergence of predicted distribution and ground truth distribution. Because the domain of trajectory distribution is over the geometric locations, MLE makes these two distributions ”close” to each other geometrically. However, we argue that maintaining the geometrical nearness only is not good enough for motion forecasting task in autonomous driving. In complex traffic scenarios, there can be many potential trajectories close enough to the ground truth geometrically but are very unlikely to happen. For example, if the ground truth trajectory $Y _ { i , g t }$ is on the outermost lane, a trajectory that is close to $Y _ { i , g t }$ but outside the drivable region is unlike to happen in the real world. However, MLE loss will not impose a significant enough penalty on such a case to avoid such a prediction. Fig.1 demonstrates a prediction example from Trajectron $^ { + + }$ (Salzmann et al., 2020) where part of the distribution is outside of the derivable region or on the lane with the wrong direction. The MLE-based loss ",
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+    {
+        "type": "image",
+        "img_path": "images/1847de5db5a5b37e5aedcaf818bf8b15925a84c83024d377c47a7cf211c8cb7d.jpg",
+        "image_caption": [
+            "Figure 1: Examples of predicted distribution from Trajectron $^ { + + }$ (Salzmann et al., 2020). White points denote the ground truth trajectory $\\mathbf { Y } _ { i , g t }$ and the color region indicates the predicted distribution. Some of the prediction go outside of the drivable region or go to the lane in opposite direction. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "only offers learning signals that contain the geometric location information of the ground truth trajectories. All the other contextual information, like the drivable region and the lane direction, are missing in the learning signals. While the inputs to a model contain rich contextual information, the model cannot use it to avoid the prediction that is geometrically close to ground truth but violates context. In contrast, this is quite a simple task for humans. ",
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+    {
+        "type": "text",
+        "text": "3.2 UNLIKELIHOOD LOSS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To mitigate this problem, we design a new loss term that encourages the model to consider the contextual information. Inspired by contrastive learning and unlikelihood training, we additionally train our model to minimize the likelihood of trajectories that violate the contextual information given input $X _ { i }$ . We denote them as negative trajectories ${ Y _ { i , n e g } }$ . Let’s first assume that we already have a distribution of negative trajectories $p _ { \\mathrm { n e g } } ( \\bar { \\mathbf { Y } } _ { i } \\mid X _ { i } )$ . One intuitive way is to directly minimize the log likelihood of ${ Y _ { i , n e g } }$ in our predicted distribution, similar to MLE but in an opposite manner ",
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+        "text": "",
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+        "img_path": "images/3d24f7b916935282c2cf06d875908b1feeff02041447abe1c5cef8f4782d3140.jpg",
+        "text": "$$\nL _ { \\mathrm { u n l i k e } } = \\mathbb { E } _ { X _ { i } , \\sim \\mathbb { D } , Y _ { i , n e g } \\sim p _ { \\mathrm { n e g } } ( Y _ { i } | X _ { i } ) } [ \\log p _ { \\theta } ( Y _ { i , n e g } \\mid X _ { i } ) ]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "However, the gradient of log function tends to infinity when the input tends to 0, which leads to unstable training since the model are optimized to minimize $p _ { \\theta } ( \\bar { \\mathbf { Y } _ { i , n e g } } \\mid \\mathbf { X } _ { i } )$ and $p _ { \\theta } ( \\mathbf { Y } _ { i , n e g } \\mid$ $\\mathbf { X } _ { i } ) \\geq 0$ . To avoid the infinity gradient region of log function, we add a small constant $\\epsilon$ to the likelihood. The final loss term we propose is ",
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+    {
+        "type": "equation",
+        "img_path": "images/5c3382e3c245247bc8187e6b58792cfff5d403acb4ba059dbadb41796f445159.jpg",
+        "text": "$$\nL _ { \\mathrm { u n l i k e } } = \\mathbb { E } _ { X _ { i } , \\sim \\mathbb { D } , Y _ { i , n e g } \\sim p _ { \\mathrm { n e g } } ( \\mathbf { Y } _ { i } \\mid \\mathbf { X } ) } [ \\log ( p _ { \\theta } ( { Y } _ { i , n e g } \\mid X _ { i } ) + \\epsilon ) ) ]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "We call it unlikelihood loss. We use a coefficient $\\gamma$ to balance $L _ { \\mathrm { u n l i k e } }$ . The final training objective in case we combine our method with Trajectron $^ { + + }$ is ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/b7231758c98e06b77e9a1ebd6525d40825bbfa3d18b205afb803529af5f61efc.jpg",
+        "text": "$$\nL = L _ { \\mathrm { t r a j + + } } + \\gamma L _ { \\mathrm { u n l i k e } }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Eq.4 is also easily adapted to combine with any other models that predict trajectory distribution as output. With the help of $L _ { \\mathrm { u n l i k e } }$ , we inject the contextual information into the learning signal, force the model to better extract and use contextual information in $\\mathbf { X } _ { i }$ , and generate more reasonable predicted distribution to avoid high $L _ { \\mathrm { u n l i k e } }$ . ",
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+    {
+        "type": "text",
+        "text": "3.3 NEGATIVE TRAJECTORIES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our proposed loss term is highly dependent on the negative samples ${ Y } _ { i , n e g }$ from the distribution $p _ { \\mathrm { n e g } } ( Y _ { i } \\mid X _ { i } )$ . However, these are not given in the dataset. To solve this issue, we approximate the samples by directly drawing a set of trajectories from the predicted distribution and select the trajectories that violate the contextual information out by a context checker. Note that this checker does not need to be differentiable and it can be as complex and advanced as necessary. The type of unlike predictions the model learns to avoid by our method depends on the type of unlike trajectories the checker can detect. ",
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+    {
+        "type": "text",
+        "text": "Design of Our Checker We implement a map-based checker to judge whether a given trajectory suits the context or not. In detail, the checker examines whether the trajectory goes into the lane in the opposite direction or out of the road. We create a map that stores the lane direction at every location of lanes and the drivable region. Two examples are shown in Fig.2. We first check whether all the locations of a given trajectory are in the drivable region. If so, we further calculate angles between velocity and the lane direction at each time step to see whether they are all inside a 90 degree. The velocities are approximated by differentiating the trajectory. The trajectories that fail to pass the exam are judged as negative trajectories ${ Y _ { i , n e g } }$ . Note that the lane direction information originally comes from the dataset and is usually incomplete or invalid like in the intersection. In this case, we only use the drivable region information. In addition, there are also a small part of ground truth trajectories in the dataset that violates the lane direction or drivable region. In this case, our checker skips the checking and we train it without $L _ { \\mathrm { u n l i k e } }$ to allow similar prediction. Note that this checker is not perfect due to the incomplete information and simple checking mechanism. A trajectory that passes the exam of the checker doesn’t mean it is $100 \\%$ compatible with the context. However, our method can still work properly, because our method only depends on the negative trajectories that do not pass the exam and has nothing to do with the passing ones. Our checker design offers reasonable negative trajectories to support our approach. But of course, a more advanced checker helps the model to avoid more complex unlikely prediction. We leave it open for future research. ",
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+    {
+        "type": "text",
+        "text": "3.4 ALGORITHM ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The finial algorithm is shown in Alg.1. At each iteration, we first run the forward pass of the model to get the output distribution $p _ { \\theta } ( \\pmb { Y } _ { i } \\mid \\pmb { X } _ { i } )$ given the input $X _ { i }$ . Then, $K$ negative candidate trajectories are drawn from this distribution and we select the negative trajectories out ${ Y _ { i , n e g } }$ via our checker. After that, ground truth trajectory and negative trajectories are used to calculate the loss function, update the model, and go to next iteration. Note that if there are no ${ Y _ { i , n e g } }$ in the $K$ negative candidates judged by our checker for data $i$ , we don’t apply $L _ { \\mathrm { u n l i k e } }$ on $i$ . ",
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+        "img_path": "images/3cc40cce52f70c1a0954a903aa911a5f9324ffbf776ff889b682efe3fa35338e.jpg",
+        "image_caption": [
+            "Figure 2: Examples of maps used in the checker in nuScences (Caesar et al., 2019a). Green and blue region together denote the drivable region and the blue means that we have lane direction information here. Random locations are sampled and their lane directions are plotted as red arrows to show the concrete directions. Blue line denotes the trajectories to check and the velocity directions are represented as yellow arrows. (a) shows a negative trajectory that goes out of the road. (b) is a passing case. "
+        ],
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+    {
+        "type": "text",
+        "text": "3.5 GRADIENT ANALYSIS ",
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+    {
+        "type": "text",
+        "text": "Let’s assume a single-mode prediction case that the future position $_ { { \\mathbf { \\mathit { y } } } _ { i , g t , t } }$ at time step $t$ of agent $i$ is modeled by a simple Gaussian distribution $\\mathcal { N } ( \\boldsymbol { y } _ { i , t } ; \\bar { \\boldsymbol { \\mu } } _ { i , t } , \\hat { \\sigma } _ { i , t } \\bar { \\boldsymbol { I } } )$ . $\\hat { \\pmb { \\mu } }$ and $\\hat { \\sigma } _ { i , t }$ are calculated by the model. With a single negative position ${ \\bf { \\it { y } } } _ { i , n e g , t }$ we define a simple loss for step $t$ $L _ { t } = - \\log \\mathcal { N } ( y _ { g t , t } ; \\hat { \\mu } _ { t } , \\hat { \\sigma } _ { t } I ) + \\log \\bar { \\mathcal { N } } ( y _ { n e g , t } ; \\hat { \\mu } _ { t } , \\hat { \\sigma } _ { t } I )$ and omit the subscript $i$ for brevity. The gradient of $L _ { t }$ with respect to $\\hat { \\pmb { \\mu } } _ { t }$ and $\\hat { \\sigma } _ { t }$ in this case is (Derivation in Appx.A): ",
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+        "type": "equation",
+        "img_path": "images/8f5f26cfdaaed43d7bd1b48fd77525772eee4fd14f4940ed5aeb8a0ae445ab8c.jpg",
+        "text": "$$\n\\frac { \\partial { { L } _ { t } } } { \\partial { { \\hat { \\pmb { \\mu } } } _ { t } } } = - \\frac { 1 } { { { \\hat { \\sigma } } _ { t } ^ { 2 } } } ( ( \\pmb { y } _ { g t , t } - { { \\hat { \\pmb { \\mu } } } _ { t } } ) + ( \\pmb { { \\hat { \\mu } } } _ { t } - \\pmb { y } _ { n e g , t } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b9b5b6e1f7074b8bf23a58805170f97d89b0c0ad5254f61d8373334c66f76382.jpg",
+        "text": "$$\n\\frac { \\partial L } { \\partial \\hat { \\sigma } _ { t } } = - \\frac { 1 } { { \\hat { \\sigma } _ { t } } ^ { 3 } } ( | | y _ { g t , t } - \\hat { \\mu } _ { t } | | ^ { 2 } - | | y _ { n e g , t } - \\hat { \\mu } _ { t } | | ^ { 2 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            648,
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+            683
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Eq.5 shows that the center of the predicted distribution $\\hat { \\pmb { \\mu } } _ { t }$ is pushed towards ${ \\bf \\nabla } _ { { \\bf { y } } _ { g t , t } }$ and pushed away from yneg,t by this learning objective. In Eq.6, when ygt,t is closer to the center than yneg,t, ∂L∂σˆt is positive and $\\hat { \\sigma } _ { t }$ is decreased. Note that ${ { y } _ { n e g , t } }$ is selected out from samples of $\\mathcal { N } ( \\hat { \\mu } _ { t } , \\hat { \\sigma } _ { t } \\pmb { I } ) )$ , this means when $\\mathcal { N } ( \\hat { \\mu } _ { t } , \\hat { \\sigma } _ { t } \\pmb { I } ) )$ ) covers context-violated region and this region is farther than ground truth region, $\\mathcal { N } ( \\hat { \\mu } _ { t } , \\hat { \\sigma } _ { t } \\pmb { I } ) )$ will shrink to exclude the negative region and become a better estimation to the true data distribution. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "When the prediction is not so accurate (e.g. at the beginning of training), our ground truth $\\scriptstyle { \\boldsymbol { \\mathbf { \\mathit { y } } } } _ { \\mathit { g t , t } }$ may be farther than the negative location ${ \\bf { \\mathscr { y } } } _ { n e g , t }$ . In this case, $\\mathcal { N } ( \\hat { \\mu } _ { t } , \\hat { \\sigma } _ { t } \\pmb { I } )$ will expand to better cover the ground truth and make the prediction more uncertain. A simple approach to alleviate this issue is turning off our unlikelihood loss $L _ { \\mathrm { u n l i k e } }$ in the first few training epochs. We implement this by making $\\gamma$ in Eq.4 as a sigmoid function centered at a specified epoch. By this way, we smoothly turn on $L _ { \\mathrm { u n l i k e } }$ during training. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present the experimental results of our method to demonstrate our performance. Our method is easily to applied on state of the art models that generate a future trajectory distribution and can further improve their performance. In our experiments, we select Trajectron $^ { + + }$ (Salzmann et al., 2020), one of the state-of-the-art methods on NuScenes dataset (Caesar et al., 2019b) with open-source implementation, as our base model. We extend the implementation to work on Argoverse dataset (Chang et al., 2019). We evaluate our approach on these two motion forecasting datasets. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/fcefd29c84478312f7cd963d3ee6f0b6c08b1e0eb0e4e72b4362a6fcfd3f0422.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm 1: Training process (Use Trajec- tron++ as base model)</td></tr><tr><td>Initialize the model parameters 0; Initialize learning rate α and coeifficient γ; while not converge do X,Yi,gt ~ D run forward pass to compute pe(Yi | Xi) draw K trajectotries Yi,k ~ pe(Yi |Xi) select Yi,neg via checker Compute Ltraj++ using Eq.1 Compute Lunlike using Eq.3 L = Ltraj++ + γLunlike 0=0-αVθ(L) end</td></tr></table>",
+        "bbox": [
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+            313
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Test Model Trajectron $^ { + + }$ (Salzmann et al.,   \n2020) is a CVAE-based (Sohn et al., 2015) model. Its input $X _ { i }$ contains positions, velocities, heading of the predicted and surrounding vehicles, and a map patch. The output distribution $\\begin{array} { r } { p _ { \\theta } ( \\boldsymbol { Y _ { i } } ~ \\{ \\textbf {  { X } } _ { i } ) ~ = ~ \\sum _ { z } p _ { \\theta 1 } ( z ~ \\lvert ~ \\boldsymbol { X } _ { i } ) p _ { \\theta 2 } ( \\boldsymbol { Y _ { i } } ~ \\{ \\textbf {  { X } } _ { i } , { z } \\} | } \\end{array}$ is a Gaussian mixture model with 25 components and modeled by an encoder net $p _ { \\theta 1 } ( z \\mid X _ { i } )$ and a decoder net $p _ { \\theta 1 } ( Y _ { i } \\mid X _ { i } , z )$ . In addition, it has another encoder net ${ q _ { \\theta 3 } } ( z \\mid X _ { i } , Y _ { i , g t } )$ used only in training. The original learning objective is shown in Eq.1. $I _ { q }$ denotes the mutual information. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Evaluation Metrics we use average $l _ { 2 }$ displacement error (ADE) and final $l _ { 2 }$ displacement error (FDE) to evaluate the prediction performance. Each of them contains some sub-versions. FDE-1 is the FDE calculated using only 1 predicted trajectory. In both original Trajectron $^ { + + }$ and our approach, this single trajectory is drawn from predicted distribution by greedy search step by step. ADE-Full/FDE-Full represents the quality of the whole output distribution. To compute ADEFull/FDE-Full, we randomly sample 200 trajectories and calculate the average performance as the reported scores. In addition, we use our context checker to measure the context-violation rate in these 200 trajectories as a metric to show the context-related performance. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1 NUSCENES DATASET ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "nuScenes dataset (Caesar et al., 2019b) contains 1000 city driving scenes from both left-hand (Singapore) and right-hand (Boston) traffic regions. Each scene is 20s long and recorded in $2 \\mathrm { H z }$ . It is one of the biggest open-source motion forecasting datasets with detailed semantic maps. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Experiments The batch size is set to 1024. Models are trained for 35 epochs and we test the weights from the best epoch measured by average ADE on validation set. The coefficient $\\gamma$ in Alg.1 increases gradually from 0 to 1 as a sigmoid function centered at 24th epoch. Initial learning rate is 3e-3 and it decays exponentially by 0.9995 per iteration. These hyperparameters except $\\gamma$ are optimized for Trajectron $^ { + + }$ and lead to better performance than that in original paper. In addition, we rotate the scenes randomly from $1 5 ^ { \\circ }$ to $3 4 5 ^ { \\circ }$ in the training set for data augmentation following the setting of original Trajectron $^ { + + }$ . For each model, we run 5 experiments and report the mean and standard deviation of the measured metrics. The models are trained to predict 3 seconds into the future. To evaluate on generalization beyond the training horizon, we test on both 3 second and 4 second prediction horizons. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Tab.1 shows the quantitative results 3s and 4s prediction measured by the FDE/ADE-Full metrics and the context-violation rate. Our unlikelihood loss improves Trajectron $^ { + + }$ by about $8 \\%$ for both metrics in both 3 and 4 second prediction horizons. Results indicate that our method helps improve the accuracy of the predicted distribution. This is also demonstrated in the qualitative comparison in Fig.3. The predicted distribution from our methods covers the not-drivable region less compared to original Trajectron $^ { + + }$ without our proposed loss. In contrast, original Trajectron $^ { + + }$ tends to violate the contextual information when prediction horizon is long. This shows that our method encourages the model to be more sensitive to the road boundary and the lane direction. In addition, we observe a reduction of the performance variance measured by standard deviation in Tab1, indicates that our ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/17f5926890a780be693cd005263d9524142bd06513d45c048d507eb89c94f5b1.jpg",
+        "table_caption": [
+            "Table 1: nuScenes: Experiment results on nuScenes dataset (Caesar et al., 2019b) with Trajectron $^ { + + }$ (Salzmann et al., 2020). FDE and ADE are averaged over 200 trajectories drawn from the predicted distribution. Our proposed loss improves the performance of Trajectron $^ { + + }$ by about $8 \\%$ , and avoid $16 \\%$ context-violated prediction compared to Trajectron $^ { + + }$ , which indicates a better predicted distribution. Mean and standard deviation are calculated over 5 runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td colspan=\"2\">FDE-Full</td><td colspan=\"2\">ADE-Full</td><td colspan=\"2\">Context-Violation-Rate</td></tr><tr><td></td><td>3s</td><td>4s</td><td>3s</td><td>4s</td><td>3s</td><td>4s</td></tr><tr><td>Trajectron++</td><td>1.46±0.07</td><td>2.74±0.10</td><td>0.59±0.04</td><td>1.04±0.05</td><td>7.29%±0.22%</td><td>10.59%±0.54%</td></tr><tr><td>Ours</td><td>1.34±0.04</td><td>2.51±0.06</td><td>0.54±0.02</td><td>0.95±0.03</td><td>6.57 %±0.15%</td><td>8.85% ± 0.32%</td></tr></table>",
+        "bbox": [
+            176,
+            190,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 2: nuScenes: Experimental results on the nuScenes dataset for single prediction. FDE and ADE are computed by only one predicted trajectory. For both Trajectron $^ { + + }$ and our method, this trajectory is sampled by greedy search. Our method helps to improve the predicted accuracy. Mean and standard deviation are calculated over 5 runs. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/fdf315ac978add79ad882216361041726f6756967b1f40ed84c1538ca8807852.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td colspan=\"4\">FDE-1</td></tr><tr><td></td><td>1s</td><td>2s</td><td>3s</td><td>4s</td></tr><tr><td>Const. Velocity (Salzmann et al., 2020)</td><td>0.32</td><td>0.89</td><td>1.70</td><td>2.73</td></tr><tr><td>S-LSTM (Alahi et al., 2016)</td><td>0.47</td><td></td><td>1.61</td><td>-</td></tr><tr><td>CSP (Deo &amp; Trivedi,2018)</td><td>0.46</td><td>2.35</td><td>1.50</td><td></td></tr><tr><td>CAR-Net (Sadeghian et al., 2018)</td><td>0.38</td><td>-</td><td>1.35</td><td></td></tr><tr><td>SpAGNN (Casas et al., 2019)</td><td>0.36</td><td>=</td><td>1.23</td><td>-</td></tr><tr><td>Trajectron++ (Salzmann et al., 2020)</td><td>0.07</td><td>0.45</td><td>1.14</td><td>2.20</td></tr><tr><td>Trajectron++ (our hyperparameters)</td><td>0.06±0.01</td><td>0.43±0.01</td><td>1.08±0.04</td><td>2.05±0.08</td></tr><tr><td>Ours</td><td>0.05±0.00</td><td>0.42±0.01</td><td>1.05±0.02</td><td>1.99±0.05</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "method helps to stable the training process. Comparison with other methods is shown in Tab.2. The FDE for single predicted trajectory is also improved by our method. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 ARGOVERSE DATASET ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Argoverse dataset (Chang et al., 2019) contains 300,000 5-second tracked scenarios in 2 American cities Miami and Pittsburgh. The data is recorded in $1 0 \\ \\mathrm { H z }$ . The first 2 seconds are used as input to predict the next 3 seconds future. It is also one of the biggest open-source motion forecasting datasets that offer semantic maps. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Experiments We downsample the data from $1 0 \\mathrm { H z }$ to $2 \\mathrm { H z }$ to make the setting similar to nuScenes following the setting of (Park et al., 2020). Argoverse does not release the ground truth future trajectories for the test dataset. Therefore, we use the original validation set as our test set in this experiments and randomly split the original training set into our training set with $9 5 \\%$ data and our validation set with $5 \\%$ data. Batch size is 256. Initial learning rate is 3e-3 and it decays exponentially by 0.9995 per iteration. The model is trained for maximal 60 epochs and we select the weights from the best epoch measured by average ADE in validation set. The coeffient $\\gamma$ in Alg.1 increases gradually from 0 to 1 as a sigmoid function centered at 18th epoch. Trajectron $^ { + + }$ is not designed for Argoverse. To make the experiments runnable, we made some small modifications that are explained in Appx.D. We execute 6 training instances for both original Trajectron $^ { + + }$ and our method, report the average performance and the standard deviation. The results are listed in Tab.3. Compared to Trajectron $^ { + + }$ without our method, our model improves the accuracy of the prediction by about $8 \\%$ with our simple loss and reduces the performance variance by about $50 \\%$ measured by standard deviation. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Prediction horizon for negative candidates Assume we have a predicted trajectory with a length six, and it is on the drivable region and obeys the lane direction. However, the trajectory tends to hit the road boundary in the near future (e.g., in 1 second). Such a trajectory can pass our checker’s exam, but it is still unlikely to happen in the real world. We can easily select out such a trajectory by extending the prediction horizon for the candidate trajectories and examining our checker’s extended version. To verify whether this helps us build a better checker, we extend the prediction horizon for negative candidates from 3 seconds to 4, 5, 6 seconds, respectively, and examine them by our original checker. The selected negative trajectories are truncated back to 3 seconds for computing $L _ { \\mathrm { u n l i k e } }$ . We can see in table 4 that the model benefits from an adequately extended prediction horizon. The prediction horizon for ground truth trajectory is 3s. By extending the negative trajectories 1 second more, we improve the prediction accuracy. Numbers are averaged over two training instances. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/10373648cea99d219d79d7cb06ab1e5382c63c59f66797cc160ed58c20678d7d.jpg",
+        "image_caption": [
+            "Figure 3: Qualitative results of our method and Trajectron $^ { + + }$ in a complex scenario. Some of the predicted distribution of Trajectron $^ { + + }$ are out of the road or cover the lane with wrong direction. Our method helps alleviate this issue. Predicted distribution is plotted as colored region and white points denotes the ground truth trajectories. More results are in Appx.F "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            227,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/d4905741ac7e4cad7bef44291056849b37a8da35de82b911ac4086575944fe9a.jpg",
+        "table_caption": [
+            "Table 3: Argoverse: Experimental results on Argoverse dataset. Compared to our base model Trajectron $^ { + + }$ , our proposed method helps increase the accuracy and stable the performance. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>ADE-1</td><td>FDE-1</td><td>ADE-Full</td><td>FDE-Full</td></tr><tr><td>Trajectron++</td><td>1.15±0.14</td><td>2.73±0.24</td><td>1.43±0.15</td><td>3.40±0.20</td></tr><tr><td>Ours</td><td>1.06±0.08</td><td>2.58±0.14</td><td>1.34±0.06</td><td>3.25±0.10</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/2f755b758385b03a110c38ce7547097b948b5982ac7dd664cd75bd8888930dea.jpg",
+        "table_caption": [
+            "Table 4: nuScenes: Ablation study on prediction horizon of negative candidates on nuScenes dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td colspan=\"5\">FDE Full</td><td colspan=\"3\">FDE ML</td><td colspan=\"5\">B.Violations 3s</td></tr><tr><td>Model</td><td>PHn</td><td>1s</td><td>2s</td><td>3s</td><td>4s</td><td>1s</td><td>2s</td><td>3s</td><td>4s</td><td>1s</td><td>2s</td><td></td><td></td><td>4s</td></tr><tr><td>Trajectron++</td><td>-</td><td>0.107</td><td>0.560</td><td>1.378</td><td>2.621</td><td>0.052</td><td>0.396</td><td>1.010</td><td></td><td>1.950</td><td>9.169%</td><td>9.710%</td><td>13.048%</td><td>21.369%</td></tr><tr><td>Ours</td><td>3s</td><td>0.105</td><td>0.538</td><td>1.320</td><td></td><td>2.494</td><td>0.054</td><td>0.414</td><td>1.029</td><td>1.960</td><td>9.188%</td><td>9.619%</td><td>11.818%</td><td>17.460%</td></tr><tr><td>Ours</td><td>4s</td><td>0.087</td><td>0.498</td><td>1.238</td><td>2.339</td><td></td><td>0.050</td><td>0.387</td><td>0.981</td><td>1.869</td><td>9.183%</td><td>9.620%</td><td>11.872%</td><td>17.551%</td></tr><tr><td>Ours</td><td>5s</td><td>0.087</td><td>0.497</td><td>1.259</td><td>2.406</td><td></td><td>0.056</td><td>0.397</td><td>1.002</td><td>1.889</td><td>9.174%</td><td>9.667%</td><td>12.106%</td><td>17.992%</td></tr><tr><td>Ours</td><td>6s</td><td>0.099</td><td>0.521</td><td>1.297</td><td>2.474</td><td></td><td>0.059</td><td>0.405</td><td>1.004</td><td>1.912</td><td>9.170%</td><td>9.636%</td><td>12.046%</td><td>17.761%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            819,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We present unlikelihood guided trajectory prediction method, that minimizes the probability of unlikely trajectories. During training, our context checker detects predicted unlikely trajectories and their probabilities are reduced through an unlikelihood loss. Our method can be incorporated into state-of-the-art models with a maximum likelihood estimation objective. Our experimental results demonstrate that our method significantly improves state-of-the-art trajectory prediction models. ",
+        "bbox": [
+            174,
+            853,
+            823,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We hope that our work may encourage future work on exploring better unlikelihood methods for trajectory prediction and improved context checker models. ",
+        "bbox": [
+            171,
+            103,
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+            132
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            154,
+            287,
+            167
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Abubakar Abid and James Zou. Contrastive variational autoencoder enhances salient features. arXiv preprint arXiv:1902.04601, 2019. ",
+        "bbox": [
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+        ],
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+    },
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Holger Caesar, Varun Bankiti, Alex H Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom. nuscenes: A multimodal dataset for autonomous driving. arXiv preprint arXiv:1903.11027, 2019a. ",
+        "bbox": [
+            173,
+            263,
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+        ],
+        "page_idx": 8
+    },
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+        "text": "Tim Salzmann, Boris Ivanovic, Punarjay Chakravarty, and Marco Pavone. Trajectron $^ { + + }$ : Multiagent generative trajectory forecasting with heterogeneous data for control. arXiv preprint arXiv:2001.03093, 2020. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Tianyang Zhao, Yifei Xu, Mathew Monfort, Wongun Choi, Chris Baker, Yibiao Zhao, Yizhou Wang, and Ying Nian Wu. Multi-agent tensor fusion for contextual trajectory prediction. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 12126–12134, 2019. ",
+        "bbox": [
+            176,
+            540,
+            823,
+            583
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A DERIVATION OF GRADIENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            437,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Here we show how to obtain Eq.5 and Eq.6 ",
+        "bbox": [
+            174,
+            133,
+            459,
+            148
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d1ad1debceb4d8457896ed20b250ac7b33c1f8d93c6bc05ce71e1f090636c18d.jpg",
+        "text": "$$\n\\begin{array} { r l } & { L _ { t } = - \\log \\mathcal { N } ( y _ { g t , t } ; \\hat { \\mu } _ { t } , \\hat { \\sigma } _ { t } I ) + \\log \\mathcal { N } ( y _ { n e g , t } ; \\hat { \\mu } _ { t } , \\hat { \\sigma } _ { t } I ) } \\\\ & { \\quad = \\displaystyle \\frac { 1 } { 2 } ( \\log 2 \\pi + \\log \\hat { \\sigma } _ { t } ^ { 2 } + \\frac { | | y _ { g t , t } - \\hat { \\mu } _ { t } | | ^ { 2 } } { \\hat { \\sigma } _ { t } ^ { 2 } } ) - \\frac { 1 } { 2 } ( \\log 2 \\pi + \\log \\hat { \\sigma } _ { t } ^ { 2 } + \\frac { | | y _ { n e g , t } - \\hat { \\mu } _ { t } | | ^ { 2 } } { \\hat { \\sigma } _ { t } ^ { 2 } } ) } \\\\ & { \\quad = \\displaystyle \\frac { 1 } { 2 \\hat { \\sigma } _ { t } ^ { 2 } } ( | | y _ { g t , t } - \\hat { \\mu } _ { t } | | ^ { 2 } - | | y _ { n e g , t } - \\hat { \\mu } _ { t } | | ^ { 2 } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            174,
+            782,
+            265
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/000e146ee845453f2a4026a4594c3144fd5b0b0630e1bfd2a324fbf08b521486.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { \\frac { \\partial L _ { t } } { \\partial \\hat { \\pmb { \\mu } } _ { t } } = \\frac { \\partial } { \\partial \\hat { \\pmb { \\mu } } _ { t } } \\frac { 1 } { 2 \\hat { \\sigma } _ { t } ^ { 2 } } ( | | \\pmb { y } _ { g t , t } - \\hat { \\pmb { \\mu } } _ { t } | | ^ { 2 } - | | \\pmb { y } _ { n e g , t } - \\hat { \\pmb { \\mu } } _ { t } | | ^ { 2 } ) } } \\\\ & { } & { = - \\frac { 1 } { \\hat { \\sigma } _ { t } ^ { 2 } } ( ( \\pmb { y } _ { g t , t } - \\hat { \\pmb { \\mu } } _ { t } ) + ( \\hat { \\pmb { \\mu } } _ { t } - \\pmb { y } _ { n e g , t } ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            289,
+            671,
+            356
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/517d1a63829cb006149791b540ee5275a091643fdcc7764972ee557e0595ec1f.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { } & { \\displaystyle \\frac { \\partial L } { \\partial \\hat { \\sigma } _ { t } } = \\frac { \\partial } { \\partial \\hat { \\sigma } _ { t } } \\frac { 1 } { 2 \\hat { \\sigma } _ { t } ^ { 2 } } ( | | y _ { g t , t } - \\hat { \\pmb { \\mu } } _ { t } | | ^ { 2 } - | | y _ { n e g , t } - \\hat { \\pmb { \\mu } } _ { t } | | ^ { 2 } ) } \\\\ & { } & { \\mathrm { ~ \\ ~ \\ } = - \\frac { 1 } { \\hat { \\sigma } _ { t } ^ { 3 } } ( | | y _ { g t , t } - \\hat { \\pmb { \\mu } } _ { t } | | ^ { 2 } - | | y _ { n e g , t } - \\hat { \\pmb { \\mu } } _ { t } | | ^ { 2 } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            380,
+            669,
+            448
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B GRADIENT ANALYSIS IN GAUSSIAN MIXTURE MODEL ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            460,
+            661,
+            478
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In case we model the output distribution as a Gaussian mixture model $\\begin{array} { r l } { p _ { \\mathrm { G M M } } ( \\pmb { y } _ { t } ) } & { { } = } \\end{array}$ $\\begin{array} { r l } { \\sum _ { i } \\phi _ { i } \\mathcal { N } ( \\pmb { y } _ { t } ; \\hat { \\pmb { \\mu } } _ { i , t } , \\hat { \\sigma } _ { i , t } \\pmb { I } ) } & { { } } \\end{array}$ , the gradient of $L _ { t }$ w.r.t. the mean of component $i$ is ",
+        "bbox": [
+            173,
+            492,
+            825,
+            522
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/864bda383cab8def554c6657cbcb21cdf49f68f13d9003cd5a5b68bcee686ab4.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\frac { \\partial L _ { t } } { \\partial \\hat { \\mu } _ { i , t } } = \\frac { \\partial } { \\partial \\hat { \\mu } _ { i , t } } ( - \\log \\frac { p _ { \\mathrm { G M M } } ( y _ { g , t , t } ) } { p _ { \\mathrm { G M M } } ( y _ { n e g , t } ) } ) } \\\\ & { \\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 \\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 \\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 } \\\\ & { \\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 \\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 \\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 \\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 \\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 \\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 \\quad \\quad \\quad \\ \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            194,
+            545,
+            805,
+            655
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This gradient shows that the center $\\hat { \\mu } _ { i , t }$ of component $i$ will be pushed towards the ground truth location ${ \\bf \\nabla } _ { { \\bf { y } } _ { g t , t } }$ and way from the negative location ${ \\bf { \\mathscr { y } } } _ { n e g , t }$ . For $\\hat { \\sigma } _ { i , t }$ we have ",
+        "bbox": [
+            173,
+            660,
+            823,
+            689
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1863c1e7317df6a7a2214b739a72300d02382bb842aa9e5c48edeec7c2256da6.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { \\frac { \\partial L } { \\partial \\hat { \\sigma } _ { i , t } } = - \\phi _ { i } ( \\frac { 1 } { p _ { \\mathrm { G M M } } ( y _ { g t , t } ) } \\frac { \\partial \\mathcal { N } ( y _ { g t , t } ; \\hat { \\mu } _ { i , t } , \\hat { \\sigma } _ { i , t } I ) } { \\partial \\hat { \\sigma } _ { i , t } } - \\frac { 1 } { p _ { \\mathrm { G M M } } ( y _ { n e g , t } ) } \\frac { \\partial \\mathcal { N } ( y _ { n e g , t } ; \\hat { \\mu } _ { i , t } , \\hat { \\sigma } _ { i , t } I ) } { \\partial \\hat { \\sigma } _ { i , t } } ) } } \\\\ & { } & { = - \\frac { \\phi _ { i } } { \\sigma _ { i , t } ^ { 3 } } ( \\frac { \\mathcal { N } ( y _ { g t , t } ; \\hat { \\mu } _ { i , t } , \\hat { \\sigma } _ { i , t } I ) } { p _ { \\mathrm { G M M } } ( y _ { g t , t } ) } ( \\vert \\vert y _ { g t , t } - \\hat { \\mu } _ { i , t } \\vert \\vert ^ { 2 } - \\sigma _ { i , t } ^ { 2 } ) } \\\\ & { } & { \\quad - \\frac { \\mathcal { N } ( y _ { n e g , t } ; \\hat { \\mu } _ { i , t } , \\hat { \\sigma } _ { i , t } I ) } { p _ { \\mathrm { G M M } } ( y _ { n e g , t } ) } ( \\vert \\vert y _ { n e g , t } - \\hat { \\mu } _ { i , t } \\vert \\vert ^ { 2 } - \\sigma _ { i , t } ^ { 2 } ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            196,
+            712,
+            802,
+            823
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C THE INFLUENCE OF OUR LOSS WITHOUT MAP INPUT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            834,
+            656,
+            853
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this experiment, we remove the map input to the model and demonstrate the influence of our unlikelihood loss in this case. Results are shown in Tab.5 Interestingly, our loss helps improve the performance from 1.52 to 1.42 measured by the FDE-Full in 3 seconds case although the model doesn’t see the context during inference. We think the reason is that compared to the Trajectron $^ { + + }$ without map input, our unlikelihood loss still offers context information to support the training since this loss is calculated using the context. Therefore, our model receives more information during training and performs better. Besides, our method without map input even achieves a comparable result (FDE-FUll 1.42) compared to Trajectron $^ { + + }$ with map input (FDE-Full 1.46). This experiment shows that our loss can inject context information into learning signals. In addition, map input improves FDE-Full of Trajectron $^ { + + }$ in 3s prediction from 1.52 to 1.46, which is about $6 \\mathrm { c m }$ . However, when we further add our unlikelihood loss, performance improved from 1.46 to 1.34, which is 12 cm. Our loss triple the contribution of context information, which is significant. ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/d4e5d674ac9dc901b245c9d00ad6987aad549d132494ff5bf9184b5a1f5da906.jpg",
+        "table_caption": [
+            "Table 5: The influence of our loss without map input to the model on nuScenes. Compared to Trajectron $^ { + + }$ without map input, our loss performs better, which indicates that our loss can inject context information into learning signals "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>FDE-Full 3s</td><td>ADE-Full 3s</td></tr><tr><td>Trajectron++ without map input</td><td>1.52±0.04</td><td>0.61±0.02</td></tr><tr><td>Trajectron++</td><td>1.46±0.07</td><td>0.59±0.04</td></tr><tr><td>Ours without map input</td><td>1.42±0.03</td><td>0.57±0.01</td></tr><tr><td>Ours</td><td>1.34±0.04</td><td>0.54±0.02</td></tr></table>",
+        "bbox": [
+            290,
+            161,
+            707,
+            239
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            270,
+            825,
+            382
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D MODIFICATION ON TRAJECTRON $^ { + + }$ FOR ARGOVERSE EVALUATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            401,
+            769,
+            417
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The original Trajectron+ $^ { - + }$ (Salzmann et al., 2020) is evaluated on nuScenes dataset (Caesar et al., 2019b) but not on Argoverse (Chang et al., 2019). To make it runnable on Argoverse, we make some modifications on Trajectron $^ { + + }$ . The input states of the full version of Trajectron $^ { + + }$ consist of the positions, velocities, accelerations, heading angles and heading angular velocities. However, Argoverse doesn’t offer the data for heading angles and heading angular velocities. Therefore, we remove them from the input states. In addition, Trajectron $^ { + + }$ has a unicycle physics model to convert the predicted velocity from neural networks to locations. The unicycle physics model requires also the angular velocities information. Therefore, we replace the unicycle model by the single integrator model that requires only velocities and initial positions. The maps offered by nuScens and Argoverse are different, too. We stack the drivable region and region of interest offered by Argoverse and upsample them to the same resolution as nuScenes as the input map. Please refer to Argoverse for the details. Lastly, Trajectron $^ { + + }$ use separate modules to process surrounding vehicles and surrounding pedestrians in nuScenes. However, Argoverse doesn’t offer the labels for the surrouding agents and just simply denote them as ”others”. Therefore, we remove the pedestrian modules in Trajectron $^ { + + }$ and use the vehicle modules to process both surrounding pedestrians and vehicles. ",
+        "bbox": [
+            174,
+            434,
+            825,
+            655
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "E COMPARISON WITH OTHER METHODS ON ARGOVERSE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            658,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Our experiment setting on Argoverse follows AttGlobal-CAM-Nf (Park et al., 2020). Here we list the detailed comparison with their numbers using their evaluation metrics minADE-12 and minFDE12, which are the minimal ADE and FDE over 12 prediction candidates, in Tab.6. Both Trajectron++ and our method outperform their numbers. In addition, our unlikelihood loss helps improve the performance of Trajectron $^ { + + }$ . Our scores and the standard deviation are calculated over 4 training instances. ",
+        "bbox": [
+            174,
+            132,
+            825,
+            217
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/991856beba73346ce3e9cf3f71900053382ce2477872b97e8ab13fb5db2db039.jpg",
+        "table_caption": [
+            "Table 6: Experimental results on Argoverse dataset "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>minADE-12</td><td>minFDE-12</td></tr><tr><td>CSP (Deo &amp; Trivedi,2018; Park et al.,2020)</td><td>1.39</td><td>2.57</td></tr><tr><td>DESIRE (Lee et al.,2017; Park et al.,2020)</td><td>0.90</td><td>1.45</td></tr><tr><td>MATF-GAN (Zhao et al.,2019;Park et al., 2020)</td><td>1.26</td><td>2.31</td></tr><tr><td>R2P2-MA(Rhinehart et al.,2019;Park etal.,2020)</td><td>1.11</td><td>1.77</td></tr><tr><td>AttGlobal-CAM-Nf (Park et al., 2020)</td><td>0.73</td><td>1.12</td></tr><tr><td>Trajectron++ (Salzmann et al., 2020)</td><td>0.65±0.09</td><td>1.08±0.07</td></tr><tr><td>Ours</td><td>0.63±0.04</td><td>1.08±0.06</td></tr></table>",
+        "bbox": [
+            274,
+            253,
+            722,
+            356
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "F QUALITATIVE RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            386,
+            401,
+            404
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Here, we demonstrate our method’s qualitative results compared with Trajectron $^ { + + }$ for 3 seconds prediction. We randomly sample 50 trajectories from the predicted prediction, use kernel density estimation (KDE) to approximate the total output distribution from the samples, and print it out in Fig.4. White points represent the ground truth trajectories. Compared to Trajectron $^ { + + }$ , our method suits the contextual information more and therefore is more accurate and plausible. ",
+        "bbox": [
+            173,
+            417,
+            825,
+            488
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/4a3f8749352651ddd592c8b50fb69586771e280e08d9c3215cbe03f98ac3c4c4.jpg",
+        "image_caption": [
+            "Figure 4: Qualitative results of our method and Trajectron $^ { + + }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            259,
+            145,
+            735,
+            851
+        ],
+        "page_idx": 13
+    }
+]

parse/train/5k8F6UU39V/5k8F6UU39V.md

@@ -0,0 +1,376 @@
+# AUTOREGRESSIVE ENTITY RETRIEVAL
+
+Nicola De $\mathbf { C a o ^ { 1 , 2 * } }$ , Gautier Izacard $^ { 2 , 3 , 4 }$ , Sebastian Riedel2,5, Fabio Petroni2
+
+1University of Amsterdam, 2Facebook AI Research   
+3ENS, PSL University, 4Inria, 5University College London   
+nicola.decao@gmail.com, {gizacard, sriedel, fabiopetroni}@fb.com
+
+# ABSTRACT
+
+Entities are at the center of how we represent and aggregate knowledge. For instance, Encyclopedias such as Wikipedia are structured by entities (e.g., one per Wikipedia article). The ability to retrieve such entities given a query is fundamental for knowledge-intensive tasks such as entity linking and open-domain question answering. One way to understand current approaches is as classifiers among atomic labels, one for each entity. Their weight vectors are dense entity representations produced by encoding entity meta information such as their descriptions. This approach leads to several shortcomings: (i) context and entity affinity is mainly captured through a vector dot product, potentially missing fine-grained interactions between the two; (ii) a large memory footprint is needed to store dense representations when considering large entity sets; (iii) an appropriately hard set of negative data has to be subsampled at training time. In this work, we propose GENRE, the first system that retrieves entities by generating their unique names, left to right, token-by-token in an autoregressive fashion and conditioned on the context. This enables us to mitigate the aforementioned technical issues since: (i) the autoregressive formulation allows us to directly capture relations between context and entity name, effectively cross encoding both; (ii) the memory footprint is greatly reduced because the parameters of our encoder-decoder architecture scale with vocabulary size, not entity count; (iii) the exact softmax loss can be efficiently computed without the need to subsample negative data. We show the efficacy of the approach, experimenting with more than 20 datasets on entity disambiguation, end-to-end entity linking and document retrieval tasks, achieving new state-of-the-art or very competitive results while using a tiny fraction of the memory footprint of competing systems. Finally, we demonstrate that new entities can be added by simply specifying their unambiguous name. Code and pre-trained models at https://github.com/facebookresearch/GENRE.
+
+# 1 INTRODUCTION
+
+The ability to retrieve the correct entity from large Knowledge Bases (KBs) given a textual input is a fundamental building block for several applications (Ferrucci, 2012; Slawski, 2015; Yang et al., 2018a). Most commercial recommendation systems, for instance, include in their pipelines components to detect and disambiguate entity mentions in open text, in order to isolate relevant concepts from non-meaningful data (Slawski, 2015; Yang et al., 2018a). Another example are chat-bots and question answering systems, that are often equipped with retrieval components to surface specific KB entries (e.g., Wikipedia articles) to find knowledge for sustaining a conversation or answering a question (Ferrucci, 2012; Chen et al., 2017; Lewis et al., 2020b; Roller et al., 2020).
+
+Although there has been extensive previous work on entity retrieval (e.g. Hoffart et al., 2011; Piccinno & Ferragina, 2014; Huang et al., 2015; Le & Titov, 2018; Logeswaran et al., 2019; Broscheit, 2019; Wu et al., 2020, to name just a few) there is a common design choice to most current solutions: entities are associated with a unique atomic label and the retrieval problem can be interpreted as multi-class classification across these labels. The match between input and label is calculated through a bi-encoder (Wu et al., 2020; Karpukhin et al., 2020): a dot product between dense vector encodings of the input and the entity’s meta information (such as title and description).
+
+![](images/afb95d39a564faccda97aa93096798fa9caab47325ae274eef5e4e11ed69ead7.jpg)  
+Figure 1: Examples of entities correctly retrieved from GENRE (we show only the top-3 rank). On the top three entity disambiguation instances and on the bottom three document retrieval instances, two for open-domain question answering and one for fact checking. All of them are cast as sequenceto-sequence problems while inference is done using constrained beam search. Gold entities in bold. Sub-captions indicate the type of interaction between the input context and the entity names required.
+
+Critically, this formulation enables sub-linear search using modern maximum-inner-product-search libraries (Johnson et al., 2019) and hence supports retrieving from large entity databases.
+
+Unfortunately, the classifier approach to entity retrieval also has several shortcomings. First, unless a costly cross-encoder is used for re-ranking (Wu et al., 2020), the dot-product can miss fine-grained interactions between input and entity meta information (Humeau et al., 2020). Second, storing dense vectors for the whole KB requires a large memory footprint, especially in real-world scenarios (i.e., ${ \sim } 2 4 \mathrm { G B }$ to store 1024-dimensional vectors for all of the ${ \sim } 6 \mathbf { M }$ Wikipedia pages), and the size linearly grows with the addition of new entities. Third, computing an exact softmax over all entities is very expensive, hence current solutions need to subsample negative data (Logeswaran et al., 2019; Karpukhin et al., 2020) at training time. Tuning an appropriately hard set of negative instances can be challenging and time-consuming. Finally, existing systems can suffer from a cold-start problem since they cannot represent entities about which they have not yet gathered sufficient information, in the form, for instance, of a textual description or a set of relations with the existing entities.
+
+The treatment of entity identifiers as atomic labels in a classifier ignores the fact that we often have unambiguous, highly structured and compositional entity names. Wikipedia, for instance, associates unique titles to articles,1 that may be the name of the subject or a description of its topic, as well as potential distinctive information to disambiguate 2 (see Figure 1 for some examples). These entity names often interact with mention contexts in a predictable and regular fashion. For example, often entity names are identical with the mention strings that refer to them (e.g., Fig. 1f). When this is not possible, they might be composed of tokens in the context (e.g., Fig. 1b), include a type specification that can inferred (e.g., Fig. 1a), be the translation of the string mention (e.g., Fig. 1c), require ‘normalization’ such as referring to the correct alias of a mention (e.g., Fig. 1d), or require factual knowledge that might be stored in the parameters of a model (e.g., Fig. 1e). These observations suggest that inputs could be translated into unique entity names, word by word, instead of being classified among a huge set of options.
+
+In this paper, we propose GENRE (for Generative ENtity REtrieval), the first entity retriever that exploits a sequence-to-sequence architecture to generate entity names in an autoregressive fashion conditioned on the context. Concretely, GENRE uses a transformer-based architecture, pre-trained with a language modeling objective (i.e., we use BART weights from Lewis et al. (2020a)) and fine-tuned to generate entity names. This architecture has been shown to retain factual knowledge to some extent (Petroni et al., 2019) and language translation skills (Radford et al., 2019) among other things, both desirable properties for an entity retriever. Naturally, the generated output might not always be a valid entity name. To solve this problem, GENRE employs a constrained decoding strategy that forces each generated name to be in a predefined candidate set.
+
+The autoregressive formulation allows us to directly capture the aforementioned relations between context and entity name, effectively cross encoding both. Also, the memory footprint required is orders of magnitude smaller than current systems, since the parameters of a sequence-to-sequence model scale linearly with the vocabulary size, not entity count. Moreover, the exact softmax can be computed efficiently for each output token (i.e., all non-gold tokens are considered negative), thereby eliminating the need for negative data downsampling. Finally, our model never accesses any explicit meta-information about the entity beyond their title, hence new entities can be added by simply appending their unambiguous name to the candidate set (e.g., Fig. 1b refers to an entity added after training).
+
+We empirically evaluate the performance of GENRE on more than 20 datasets, spanning three families of tasks: (i) entity disambiguation, using popular datasets and settings (both in and out-of– domain); (ii) end-to-end entity linking, with the GERBIL benchmarking tool (Roder et al., 2018), by ¨ using a novel dynamically markup-constrained decoding strategy; (iii) document retrieval, with the recently proposed KILT benchmark (Petroni et al., 2020b) which spans 5 different sub-tasks. Our models achieve state-of-the-art or very competitive results on nearly all datasets, often with substantial improvement $( + 1 3 . 7$ precision points on KILT for retrieval on average). Further, we show that compared with recent models, GENRE requires substantially less memory ${ \sim } 2 0$ times smaller footprint on average). Finally, we demonstrate that our model can be applied in scenarios where the only entity information available is its name.
+
+We organize the paper as follows: in Section 2 we describe our problem formulation. Then, in Section 3 we present GENRE and eventually in Section 4 we extensively evaluate our method on the aforementioned settings. We will release code and pre-processed data to reproduce our experiments.
+
+# 2 ENTITY RETRIEVAL
+
+We assume to have a collection of entities $\mathcal { E }$ (e.g., Wikipedia articles) where each entity is an entry in a Knowledge Base (KB) such as Wikipedia. We want to approach the following retrieval problem: given a textual input source $x$ (e.g., question), a model has to return the most relevant entities from $\mathcal { E }$ with respect to $x$ . We assume that each $e \in { \mathcal { E } }$ is uniquely assigned to a textual representation (i.e., its name): a sequence of tokens $y$ (e.g., Wikipedia pages are identified by their titles).
+
+A particular instance of this problem is Entity Disambiguation (ED) (see Figure 1 for an example) where an input $x$ is annotated with a mention and a system has to select either its corresponding entity from $\mathcal { E }$ , or to predict that there is no corresponding entry in the KB. Another instance is pagelevel Document Retrieval (DR) where the input $x$ is intended as a query and $\mathcal { E }$ as a collection of documents identified by their unique titles (e.g., Wikipedia articles).
+
+# 3 METHOD
+
+We address the retrieval problem with an sequence-to-sequence model that generates textual entity identifiers (i.e., entity names). Concretely, GENRE ranks each $e \in { \mathcal { E } }$ by computing a score with an autoregressive formulation: $\begin{array} { r } { \mathrm { s c o r e } ( e | x ) = p _ { \theta } ( y | x ) = \prod _ { i = 1 } ^ { N } p _ { \theta } ( y _ { i } | y _ { < i } , x ) } \end{array}$ where $y$ is the set of $N$ tokens in the identifier of $e$ , and $\theta$ the parameters of the model. We take advantage of fine-tuning the BART (Lewis et al., 2020a) pre-trained language model. We train GENRE using a standard seq2seq objective, i.e., maximizing the output sequence likelihood with teacher forcing (Sutskever et al., 2011; 2014) and regularized with dropout (Srivastava et al., 2014) and label smoothing (Szegedy et al., 2016). Concretely, we use the objective that is typically used for neural machine translation (NMT, Wu et al., 2016), that is maximizing $\log p \theta ( y | x )$ with respect to model’s parameters $\theta$ which, due to the factorized formulation, can be calculated exactly. We do not need negative sampling to approximate the loss normalizer.
+
+![](images/a276253b4e90bdecbab89682f15bf698407030d633624853b2cfcf3474f42883.jpg)
+
+In	1503,	[Leonardo_
+
+![](images/926f6ddadda3670b8a72c5e09335699a108d0229b58fab5b7e0915f68f934483.jpg)
+
+In 1503, [Leonardo](Leonardo_ (a) Outside: we can either continue to generate the input or start a new mention.
+
+![](images/ddbc51f005c4a29a1a69f2e29e36fc9da7a0858fff6d58723c847f4bc6b818e6.jpg)
+
+(b) Inside a mention: we can either continue to generate the input or end the current mention.
+
+(c) Inside an entity link: we can either generate from the entities prefix trie or close if the generated prefix is a valid entity.
+
+Figure 2: Example of dynamically constrained Markup decoding for entity linking using “In 1503, Leonardo began painting the Mona Lisa.” as input. There are 3 cases: when we are outside a mention/entity (a), inside a mention generation step (b), and inside an entity link generation step (c). The model is supposed to output the input source annotating mentions and pointing them to the respective entities: “In 1503, [Leonardo](Leonardo da Vinci) began painting the [Mona Lisa](Mona Lisa)”.
+
+# 3.1 INFERENCE WITH CONSTRAINED BEAM SEARCH
+
+Naturally, at test time, we could compute a score for every element in $\mathcal { E }$ and then sort them. Unfortunately, this might be prohibitively expensive when $\mathcal { E }$ is very large (e.g., Wikipedia has ${ \sim } 6 \mathbf { M }$ entities). Hence, we exploit Beam Search (BS, Sutskever et al., 2014), an established approximate decoding strategies to efficiently navigate the search space. Instead of explicitly scoring all entities in $\mathcal { E }$ , we search for the top- $k$ entities in $\mathcal { E }$ decoding from our model using BS with $k$ beams. Note that using BS implies that the time cost of our retriever does not depend on the size of $\mathcal { E }$ , but only on the size of the beams and the average length of entity representations as we do autoregressive generation. The average length of entity representations is tractable (e.g., Wikipedia titles have 6 BPE tokens on average) and we follow standard NMT settings where $k$ is small (e.g., 10).
+
+Since we want to output only entities from $\mathcal { E }$ we cannot use traditional BS while decoding. Indeed, allowing to generate any token from the vocabulary at every decoding step might lead the model to generate output strings that are not valid identifiers. Hence, we resort to Constrained BS, forcing to only decode valid entity identifiers. BS only considers one step ahead during decoding so we can only constrain the generation of a single next token conditioned on the previous ones. Thus, we define our constrain in terms of a prefix tree $\tau$ (aka trie) (Cormen et al., 2009) where nodes are annotated with tokens from the vocabulary. For each node $t \in \tau$ , its children indicate all the allowed continuations from the prefix defined traversing the trie from the root to $t$ .
+
+See Figure 9 in Appendix C for an exampled of a trie. When the number of allowed outputs is tractable (e.g., generating a Wikipedia title among ${ \sim } 6 \mathbf { M } )$ ) the trie is relatively small it can be precomputed and stored into memory (e.g., constraining on Wikipedia titles using the BART tokenizer produces a trie with ${ \sim } 6 \mathbf { M }$ leaves, ${ \sim } 1 7 \mathbf { M }$ internal nodes that occupied ${ \sim } 6 0 0 \mathrm { M B }$ of disk space). We employed the constraints masking the log-probabilities of the invalid tokens and not their logits (i.e., we do not re-normalize the probability over the vocabulary).3
+
+# 3.2 AUTOREGRESSIVE END-TO-END ENTITY LINKING
+
+We additionally extend our autoregressive framework to address end-to-end Entity Linking (EL) where, given a document, a system has to both detect entity mentions and link those mentions to their respective KB entities. In this setting, we train the model to predict the source input again but with annotated spans. We use a Markup annotation where spans boundaries are flagged with special tokens and accompanied by their corresponding entity identifiers.
+
+Differently from a setting where the output space is relatively small (e.g., a pre-defined set $\mathcal { E }$ ), the space of annotated outputs is exponentially large. Hence, it is intractable to pre-compute a trie for decoding, and we compute it dynamically instead. In Figure 2 we show an example. At each generation step, the decoder is either generating a mention span, generating a link to a mention, or continuing from the input source. When outside a mention/entity step the decoder has only two options: (i) to continue by copying the next token from the input source, or (ii) to generate the start of mention token (i.e., ‘[’) which makes the decoder enter the mention generating phase. While generating a mention, the decoder has either to continue with the next token in the input source or to generate the end of mention token (i.e., ‘]’) which makes the decoder enter the entity generating phase. Finally, when generating an entity, the decoder employs the entities trie such that it can only output a valid entity identifier as in Constrained Beam Search explained above.
+
+# 4 EXPERIMENTS
+
+We extensively evaluate GENRE on more than 20 datasets across 3 tasks: Entity Disambiguation, end-to-end Entity Linking (EL), and page-level Document Retrieval. We describe the experimental settings in Section 4.1 where we discuss results in Section 4.2. All experiments are in English.
+
+# 4.1 SETTINGS
+
+Entity Disambiguation (ED) We reproduce the setting of Le & Titov (2018) using the same candidate sets, in-domain and out-of-domain datasets, and evaluating using the InKB micro- $F _ { 1 }$ . We train GENRE feeding each document where a single mention is flagged with two special start and end tokens and the target output is the textual representation of the corresponding entity. At test time, we decode using constrained beam search with a trie obtained using the provided candidate set (i.e., a subset of $\mathcal { E }$ ). As large generative models benefit from large amount of data, we first pre-train GENRE on the BLINK data (Wu et al., 2020), i.e., 9M unique triples document-mention-entity from Wikipedia. Then, for the in-domain scenario, we fine-tune using the AIDA-CoNLL dataset (Hoffart et al., 2011). For the out-of-domain scenario, we evaluate on five test sets: MSNBC, AQUAINT, ACE2004, WNED-CWEB (CWEB) and WNED-WIKI (WIKI) (Gabrilovich et al., 2013; Guo & Barbosa, 2018). More task details and hyperparameters setting are reported in Appendix A.1.
+
+End-to-End Entity Linking (EL) For EL, we reproduce the setting of Kolitsas et al. (2018) using the same in-domain and out-of-domain datasets as well as evaluating the InKB micro- $F _ { 1 }$ on the GERBIL benchmark platform (Roder et al., 2018). Similarly to the ED setting, we first pre-traine ¨ our model on all abstract sections from Wikipedia4 enriched by a string matching heuristic to solve co-references (i.e., if there is a string that matches exactly with another hyperlink we also add it to the dataset as a mention/entity pairs). Then, for the in-domain scenario, we fine-tune using the AIDACoNLL dataset. We evaluate on seven out-of-domain test sets: MSNBC, Derczynski (Der) (Derczynski et al., 2015), KORE 50 (K50) (Hoffart et al., 2012), N3-Reuters-128 (R128), N3-RSS-500 (R500) (Roder et al., 2014), and OKE challenge 2015 and 2016 (OKE15 and OKE16) (Nuzzolese ¨ et al., 2015). More task details and hyperparameters setting are reported in Appendix A.2.
+
+Page-level Document Retrieval (DR) For this setting, we test GENRE on all the KILT benchmark tasks (Petroni et al., 2020b). Here, whole Wikipedia is used as the candidate set and we evaluate using R-precision (Beitzel et al., 2009). KILT consists of five tasks that use the same Wikipedia dump as a knowledge source: fact checking with FEVER (Thorne et al., 2018); open domain question answering using Natural Questions (Kwiatkowski et al., 2019), HotpotQA (Yang et al., 2018b), TriviaQA (Joshi et al., 2017), ELI5 (Fan et al., 2019); slot filling with T-REx (Elsahar et al., 2018), Zero Shot RE (Levy et al., 2017); entity disambiguation on AIDA CoNLL-YAGO, WNED-WIKI and WNED-CWEB; dialogue with Wizard of Wikipedia (Dinan et al., 2019). We train GENRE on BLINK and all KILT data simultaneously with a single model.5 More details on the hyperparameter setting are reported in Appendix A.3.
+
+# 4.2 RESULTS
+
+Overall, GENRE achieves very competitive results in all of the three settings being the best performing system on average across all of them. See Appendix C for examples of inputs, ground truth and model predictions for all of the three tasks. In the following, we discuss how GENRE compares to SOTA systems as well as showing some quantitative analysis on its memory footprint, how it exploits the structured of the entity name space, and how it behaves on a cold-start scenario where new unseen entities are added to the KB (descriptions of those entities are unobserved).
+
+Table 1: Micro $F _ { 1 }$ (InKB) on the in-domain test set and five out-of-domain test sets for the named entity disambiguation task. Bold indicates best model and underline indicates second best (not for ablations). \*WIKI is usually considered out-of-domain but note that all methods use a part of Wikipedia to train. \*\*results taken from https://github.com/facebookresearch/ BLINK and normalized to accommodate entities not in KB.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">In-domain AIDA</td><td colspan="6">Out-of-domain</td></tr><tr><td>MSNBC</td><td>AQUAINT</td><td>ACE2004</td><td>CWEB</td><td>WIKI*</td><td>Avg.</td></tr><tr><td>Ganea &amp; Hofmann (2017)</td><td>92.2</td><td>93.7</td><td>88.5</td><td>88.5</td><td>77.9</td><td>77.5</td><td>86.4</td></tr><tr><td>Guo &amp; Barbosa (2018)</td><td>89</td><td>92</td><td>87</td><td>88</td><td>77</td><td>84.5</td><td>86.2</td></tr><tr><td>Yang et al. (2018a)</td><td>95.9</td><td>92.6</td><td>89.9</td><td>88.5</td><td>81.8</td><td>79.2</td><td>88.0</td></tr><tr><td>Shahbazi et al. (2019)</td><td>93.5</td><td>92.3</td><td>90.1</td><td>88.7</td><td>78.4</td><td>79.8</td><td>87.1</td></tr><tr><td>Yang et al. (2019)</td><td>93.7</td><td>93.8</td><td>88.2</td><td>90.1</td><td>75.6</td><td>78.8</td><td>86.7</td></tr><tr><td>Le &amp; Titov (2019)</td><td>89.6</td><td>92.2</td><td>90.7</td><td>88.1</td><td>78.2</td><td>81.7</td><td>86.8</td></tr><tr><td>Fang et al. (2019)</td><td>94.3</td><td>92.8</td><td>87.5</td><td>91.2</td><td>78.5</td><td>82.8</td><td>87.9</td></tr><tr><td>BLINK w/o candidate set**</td><td>79.6</td><td>80.0</td><td>80.3</td><td>82.5</td><td>64.2</td><td>75.5</td><td>77.0</td></tr><tr><td>GENRE</td><td>93.3</td><td>94.3</td><td>89.9</td><td>90.1</td><td>77.3</td><td>87.4</td><td>88.8</td></tr><tr><td colspan="8">Ablations</td></tr><tr><td>GENRE only AIDA data</td><td>88.6</td><td>88.1</td><td>77.1</td><td>82.3</td><td>71.9</td><td>71.7</td><td>80.0</td></tr><tr><td>GENRE only BLINK data</td><td>89.3</td><td>93.3</td><td>90.9</td><td>91.1</td><td>76.0</td><td>87.9</td><td>88.1</td></tr><tr><td>GENRE w/o candidate set</td><td>91.2</td><td>86.9</td><td>87.2</td><td>87.5</td><td>71.1</td><td>86.4</td><td>85.1</td></tr><tr><td>GENRE w/o constraints</td><td>86.4</td><td>80.0</td><td>81.7</td><td>82.1</td><td>66.0</td><td>81.1</td><td>79.6</td></tr></table>
+
+Comparing GENRE to SOTA systems In ED the difference in average $F _ { 1 }$ score between GENRE and the second best performing system is small (i.e., $+ 0 . 8 )$ however, ED is an established task with more than a decade of research that benchmarked on those datasets. Indeed all systems reported in Table 1 achieved high and similar results even if they were taken from three years back.
+
+The improvements on EL are instead more evident. GENRE is the best in-domain system for AIDA while performing remarkably well also on the out-of-domain setting (e.g., $+ 1 3$ $F _ { 1 }$ points on Derczynski, and $+ 4 . 7$ on KORE50). Noticeably, in two datasets (OKE15 and OKE16) our model performs poorly. However, these datasets are annotated with coreference (pronouns and common nouns are linked to entities) while our model was not specifically trained for that. Conversely, most of the other systems, have a mention detection component in their pipelines that can be trained or biased to also solve these cases. We considered out of the aim of this work to additional train and evaluate on coreference and we leave it for future work.
+
+On page-level DR, the superiority of GENRE is remarkable. Our model is the best performing system across all 5 KILT tasks and all datasets except on Natural Questions where it is the second best. We achieve $+ 1 3 . 7$ R-precision points on average with respect to the best performing baseline. In Table 3 we compare GENRE against all methods reported in the public leaderboard: DPR (Karpukhin et al., 2020), DPR $+$ BERT (Devlin et al., 2019), DPR $+$ BART, tf-idf (Leskovec et al., 2014), RAG (Lewis et al., 2020b), and BLINK $^ +$ flair (Wu et al., 2020; Akbik et al., 2019). No model except ours was trained on the entire KILT dataset at the same time. A RAG model was trained for every single task as well as for DPR $+$ BERT. Note that this gives and advantage to RAG and DPR $+$ BERT to specialize on single tasks where we have only a single model to solve all of them which still performs better. We speculate that multi-task training could have helped since the all tasks share a common objective to retrieve entities. Both DPR and BLINK $^ +$ flair were not trained specifically on KILT. However, DPR was trained using several QA datasets which include Natural Question and TriviaQA. In Appendix B we report additional results where we do not pre-train or fine-tune our models for both the ED and retrieval setting in Table 1 and 8 respectively. When we train GENRE only in the DPR or BLINK data, our model still outperforms them.
+
+<table><tr><td></td><td colspan="9">In-domain</td></tr><tr><td>Method</td><td>AIDA</td><td>MSNBC</td><td>Der</td><td>K50</td><td>R128</td><td>Out-of-domain R500</td><td>OKE15*</td><td>OKE16*</td><td>Avg.</td></tr><tr><td>Hoffart et al. (2011)</td><td>72.8</td><td>65.1</td><td>32.6</td><td>55.4</td><td>46.4</td><td>42.4</td><td>63.1</td><td>0.0</td><td>47.2</td></tr><tr><td>Steinmetz &amp; Sack (2013)</td><td>42.3</td><td>30.9</td><td>26.5</td><td>46.8</td><td>18.1</td><td>20.5</td><td>46.2</td><td>46.4</td><td>34.7</td></tr><tr><td>Moro et al. (2014)</td><td>48.5</td><td>39.7</td><td>29.8</td><td>55.9</td><td>23.0</td><td>29.1</td><td>41.9</td><td>37.7</td><td>38.2</td></tr><tr><td>Kolitsas et al. (2018)</td><td>82.4</td><td>72.4</td><td>34.1</td><td>35.2</td><td>50.3</td><td>38.2</td><td>61.9</td><td>52.7</td><td>53.4</td></tr><tr><td>Broscheit (2019)</td><td>79.3</td><td>1</td><td>1</td><td>-</td><td>1</td><td>1</td><td>-</td><td>1</td><td></td></tr><tr><td>Martins et al. (2019)</td><td>81.9</td><td>-</td><td>-</td><td>-</td><td>1</td><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td>van Hulst et al. (2020)†</td><td>80.5</td><td>72.4</td><td>41.1</td><td>50.7</td><td>49.9</td><td>35.0</td><td>63.1</td><td>58.3</td><td>56.4</td></tr><tr><td>GENRE</td><td>83.7</td><td>73.7</td><td>54.1</td><td>60.7</td><td>46.7</td><td>40.3</td><td>56.1</td><td>50.0</td><td>58.2</td></tr></table>
+
+Table 2: Micro $F _ { 1 }$ (InKB) on the in-domain test set and four out-of-domain test sets for the entity linking task. Bold indicates best model and underline indicates second best. \*annotated with coreference (note that we do not train/evaluate our model to link pronouns and common nouns). †results from the Wikipedia 2019 setting as opposed to the 2014 setting (older dump and fewer entities).
+
+Memory Footprint GENRE is not only performing better than other SOTA models on DR but it has a significant reduction of memory footprint (disk space). In Figure 4 we compare the number of model/index parameter against DPR, RAG, and BLINK. GENRE uses an order of magnitude less parameters (millions instead of billions) to store the entity index because it just has to use a prefix tree of the entity names as opposed to a dense vector for each entity. Concretely, GENRE occupied 14 times less memory than BLINK and 34 times less memory than DPR.
+
+Exploiting the Structured Name Space We investigated some properties of GENRE, comparing two variants of our model to BLINK on the ED task (using WNED-KILT validation set): one trained to generate entity names and another to generate numerical identifiers (IDs). All models are trained on the same data and we report results in Figure 5. When there is an exact match between a mention and its entity name, both BLINK and GENRE almost always make an accurate prediction. Different is the case of partial and no match: GENRE performance is much higher suggesting that our model uses the context more effectively, as the autoregressive formulation allows to cross-encode mention context and entity candidates directly capturing fine-grained interactions between the two. Moreover, when we switch to predicting IDs, the performance drops drastically (-20.3 points on average) indicating that it is important that entity names are meaningful, structured and compositional (as they are in Wikipedia) conversely to atomic IDs. Surprisingly, when there is no overlap between a mention-entity pair, performance are still relatively high by using IDs. This suggests that the model is good at memorizing and recalling identifiers even if numeric.
+
+Ablation study We here discuss an ablation study on the entity disambiguation task (see Table 1). Due to space limitation, we discuss an ablation study on document retrieval in Appendix B.2. In Table 1, GENRE only AIDA or BLINK data indicates the ablation for which we only train on one of the two datasets (i.e., only fine-tuning). GENRE (full) is also used with constrained decoding (see Section 3) and in combination with a candidate set (as provided by Le & Titov, 2018). GENRE without candidate set denotes ablating the provided (and small) candidate set and therefore using all the entities in the KB (in our case Wikipedia) as candidates. GENRE without constraints indicates ablating constrained decoding which implies no use of the provided candidates set but also unconstrained generation (i.e., the model may generate entity names that are not in the KB). Eventually, using constrained generation and exploiting the candidate sets proved useful. Training only on AIDA data is insufficient to get high $F _ { 1 }$ (but AIDA is quite small compared to the 9M datapoints of BLINK data).
+
+Entity frequency The performance of a model naturally depends on how many times entities appear in the training data. We show the data distribution of the mention-entity frequency in Figure 3. Most of the pairs appears in Wikipedia $( 1 0 9 3 1 / 1 3 3 5 4 )$ where 2423 do not (first bin). The average accuracy is $8 2 . 5 \%$ but noticeable it is higher for mention-entity pairs that are more frequent (right side of the plot). The accuracy for pairs that do not appear in Wikipedia is substantially lower than the average suggesting that those are harder cases (the very end tail of the distribution). The degradation in performance is minimal indicating that our model is good at predicting rare entities.
+
+Table 3: R-Precision for page-level retrieval on KILT test data. Bold indicates the best model and underline indicates the second best. For our model, we indicated what datasets we used for training.   
+
+<table><tr><td>Model</td><td rowspan="2">Fact Check. FEV</td><td colspan="3">Entity Disambiguation</td><td colspan="2">Slot Filling</td><td colspan="3">Open Domain QA</td><td colspan="3">Dial.</td></tr><tr><td></td><td>AY2</td><td>WnWi</td><td>WnCw</td><td>T-REx</td><td>zsRE</td><td>NQ</td><td>HoPo</td><td>TQA</td><td>ELI5</td><td>WoW</td><td>Avg.</td></tr><tr><td>DPR+BERT</td><td>72.9</td><td>-</td><td>-</td><td>-</td><td>1</td><td>40.1</td><td>60.7</td><td>25.0</td><td>43.4</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DPR</td><td>55.3</td><td>1.8</td><td>0.3</td><td>0.5</td><td>13.3</td><td>28.9</td><td>54.3</td><td>25.0</td><td>44.5</td><td>10.7</td><td>25.5</td><td>23.6</td></tr><tr><td>tf-idf</td><td>50.9</td><td>3.7</td><td>0.24</td><td>2.1</td><td>44.7</td><td>60.8</td><td>28.1</td><td>34.1</td><td>46.4</td><td>13.7</td><td>49.0</td><td>30.5</td></tr><tr><td>DPR +BART</td><td>55.3</td><td>75.5</td><td>45.2</td><td>46.9</td><td>13.3</td><td>28.9</td><td>54.3</td><td>25.0</td><td>44.4</td><td>10.7</td><td>25.4</td><td>38.6</td></tr><tr><td>RAG</td><td>61.9</td><td>72.6</td><td>48.1</td><td>47.6</td><td>28.7</td><td>53.7</td><td>59.5</td><td>30.6</td><td>48.7</td><td>11.0</td><td>57.8</td><td>47.3</td></tr><tr><td>BLINK + fair</td><td>63.7</td><td>81.5</td><td>80.2</td><td>68.8</td><td>59.6</td><td>78.8</td><td>24.5</td><td>46.1</td><td>65.6</td><td>9.3</td><td>38.2</td><td>56.0</td></tr><tr><td>GENRE</td><td>83.6</td><td>89.9</td><td>87.4</td><td>71.2</td><td>79.4</td><td>95.8</td><td>60.3</td><td>51.3</td><td>69.2</td><td>15.8</td><td>62.9</td><td>69.7</td></tr></table>
+
+Table 4: Comparison between retrieval models on memory (disk space) footprint and number of model/index parameters.   
+
+<table><tr><td>Model</td><td>Memory</td><td>Param.</td><td>Index</td></tr><tr><td>DPR</td><td>70.9GB</td><td>220M</td><td>15B</td></tr><tr><td>RAG</td><td>40.4GB</td><td>626M</td><td>15B</td></tr><tr><td>BLINK</td><td>30.1GB</td><td>680M</td><td>6B</td></tr><tr><td>GENRE</td><td>2.1GB</td><td>406M</td><td>17M</td></tr></table>
+
+Table 5: Different types of matches between mentions and their entity names on the WNED-KILT. \*indicates GENRE trained on numerical identifiers.   
+
+<table><tr><td>Type (support)</td><td>BLINK</td><td>GENRE</td><td>IDs*</td></tr><tr><td>Exact match (1543)</td><td>97.8</td><td>96.6</td><td>76.0</td></tr><tr><td>Partial match (1531)</td><td>70.7</td><td>86.9</td><td>63.8</td></tr><tr><td>No match (322)</td><td>49.4</td><td>59.9</td><td>55.0</td></tr><tr><td>Total (3396)</td><td>81.0</td><td>88.8</td><td>68.5</td></tr></table>
+
+Cold-start We manually collect 50 Wikipedia articles that were created in $2 0 2 0 ^ { 6 }$ to simulate a cold-start setting where new entities are added to the KB and the only entity information available is their names. To create ED instances we resort to hyperlinks pointing to those entities in other Wikipedia articles. 19 out of 50 mentions have an exact match with their respective entity names and all of them were correctly classified by GENRE. In combination with the results from Table 5 we can conclude that GENRE has a bias on exactly copying the mention, and this helps on unseen data. GENRE also correctly classified 14/31 of the remaining mentions $( 4 5 . 2 \% )$ . This demonstrates the ability of our solution to be applied in scenarios where entity metadata is unavailable (apart his name), a setting where, to the best of our knowledge, no existing system is capable to operate.
+
+We additionally test how GENRE performs on unseen mention-entity pairs on WikilinksNED Unseen-Mentions data (Onoe & Durrett, 2020) and we report all results in Table 6 in Appendix B.1. Surprisingly, GENRE performs almost the same for seen and unseen entity pairs (64.4 vs 63.2 accuracy) However, in the Onoe & Durrett (2020) setting we cannot guarantee entity descriptions have not been seen by BART during pre-training (given his training data contains Wikipedia).
+
+# 5 RELATED WORKS
+
+Casting NLP tasks with a structured input or output into sequence-to-sequence problems has been explored for different problems, including semantic parsing (Rongali et al., 2020), semantic role labelling (Daza & Frank, 2018), discourse representation structure parsing (Liu et al., 2018), generation of fluent natural language responses from structured semantic representations (Balakrishnan et al., 2019), generation and parsing of abstract meaning representation (Konstas et al., 2017). In these works a structured representation, a tree or a graph for instance, is linearized into a sequence of symbols compatible with a seq2seq architecture. To the best of our knowledge, we are the first to cast entity retrieval as a sequence-to-sequence problem while decoding with an autoregressive formulation during inference.
+
+Related to our constrained generation mechanism, Daza & Frank (2018); Rongali et al. (2020) use a copying mechanism in order to limit lexical deviations between the input and output strings. In these tasks, as well as for our problem, it is natural to promote a copying mechanism due to the input and the output proximity. A different type of constraint, a structural constraint, is used in Balakrishnan et al. (2019) to maintain a valid tree structure. Our constrained beam search encompasses both aspects, a copying mechanism that restrains the vocabulary and a structural constraint to obtain a well-formed annotated output. In addition to these tasks with close input and output, the integration of a mechanism to guide the output of neural networks has been explored in various settings. Lexically constrained decoding has been used to force the inclusion of pre-specified words for machine translation (Hokamp & Liu, 2017; Post & Vilar, 2018), and image captioning (Anderson et al., 2017). To the best of our knowledge, we are the first to exploit constrained generation for entity disambiguation, end-to-end entity linking, and query-based entity retrieval.
+
+![](images/0066823b752c9e840cd614f20e83617c9ee824858c21c15a0cbccc2584a48ad1.jpg)  
+Figure 3: Accuracy per mention-entity pair frequency (in Wikipedia) on the validation sets of all Entity Disambiguation tasks in KILT.
+
+Nogueira et al. (2020) propose to use a sequence-to-sequence model to re-rank document. Given a query and a document the model is trained to output the words ”true” or ”false” depending on whether the document is relevant or not. Differently from our approach for entity retrieval, it requires a limited list of candidates documents, obtained with BM25 for instance, in order to be computationally possible. Massarelli et al. (2019); Petroni et al. (2020a) explore the idea of using an autoregressive language model as neural retriever, by exploiting the implicit knowledge stored in their parameters to generate relevant sentences given a query. While intriguing, such solutions still lag behind retrievers with an explicit knowledge access (e.g., an explicit Wikipedia index). The idea of using a generative model for entity disambiguation was proposed in Petroni et al. (2020b) as they trained both BART and T5 in a seq2seq fashion on all KILT tasks (including ED). We expanded that intuition generalizing on multiple tasks (end-to-end EL and page-level retrieval) as well as introducing constrained decoding for an efficient and effective search.
+
+# 6 CONCLUSIONS
+
+In this work, we propose GENRE, a novel paradigm to addresses entity retrieval: generate entity names autoregressively. Entity names have several properties that might help (even humans) retrieving them, including a compositional structure and a predictable interaction with the context. The autoregressive formulation allows us to directly capture some of these properties, leading to several advantages with respect to current solutions, including an efficient way to cross encode mention context and entity candidates, a much smaller memory footprint, and the ability to compute an exact softmax without the need to subsample negative data. We empirically show that these characteristics, combined with constrained decoding strategies, led to state-of-the-art performance on a plethora of entity retrieval datasets, spanning entity disambiguation, end-to-end entity linking, and page-level document retrieval, while resulting in systems with a remarkably contained memory footprint, a space reduction by a factor of twenty on average. We additionally demonstrate that new entities can be effectively considered in our system by simply appending their unambiguous name to the candidate set.
+
+# ACKNOWLEDGMENTS
+
+Authors thank Patrick Lewis, Aleksandra Piktus, Michael Schlichtkrull, Ivan Titov, Jean Maillard, Edouard Grave, Sergio De Cao, Luisa Quarta for helpful discussions and technical support.
+
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+
+# A EXPERIMENTAL DETAILS
+
+We implemented, trained, and evaluate our model using the fariseq library (Ott et al., 2019). We trained GENRE for every task using Adam (Kingma & Ba, 2014) with a learning rate $3 \cdot 1 0 ^ { - 5 }$ with a linear warm-up for 500 steps and then liner decay. The objective is sequence-to-sequence categorical cross-entropy loss with 0.1 of label smoothing.
+
+# A.1 NAMED ENTITY DISAMBIGUATION
+
+Setting Given a document $d _ { j }$ (e.g., a sentence) containing a set of entity mentions $\displaystyle m _ { 1 } , \ldots , m _ { N }$ , a system either has to assign, to each mention $m _ { i }$ , either a KB entity (i.e., $e _ { i } \in \mathcal { E } _ { i }$ ), or predicts that there is no corresponding entry in the KB (i.e., $e _ { i } = \mathrm { N L L } )$ . Moreover, a restricted candidates set $C _ { i } = \{ \hat { e } _ { i 1 } , \dots , \hat { e } _ { i K } \} \subseteq \bar { \mathcal { E } } \cup \{ \hat { \mathrm { N I L } } \}$ for each mention $m _ { i }$ is provided.
+
+Training We pre-trained GENRE on BLINK data for $2 0 0 \mathrm { k }$ steps and then we do model selection on the validation set. Afterward, we fine-tuned on AIDA without resetting the learning rate nor the optimizer statistics for $1 0 \mathrm { k }$ steps and we do model selection on the validation set. Following previous works (Yamada et al., 2016; Ganea & Hofmann, 2017; Le & Titov, 2018), we considered only mentions that have entities in the KB (i.e., Wikipedia). Training was done on 32 GPUs (with 32GB of memory) and it completed in ${ \sim } 2 4 \mathrm { h }$ for a total of ${ \sim } 3 2$ GPU/day.
+
+Inference At test time, we use Constrained Beam Search with 10 beams, and maximum decoding steps of 15. We restrict the input sequence to be at most 384 tokens cutting the left, right, or both parts of the context around a mention. We normalize the log-probabilities by sequence length.
+
+# A.2 ENTITY LINKING
+
+Setting Given a document $d _ { j }$ (e.g., a sentence) a system has to return a set of tuples $\langle m _ { i } , e _ { i } \rangle$ where $m _ { i }$ is a entity mentions (a span contained in $d _ { j }$ ) and $e _ { i } \in \mathcal { E }$ its corresponding entity in the KB. Following Kolitsas et al. (2018), we considered only mentions that have entities in the KB (i.e., Wikipedia) and we used their candidate sets with the additions of the table computed by Hoffart et al. (2011).
+
+Training We pre-trained GENRE on all abstract sections from Wikipedia7 enriched by a string matching heuristic to solve co-references (i.e., if there is a string that matches exactly with another hyperlink we also add it to the dataset as a mention/entity pairs) data for $2 0 0 \mathrm { k }$ steps. Then we do model selection on the validation set. Afterward, we fine-tuned on AIDA resetting the learning rate and the optimizer statistics for $1 0 \mathrm { k }$ steps and we do model selection on the validation set. Again, following previous works (Kolitsas et al., 2018), we considered only mentions that have entities in Wikipedia. Training was done on 64 GPUs (with 32GB of memory) and it completed in ${ \sim } 3 0 \mathrm { h }$ for a total of ${ \sim } 8 0$ GPU/day.
+
+Inference At test time, we use Constrained Beam Search with 6 beams, and a maximum decoding step of 384. When the input sequence is too long, we split the input into multiple chunks of equal size. We normalize the log-probabilities by sequence length.
+
+# A.3 PAGE-LEVEL DOCUMENT RETRIEVAL
+
+Setting Given a query $q$ (e.g., a question) and a collection of documents $\mathcal { D }$ (in KILT are Wikipedia pages), a system has to rank documents in $\mathcal { D }$ based on their relevance to $q$ .
+
+Training We trained GENRE on all KILT data simultaneously for $2 0 0 \mathrm { k }$ steps and we do model selection on the validation set averaging the score across tasks. Training was done on 128 GPUs (with 32GB of memory) and it completed in ${ \sim } 3 3 \mathrm { h }$ for a total of ${ \sim } 1 7 6$ GPU/day.
+
+Inference At test time, we use Constrained Beam Search with 10 beams. For the ED sub-task, we restrict the input sequence to be at most 384 tokens cutting the left, right, or both parts of the context around a mention. We normalize the log-probabilities by sequence length.
+
+# B ADDITIONAL RESULTS
+
+# B.1 NAMED ENTITY DISAMBIGUATION
+
+Table 6 reports evaluation of GENRE on on WikilinksNED Unseen-Mentions data (Onoe & Durrett, 2020). We also report additional results on AIDA from the literature in Table 7.   
+
+<table><tr><td></td><td>Seen</td><td>Unseen</td><td>Total</td></tr><tr><td>Exact match</td><td>87.48 (751)</td><td>70.36 (2227)</td><td>74.68 (2978)</td></tr><tr><td>Partial match</td><td>56.39 (1566)</td><td>61.47 (4838)</td><td>60.23 (6404)</td></tr><tr><td>No match</td><td>41.46 (205)</td><td>45.04 (413)</td><td>43.85 (618)</td></tr><tr><td>Total</td><td>64.43 (2522)</td><td>63.21 (7478)</td><td>63.52 (10k)</td></tr></table>
+
+Table 6: Evaluation of GENRE on WikilinksNED Unseen-Mentions data (Onoe & Durrett, 2020). We train on the provided train set and we report accuracy scores (i.e., precision at 1) alongside with the number of supporting datapoints. We report scores splitting the test set in seen and unseen entities as well as in three different matchings between a mention and its gold entity.   
+Table 7: Additional results on AIDA. We report Micro InKB $F _ { 1 }$ on test sets.   
+
+<table><tr><td>Methods</td><td>micro-F1</td></tr><tr><td>Guo &amp; Barbosa (2018)</td><td>89</td></tr><tr><td>Le &amp; Titov (2019)</td><td>89.6</td></tr><tr><td>Yamada et al. (2016)</td><td>91.5</td></tr><tr><td>Ganea &amp; Hofmann (2017)</td><td>92.2</td></tr><tr><td>Shahbazi et al. (2019)</td><td>93.5</td></tr><tr><td>Chen et al. (2020)</td><td>93.5</td></tr><tr><td>Yang et al. (2019)</td><td>93.7</td></tr><tr><td>Fang et al. (2019) Raiman &amp; Raiman (2018)</td><td>94.3</td></tr><tr><td>Mulang&#x27; et al. (2020)</td><td>94.9</td></tr><tr><td></td><td>94.9</td></tr><tr><td>Yang et al. (2018a)</td><td>95.9</td></tr><tr><td>GENRE</td><td>93.3</td></tr></table>
+
+# B.2 DOCUMENT RETRIEVAL
+
+Table 8 extends Table 3 with additional results (i.e., training GENRE on the numerical identifiers) and an ablation study on the document retrieval task. The purpose of the experiment is to see whether GENRE benefits from the entity names to be meaningful as well as compositional. Numerical IDs do not have that property. In both cases, the model uses its memorizing capabilities but when using IDs the performance is significantly low. Indeed, with IDs the model has no way to generalize nor to use “implicit knowledge” acquired during the unsupervised pre-training. We also ablate the training data. DPR data corresponds to training only on Natural Questions (NQ) and TriviaQA (TQA) as DPR was trained only for QA tasks on those datasets and two extra ones. Note that training on BLINK data corresponds to only training for entity disambiguation. However, every other task share similarities with entity disambiguation and thus the model is also capable to address the other tasks with non-zero performance. For the ablations, underlined cells indicate what are the results on the respective task on which a model was trained for (i.e., GENRE only BLINK data was trained only for ED where GENRE only DPR data was trained only for QA). The ablation on data suggests that it is beneficial to train on all tasks simultaneously. GENRE without constraints indicates ablating constrained decoding which implies unconstrained generation (i.e., the model may generate entity names that are not in the KB).
+
+<table><tr><td></td><td rowspan="2">Fact Check. FEV</td><td colspan="3">Entity Disambiguation</td><td colspan="3">Slot Filling</td><td colspan="3">Open Domain QA</td><td colspan="3">Dial.</td></tr><tr><td>Model</td><td></td><td>AY2</td><td>WnWi</td><td>WnCw</td><td>T-REx</td><td>zsRE</td><td>NQ</td><td>HoPo</td><td>TQA</td><td>ELI5</td><td>WoW</td><td>|Avg.</td></tr><tr><td>DPR + BERT</td><td>72.9</td><td>1</td><td>1</td><td>1</td><td>1</td><td>40.1</td><td>60.7</td><td>25.0</td><td>43.4</td><td></td><td>1</td><td></td><td>1</td></tr><tr><td>DPR</td><td>55.3</td><td>1.8</td><td>0.3</td><td>0.5</td><td>13.3</td><td>28.9</td><td></td><td>54.3</td><td>25.0</td><td>44.5</td><td>10.7</td><td>25.5</td><td>23.6</td></tr><tr><td>tf-idf</td><td>50.9</td><td>3.7</td><td>0.24</td><td>2.1</td><td>44.7</td><td>60.8</td><td>28.1</td><td></td><td>34.1</td><td>46.4</td><td>13.7</td><td>49.0</td><td>30.5</td></tr><tr><td>DPR + BART</td><td>55.3</td><td>75.5</td><td>45.2</td><td>46.9</td><td>13.3</td><td>28.9</td><td>54.3</td><td></td><td>25.0</td><td>44.4</td><td>10.7</td><td>25.4</td><td>38.6</td></tr><tr><td>RAG</td><td>61.9</td><td>72.6</td><td>48.1</td><td>47.6</td><td>28.7</td><td>53.7</td><td></td><td>59.5</td><td>30.6</td><td>48.7</td><td>11.0</td><td>57.8</td><td>47.3</td></tr><tr><td>BLINK + flair</td><td>63.7</td><td>81.5</td><td>80.2</td><td>68.8</td><td>59.6</td><td>78.8</td><td>24.5</td><td></td><td>46.1</td><td>65.6</td><td>9.3</td><td>38.2</td><td>56.0</td></tr><tr><td>GENRE only BLINK IDs</td><td>1.8</td><td>65.0</td><td>63.5</td><td>58.6</td><td>0.1</td><td>0.2</td><td>0.4</td><td></td><td>0.3</td><td>5.4</td><td>0.3</td><td>13.3</td><td>19.0</td></tr><tr><td>GENRE only DPR data</td><td>70.8</td><td>9.7</td><td>1.9</td><td>7.3</td><td>60.0</td><td>79.7</td><td>58.3</td><td></td><td>40.3</td><td>69.6</td><td>13.2</td><td>52.6</td><td>42.1</td></tr><tr><td>GENRE only BLINK data</td><td>28.1</td><td>82.5</td><td>88.1</td><td>69.9</td><td>44.8</td><td>66.1</td><td>15.0</td><td></td><td>16.4</td><td>25.6</td><td>6.8</td><td>38.7</td><td>43.8</td></tr><tr><td>GENRE w/o constraints</td><td>78.9</td><td>87.2</td><td>83.2</td><td>36.5</td><td>74.4</td><td>93.6</td><td></td><td>53.3</td><td>45.2</td><td>63.7</td><td>14.3</td><td>62.7</td><td>63.0</td></tr><tr><td>GENRE full</td><td>83.6</td><td>89.9</td><td>87.4</td><td>71.2</td><td>79.4</td><td></td><td>95.8</td><td>60.3</td><td>51.3</td><td>69.2</td><td>15.8</td><td>62.9</td><td>69.7</td></tr></table>
+
+Table 8: Ablation study on KILT retrieval. We report R-Precision. GENRE only BLINK IDs denotes training on BLINK (Wu et al., 2020) data where instead of using the textual entity representation as target we used a numerical ID.
+
+![](images/9ab3e75d2df00730453e67b6ddf05d2ae034fdf27788fb5f49e7d1aa59bbcdd5.jpg)  
+Figure 4: Accuracy per number of BPE tokens of the Wikipedia title to generate on the validation sets of all KILT datasets except ELI5 (as it is fundamentally different from the others). We also show the data distribution of token lengths. Most of the titles have less than 15 BPE tokens while the mode of the distribution is 5. Here GENRE has an average accuracy of $7 8 . 6 \%$ but it is higher for short titles (e.g., ${ < } 1 0 _ { \circ }$ ) and it is lower for long titles (e.g., ${ \geq } 1 0 _ { \cdot }$ ). Degradation in performance does not directly follow the data distribution of the token lengths. Indeed, even if long titles are rare performance is not heavily affected (e.g., for length ${ > } 1 5$ ).
+
+![](images/236f52df4bd2f6823c5c6b139a3c444d90284492b7e06ad60694700cb1d27c7d.jpg)  
+Figure 5: Accuracy per number of incoming links in Wikipedia on the validation sets of all KILT datasets except ELI5 (as it is fundamentally different from the others). We also show the data distribution of the number of incoming links. Intuitively, a page/entity with few incoming links has been observed less than highly connected pages/entities. Indeed, for pages/entities never linked (first bin on the left) the average accuracy is $20 \%$ lower than the global average $( 7 8 . 6 \% )$ . However, for pages/entities linked at least once it is above the global average. This indicates that GENRE seems effective on linking rare entities.
+
+# C EXAMPLES
+
+1 ID : ’ 87d95287−707e−4bd9−9633−ca0c611a4a3a World Without Superma : 8 ’   
+2 i n p u t s : ’ [ . . ] When Superman l e a v e s E a r t h f o r New K r y p t o n , he a p p o i n t s , n e w l y f r e e d f r o m & the Phantom Zone , t o ta ke h i s pla ce as gua rdian of [ START ENT ] M e t r o p o l i s [ END ENT& Mon−E l assumes t h e s e c r e t i d e n t i t y o f J o h n a t h a n Kent as a t r i b u t e t o C l a r k \ ’ s & a d o p t i v e f a t h e r , p o s i n g as C l a r k \ ’ s c o u s i n . [ . . ]   
+3 g o l d o u t p u t : ’ M e t r o p o l i s ( c o m i c s ) ’   
+4 p r e d i c t e d o u t p u t s :   
+5 ( ’ M e t r o p o l i s ( c o m i c s ) ’ , −0.09) ,   
+6 ( ’ T h e m y s c i r a ( DC Comics ) ’ , −1.09) ,   
+7 ( ’ M e t r o p o l i s ( d i s a m b i g u a t i o n ) ’ −1.27) ,   
+8 ’ Superman ( comic book ) , −1.51) ,   
+9 ( ’ Superman ( E a r t h−Two ) ’ , −1.52)   
+10 ]
+
+![](images/319f43317fe610338f1634c083374b1944e125f86103f7f64b378d3b57ec75e0.jpg)  
+Figure 6: Example of a GENRE prediction for named entity disambiguation on KILT WNED. The input is plain text where a mention is flagged with two special start and end tokens [START ENT] and [END ENT]. The output is a ranked list of entity (where we report the log-likelihood as well).   
+Figure 7: Example of GENRE predictions for the retrieval task on KILT. The input is a query and the output is a ranked list of Wikipedia article titles (we also report the log-likelihood of the solutions).
+
+![](images/70f7bd3e4d3c2e209d178e29e65ee7fd4eec572294849bb44de8e62645c1a7f6.jpg)  
+Figure 8: Example of a GENRE prediction for end-to-end entity linking on AIDA. The input is plain text and the output is a Markup string where the links are Wikipedia titles. Spans are in the format $\left. s _ { i } , l _ { i } , t _ { i } \right.$ : start of the mention, length of the mention, and title respectively.
+
+![](images/184990c507b796d30aeba4523f2a47444d1e1046993c59271add350605b41ea0.jpg)  
+Figure 9: Example of prefix tree (trie) structure where the allowed entities identifiers are ‘English language’, ‘English literature’ and ‘France’. Note that at the root there is the start-of-sequence token SOS and all leaves are end-of-sequence tokens EOS. Since more that one sequence has the same prefix (i.e., ‘English’), this end up being an internal node where branches are the possible continuations.

parse/train/B1QRgziT-/B1QRgziT-.md

@@ -0,0 +1,548 @@
+# SPECTRAL NORMALIZATION FOR GENERATIVE ADVERSARIAL NETWORKS
+
+Takeru Miyato1, Toshiki Kataoka1, Masanori Koyama2, Yuichi Yoshida3
+
+{miyato, kataoka}@preferred.jp   
+koyama.masanori@gmail.com   
+yyoshida@nii.ac.jp   
+1Preferred Networks, Inc. 2Ritsumeikan University 3National Institute of Informatics
+
+# ABSTRACT
+
+One of the challenges in the study of generative adversarial networks is the instability of its training. In this paper, we propose a novel weight normalization technique called spectral normalization to stabilize the training of the discriminator. Our new normalization technique is computationally light and easy to incorporate into existing implementations. We tested the efficacy of spectral normalization on CIFAR10, STL-10, and ILSVRC2012 dataset, and we experimentally confirmed that spectrally normalized GANs (SN-GANs) is capable of generating images of better or equal quality relative to the previous training stabilization techniques. The code with Chainer (Tokui et al., 2015), generated images and pretrained models are available at https://github.com/pfnet-research/sngan_ projection.
+
+# 1 INTRODUCTION
+
+Generative adversarial networks (GANs) (Goodfellow et al., 2014) have been enjoying considerable success as a framework of generative models in recent years, and it has been applied to numerous types of tasks and datasets (Radford et al., 2016; Salimans et al., 2016; Ho & Ermon, 2016; Li et al., 2017). In a nutshell, GANs are a framework to produce a model distribution that mimics a given target distribution, and it consists of a generator that produces the model distribution and a discriminator that distinguishes the model distribution from the target. The concept is to consecutively train the model distribution and the discriminator in turn, with the goal of reducing the difference between the model distribution and the target distribution measured by the best discriminator possible at each step of the training. GANs have been drawing attention in the machine learning community not only for its ability to learn highly structured probability distribution but also for its theoretically interesting aspects. For example, (Nowozin et al., 2016; Uehara et al., 2016; Mohamed & Lakshminarayanan, 2017) revealed that the training of the discriminator amounts to the training of a good estimator for the density ratio between the model distribution and the target. This is a perspective that opens the door to the methods of implicit models (Mohamed & Lakshminarayanan, 2017; Tran et al., 2017) that can be used to carry out variational optimization without the direct knowledge of the density function.
+
+A persisting challenge in the training of GANs is the performance control of the discriminator. In high dimensional spaces, the density ratio estimation by the discriminator is often inaccurate and unstable during the training, and generator networks fail to learn the multimodal structure of the target distribution. Even worse, when the support of the model distribution and the support of the target distribution are disjoint, there exists a discriminator that can perfectly distinguish the model distribution from the target (Arjovsky & Bottou, 2017). Once such discriminator is produced in this situation, the training of the generator comes to complete stop, because the derivative of the so-produced discriminator with respect to the input turns out to be 0. This motivates us to introduce some form of restriction to the choice of the discriminator.
+
+In this paper, we propose a novel weight normalization method called spectral normalization that can stabilize the training of discriminator networks. Our normalization enjoys following favorable properties.
+
+• Lipschitz constant is the only hyper-parameter to be tuned, and the algorithm does not require intensive tuning of the only hyper-parameter for satisfactory performance. • Implementation is simple and the additional computational cost is small.
+
+In fact, our normalization method also functioned well even without tuning Lipschitz constant, which is the only hyper parameter. In this study, we provide explanations of the effectiveness of spectral normalization for GANs against other regularization techniques, such as weight normalization (Salimans & Kingma, 2016), weight clipping (Arjovsky et al., 2017), and gradient penalty (Gulrajani et al., 2017). We also show that, in the absence of complimentary regularization techniques (e.g., batch normalization, weight decay and feature matching on the discriminator), spectral normalization can improve the sheer quality of the generated images better than weight normalization and gradient penalty.
+
+# 2 METHOD
+
+In this section, we will lay the theoretical groundwork for our proposed method. Let us consider a simple discriminator made of a neural network of the following form, with the input $_ { \textbf { \em x } }$ :
+
+$$
+f ( \pmb { x } , \pmb { \theta } ) = W ^ { L + 1 } a _ { L } ( W ^ { L } ( a _ { L - 1 } ( W ^ { L - 1 } ( \dots a _ { 1 } ( W ^ { 1 } \pmb { x } ) \dots ) ) ) ) ,
+$$
+
+where $\theta : = \{ W ^ { 1 } , \ldots , W ^ { L } , W ^ { L + 1 } \}$ is the learning parameters set, $W ^ { l } ~ \in ~ \mathbb { R } ^ { d _ { l } \times d _ { l - 1 } }$ , $W ^ { L + 1 } \in$ $\mathbb { R } ^ { 1 \times d _ { L } }$ , and $a _ { l }$ is an element-wise non-linear activation function. We omit the bias term of each layer for simplicity. The final output of the discriminator is given by
+
+$$
+D ( \pmb { x } , \theta ) = \mathcal { A } ( f ( \pmb { x } , \theta ) ) ,
+$$
+
+where $\mathcal { A }$ is an activation function corresponding to the divergence of distance measure of the user’s choice. The standard formulation of GANs is given by
+
+$$
+\operatorname* { m i n } _ { G } \operatorname* { m a x } _ { D } V ( G , D )
+$$
+
+where min and max of $G$ and $D$ are taken over the set of generator and discriminator functions, respectively. The conventional form of $V ( G , D )$ (Goodfellow et al., 2014) is given by $\mathrm { E } _ { { \pmb x } \sim q _ { \mathrm { d a t a } } } [ \log D ( { \pmb x } ) ] + \mathrm { E } _ { { \pmb x } ^ { \prime } \sim p _ { G } } [ \log ( 1 - D ( { \pmb x } ^ { \prime } ) ) ]$ , where $q _ { \mathrm { d a t a } }$ is the data distribution and $p _ { G }$ is the (model) generator distribution to be learned through the adversarial min-max optimization. The activation function $\mathcal { A }$ that is used in the $D$ of this expression is some continuous function with range $[ 0 , 1 ]$ (e.g, sigmoid function). It is known that, for a fixed generator $G$ , the optimal discriminator for this form of $V ( G , D )$ is given by $D _ { G } ^ { * } ( { \pmb x } ) : = q _ { \mathrm { d a t a } } ( { \pmb x } ) / ( q _ { \mathrm { d a t a } } ^ { - } ( { \pmb x } ) + p _ { G } ( { \pmb x } ) )$ .
+
+The machine learning community has been pointing out recently that the function space from which the discriminators are selected crucially affects the performance of GANs. A number of works (Uehara et al., 2016; Qi, 2017; Gulrajani et al., 2017) advocate the importance of Lipschitz continuity in assuring the boundedness of statistics. For example, the optimal discriminator of GANs on the above standard formulation takes the form
+
+$$
+D _ { G } ^ { * } ( x ) = \frac { q _ { \mathrm { d a t a } } ( x ) } { q _ { \mathrm { d a t a } } ( x ) + p _ { G } ( x ) } = \mathrm { s i g m o i d } ( f ^ { * } ( x ) ) \mathrm { , w h e r e ~ } f ^ { * } ( x ) = \log q _ { \mathrm { d a t a } } ( x ) - \log p _ { G } ( x ) \mathrm { , }
+$$
+
+and its derivative
+
+$$
+\nabla _ { \pmb { x } } { f } ^ { * } ( \pmb { x } ) = \frac { 1 } { q _ { \mathrm { d a t a } } ( \pmb { x } ) } \nabla _ { \pmb { x } } q _ { \mathrm { d a t a } } ( \pmb { x } ) - \frac { 1 } { p _ { G } ( \pmb { x } ) } \nabla _ { \pmb { x } } p _ { G } ( \pmb { x } )
+$$
+
+can be unbounded or even incomputable. This prompts us to introduce some regularity condition to the derivative of $f ( { \pmb x } )$ .
+
+A particularly successful works in this array are (Qi, 2017; Arjovsky et al., 2017; Gulrajani et al., 2017), which proposed methods to control the Lipschitz constant of the discriminator by adding regularization terms defined on input examples $_ { \textbf { \em x } }$ . We would follow their footsteps and search for the discriminator $D$ from the set of $K$ -Lipschitz continuous functions, that is,
+
+$$
+\operatorname { a r g m a x } _ { \ell \| \mathbf { L i p } \leq K } V ( G , D ) ,
+$$
+
+where we mean by $\Vert f \Vert _ { \mathrm { L i p } }$ the smallest value $M$ such that $\| f ( \pmb { x } ) - f ( \pmb { x } ^ { \prime } ) \| / \| \pmb { x } - \pmb { x } ^ { \prime } \| \leq M$ for any ${ \mathbf { } } x , x ^ { \prime }$ , with the norm being the $\ell _ { 2 }$ norm.
+
+While input based regularizations allow for relatively easy formulations based on samples, they also suffer from the fact that, they cannot impose regularization on the space outside of the supports of the generator and data distributions without introducing somewhat heuristic means. A method we would introduce in this paper, called spectral normalization, is a method that aims to skirt this issue by normalizing the weight matrices using the technique devised by Yoshida & Miyato (2017).
+
+# 2.1 SPECTRAL NORMALIZATION
+
+Our spectral normalization controls the Lipschitz constant of the discriminator function $f$ by literally constraining the spectral norm of each layer $g : h _ { i n } \mapsto h _ { o u t }$ . By definition, Lipschitz norm $\| g \| _ { \mathrm { L i p } }$ is equal to $\mathrm { { \bar { s u p } } } _ { h } \sigma ( \nabla g ( h ) )$ , where $\sigma ( A )$ is the spectral norm of the matrix $A$ ( $L _ { 2 }$ matrix norm of $A$ )
+
+$$
+\sigma ( A ) : = \operatorname* { m a x } _ { \pmb { h } : \pmb { h } \neq \mathbf { 0 } } \frac { \| A \pmb { h } \| _ { 2 } } { \| \pmb { h } \| _ { 2 } } = \operatorname* { m a x } _ { \| \pmb { h } \| _ { 2 } \leq 1 } \| A \pmb { h } \| _ { 2 } ,
+$$
+
+which is equivalent to the largest singular value of $A$ . Therefore, for a linear layer $g ( h ) = W h$ , the norm is given by $\| g \| _ { \mathrm { L i p } } = \operatorname* { s u p } _ { h } \sigma ( { \bar { \nabla } } g ( h ) ) = \operatorname* { s u p } _ { h } \sigma ( W ) = \sigma ( W )$ . If the Lipschitz norm of the activation function $\Vert a _ { l } \Vert _ { \mathrm { L i p } }$ is equal to $1 ^ { 1 }$ , we can use the inequality $\| g _ { 1 } \circ g _ { 2 } \| _ { \mathrm { L i p } } \leq \| g _ { 1 } \| _ { \mathrm { L i p } } \cdot \| g _ { 2 } \| _ { \mathrm { L i p } }$ to observe the following bound on $\Vert f \Vert _ { \mathrm { L i p } }$ :
+
+$$
+\begin{array} { r l } & { \| f \| _ { \mathrm { L i p } } \leq \| ( h _ { L } \mapsto W ^ { L + 1 } h _ { L } ) \| _ { \mathrm { L i p } } \cdot \| a _ { L } \| _ { \mathrm { L i p } } \cdot \| ( h _ { L - 1 } \mapsto W ^ { L } h _ { L - 1 } ) \| _ { \mathrm { L i p } } } \\ & { \qquad \cdots \| a _ { 1 } \| _ { \mathrm { L i p } } \cdot \| ( h _ { 0 } \mapsto W ^ { 1 } h _ { 0 } ) \| _ { \mathrm { L i p } } = \displaystyle \prod _ { l = 1 } ^ { L + 1 } \| ( h _ { l - 1 } \mapsto W ^ { l } h _ { l - 1 } ) \| _ { \mathrm { L i p } } = \displaystyle \prod _ { l = 1 } ^ { L + 1 } \sigma ( W ^ { l } ) . } \end{array}
+$$
+
+Our spectral normalization normalizes the spectral norm of the weight matrix $W$ so that it satisfies the Lipschitz constraint $\sigma ( W ) = 1$ :
+
+$$
+\bar { W } _ { \mathrm { S N } } ( W ) : = W / \sigma ( W ) .
+$$
+
+If we normalize each $W ^ { l }$ using (8), we can appeal to the inequality (7) and the fact that $\sigma \left( \bar { W } _ { \mathrm { S N } } ( W ) \right) = 1$ to see that $\Vert f \Vert _ { \mathrm { L i p } }$ is bounded from above by 1.
+
+Here, we would like to emphasize the difference between our spectral normalization and spectral norm ”regularization” introduced by Yoshida & Miyato (2017). Unlike our method, spectral norm ”regularization” penalizes the spectral norm by adding explicit regularization term to the objective function. Their method is fundamentally different from our method in that they do not make an attempt to ‘set’ the spectral norm to a designated value. Moreover, when we reorganize the derivative of our normalized cost function and rewrite our objective function (12), we see that our method is augmenting the cost function with a sample data dependent regularization function. Spectral norm regularization, on the other hand, imposes sample data independent regularization on the cost function, just like L2 regularization and Lasso.
+
+# 2.2 FAST APPROXIMATION OF THE SPECTRAL NORM $\sigma ( W )$
+
+As we mentioned above, the spectral norm $\sigma ( W )$ that we use to regularize each layer of the discriminator is the largest singular value of $W$ . If we naively apply singular value decomposition to compute the $\sigma ( W )$ at each round of the algorithm, the algorithm can become computationally heavy. Instead, we can use the power iteration method to estimate $\sigma ( W )$ (Golub & Van der Vorst, 2000; Yoshida & Miyato, 2017). With power iteration method, we can estimate the spectral norm with very small additional computational time relative to the full computational cost of the vanilla GANs. Please see Appendix A for the detail method and Algorithm 1 for the summary of the actual spectral normalization algorithm.
+
+# 2.3 GRADIENT ANALYSIS OF THE SPECTRALLY NORMALIZED WEIGHTS
+
+The gradien $^ { - 2 }$ of $\bar { W } _ { \mathrm { S N } } ( W )$ with respect to $W _ { i j }$ is:
+
+$$
+\begin{array} { r l r } { \displaystyle \frac { \partial \bar { W } _ { \mathrm { S N } } ( W ) } { \partial W _ { i j } } = \frac { 1 } { \sigma ( W ) } E _ { i j } - \frac { 1 } { \sigma ( W ) ^ { 2 } } \frac { \partial \sigma ( W ) } { \partial W _ { i j } } W = \frac { 1 } { \sigma ( W ) } E _ { i j } - \frac { [ { \bf u } _ { 1 } { \bf v } _ { 1 } ^ { \mathrm { T } } ] _ { i j } } { \sigma ( W ) ^ { 2 } } W } & { } & \\ { = \frac { 1 } { \sigma ( W ) } \left( E _ { i j } - [ { \bf u } _ { 1 } { \bf v } _ { 1 } ^ { \mathrm { T } } ] _ { i j } \bar { W } _ { \mathrm { S N } } \right) , } & { } & \end{array}
+$$
+
+where $E _ { i j }$ is the matrix whose $( i , j )$ -th entry is 1 and zero everywhere else, and $\mathbf { \delta u } _ { 1 }$ and ${ \pmb v } _ { 1 }$ are respectively the first left and right singular vectors of $W$ . If $^ { h }$ is the hidden layer in the network to be transformed by $\bar { W } _ { S N }$ , the derivative of the $V ( G , D )$ calculated over the mini-batch with respect to $W$ of the discriminator $D$ is given by:
+
+$$
+\begin{array} { r l } & { \displaystyle \frac { \partial V ( \boldsymbol { G } , \boldsymbol { D } ) } { \partial W } = \frac { 1 } { \sigma ( W ) } \left( \hat { \mathrm { E } } \left[ \delta \boldsymbol { h } ^ { \mathrm { T } } \right] - \left( \hat { \mathrm { E } } \left[ \delta ^ { \mathrm { T } } \bar { W } _ { \mathrm { S N } } \boldsymbol { h } \right] \right) \boldsymbol { u } _ { 1 } \boldsymbol { v } _ { 1 } ^ { \mathrm { T } } \right) } \\ & { \displaystyle \quad \quad = \frac { 1 } { \sigma ( W ) } \left( \hat { \mathrm { E } } \left[ \delta \boldsymbol { h } ^ { \mathrm { T } } \right] - \lambda \boldsymbol { u } _ { 1 } \boldsymbol { v } _ { 1 } ^ { \mathrm { T } } \right) } \end{array}
+$$
+
+where $\delta : = \left( \partial V ( G , D ) / \partial \left( \bar { W } _ { \mathrm { S N } } h \right) \right) ^ { \mathrm { T } } , \lambda : = \hat { \mathrm { E } } \left[ \delta ^ { \mathrm { T } } \left( \bar { W } _ { \mathrm { S N } } h \right) \right]$ , and $\hat { \mathrm { E } } [ \cdot ]$ represents empirical expectation over the mini-batch. $\begin{array} { r } { \frac { \partial V } { \partial W } = 0 } \end{array}$ when $\hat { \mathrm { E } } [ \delta \pmb { h } ^ { \mathrm { T } } ] = k \pmb { u } _ { 1 } \pmb { v } _ { 1 } ^ { T }$ for some $k \in \mathbb { R }$ .
+
+We would like to comment on the implication of (12). The first term $\hat { \mathrm { ~ E ~ } } ^ { \left[ \delta h ^ { \operatorname { T } } \right] }$ is the same as the derivative of the weights without normalization. In this light, the second term in the expression can be seen as the regularization term penalizing the first singular components with the adaptive regularization coefficient $\lambda$ . $\lambda$ is positive when $\delta$ and $\bar { W } _ { \mathrm { S N } } h$ are pointing in similar direction, and this prevents the column space of $W$ from concentrating into one particular direction in the course of the training. In other words, spectral normalization prevents the transformation of each layer from becoming to sensitive in one direction. We can also use spectral normalization to devise a new parametrization for the model. Namely, we can split the layer map into two separate trainable components: spectrally normalized map and the spectral norm constant. As it turns out, this parametrization has its merit on its own and promotes the performance of GANs (See Appendix E).
+
+# 3 SPECTRAL NORMALIZATION VS OTHER REGULARIZATION TECHNIQUES
+
+The weight normalization introduced by Salimans & Kingma (2016) is a method that normalizes the $\ell _ { 2 }$ norm of each row vector in the weight matrix. Mathematically, this is equivalent to requiring the weight by the weight normalization $\bar { W } _ { \mathrm { W N } }$ :
+
+$$
+\sigma _ { 1 } ( \bar { W } _ { \mathrm { W N } } ) ^ { 2 } + \sigma _ { 2 } ( \bar { W } _ { \mathrm { W N } } ) ^ { 2 } + \cdot \cdot \cdot + \sigma _ { T } ( \bar { W } _ { \mathrm { W N } } ) ^ { 2 } = d _ { o } , \mathrm { w h e r e } T = \mathrm { m i n } ( d _ { i } , d _ { o } ) ,
+$$
+
+where $\sigma _ { t } ( A )$ is a $t { \cdot }$ -th singular value of matrix $A$ . Therefore, up to a scaler, this is same as the Frobenius normalization, which requires the sum of the squared singular values to be 1. These normalizations, however, inadvertently impose much stronger constraint on the matrix than intended. If $\bar { W } _ { \mathrm { W N } }$ is the weight normalized matrix of dimension $d _ { i } \times d _ { o }$ , the norm $\| \bar { W } _ { \mathrm { W N } } h \| _ { 2 }$ for a fixed unit vector $^ { h }$ is maximized at $\lVert \bar { W } _ { \mathrm { W N } } \underline { { h } } \rVert _ { 2 } ~ = ~ \sqrt { \ d _ { o } }$ when $\sigma _ { 1 } ( \bar { W } _ { \mathrm { W N } } ) ~ { = } ~ \sqrt { d _ { o } }$ and $\sigma _ { t } ( \bar { W } _ { \mathrm { W N } } ) \ : = \ : 0$ for $t = 2 , \dots , T$ , which means that $\bar { W } _ { \mathrm { W N } }$ is of rank one. Similar thing can be said to the Frobenius normalization (See the appendix for more details). Using such $\bar { W } _ { \mathrm { W N } }$ corresponds to using only one feature to discriminate the model probability distribution from the target. In order to retain as much norm of the input as possible and hence to make the discriminator more sensitive, one would hope to make the norm of $\hat { W } _ { \mathrm { W N } } h$ large. For weight normalization, however, this comes at the cost of reducing the rank and hence the number of features to be used for the discriminator. Thus, there is a conflict of interests between weight normalization and our desire to use as many features as possible to distinguish the generator distribution from the target distribution. The former interest often reigns over the other in many cases, inadvertently diminishing the number of features to be used by the discriminators. Consequently, the algorithm would produce a rather arbitrary model distribution that matches the target distribution only at select few features. Weight clipping (Arjovsky et al., 2017) also suffers from same pitfall.
+
+Our spectral normalization, on the other hand, do not suffer from such a conflict in interest. Note that the Lipschitz constant of a linear operator is determined only by the maximum singular value. In other words, the spectral norm is independent of rank. Thus, unlike the weight normalization, our spectral normalization allows the parameter matrix to use as many features as possible while satisfying local 1-Lipschitz constraint. Our spectral normalization leaves more freedom in choosing the number of singular components (features) to feed to the next layer of the discriminator.
+
+Brock et al. (2016) introduced orthonormal regularization on each weight to stabilize the training of GANs. In their work, Brock et al. (2016) augmented the adversarial objective function by adding the following term:
+
+$$
+\| W ^ { \mathrm { T } } W - I \| _ { F } ^ { 2 } .
+$$
+
+While this seems to serve the same purpose as spectral normalization, orthonormal regularization are mathematically quite different from our spectral normalization because the orthonormal regularization destroys the information about the spectrum by setting all the singular values to one. On the other hand, spectral normalization only scales the spectrum so that the its maximum will be one.
+
+Gulrajani et al. (2017) used Gradient penalty method in combination with WGAN. In their work, they placed $K$ -Lipschitz constant on the discriminator by augmenting the objective function with the regularizer that rewards the function for having local 1-Lipschitz constant (i.e. $\| \nabla _ { \hat { \pmb { x } } } f \| _ { 2 } = 1 )$ ) at discrete sets of points of the form $\hat { \pmb { x } } : = \epsilon \tilde { \pmb { x } } + \bar { ( 1 - \epsilon ) } \pmb { x }$ generated by interpolating a sample $\tilde { \pmb x }$ from generative distribution and a sample $_ { \textbf { \em x } }$ from the data distribution. While this rather straightforward approach does not suffer from the problems we mentioned above regarding the effective dimension of the feature space, the approach has an obvious weakness of being heavily dependent on the support of the current generative distribution. As a matter of course, the generative distribution and its support gradually changes in the course of the training, and this can destabilize the effect of such regularization. In fact, we empirically observed that a high learning rate can destabilize the performance of WGAN-GP. On the contrary, our spectral normalization regularizes the function the operator space, and the effect of the regularization is more stable with respect to the choice of the batch. Training with our spectral normalization does not easily destabilize with aggressive learning rate. Moreover, WGAN-GP requires more computational cost than our spectral normalization with single-step power iteration, because the computation of $\| \nabla _ { \hat { \pmb { x } } } f \| _ { 2 }$ requires one whole round of forward and backward propagation. In the appendix section, we compare the computational cost of the two methods for the same number of updates.
+
+# 4 EXPERIMENTS
+
+In order to evaluate the efficacy of our approach and investigate the reason behind its efficacy, we conducted a set of extensive experiments of unsupervised image generation on CIFAR-10 (Torralba et al., 2008) and STL-10 (Coates et al., 2011), and compared our method against other normalization techniques. To see how our method fares against large dataset, we also applied our method on ILSVRC2012 dataset (ImageNet) (Russakovsky et al., 2015) as well. This section is structured as follows. First, we will discuss the objective functions we used to train the architecture, and then we will describe the optimization settings we used in the experiments. We will then explain two performance measures on the images to evaluate the images produced by the trained generators. Finally, we will summarize our results on CIFAR-10, STL-10, and ImageNet.
+
+As for the architecture of the discriminator and generator, we used convolutional neural networks. Also, for the evaluation of the spectral norm for the convolutional weight $W \in \mathbb { R } ^ { d _ { \mathrm { o u t } } \times d _ { \mathrm { i n } } \times h \times w }$ , we treated the operator as a 2-D matrix of dimension $d _ { \mathrm { o u t } } \times ( d _ { \mathrm { i n } } h w ) ^ { 3 }$ . We trained the parameters of the generator with batch normalization (Ioffe & Szegedy, 2015). We refer the readers to Table 3 in the appendix section for more details of the architectures.
+
+For all methods other than WGAN-GP, we used the following standard objective function for the adversarial loss:
+
+$$
+V ( G , D ) : = \underset { x \sim q _ { \mathrm { d a t a } } ( \mathbf { x } ) } { \mathrm { ~ E ~ } } [ \log D ( \pmb { x } ) ] + \underset { z \sim p ( z ) } { \mathrm { ~ E ~ } } [ \log ( 1 - D ( G ( z ) ) ) ] ,
+$$
+
+where $z \in \mathbb { R } ^ { d _ { z } }$ is a latent variable, $p ( z )$ is the standard normal distribution $\mathcal { N } ( 0 , I )$ , and $G : \mathbb { R } ^ { d _ { z } } $ $\mathbb { R } ^ { d _ { 0 } }$ is a deterministic generator function. We set $d _ { z }$ to 128 for all of our experiments. For the updates of $G$ , we used the alternate cost proposed by Goodfellow et al. $( 2 0 1 4 ) - \mathrm { E } _ { z \sim p ( z ) } [ \log ( D ( G ( z ) ) ) ]$ as used in Goodfellow et al. (2014) and Warde-Farley & Bengio (2017). For the updates of $D$ , we used the original cost defined in (15). We also tested the performance of the algorithm with the so-called hinge loss, which is given by
+
+$$
+\begin{array} { r l } & { V _ { D } ( \hat { G } , D ) = \underset { x \sim q _ { \mathrm { d a t a } } ( x ) } { \mathrm { E } } \left[ \operatorname* { m i n } \left( 0 , - 1 + D ( x ) \right) \right] + \underset { z \sim p ( z ) } { \mathrm { E } } \left[ \operatorname* { m i n } \left( 0 , - 1 - D \left( \hat { G } ( z ) \right) \right) \right] } \\ & { V _ { G } ( G , \hat { D } ) = - \underset { z \sim p ( z ) } { \mathrm { E } } \left[ \hat { D } \left( G ( z ) \right) \right] , } \end{array}
+$$
+
+respectively for the discriminator and the generator. Optimizing these objectives is equivalent to minimizing the so-called reverse KL divergence : $\mathrm { K L } [ p _ { g } ] | q _ { \mathrm { d a t a } } ]$ . This type of loss has been already proposed and used in Lim & Ye (2017); Tran et al. (2017). The algorithm based on the hinge loss also showed good performance when evaluated with inception score and FID. For Wasserstein GANs with gradient penalty (WGAN-GP) (Gulrajani et al., 2017), we used the following objective function: $V ( G , D ) : = \operatorname { E } _ { x \sim q _ { \mathrm { d a t a } } } [ D ( \pmb { x } ) ] - \operatorname { E } _ { \pmb { z } \sim p ( \pmb { z } ) } [ D ( G ( \pmb { z } ) ) ] \underset { \ r { \ r { \ r { \ r { \alpha } } } } } { - } \lambda \operatorname { E } _ { \hat { \pmb { x } } \sim p _ { \hat { \pmb { x } } } } [ ( \lVert \nabla _ { \hat { \pmb { x } } } D ( \hat { \pmb { x } } ) \rVert _ { 2 } - 1 ) ^ { 2 } ]$ , where the regularization term is the one we introduced in the appendix section D.4.
+
+For quantitative assessment of generated examples, we used inception score (Salimans et al., 2016) and Frechet inception distance ´ (FID) (Heusel et al., 2017). Please see Appendix B.1 for the details of each score.
+
+# 4.1 RESULTS ON CIFAR10 AND STL-10
+
+In this section, we report the accuracy of the spectral normalization (we use the abbreviation: SNGAN for the spectrally normalized GANs) during the training, and the dependence of the algorithm’s performance on the hyperparmeters of the optimizer. We also compare the performance quality of the algorithm against those of other regularization/normalization techniques for the discriminator networks, including: Weight clipping (Arjovsky et al., 2017), WGAN-GP (Gulrajani et al., 2017), batch-normalization (BN) (Ioffe & Szegedy, 2015), layer normalization (LN) (Ba et al., 2016), weight normalization (WN) (Salimans & Kingma, 2016) and orthonormal regularization (orthonormal) (Brock et al., 2016). In order to evaluate the stand-alone efficacy of the gradient penalty, we also applied the gradient penalty term to the standard adversarial loss of GANs (15). We would refer to this method as ‘GAN-GP’. For weight clipping, we followed the original work Arjovsky et al. (2017) and set the clipping constant $c$ at 0.01 for the convolutional weight of each layer. For gradient penalty, we set $\lambda$ to 10, as suggested in Gulrajani et al. (2017). For orthonormal, we initialized the each weight of $D$ with a randomly selected orthonormal operator and trained GANs with the objective function augmented with the regularization term used in Brock et al. (2016). For all comparative studies throughout, we excluded the multiplier parameter $\gamma$ in the weight normalization method, as well as in batch normalization and layer normalization method. This was done in order to prevent the methods from overtly violating the Lipschitz condition. When we experimented with different multiplier parameter, we were in fact not able to achieve any improvement.
+
+For optimization, we used the Adam optimizer Kingma & Ba (2015) in all of our experiments. We tested with 6 settings for (1) $n _ { \mathrm { d i s } }$ , the number of updates of the discriminator per one update of the generator and (2) learning rate $\alpha$ and the first and second order momentum parameters $( \beta _ { 1 } , \beta _ { 2 } )$ of Adam. We list the details of these settings in Table 1 in the appendix section. Out of these 6 settings, A, B, and C are the settings used in previous representative works. The purpose of the settings D, E, and F is to the evaluate the performance of the algorithms implemented with more aggressive learning rates. For the details of the architectures of convolutional networks deployed in the generator and the discriminator, we refer the readers to Table 3 in the appendix section. The number of updates for GAN generator were 100K for all experiments, unless otherwise noted.
+
+Firstly, we inspected the spectral norm of each layer during the training to make sure that our spectral normalization procedure is indeed serving its purpose. As we can see in the Figure 9 in the C.1, the spectral norms of these layers floats around 1–1.05 region throughout the training. Please see Appendix C.1 for more details.
+
+Table 1: Hyper-parameter settings we tested in our experiments. $\dag , \ddag$ and $\star$ are the hyperparameter settings following Gulrajani et al. (2017), Warde-Farley & Bengio (2017) and Radford et al. (2016), respectively.   
+
+<table><tr><td>Setting</td><td>α</td><td>β</td><td>β</td><td>ndis</td></tr><tr><td>At</td><td>0.0001</td><td>0.5</td><td>0.9</td><td>5</td></tr><tr><td>B</td><td>0.0001</td><td>0.5</td><td>0.999</td><td>1</td></tr><tr><td>C*</td><td>0.0002</td><td>0.5</td><td>0.999</td><td>1</td></tr><tr><td>D</td><td>0.001</td><td>0.5</td><td>0.9</td><td>5</td></tr><tr><td>E</td><td>0.001</td><td>0.5</td><td>0.999</td><td>5</td></tr><tr><td>F</td><td>0.001</td><td>0.9</td><td>0.999</td><td>5</td></tr></table>
+
+![](images/0679674629753ae35ffe4c79d892afcc2435d4ac00bb331302ee5682e726268b.jpg)  
+Figure 1: Inception scores on CIFAR-10 and STL-10 with different methods and hyperparameters (higher is better).
+
+In Figures 1 and 2 we show the inception scores of each method with the settings A–F. We can see that spectral normalization is relatively robust with aggressive learning rates and momentum parameters. WGAN-GP fails to train good GANs at high learning rates and high momentum parameters on both CIFAR-10 and STL-10. Orthonormal regularization performed poorly for the setting E on the STL-10, but performed slightly better than our method with the optimal setting. These results suggests that our method is more robust than other methods with respect to the change in the setting of the training. Also, the optimal performance of weight normalization was inferior to both WGAN-GP and spectral normalization on STL-10, which consists of more diverse examples than CIFAR-10. Best scores of spectral normalization are better than almost all other methods on both CIFAR-10 and STL-10.
+
+In Tables 2, we show the inception scores of the different methods with optimal settings on CIFAR10 and STL-10 dataset. We see that SN-GANs performed better than almost all contemporaries on the optimal settings. SN-GANs performed even better with hinge loss (17).4. For the training with same number of iterations, SN-GANs fell behind orthonormal regularization for STL-10. For more detailed comparison between orthonormal regularization and spectral normalization, please see section 4.1.2.
+
+In Figure 6 we show the images produced by the generators trained with WGAN-GP, weight normalization, and spectral normalization. SN-GANs were consistently better than GANs with weight normalization in terms of the quality of generated images. To be more precise, as we mentioned in Section 3, the set of images generated by spectral normalization was clearer and more diverse than the images produced by the weight normalization. We can also see that WGAN-GP failed to train good GANs with high learning rates and high momentums (D,E and F). The generated images with GAN-GP, batch normalization, and layer normalization is shown in Figure 12 in the appendix section.
+
+![](images/34bfed54b3a7c3a53dd749885543ecaeb015b22db3a4b3a3bc9ad66f2531c20e.jpg)  
+Figure 2: FIDs on CIFAR-10 and STL-10 with different methods and hyperparameters (lower is better).
+
+Table 2: Inception scores and FIDs with unsupervised image generation on CIFAR-10. $\dagger$ (Radford et al., 2016) (experimented by Yang et al. (2017)), $^ \ddag$ (Yang et al., 2017), $^ *$ (Warde-Farley & Bengio, 2017), †† (Gulrajani et al., 2017)   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Inception score</td><td colspan="2">FID</td></tr><tr><td>CIFAR-10</td><td>STL-10</td><td>CIFAR-10</td><td>STL-10</td></tr><tr><td>Real data</td><td>11.24±.12</td><td>26.08±.26</td><td>7.8</td><td>7.9</td></tr><tr><td colspan="5">-Standard CNN-</td></tr><tr><td>Weight clipping</td><td>6.41±.11</td><td>7.57±.10</td><td>42.6</td><td>64.2</td></tr><tr><td>GAN-GP</td><td>6.93±.08</td><td></td><td>37.7</td><td></td></tr><tr><td>WGAN-GP</td><td>6.68±.06</td><td>8.42±.13</td><td>40.2</td><td>55.1</td></tr><tr><td>Batch Norm.</td><td>6.27±.10</td><td></td><td>56.3</td><td></td></tr><tr><td>Layer Norm.</td><td>7.19±.12</td><td>7.61±.12</td><td>33.9</td><td>75.6</td></tr><tr><td>Weight Norm.</td><td>6.84±.07</td><td>7.16±.10</td><td>34.7</td><td>73.4</td></tr><tr><td>Orthonormal</td><td>7.40±.12</td><td>8.56±.07</td><td>29.0</td><td>46.7</td></tr><tr><td>(ours) SN-GANs</td><td>7.42±.08</td><td>8.28±.09</td><td>29.3</td><td>53.1</td></tr><tr><td>Orthonormal (2x updates)</td><td></td><td>8.67±.08</td><td></td><td>44.2</td></tr><tr><td>(ours) SN-GANs (2x updates)</td><td></td><td>8.69±.09</td><td></td><td>47.5</td></tr><tr><td>(ours) SN-GANs,Eq.(17)</td><td>7.58±.12</td><td></td><td>25.5</td><td></td></tr><tr><td>(ours) SN-GANs,Eq.(17) (2x updates)</td><td></td><td>8.79±.14</td><td></td><td>43.2</td></tr><tr><td colspan="5">-ResNet-5</td></tr><tr><td>Orthonormal,Eq.(17)</td><td>7.92±.04</td><td>8.72±.06</td><td></td><td></td></tr><tr><td>(ours) SN-GANs,Eq.(17)</td><td>8.22±.05</td><td>9.10±.04</td><td>23.8±.58 21.7±.21</td><td>42.4±.99 40.1±.50</td></tr><tr><td>DCGANt</td><td>6.64±.14</td><td>7.84±.07</td><td></td><td></td></tr><tr><td>LR-GANs‡</td><td>7.17±.07</td><td></td><td></td><td></td></tr><tr><td>Warde-Farley et al.*</td><td>7.72±.13</td><td>8.51±.13</td><td></td><td></td></tr><tr><td>WGAN-GP (ResNet)++</td><td>7.86±.08</td><td></td><td></td><td></td></tr></table>
+
+We also compared our algorithm against multiple benchmark methods ans summarized the results on the bottom half of the Table 2. We also tested the performance of our method on ResNet based GANs used in Gulrajani et al. (2017). Please note that all methods listed thereof are all different in both optimization methods and the architecture of the model. Please see Table 4 and 5 in the appendix section for the detail network architectures. Our implementation of our algorithm was able to perform better than almost all the predecessors in the performance.
+
+![](images/0fe08b80ae4b9ca468b508e150744d1efd2c8a94434418ec4387d95d307d0538.jpg)  
+Figure 3: Squared singular values of weight matrices trained with different methods: Weight clipping (WC), Weight Normalization (WN) and Spectral Normalization (SN). We scaled the singular values so that the largest singular values is equal to 1. For WN and SN, we calculated singular values of the normalized weight matrices.
+
+# 4.1.1 ANALYSIS OF SN-GANS
+
+Singular values analysis on the weights of the discriminator $D$ In Figure 3, we show the squared singular values of the weight matrices in the final discriminator $D$ produced by each method using the parameter that yielded the best inception score. As we predicted in Section 3, the singular values of the first to fifth layers trained with weight clipping and weight normalization concentrate on a few components. That is, the weight matrices of these layers tend to be rank deficit. On the other hand, the singular values of the weight matrices in those layers trained with spectral normalization is more broadly distributed. When the goal is to distinguish a pair of probability distributions on the low-dimensional nonlinear data manifold embedded in a high dimensional space, rank deficiencies in lower layers can be especially fatal. Outputs of lower layers have gone through only a few sets of rectified linear transformations, which means that they tend to lie on the space that is linear in most parts. Marginalizing out many features of the input distribution in such space can result in oversimplified discriminator. We can actually confirm the effect of this phenomenon on the generated images especially in Figure 6b. The images generated with spectral normalization is more diverse and complex than those generated with weight normalization.
+
+Training time On CIFAR-10, SN-GANs is slightly slower than weight normalization (about 110 $\sim 1 2 0 \%$ computational time), but significantly faster than WGAN-GP. As we mentioned in Section 3, WGAN-GP is slower than other methods because WGAN-GP needs to calculate the gradient of gradient norm $\| \nabla _ { \pmb { x } } D \| _ { 2 }$ . For STL-10, the computational time of SN-GANs is almost the same as vanilla GANs, because the relative computational cost of the power iteration (18) is negligible when compared to the cost of forward and backward propagation on CIFAR-10 (images size of STL-10 is larger $( 4 8 \times 4 8 )$ ). Please see Figure 10 in the appendix section for the actual computational time.
+
+# 4.1.2 COMPARISON BETWEEN SN-GANS AND ORTHONORMAL REGULARIZATION
+
+In order to highlight the difference between our spectral normalization and orthonormal regularization, we conducted an additional set of experiments. As we explained in Section 3, orthonormal regularization is different from our method in that it destroys the spectral information and puts equal emphasis on all feature dimensions, including the ones that ’shall’ be weeded out in the training process. To see the extent of its possibly detrimental effect, we experimented by increasing the dimension of the feature space 6, especially at the final layer (7th conv) for which the training with our spectral normalization prefers relatively small feature space (dimension $< 1 0 0$ ; see Figure 3b). As for the setting of the training, we selected the parameters for which the orthonormal regularization performed optimally. The figure 4 shows the result of our experiments. As we predicted, the performance of the orthonormal regularization deteriorates as we increase the dimension of the feature maps at the final layer. Our SN-GANs, on the other hand, does not falter with this modification of the architecture. Thus, at least in this perspective, we may such that our method is more robust with respect to the change of the network architecture.
+
+![](images/3835c35b2adb678885bda11615c5ee28671aa080f9f713aeb097adb1f7a2405d.jpg)  
+Figure 4: The effect on the performance on STL-10 induced by the change of the feature map dimension of the final layer. The width of the highlighted region represents standard deviation of the results over multiple seeds of weight initialization. The orthonormal regularization does not perform well with large feature map dimension, possibly because of its design that forces the discriminator to use all dimensions including the ones that are unnecessary. For the setting of the optimizers’ hyper-parameters, We used the setting C, which was optimal for “orthonormal regularization”
+
+![](images/8040f692f6b49e627d307739c23a5985a583a57103d09e7349be855ee6209f5a.jpg)  
+Figure 5: Learning curves for conditional image generation in terms of Inception score for SNGANs and GANs with orthonormal regularization on ImageNet.
+
+# 4.2 IMAGE GENERATION ON IMAGENET
+
+To show that our method remains effective on a large high dimensional dataset, we also applied our method to the training of conditional GANs on ILRSVRC2012 dataset with 1000 classes, each consisting of approximately 1300 images, which we compressed to $1 2 8 \times 1 2 8$ pixels. Regarding the adversarial loss for conditional GANs, we used practically the same formulation used in Mirza & Osindero (2014), except that we replaced the standard GANs loss with hinge loss (17). Please see Appendix B.3 for the details of experimental settings.
+
+GANs without normalization and GANs with layer normalization collapsed in the beginning of training and failed to produce any meaningful images. GANs with orthonormal normalization Brock et al. (2016) and our spectral normalization, on the other hand, was able to produce images. The inception score of the orthonormal normalization however plateaued around 20Kth iterations, while SN kept improving even afterward (Figure 5.) To our knowledge, our research is the first of its kind in succeeding to produce decent images from ImageNet dataset with a single pair of a discriminator and a generator (Figure 7). To measure the degree of mode-collapse, we followed the footstep of Odena et al. (2017) and computed the intra MS-SSIM Odena et al. (2017) for pairs of independently generated GANs images of each class. We see that our SN-GANs ((intra MS-SSIM) $\scriptstyle  \begin{array} { l l l } { \scriptstyle  - 0 . 1 0 1 } \end{array} $ ) is suffering less from the mode-collapse than AC-GANs ((intra MS-SSIM) ${ \sim } 0 . 2 5$ ).
+
+To ensure that the superiority of our method is not limited within our specific setting, we also compared the performance of SN-GANs against orthonormal regularization on conditional GANs with projection discriminator (Miyato & Koyama, 2018) as well as the standard (unconditional) GANs. In our experiments, SN-GANs achieved better performance than orthonormal regularization for the both settings (See Figure 13 in the appendix section).
+
+# 5 CONCLUSION
+
+This paper proposes spectral normalization as a stabilizer of training of GANs. When we apply spectral normalization to the GANs on image generation tasks, the generated examples are more diverse than the conventional weight normalization and achieve better or comparative inception scores relative to previous studies. The method imposes global regularization on the discriminator as opposed to local regularization introduced by WGAN-GP, and can possibly used in combinations. In the future work, we would like to further investigate where our methods stand amongst other methods on more theoretical basis, and experiment our algorithm on larger and more complex datasets.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank the members of Preferred Networks, Inc., particularly Shin-ichi Maeda, Eiichi Matsumoto, Masaki Watanabe and Keisuke Yahata for insightful comments and discussions. We also would like to thank anonymous reviewers and commenters on the OpenReview forum for insightful discussions.
+
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+
+![](images/8cf255f3b58a6bc43644532566b02fd7e738201692f776d8f6a0ab38e5d25b26.jpg)  
+Figure 6: Generated images on different methods: WGAN-GP, weight normalization, and spectral normalization on CIFAR-10 and STL-10.
+
+![](images/bb0328547611a1ee65156172a7f8bc5bdb60d8ce966ebdaf79fa39d9944e929c.jpg)  
+Figure 7: $1 2 8 \mathrm { x } 1 2 8$ pixel images generated by SN-GANs trained on ILSVRC2012 dataset. The inception score is $2 1 . 1 { \pm } . 3 5$ .
+
+# A THE ALGORITHM OF SPECTRAL NORMALIZATION
+
+Let us describe the shortcut in Section 2.1 in more detail. We begin with vectors $\tilde { \mathbf { \pmb { u } } }$ that is randomly initialized for each weight. If there is no multiplicity in the dominant singular values and if $\tilde { \textbf { \em u } }$ is not orthogonal to the first left singular vectors7, we can appeal to the principle of the power method and produce the first left and right singular vectors through the following update rule:
+
+$$
+\begin{array} { r } { \tilde { \pmb { v } }  W ^ { \mathrm { T } } \tilde { \pmb { u } } / \| W ^ { \mathrm { T } } \tilde { \pmb { u } } \| _ { 2 } , \tilde { \pmb { u } }  W \tilde { \pmb { v } } / \| W \tilde { \pmb { v } } \| _ { 2 } . } \end{array}
+$$
+
+We can then approximate the spectral norm of $W$ with the pair of so-approximated singular vectors:
+
+$$
+\sigma ( W ) \approx \tilde { \pmb { u } } ^ { \mathrm { T } } W \tilde { \pmb { v } } .
+$$
+
+If we use SGD for updating $W$ , the change in $W$ at each update would be small, and hence the change in its largest singular value. In our implementation, we took advantage of this fact and reused the $\tilde { \mathbf { \pmb { u } } }$ computed at each step of the algorithm as the initial vector in the subsequent step. In fact, with this ‘recycle’ procedure, one round of power iteration was sufficient in the actual experiment to achieve satisfactory performance. Algorithm 1 in the appendix summarizes the computation of the spectrally normalized weight matrix $\bar { W }$ with this approximation. Note that this procedure is very computationally cheap even in comparison to the calculation of the forward and backward propagations on neural networks. Please see Figure 10 for actual computational time with and without spectral normalization.
+
+# Algorithm 1 SGD with spectral normalization
+
+• Initialize $\tilde { \mathbf { u } } _ { l } \in \mathcal { R } ^ { d _ { l } }$ for $l = 1 , \ldots , L$ with a random vector (sampled from isotropic distribution).
+
+• For each update and each layer $l$ :
+
+1. Apply power iteration method to a unnormalized weight $W ^ { l }$ :
+
+$$
+\begin{array} { r l } & { \tilde { \pmb { v } } _ { l }  ( W ^ { l } ) ^ { \mathrm { T } } \tilde { \pmb { u } } _ { l } / \| ( W ^ { l } ) ^ { \mathrm { T } } \tilde { \pmb { u } } _ { l } \| _ { 2 } } \\ & { \tilde { \pmb { u } } _ { l }  W ^ { l } \tilde { \pmb { v } } _ { l } / \| W ^ { l } \tilde { \pmb { v } } _ { l } \| _ { 2 } } \end{array}
+$$
+
+2. Calculate $\bar { W } _ { \mathrm { S N } }$ with the spectral norm:
+
+$$
+\bar { W } _ { \mathrm { S N } } ^ { l } ( W ^ { l } ) = W ^ { l } / \sigma ( W ^ { l } ) , \mathrm { w h e r e } \sigma ( W ^ { l } ) = \tilde { \mathbf { u } } _ { l } ^ { \mathrm { T } } W ^ { l } \tilde { \mathbf { v } } _ { l }
+$$
+
+3. Update $W ^ { l }$ with SGD on mini-batch dataset $\mathcal { D } _ { M }$ with a learning rate $\alpha$ :
+
+$$
+W ^ { l } \gets W ^ { l } - \alpha \nabla _ { W ^ { l } } \ell ( \bar { W } _ { \mathrm { S N } } ^ { l } ( W ^ { l } ) , { \mathcal D } _ { M } )
+$$
+
+# B EXPERIMENTAL SETTINGS
+
+# B.1 PERFORMANCE MEASURES
+
+Inception score is introduced originally by Salimans et al. (2016): $\begin{array} { r l } { I ( \{ x _ { n } \} _ { n = 1 } ^ { N } ) } & { { } : = } \end{array}$ $\exp ( \mathrm { E } [ D _ { \mathrm { K L } } [ p ( y | \mathbf { x } ) | | p ( y ) ] ] )$ , where $p ( y )$ is approximated by $\begin{array} { r } { { \frac { 1 } { N } } \sum _ { n = 1 } ^ { N } p ( y | \mathbf { x } _ { n } ) } \end{array}$ and $p ( y | x )$ is the trained Inception convolutional neural network (Szegedy et al., 2015), which we would refer to Inception model for short. In their work, Salimans et al. (2016) reported that this score is strongly correlated with subjective human judgment of image quality. Following the procedure in Salimans et al. (2016); Warde-Farley & Bengio (2017), we calculated the score for randomly generated 5000 examples from each trained generator to evaluate its ability to generate natural images. We repeated each experiment 10 times and reported the average and the standard deviation of the inception scores.
+
+Frechet inception distance (Heusel et al., 2017) is another measure for the quality of the generated ´ examples that uses 2nd order information of the final layer of the inception model applied to the examples. On its own, the Frechet distance ´ Dowson $\&$ Landau (1982) is 2-Wasserstein distance between two distribution $p _ { 1 }$ and $p _ { 2 }$ assuming they are both multivariate Gaussian distributions:
+
+$$
+F ( p _ { 1 } , p _ { 2 } ) = \| { \pmb \mu } _ { p _ { 1 } } - { \pmb \mu } _ { p _ { 2 } } \| _ { 2 } ^ { 2 } + \mathrm { t r a c e } \left( C _ { p _ { 1 } } + C _ { p _ { 2 } } - 2 ( C _ { p _ { 1 } } C _ { p _ { 2 } } ) ^ { 1 / 2 } \right) ,
+$$
+
+where $\{ \mu _ { p _ { 1 } } , C _ { p _ { 1 } } \}$ , $\{ \mu _ { p _ { 2 } } , C _ { p _ { 2 } } \}$ are the mean and covariance of samples from $q$ and $p$ , respectively. If $f _ { \ominus }$ is the output of the final layer of the inception model before the softmax, the Frechet inception ´ distance (FID) between two distributions $p _ { 1 }$ and $p _ { 2 }$ on the images is the distance between $f _ { \ominus } \circ p _ { 1 }$ and $f _ { \ominus } \circ p _ { 2 }$ . We computed the Frechet inception distance between the true distribution and the generated ´ distribution empirically over 10000 and 5000 samples. Multiple repetition of the experiments did not exhibit any notable variations on this score.
+
+# B.2 IMAGE GENERATION ON CIFAR-10 AND STL-10
+
+For the comparative study, we experimented with the recent ResNet architecture of Gulrajani et al. (2017) as well as the standard CNN. For this additional set of experiments, we used Adam again for the optimization and used the very hyper parameter used in Gulrajani et al. (2017) $( \alpha = 0 . 0 0 0 2 , \beta _ { 1 } = 0 , \beta _ { 2 } = 0 . 9 , n _ { d i s } = 5 )$ . For our SN-GANs, we doubled the feature map in the generator from the original, because this modification achieved better results. Note that when we doubled the dimension of the feature map for the WGAN-GP experiment, however, the performance deteriorated.
+
+# B.3 IMAGE GENERATION ON IMAGENET
+
+The images used in this set of experiments were resized to $1 2 8 \times 1 2 8$ pixels. The details of the architecture are given in Table 6. For the generator network of conditional GANs, we used conditional batch normalization (CBN) (Dumoulin et al., 2017; de Vries et al., 2017). Namely we replaced the standard batch normalization layer with the CBN conditional to the label information $y \in \{ 1 , \ldots , 1 0 0 0 \}$ . For the optimization, we used Adam with the same hyperparameters we used for ResNet on CIFAR-10 and STL-10 dataset. We trained the networks with 450K generator updates, and applied linear decay for the learning rate after 400K iterations so that the rate would be 0 at the end.
+
+# B.4 NETWORK ARCHITECTURES
+
+Table 3: Standard CNN models for CIFAR-10 and STL-10 used in our experiments on image Generation. The slopes of all lReLU functions in the networks are set to 0.1.   
+
+<table><tr><td rowspan=1 colspan=1>RGB image x ∈ RM×M×3</td></tr><tr><td rowspan=1 colspan=1>3×3, stride=1 conv 64 IReLU4×4, stride=2 conv 64 IReLU</td></tr><tr><td rowspan=1 colspan=1>3×3, stride=1 conv 128 lReLU4×4, stride=2 conv 128 lReLU</td></tr><tr><td rowspan=1 colspan=1>3×3,stride=1 conv 256 lReLU4×4, stride=2 conv 256 lReLU</td></tr><tr><td rowspan=1 colspan=1>3×3, stride=1 conv. 512 IReLU</td></tr><tr><td rowspan=1 colspan=1>dense →1</td></tr></table>
+
+(b) Discriminator, $M = 3 2$ for SVHN and CIFAR10, and $M = 4 8$ for STL-10
+
+<table><tr><td rowspan=1 colspan=1>z ∈ R128 ~N(0,I)</td></tr><tr><td rowspan=1 colspan=1>dense →Mg × Mg × 512</td></tr><tr><td rowspan=1 colspan=1>4×4, stride=2 deconv. BN 256 ReLU</td></tr><tr><td rowspan=1 colspan=1>4×4, stride=2 deconv. BN 128 ReLU</td></tr><tr><td rowspan=1 colspan=1>4×4, stride=2 deconv. BN 64 ReLU</td></tr><tr><td rowspan=1 colspan=1>3×3,stride=1 conv.3 Tanh</td></tr></table>
+
+(a) Generator, $M _ { g } = 4$ for SVHN and CIFAR10, and $M _ { g } = 6$ for STL-10
+
+Table 4: ResNet architectures for CIFAR10 dataset. We use similar architectures to the ones used in Gulrajani et al. (2017).   
+
+<table><tr><td rowspan=1 colspan=1>RGB image x ∈ R32×32×3</td></tr><tr><td rowspan=1 colspan=1>ResBlock down 128</td></tr><tr><td rowspan=1 colspan=1>ResBlock down128</td></tr><tr><td rowspan=1 colspan=1>ResBlock 128</td></tr><tr><td rowspan=1 colspan=1>ResBlock 128</td></tr><tr><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Global sum pooling</td></tr><tr><td rowspan=1 colspan=1>dense →1</td></tr><tr><td rowspan=1 colspan=1>(b)Discriminator</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>z ∈ R128 ~ N(0,I)</td></tr><tr><td rowspan=1 colspan=1>dense,4 × 4 × 256</td></tr><tr><td rowspan=1 colspan=1>ResBlock up 256</td></tr><tr><td rowspan=1 colspan=1>ResBlock up 256</td></tr><tr><td rowspan=1 colspan=1>ResBlock up 256</td></tr><tr><td rowspan=1 colspan=1>BN,ReLU,3×3 conv,3 Tanh</td></tr></table>
+
+(a) Generator
+
+Figure 8: ResBlock architecture. For the discriminator we removed BN layers in ResBlock.
+
+Table 5: ResNet architectures for STL-10 dataset.   
+
+<table><tr><td rowspan=1 colspan=1>RGB image x ∈ R48×48×3</td></tr><tr><td rowspan=1 colspan=1>ResBlock down 64</td></tr><tr><td rowspan=1 colspan=1>ResBlock down128</td></tr><tr><td rowspan=1 colspan=1>ResBlock down1256</td></tr><tr><td rowspan=1 colspan=1>ResBlock down 512</td></tr><tr><td rowspan=1 colspan=1>ResBlock 1024</td></tr><tr><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Global sum pooling</td></tr><tr><td rowspan=1 colspan=1>dense → 1</td></tr><tr><td rowspan=1 colspan=1>(b) Discriminator</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>z ∈ R128 ~ N(0,I)</td></tr><tr><td rowspan=1 colspan=1>dense,6 ×6 × 512</td></tr><tr><td rowspan=1 colspan=1>ResBlock up 256</td></tr><tr><td rowspan=1 colspan=1>ResBlock up 128</td></tr><tr><td rowspan=1 colspan=1>ResBlock up 64</td></tr><tr><td rowspan=1 colspan=1>BN,ReLU,3×3 conv,3 Tanh</td></tr></table>
+
+(a) Generator
+
+<table><tr><td rowspan=1 colspan=1>z ∈ R128~ N(0,I)</td></tr><tr><td rowspan=1 colspan=1>dense,4 × 4 × 1024</td></tr><tr><td rowspan=1 colspan=1>ResBlock up 1024</td></tr><tr><td rowspan=1 colspan=1>ResBlock up 512</td></tr><tr><td rowspan=1 colspan=1>ResBlock up 2256</td></tr><tr><td rowspan=1 colspan=1>ResBlock up 128</td></tr><tr><td rowspan=1 colspan=1>ResBlock up 64</td></tr><tr><td rowspan=1 colspan=1>BN, ReLU, 3×3 conv 3</td></tr><tr><td rowspan=1 colspan=1>Tanh</td></tr><tr><td rowspan=1 colspan=1>(a)Generator</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>RGB image x ∈ R128×128×3</td></tr><tr><td rowspan=1 colspan=1>ResBlock down 64</td></tr><tr><td rowspan=1 colspan=1>ResBlock down 128</td></tr><tr><td rowspan=1 colspan=1>ResBlock down 256</td></tr><tr><td rowspan=1 colspan=1>ResBlock down 512</td></tr><tr><td rowspan=1 colspan=1>ResBlock down 1024</td></tr><tr><td rowspan=1 colspan=1>ResBlock 1024</td></tr><tr><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Global sum pooling</td></tr><tr><td rowspan=1 colspan=1>dense → 1</td></tr></table>
+
+(b) Discriminator for unconditional GANs.
+
+Table 6: ResNet architectures for image generation on ImageNet dataset. For the generator of conditional GANs, we replaced the usual batch normalization layer in the ResBlock with the conditional batch normalization layer. As for the model of the projection discriminator, we used the same architecture used in Miyato & Koyama (2018). Please see the paper for the details.   
+
+<table><tr><td rowspan=1 colspan=1>RGB image x ∈ R128×128×3</td></tr><tr><td rowspan=1 colspan=1>ResBlock down 64</td></tr><tr><td rowspan=1 colspan=1>ResBlock down 128</td></tr><tr><td rowspan=1 colspan=1>ResBlock down 256</td></tr><tr><td rowspan=1 colspan=1>Concat(Embed(y), h)</td></tr><tr><td rowspan=1 colspan=1>ResBlock down 512</td></tr><tr><td rowspan=1 colspan=1>ResBlock down 1024</td></tr><tr><td rowspan=1 colspan=1>ResBlock 1024</td></tr><tr><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>Global sum pooling</td></tr><tr><td rowspan=1 colspan=1>dense →1</td></tr></table>
+
+(c) Discriminator for conditional GANs. For computational ease, we embedded the integer label $\begin{array} { c c l } { y } & { \in } & { \{ 0 , \dots , 1 0 0 0 \} } \end{array}$ into 128 dimension before concatenating the vector to the output of the intermediate layer.
+
+# C APPENDIX RESULTS
+
+# C.1 ACCURACY OF SPECTRAL NORMALIZATION
+
+Figure 9 shows the spectral norm of each layer in the discriminator over the course of the training. The setting of the optimizer is C in Table 1 throughout the training. In fact, they do not deviate by more than 0.05 for the most part. As an exception, 6 and 7-th convolutional layers with largest rank deviate by more than 0.1 in the beginning of the training, but the norm of this layer too stabilizes around 1 after some iterations.
+
+![](images/4f34fb99ec89a2583d70dd633a1c123d846febd969abc8ea75a221e2987d8258.jpg)  
+Figure 9: Spectral norms of all seven convolutional layers in the standard CNN during course of the training on CIFAR 10.
+
+# C.2 TRAINING TIME
+
+![](images/ffed4dd58cad72960ea9574822d13132c01f2defac6fa0718e3d754e192746ef.jpg)  
+Figure 10: Computational time for 100 updates. We set $n _ { \mathrm { d i s } } = 5$
+
+![](images/8ef62c13cbd0f20941e9ba74918ad7fce355e703ac43509f156ddec677984038.jpg)  
+Figure 11: The effect of $n _ { d i s }$ on spectral normalization and weight normalization. The shaded region represents the variance of the result over different seeds.
+
+C.4 GENERATED IMAGES ON CIFAR10 WITH GAN-GP, LAYER NORMALIZATION AND BATCH NORMALIZATION
+
+![](images/a358835224f9b30ec7e9a3eedd494818f7845c1c66a1b8749837887072754057.jpg)  
+Figure 12: Generated images with GAN-GP, Layer Norm and Batch Norm on CIFAR-10
+
+![](images/a9e29090d88ba333220387bd4203eaac033ae69f2b1bd2230b882671a0bc1a27.jpg)  
+Figure 13: Learning curves in terms of Inception score for SN-GANs and GANs with orthonormal regularization on ImageNet. The figure (a) shows the results for the standard (unconditional) GANs, and the figure (b) shows the results for the conditional GANs trained with projection discriminator (Miyato & Koyama, 2018)
+
+# D SPECTRAL NORMALIZATION VS OTHER REGULARIZATION TECHNIQUES
+
+This section is dedicated to the comparative study of spectral normalization and other regularization methods for discriminators. In particular, we will show that contemporary regularizations including weight normalization and weight clipping implicitly impose constraints on weight matrices that places unnecessary restriction on the search space of the discriminator. More specifically, we will show that weight normalization and weight clipping unwittingly favor low-rank weight matrices. This can force the trained discriminator to be largely dependent on select few features, rendering the algorithm to be able to match the model distribution with the target distribution only on very low dimensional feature space.
+
+# D.1 WEIGHT NORMALIZATION AND FROBENIUS NORMALIZATION
+
+The weight normalization introduced by Salimans & Kingma (2016) is a method that normalizes the $\ell _ { 2 }$ norm of each row vector in the weight matrix8:
+
+$$
+\bar { W } _ { \mathrm { W N } } : = \left[ \bar { w } _ { 1 } ^ { \mathrm { T } } , \bar { w } _ { 2 } ^ { \mathrm { T } } , . . . , \bar { w } _ { d _ { o } } ^ { \mathrm { T } } \right] ^ { \mathrm { T } } , \mathrm { ~ w h e r e ~ } \bar { w } _ { i } ( w _ { i } ) : = w _ { i } / \| w _ { i } \| _ { 2 } ,
+$$
+
+where $\bar { \mathbf { \nabla } } \bar { \mathbf { w } } _ { i }$ and ${ \pmb w } _ { i }$ are the ith row vector of $\bar { W } _ { \mathrm { W N } }$ and $W$ , respectively.
+
+Still another technique to regularize the weight matrix is to use the Frobenius norm:
+
+$$
+\bar { W } _ { \mathrm { F N } } : = W / \| W \| _ { F } ,
+$$
+
+where $\begin{array} { r } { \| W \| _ { F } : = \sqrt { \mathrm { t r } ( W ^ { \mathrm { T } } W ) } = \sqrt { \sum _ { i , j } w _ { i j } ^ { 2 } } . } \end{array}$
+
+Originally, these regularization techniques were invented with the goal of improving the generalization performance of supervised training (Salimans & Kingma, 2016; Arpit et al., 2016). However, recent works in the field of GANs (Salimans et al., 2016; Xiang & Li, 2017) found their another raison d’etat as a regularizer of discriminators, and succeeded in improving the performance of the original.
+
+These methods in fact can render the trained discriminator $D$ to be $K$ -Lipschitz for a some prescribed $K$ and achieve the desired effect to a certain extent. However, weight normalization (25) imposes the following implicit restriction on the choice of $\bar { W } _ { \mathrm { W N } }$ :
+
+$$
+\sigma _ { 1 } ( \bar { W } _ { \mathrm { W N } } ) ^ { 2 } + \sigma _ { 2 } ( \bar { W } _ { \mathrm { W N } } ) ^ { 2 } + \cdot \cdot \cdot + \sigma _ { T } ( \bar { W } _ { \mathrm { W N } } ) ^ { 2 } = d _ { o } , \mathrm { w h e r e } T = \mathrm { m i n } ( d _ { i } , d _ { o } ) ,
+$$
+
+where $\sigma _ { t } ( A )$ is a $t$ -th singular value of matrix $A$ . The above equation holds because Pmin(di,do)t=1 σt(W¯ WN)2 = tr(W¯ WNW¯ TWN) = Pdoi=1 wikw k wTikw k . Under this restriction, the norm $\| \bar { W } _ { \mathrm { W N } } h \| _ { 2 }$ for a fixed unit vector $^ { h }$ is maximized at $\| \bar { W } _ { \mathrm { W N } } h \| _ { 2 } = \sqrt { d _ { o } }$ when $\sigma _ { ^ 1 } ( \bar { W } _ { \mathrm { W N } } ) =$ $\sqrt { d _ { o } }$ and $\sigma _ { t } ( \bar { W } _ { \mathrm { W N } } ) = 0$ for $t = 2 , \dots , T$ , which means that $\bar { W } _ { \mathrm { W N } }$ is of rank one. Using such $W$ corresponds to using only one feature to discriminate the model probability distribution from the target. Similarly, Frobenius normalization requires $\sigma _ { 1 } ( \bar { W } _ { \mathrm { F N } } ) ^ { 2 } + \sigma _ { 2 } ( \bar { W } _ { \mathrm { F N } } ) ^ { 2 } + \dot { \bar { \cdots } } \dot { \cdot } + \sigma _ { T } ( \bar { W } _ { \mathrm { F N } } ) ^ { 2 } = 1 .$ , and the same argument as above follows.
+
+Here, we see a critical problem in these two regularization methods. In order to retain as much norm of the input as possible and hence to make the discriminator more sensitive, one would hope to make the norm of $\hat { W } _ { \mathrm { W N } } h$ large. For weight normalization, however, this comes at the cost of reducing the rank and hence the number of features to be used for the discriminator. Thus, there is a conflict of interests between weight normalization and our desire to use as many features as possible to distinguish the generator distribution from the target distribution. The former interest often reigns over the other in many cases, inadvertently diminishing the number of features to be used by the discriminators. Consequently, the algorithm would produce a rather arbitrary model distribution that matches the target distribution only at select few features.
+
+Our spectral normalization, on the other hand, do not suffer from such a conflict in interest. Note that the Lipschitz constant of a linear operator is determined only by the maximum singular value. In other words, the spectral norm is independent of rank. Thus, unlike the weight normalization, our spectral normalization allows the parameter matrix to use as many features as possible while satisfying local 1-Lipschitz constraint. Our spectral normalization leaves more freedom in choosing the number of singular components (features) to feed to the next layer of the discriminator.
+
+To see this more visually, we refer the reader to Figure (14). Note that spectral normalization allows for a wider range of choices than weight normalization.
+
+![](images/ac47faa0c1479a5c920e45ca5c8809435766f2b6e19e29c7da0b704d2e4567ec.jpg)  
+Figure 14: Visualization of the difference between spectral normalization (Red) and weight normalization (Blue) on possible sets of singular values. The possible sets of singular values plotted in increasing order for weight normalization (Blue) and for spectral normalization (Red). For the set of singular values permitted under the spectral normalization condition, we scaled √ $\bar { W } _ { \mathrm { W N } }$ by $1 / \sqrt { d _ { o } }$ so that its spectral norm is exactly 1. By the definition of the weight normalization, the area under the blue curves are all bound to be 1. Note that the range of choice for the weight normalization is small.
+
+In summary, weight normalization and Frobenius normalization favor skewed distributions of singular values, making the column spaces of the weight matrices lie in (approximately) low dimensional vector spaces. On the other hand, our spectral normalization does not compromise the number of feature dimensions used by the discriminator. In fact, we will experimentally show that GANs trained with our spectral normalization can generate a synthetic dataset with wider variety and higher inception score than the GANs trained with other two regularization methods.
+
+# D.2 WEIGHT CLIPPING
+
+Still another regularization technique is weight clipping introduced by Arjovsky et al. (2017) in their training of Wasserstein GANs. Weight clipping simply truncates each element of weight matrices so that its absolute value is bounded above by a prescribed constant $c \in \mathbb { R } _ { + }$ . Unfortunately, weight clipping suffers from the same problem as weight normalization and Frobenius normalization. With weight clipping with the truncation value $c$ , the value $\| W x \| _ { 2 }$ for a fixed unit vector $_ { \textbf { \em x } }$ is maximized when the rank of $W$ is again one, and the training will again favor the discriminators that use only select few features. Gulrajani et al. (2017) refers to this problem as capacity underuse problem. They also reported that the training of WGAN with weight clipping is slower than that of the original DCGAN (Radford et al., 2016).
+
+# D.3 SINGULAR VALUE CLIPPING AND SINGULAR VALUE CONSTRAINT
+
+One direct and straightforward way of controlling the spectral norm is to clip the singular values (Saito et al., 2017), (Jia et al., 2017). This approach, however, is computationally heavy because one needs to implement singular value decomposition in order to compute all the singular values.
+
+A similar but less obvious approach is to parametrize $W \in \mathbb { R } ^ { d _ { o } \times d _ { i } }$ as follows from the get-go and train the discriminators with this constrained parametrization:
+
+$$
+W : = U S V ^ { \mathrm { T } } , \mathrm { ~ s u b j e c t ~ t o ~ } U ^ { \mathrm { T } } U = I , V ^ { \mathrm { T } } V = I , \operatorname* { m a x } _ { i } S _ { i i } = K ,
+$$
+
+where $U \in \mathbb { R } ^ { d _ { o } \times P }$ , $V \in \mathbb { R } ^ { d _ { i } \times P }$ , and $S \in \mathbb { R } ^ { P \times P }$ is a diagonal matrix. However, it is not a simple task to train this model while remaining absolutely faithful to this parametrization constraint. Our spectral normalization, on the other hand, can carry out the updates with relatively low computational cost without compromising the normalization constraint.
+
+# D.4 WGAN WITH GRADIENT PENALTY (WGAN-GP)
+
+Recently, Gulrajani et al. (2017) introduced a technique to enhance the stability of the training of Wasserstein GANs (Arjovsky et al., 2017). In their work, they endeavored to place $K$ -Lipschitz constraint (5) on the discriminator by augmenting the adversarial loss function with the following regularizer function:
+
+$$
+\lambda \operatorname { E } _ { \hat { \pmb { x } } \sim p _ { \hat { \pmb { x } } } } [ ( \| \nabla _ { \hat { \pmb { x } } } D ( \hat { \pmb { x } } ) \| _ { 2 } - 1 ) ^ { 2 } ] ,
+$$
+
+where $\lambda > 0$ is a balancing coefficient and $\hat { \pmb x }$ is:
+
+$$
+\begin{array} { r l } & { \hat { \pmb x } : = \epsilon { \pmb x } + ( 1 - \epsilon ) \tilde { \pmb x } } \\ & { \quad \mathrm { w h e r e } \ \epsilon \sim U [ 0 , 1 ] , \ { \pmb x } \sim p _ { \mathrm { d a t a } } , \ \tilde { \pmb x } = G ( z ) , \ z \sim p _ { z } . } \end{array}
+$$
+
+Using this augmented objective function, Gulrajani et al. (2017) succeeded in training a GAN based on ResNet (He et al., 2016) with an impressive performance. The advantage of their method in comparison to spectral normalization is that they can impose local 1-Lipschitz constraint directly on the discriminator function without a rather round-about layer-wise normalization. This suggest that their method is less likely to underuse the capacity of the network structure.
+
+At the same time, this type of method that penalizes the gradients at sample points $\hat { \textbf { \textit { x } } }$ suffers from the obvious problem of not being able to regularize the function at the points outside of the support of the current generative distribution. In fact, the generative distribution and its support gradually changes in the course of the training, and this can destabilize the effect of the regularization itself.
+
+On the contrary, our spectral normalization regularizes the function itself, and the effect of the regularization is more stable with respect to the choice of the batch. In fact, we observed in the experiment that a high learning rate can destabilize the performance of WGAN-GP. Training with our spectral normalization does not falter with aggressive learning rate.
+
+Moreover, WGAN-GP requires more computational cost than our spectral normalization with single-step power iteration, because the computation of $\| \nabla _ { \pmb { x } } D \| _ { 2 }$ requires one whole round of forward and backward propagation. In Figure 10, we compare the computational cost of the two methods for the same number of updates.
+
+Having said that, one shall not rule out the possibility that the gradient penalty can compliment spectral normalization and vice versa. Because these two methods regularizes discriminators by completely different means, and in the experiment section, we actually confirmed that combination of WGAN-GP and reparametrization with spectral normalization improves the quality of the generated examples over the baseline (WGAN-GP only).
+
+# E REPARAMETRIZATION MOTIVATED BY THE SPECTRAL NORMALIZATION
+
+We can take advantage of the regularization effect of the spectral normalization we saw above to develop another algorithm. Let us consider another parametrization of the weight matrix of the discriminator given by:
+
+$$
+\tilde { W } : = \gamma \bar { W } _ { \mathrm { S N } }
+$$
+
+where $\gamma$ is a scalar variable to be learned. This parametrization compromises the 1-Lipschitz constraint at the layer of interest, but gives more freedom to the model while keeping the model from becoming degenerate. For this reparametrization, we need to control the Lipschitz condition by other means, such as the gradient penalty (Gulrajani et al., 2017). Indeed, we can think of analogous versions of reparametrization by replacing $\bar { W } _ { \mathrm { S N } } ^ { \ }$ in (32) with $W$ normalized by other criterions. The extension of this form is not new. In Salimans & Kingma (2016), they originally introduced weight normalization in order to derive the reparametrization of the form (32) with $\bar { W } _ { \mathrm { S N } }$ replaced (32) by $W _ { \mathrm { W N } }$ and vectorized $\gamma$ .
+
+# E.1 EXPERIMENTS: COMPARISON OF REPARAMETRIZATION WITH DIFFERENTNORMALIZATION METHODS
+
+In this part of the addendum, we experimentally compare the reparametrizations derived from two different normalization methods (weight normalization and spectral normalization). We tested the reprametrization methods for the training of the discriminator of WGAN-GP. For the architecture of the network in WGAN-GP, we used the same CNN we used in the previous section. For the ResNet-based CNN, we used the same architecture provided by (Gulrajani et al., 2017) 9.
+
+Tables 7, 8 summarize the result. We see that our method significantly improves the inception score from the baseline on the regular CNN, and slightly improves the score on the ResNet based CNN.
+
+Figure 15 shows the learning curves of (a) critic losses, on train and validation sets and (b) the inception scores with different reparametrization methods. We can see the beneficial effect of spectral normalization in the learning curve of the discriminator as well. We can verify in the figure 15a that the discriminator with spectral normalization overfits less to the training dataset than the discriminator without reparametrization and with weight normalization, The effect of overfitting can be observed on inception score as well, and the final score with spectral normalization is better than the others. As for the best inception score achieved in the course of the training, spectral normalization achieved 7.28, whereas the spectral normalization and vanilla normalization achieved 7.04 and 6.69, respectively.
+
+# F THE GRADIENT OF GENERAL NORMALIZATION METHOD
+
+Let us denote ${ \bar { W } } : = W / N ( W )$ to be the normalized weight where $N ( W )$ to be a scalar normalized coefficient (e.g. Spectral norm or Frobenius norm). In general, we can write the derivative of loss
+
+Table 7: Inception scores with different reparametrization mehtods on CIFAR10 without label supervisions. $( { } ^ { * } ) \mathrm { W e }$ reported N/A for the inception score and FID of Frobenius normalization because the training collapsed at the early stage.   
+
+<table><tr><td>Method</td><td>Inception score</td><td>FID</td></tr><tr><td>WGAN-GP (Standard CNN, Baseline)</td><td>6.68±.06</td><td>40.1</td></tr><tr><td>w/Frobenius Norm.</td><td>N/A*</td><td>N/A*</td></tr><tr><td>w/ Weight Norm.</td><td>6.36±.04</td><td>42.4</td></tr><tr><td>w/ Spectral Norm.</td><td>7.20±.08</td><td>32.0</td></tr><tr><td>(WGAN-GP, ResNet, Gulrajani et al. (2017))</td><td>7.86±.08</td><td></td></tr><tr><td>WGAN-GP (ResNet, Baseline)</td><td>7.80±.11</td><td>24.5</td></tr><tr><td>w/ Spectral norm.</td><td>7.85±.06</td><td>23.6</td></tr><tr><td>w/ Spectral norm. (1.5x feature maps in D)</td><td>7.96±.06</td><td>22.5</td></tr></table>
+
+<table><tr><td>Method (ResNet)</td><td>Inception score</td><td>FID</td></tr><tr><td>(AC-WGAN-GP, Gulrajani et al. (2017))</td><td>8.42±.10</td><td></td></tr><tr><td>AC-WGAN-GP (Baseline)</td><td>8.29±.12</td><td>19.5</td></tr><tr><td>w/ Spectral norm.</td><td>8.59±.12</td><td>18.6</td></tr><tr><td>w/ Spectral norm. (1.5x feature maps in D)</td><td>8.60±.08</td><td>17.5</td></tr></table>
+
+Table 8: Inception scores and FIDs with different reparametrization methods on CIFAR10 with the label supervision, by auxiliary classifier (Odena et al., 2017).
+
+with respect to unnormalized weight $W$ as follows:
+
+$$
+\begin{array} { l } { \displaystyle \frac { \partial V ( G , D ( W ) ) } { \partial W } = \frac { 1 } { N ( W ) } \left( \frac { \partial V } { \partial \bar { W } } - \mathrm { t r a c e } \left( \left( \frac { \partial V } { \partial \bar { W } } \right) ^ { \mathrm { T } } \bar { W } \right) \frac { \partial ( N ( W ) ) } { \partial W } \right) } \\ { \displaystyle ~ = \frac { 1 } { N ( W ) } \left( \nabla _ { \bar { W } } V - \mathrm { t r a c e } \left( ( \nabla _ { \bar { W } } V ) ^ { \mathrm { T } } \bar { W } \right) \nabla _ { W } N \right) } \\ { \displaystyle ~ = \alpha \left( \nabla _ { \bar { W } } V - \lambda \nabla _ { W } N \right) , } \end{array}
+$$
+
+where $\alpha : = 1 / N ( W )$ and $\lambda : = \operatorname { t r a c e } \left( ( \nabla _ { \bar { W } } V ) ^ { \mathrm { T } } \bar { W } \right)$ . The gradient $\nabla _ { \boldsymbol { \bar { W } } } V$ is calculated by $\hat { \mathrm { ~ E ~ } } ^ { \left[ \delta h ^ { \mathrm { T } } \right] }$ where $\pmb { \delta } : = \left( \partial V ( G , D ) / \partial \left( \bar { W } \pmb { h } \right) \right) ^ { \mathrm { T } }$ , $^ { h }$ is the hidden node in the network to be transformed by $\bar { W }$ and $\hat { \mathrm { E } }$ represents empirical expectation over the mini-batch. When $N ( W ) : = \| W \| _ { F }$ , the derivative is:
+
+$$
+\frac { \partial V ( G , D ( W ) ) } { \partial W } = \frac { 1 } { \| W \| _ { F } } \left( \hat { \mathrm { E } } \left[ \delta h ^ { \mathrm { T } } \right] - \mathrm { t r a c e } \left( \hat { \mathrm { E } } \left[ \delta h ^ { \mathrm { T } } \right] ^ { \mathrm { T } } \bar { W } \right) \bar { W } \right) ,
+$$
+
+and when $N ( W ) : = \| W \| _ { 2 } = \sigma ( W )$ ,
+
+$$
+\frac { \partial V ( G , D ( W ) ) } { \partial W } = \frac { 1 } { \sigma ( W ) } \left( \hat { \mathrm { E } } \left[ \delta h ^ { \mathrm { T } } \right] - \mathrm { t r a c e } \left( \hat { \mathrm { E } } \left[ \delta h ^ { \mathrm { T } } \right] ^ { \mathrm { T } } \bar { W } \right) \boldsymbol { u } _ { 1 } \boldsymbol { v } _ { 1 } ^ { \mathrm { T } } \right) .
+$$
+
+Notice that, at least for the case $N ( W ) : = \| W \| _ { F }$ or $N ( W ) : = \| W \| _ { 2 }$ , the point of this gradient is given by :
+
+$$
+\nabla _ { \bar { W } } V = k \nabla _ { W } N .
+$$
+
+where $\exists k \in \mathbb { R }$
+
+![](images/90c73f0772cf8eedaa5318b515941da59ac9ace65c8e6f2b4e93f2d6a3b23aa3.jpg)  
+Figure 15: Learning curves of (a) critic loss and (b) inception score on different reparametrization method on CIFAR-10 ; weight normalization (WGAN-GP w/ WN), spectral normalization (WGANGP w/ SN), and parametrization free (WGAN-GP).

parse/train/b83ibRX55T/b83ibRX55T.md

@@ -0,0 +1,393 @@
+# Towards Gradient-based Bilevel Optimization with Non-convex Followers and Beyond
+
+Risheng Liu1,2 Yaohua Liu1 Shangzhi Zeng3 Jin Zhang ∗4,5
+
+1International School of Information Science & Engineering, DUT 2Pazhou Lab, Guangzhou 3Department of Mathematics and Statistics, UVic   
+4Department of Mathematics, SUSTech 5National Center for Applied Mathematics Shenzhen rsliu@dlut.edu.cn liuyaohua_918@163.com zengshangzhi@gmail.com zhangj9@sustech.edu.cn
+
+# Abstract
+
+In recent years, Bi-Level Optimization (BLO) techniques have received extensive attentions from both learning and vision communities. A variety of BLO models in complex and practical tasks are of non-convex follower structure in nature (a.k.a., without Lower-Level Convexity, LLC for short). However, this challenging class of BLOs is lack of developments on both efficient solution strategies and solid theoretical guarantees. In this work, we propose a new algorithmic framework, named Initialization Auxiliary and Pessimistic Trajectory Truncated Gradient Method (IAPTT-GM), to partially address the above issues. In particular, by introducing an auxiliary as initialization to guide the optimization dynamics and designing a pessimistic trajectory truncation operation, we construct a reliable approximate version of the original BLO in the absence of LLC hypothesis. Our theoretical investigations establish the convergence of solutions returned by IAPTT-GM towards those of the original BLO without LLC. As an additional bonus, we also theoretically justify the quality of our IAPTT-GM embedded with Nesterov’s accelerated dynamics under LLC. The experimental results confirm both the convergence of our algorithm without LLC, and the theoretical findings under LLC.
+
+# 1 Introduction
+
+Bi-Level Optimization (BLO) has been widely used to formulate problems in the field of deep learning [1, 2], especially for hyperparameter optimization [3, 4, 5], meta learning [6, 7, 8, 9], neural architecture search [10, 11, 12], adversarial learning [13], and reinforcement learning [14], etc. BLO aims to tackle nested optimization structures appearing in applications, which has emerged as a prevailing optimization technique for modern machine learning tasks with underlying hierarchy. In the last decade, a large number of BLO methods have been proposed to address different machine learning tasks. In fact, Gradient Methods (GMs), which can effectively handle BLO problems of large scale, thus gain popularity. According to different types of strategies for gradient calculations, existing GMs can be divided into two categories, i.e., the explicit approaches which aim to replace the Lower-Level (LL) problem with dynamic iterations and implicit schemes that apply the implicit function theorem to formulate the first-order optimality condition of the LL problem.
+
+Explicit Gradient Methods for BLOs. In this type, solving the LL problem is regarded as the evolution path of the dynamic system starting from a given initial point of the LL variable. The gradient of the Upper-Level (UL) variable can be directly calculated by automatic differentiation based on the trajectory of LL variable. This class of methods can be further divided into three types, namely, recurrence-based EG (e.g., [3, 15, 6, 16, 10]), initialization-based EG (e.g., [17, 18]) and proxy-based EG methods (e.g., [19, 20, 21, 22]), differing from each other in the way of accessing the gradient of constructed dynamic trajectory. While most of this type of works assume the LLC and Lower-Level Singleton (LLS) to simplify their optimization process and theoretical analysis, cases where the LLS assumption does not hold have been tackled in the recent work [23, 24]. In particular, to eliminate the LLS assumption which is too restrctive to be satisfied in real-world complex tasks, [23] first considers incorporating UL objective information into the dynamic iterations, but the more general cases where LLC does not hold remain unsolved. On the other hand, while most of the mentioned works focus on the asymptotic convergence, the progress on nonasymptotic convergence analysis has been recently witnessed see, e.g., [25, 26, 27].
+
+Implicit Gradient Methods for BLOs. This type, also known as implicit differentiation [28, 7, 29], replaces the LL problem with its first-order optimality condition and uses the implicit function theorem to calculate the gradient of the UL problem by solving a linear system. This method decouples the calculation of UL gradient from the dynamic system of LL, resulting in a significant speed increase when the dynamic system iterates many times. However, because of the burden originated from computing a Hessian matrix and its inverse, IG methods are usually computationally expensive when linear systems are ill-conditioned. To alleviate this computational issue, there are mainly two kinds of techniques, i.e., IG based on Linear System (LS) [28, 7] and Neumann Series (NS) [29]. On the theoretical side, IG methods rely on the strong convexity of LL problems heavily, which is even more restrictive than the LLC and LLS together.
+
+Initialization Optimization for Learning. In deep learning, the selection of initialization scheme has a great influence on the training speed and performance [30]. As the most representative work in recent years, Model-Agnostic Meta-Learning (MAML) [31] applies the same initialization to all tasks, and is optimized by a loss function common to the task that evaluates the effect of the initial value, resulting in an initialization that achieves good generalization performance with only a few gradient steps on new tasks. Due to its simple form this method has been widely studied and applied [32, 33, 34, 35]. [36] noticed that not all network parameters are suitable for the same initialization, and therefore proposed a strategy to apply co-initialization only on a part of parameters. In theory, [37, 38, 39] give comprehensive study on the convergence and convergence rate of MAML and some MAML-type approaches based on the meta objective function. However, the convergence theory of these existing results are given based on the loss function for evaluating initial values, and the convergence analysis of such type of methods from the perspective of each task is still lacked.
+
+Value-Function Approach. The value function based methods have also emerged as a promising branch to solve BLO problems [40]. Under the special case where the LL is jointly convex with respect to both the UL and LL variables, the BLO problem can be equivalently reformulated into a difference-of-convex program [41], which is numerically solvable. Typically, by reformulating the BLO into an Inner Single Bi-level (ISB) optimization problem with value-function approach, a gradient-based interior-point method name BVFIM [42] is proposed to solve the BLO tasks, which effectively avoids the expensive Hessian-vector and Jacobian-vector products. Generally speaking, the value-function does not admit an explicit form, and is always nonsmooth, non-convex and with jumps.
+
+# 1.1 Our Motivations and Contributions
+
+As mentioned above, some theoretical progresses have long been witnessed in diversified learning areas, but for most existing BLO methods, extra restrictive assumptions (e.g., LLS, LLC and LL strong convexity) have to be enforced. Their algorithm design and associated theoretical analysis actually are only valid for optimization with a simplified problem structure. Unfortunately, it has been well recognized that LL non-convexity frequently appears in a variety of applications, e.g., sparse $\ell _ { q }$ regularization ( $0 < q < 1$ ) for avoiding over-fitting, and learning parameters of coupled multi-layer neural networks, etc. Therefore, in challenging real-world scenarios, we are usually required to consider BLO problems where these assumptions (e.g., LLS, LLC and even LL strong convexity) are naturally violated. These fundamental theoretical issues motivate us to propose a series of new techniques to address BLO with non-convex LL problems, which have been frequently appeared in various learning applications.
+
+In particular, by introducing an Initialization Auxiliary (IA) to the LL optimization dynamics and operating a Pessimistic Trajectory Truncation (PTT) strategy during the UL approximation, we construct a Gradient-based Method (GM), named IAPTT-GM, to address BLO in challenging optimization scenarios (i.e., with non-convex follower tasks). We analyze the convergence behaviors of IAPTT-GM on BLOs without LLC and also investigate theoretical properties of the accelerated version of our algorithm on BLOs under LLC. Extensive experiments verify our theoretical results and demonstrate the effectiveness of IAPTT-GM on different learning applications. The main contributions of our IAPTT-GM are summarized as follows:
+
+• We propose IA and PTT, two new mechanisms to efficiently handle complex BLOs where the follower is facing with a non-convex task (i.e., without LLS and even LLC). IA actually paves the way for jointly optimizing both the UL variables and the dynamical initialization, while PTT adaptively reduces the complexity of backward recurrent propagation. To our best knowledge, we establish the first strict convergence guarantee for gradientbased method on BLOs with non-convex follower tasks. We also justify the quality of our IAPTT-GM embedded with Nesterov’s accelerated dynamics under LLC. • We conduct a series of experiments to verify our theoretical findings and evaluate IAPTT-GM on various challenging BLOs, in which the follower tasks are either with non-convex loss functions (e.g., few-shot learning) or coupled network structures (e.g., data hyper-cleaning).
+
+# 2 The Proposed Algorithmic Framework
+
+In this work, we consider the BLO problem in the form:
+
+$$
+\operatorname* { m i n } _ { \mathbf { x } \in \mathcal { X } , \mathbf { y } } F ( \mathbf { x } , \mathbf { y } ) , \quad s . t . \quad \mathbf { y } \in S ( \mathbf { x } ) ,
+$$
+
+where $\mathbf { x } \in \mathbb { R } ^ { n } , \mathbf { y } \in \mathbb { R } ^ { m }$ are UL and LL variables respectively, and $\boldsymbol { S } ( \mathbf { x } )$ denotes the set of solutions of the LL problem, i.e.,
+
+$$
+S ( \mathbf x ) : = \arg \operatorname* { m i n } _ { \mathbf y \in \mathcal y } f ( \mathbf x , \mathbf y ) ,
+$$
+
+where $f$ is differentiable w.r.t. y. To ensure the BLO model in Eq. (1) is well-defined, we assume that $\scriptstyle { \mathcal { S } } ( \mathbf { x } )$ is nonempty for all $\mathbf { x } \in \mathcal { X }$ . Observe further that the above BLO is structurally different from those in existing literature in the sense that no convexity assumption is required in the LL problem.
+
+In the following, we describe the proposed IAPTT-GM to solve the class of BLOs defined in Eq. (1). The mechanism of a classical dynamics-embedded gradient method, approximates the LL solution via a dynamical system drawn from optimization iterations. Choosing gradient descent as the optimization dynamics for example, the approximation ${ \bf y } _ { K } ( { \bf x } )$ is accessed by operations repeatedly performed by $K - 1$ steps parameterized by UL variable $\mathbf { x }$
+
+$$
+\begin{array} { r } { \mathbf { y } _ { k + 1 } ( \mathbf { x } ) = \mathbf { y } _ { k } ( \mathbf { x } ) - s \nabla _ { \mathbf { y } } f ( \mathbf { x } , \mathbf { y } _ { k } ( \mathbf { x } ) ) , k = 0 , \cdots , K - 1 , } \end{array}
+$$
+
+where $s$ is a step size, and $\mathbf { y } _ { 0 }$ is a fixed initial value.
+
+# 2.1 Initialization Auxiliary
+
+Embedding the dynamical iterations into the UL problem returns the approximate version $F ( \mathbf { x } , \mathbf { y } _ { K } ( \mathbf { x } ) )$ . As long as $\nabla _ { \mathbf { y } } f ( \mathbf { x } , \mathbf { y } _ { K } ( \mathbf { x } ) )$ uniformly converges to zero w.r.t. UL variable $\mathbf { x }$ varying in $\mathcal { X }$ , we call this a good approximation. To this end, usually restrictive LL strong convexity assumptions are imposed, thus the desired convergence of solutions of approximation problems towards those of the original BLO follows. By drawing inspiration from the classic dynamics-embedded gradient method which replaces the LL problem with certain optimization dynamics, hence resulting in an approximation of the bi-level problem, we propose a new gradient scheme to solve BLO in Eq. (1) without LLC restriction. Specifically, we let $K$ be a prescribed positive integer and construct the following approximation ${ \bf y } _ { K } ( { \bf x } , { \bf z } )$ of the LL solution drawn from projected gradient descent iterations
+
+$$
+\begin{array} { r l } & { \mathbf { y } _ { 0 } ( \mathbf { x } , \mathbf { z } ) = \mathbf { z } , } \\ & { \mathbf { y } _ { k + 1 } ( \mathbf { x } , \mathbf { z } ) = \operatorname* { P r o j } _ { \mathcal { V } } ( \mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) - \alpha _ { \mathbf { y } } ^ { k } \nabla _ { \mathbf { y } } f ( \mathbf { x } , \mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) ) ) , k = 0 , \cdots , K - 1 , } \end{array}
+$$
+
+where $\{ \alpha _ { \mathbf { y } } ^ { k } \}$ is a sequence of steps sizes. We next embed the dynamical iterations ${ \bf y } _ { k } ( { \bf x } , { \bf z } )$ into $\mathrm { m a x } _ { 1 \le k \le K } \{ F ( { \bf x } , { \bf y } _ { k } ( { \bf x } , { \bf z } ) ) \}$ , which can be regarded as a pessimistic trajectory truncation of the UL objective. The mechanism of the above scheme, in comparison, accesses the approximation parameterized by UL variable $\mathbf { x }$ and LL initial point $\mathbf { y } _ { 0 }$ . Our motivation for the initialization auxiliary variable $\mathbf { z }$ comes from convergence theory [42] of non-convex first-order optimization methods. In fact, when the non-convex LL problem admits multiple solutions, the gradient descent steps with a “bad” initial point $\mathbf { y } _ { 0 }$ cannot return a desired point in the LL solution set, simultaneously optimizing the UL objective. To overcome such a difficulty, instead of using a fixed initial value, we introduce an initialization auxiliary variable $\mathbf { z }$ . Therefore, when it comes to solving the UL approximation problems, together with the UL variable $\mathbf { x }$ , the auxiliary variable $\mathbf { z }$ is also updated and hence optimized. As a consequence, we may search for the “best” initial value, starting from which the gradient descent steps approach a solution to the BLO in Eq. (1), i.e., a point in the LL solution set, simultaneously minimizing the UL objective.
+
+# 2.2 Pessimistic Trajectory Truncation
+
+This is a striking feature of our algorithm that significantly differs from existing methods and leads to some new convergence results without LLC. The motivation for the design of pessimistic trajectory truncation comes again from convergence theory of non-convex firstorder optimization methods. It is understood that when LL is non-convex, $\mathcal { R } _ { \alpha } ( { \bf x } , { \bf y } _ { K } ( { \bf x } , { \bf z } ) )$ may not uniformly converge w.r.t. $\mathbf { x }$ and $\mathbf { z }$ , where $\mathcal { R } _ { \alpha } ( \mathbf { x } , \mathbf { y } )$ is the proximal gradient residual mapping defined as $\mathcal { R } _ { \alpha } ( \mathbf { x } , \mathbf { y } ) : = \textbf { y } -$ $\mathrm { P r o j } _ { \mathcal { V } } \left( \mathbf { \bar { y } } - \alpha \nabla _ { \mathbf { y } } f ( \mathbf { x } , \mathbf { y } ) \right) ^ { 1 }$ , which can be used as a measurement of the optimality of LL problem in Eq. (2). Thus a direct embedding of ${ \bf y } _ { K } ( { \bf x } , { \bf z } )$ into UL objective $F ( \mathbf { x } , \mathbf { y } )$ may not necessarily provide an appropriate approximation. Fortunately, it is also understood that, for each $\mathbf { x } \in \mathcal { X }$ , $\mathbf { z } \in \mathcal { V }$ and $K > 0$ , there exists at
+
+# Algorithm 1 The Proposed IAPTT-GM
+
+1: Initialize $\mathbf { x } ^ { 0 }$ and $\mathbf { z } ^ { 0 }$ .   
+2: for $t = 0  T - 1$ do   
+3: $\mathbf { y } _ { 0 } = \mathbf { z } ^ { t }$ .   
+4: for $k = 0  K - 1$ do   
+5: $\%$ LL Updating with $\mathbf { x } ^ { t }$ and $\mathbf { z } ^ { t }$   
+6: $\mathbf { y } _ { k + 1 } = \mathbf { \widetilde { P } r o j } _ { \mathcal { V } } ^ { - } ( \mathbf { y } _ { k } - \alpha _ { \mathbf { y } } ^ { k } \nabla _ { \mathbf { y } } f ( \mathbf { x } ^ { t } , \mathbf { y } _ { k } ) )$   
+7: end for   
+8: 9: $\%$ $\bar { k } = \arg \operatorname* { m a x } _ { k } \{ \bar { F ( \mathbf { x } , \mathbf { y } _ { k } ) } \} _ { k = 1 } ^ { K } .$ cation.   
+10: $\%$ UL Updating with ${ \bf y } _ { \bar { k } } ( { \bf x } , { \bf z } )$   
+11: $\mathbf { x } ^ { t + 1 } = \operatorname { P r o j } _ { \mathcal { X } } ( \mathbf { x } ^ { t } - \alpha _ { \mathbf { x } } \nabla _ { \mathbf { x } } F ( \mathbf { x } ^ { t } , \mathbf { y } _ { \bar { k } } ) )$ .   
+12: $\%$ Initialization Updating with ${ \bf y } _ { \bar { k } } ( { \bf x } , { \bf z } )$   
+13: $\mathbf { z } ^ { t + 1 } = \operatorname { P r o j } _ { \mathcal { V } } ( \mathbf { z } ^ { t ^ { \top } } - \alpha _ { \mathbf { z } } \bar { \nabla } _ { \mathbf { z } } F ( \mathbf { x } ^ { t } , \mathbf { y } _ { \bar { k } } ) )$ .   
+14: end for
+
+least a $\tilde { K }$ such that along the selection ${ \mathbf { y } } _ { \tilde { K } } ( { \mathbf { x } } , { \mathbf { z } } )$ , $\mathcal { R } _ { \alpha } ( \mathbf { x } , \mathbf { y } _ { \tilde { K } } ( \mathbf { x } , \mathbf { z } ) )$ uniformly converges to zero w.r.t.   
+$\mathbf { x }$ and $\mathbf { z }$ , as $K$ tending infinity.
+
+However, in general, it is too ambitious to expect an explicit identification of the exact selection ${ \bf y } _ { \tilde { K } } ( { \bf x } , { \bf z } )$ . Alternatively, we consider a pessimistic strategy, minimizing the worst case of all selections of $\{ \mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) \}$ , i.e., $\begin{array} { r } { \operatorname* { m a x } _ { 1 \leq k \leq K } \left\{ F ( \mathbf { x } , \mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) ) \right\} } \end{array}$ . By doing so, we successfully reach a good approximation. In addition to the theoretical convergence, we also benefit from this pessimistic strategy in a numerical sense. The pessimistic max operation always results in a favorable trajectory truncation smaller than $K$ . Consequently, this technique offers inexpensive computational cost for computing the hyper-gradient through back propagation, as shown in the numerical experiments.
+
+To conclude this section, we state the complete IAPTT-GM in Algorithm 1. Note that $K$ and $T$ represent the numbers of inner and outer iterations, respectively.
+
+# 3 Theoretical Investigations
+
+With the purpose of studying the convergence of dynamics-embedded gradient method for BLO without LLC, we involve two signature features in our algorithmic design, i.e., initialization auxiliary and pessimistic trajectory truncated. This section is devoted to the convergence analysis of our proposed algorithm with and without LLC assumption. Please notice that all the proofs of our theoretical results are stated in the Supplemental Material.
+
+# 3.1 Convergence Analysis of IAPTT-GM for BLO with Non-convex Followers
+
+In this part, we conduct the convergence analysis of the IAPTT-GM for solving BLO in Eq. (1) without LLC. Before presenting our main convergence results, we introduce some notations related to BLO. With introduced function $\varphi ( \mathbf x ) : = \operatorname* { i n f } _ { \mathbf y \in { \cal S } ( \mathbf x ) } { \cal F } ( \mathbf x , \mathbf y )$ , the BLO in Eq. (1) can be rewritten as
+
+$$
+\operatorname* { m i n } _ { \mathbf { x } \in \mathcal { X } } \varphi ( \mathbf { x } ) .
+$$
+
+With given $K \geq 1$ and defining $\varphi _ { K } ( \mathbf x , \mathbf z ) : = \mathrm { m a x } _ { k } \left\{ F ( \mathbf x , \mathbf y _ { k } ( \mathbf x , \mathbf z ) ) \right\}$ with $\{ \mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) \}$ defined in Eq. (4), our proposed IAPTT-GM generates sequence $\{ ( \mathbf { x } ^ { t } , \mathbf { z } ^ { t } ) \}$ for solving following approximation problem to BLO in Eq. (5),
+
+$$
+\operatorname* { m i n } _ { { \mathbf { x } } \in { \mathcal { X } } , { \mathbf { z } } \in { \mathcal { Y } } } { \varphi _ { K } ( \mathbf { x } , \mathbf { z } ) } .
+$$
+
+This section is mainly devoted to the convergence of solutions of approximation problems in Eq. (6) towards those of the original BLO in Eq. (5).
+
+Assumption 3.1 We make following standing assumptions throughout this section.
+
+(1) $F , f : \mathbb { R } ^ { n } \times \mathbb { R } ^ { m }  \mathbb { R }$ are continuous functions.   
+(2) $\nabla f$ is continuous and $\nabla _ { \mathbf y } f$ is $L _ { f }$ Lipschitz continuous with respect to y for any $\mathbf { x } \in \mathcal { X }$ .   
+(3) $\mathcal { X }$ and $\mathcal { V }$ are convex compact sets.   
+(4) $\boldsymbol { S } ( \mathbf { x } )$ is nonempty for any $\mathbf { x } \in \mathcal { X }$ .   
+(5) For any $( { \bar { \mathbf { x } } } , { \bar { \mathbf { y } } } )$ minimizing $F ( \mathbf { x } , \mathbf { y } )$ over constraints $\mathbf { x } \in \mathcal { X } , \mathbf { y } \in \mathcal { Y }$ and $\mathbf { y } \in \hat { S } ( \mathbf { x } )$ , it holds that $\bar { \mathbf { y } } \in S ( \mathbf { x } )$ .
+
+Note that $\hat { S } ( \mathbf { x } )$ denotes the set of LL stationary points, i.e., $\hat { \mathcal { S } } ( \mathbf { x } ) = \{ \mathbf { y } \in \mathcal { Y } | 0 = \nabla \mathbf { y } f ( \mathbf { x } , \mathbf { y } ) +$ $\mathcal { N } _ { \mathcal { Y } } ( \mathbf { y } ) \}$ . It should be noticed that $\mathbf { y } \in \hat { S } ( \mathbf { x } )$ if and only if $\begin{array} { r } { \mathcal { R } _ { \alpha } ( \mathbf { x } , \mathbf { y } ) = 0 } \end{array}$ . Assumption 3.1 is standard in bi-level optimziation related literature, which will be shown to be satisfied for the numerical example given in Section 4.1.
+
+As discussed in the preceding section, for each $\mathbf { x } \in \mathcal { X }$ , $\mathbf { z } \in \mathcal { V }$ and $K > 0$ , there exists at least a $\tilde { K }$ such that along the selection ${ \mathbf { y } } _ { \tilde { K } } ( { \mathbf { x } } , { \mathbf { z } } )$ , $\mathcal { R } _ { \alpha } ( \mathbf { x } , \mathbf { y } _ { \tilde { K } } ( \mathbf { x } , \mathbf { z } ) )$ uniformly converges to zero w.r.t. $\mathbf { x }$ and $\mathbf { z }$ , as $K$ tending infinity. We next specifically show the existence of such index $\tilde { K }$ . To this end, $\tilde { K }$ can be chosen by optimizing $\| \mathcal { R } _ { \alpha } ( \mathbf { x } , \mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) ) \|$ among the indices $k = 0 , 1 , . . . , K$ . In particular, as stated in the following lemma, $\| \mathcal { R } _ { \alpha } ( \mathbf { x } , \mathbf { y } _ { \tilde { K } } ( \mathbf { x } , \mathbf { z } ) ) \|$ uniformly decreases with a $\textstyle { \frac { 1 } { \sqrt { K } } }$ rate on $\mathcal { X } \times \mathcal { V }$ as $K$ increases.
+
+Lemma 3.1 Let $\{ \mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) \}$ be the sequence defined in Eq. (4) with $\begin{array} { r } { \alpha _ { \mathbf { y } } ^ { k } \in [ \underline { { \alpha } } _ { \mathbf { y } } , \overline { { \alpha } } _ { \mathbf { y } } ] \subset ( 0 , \frac { 2 } { L _ { f } } ) } \end{array}$ , there exists $C _ { f } > 0$ such that
+
+$$
+\operatorname* { m i n } _ { 0 \leq k \leq K } \| \mathcal { R } _ { \underline { { \alpha } } _ { \mathbf { y } } } ( \mathbf { x } , \mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) ) \| \leq \frac { C _ { f } } { \sqrt { K + 1 } } , \quad \forall \mathbf { x } \in \boldsymbol { \mathcal { X } } , \mathbf { z } \in \mathcal { Y } .
+$$
+
+As shown in the Appendix, the proof of Lemma 3.1 for the existence cannot offer us an explicit identification of the exact selection of $\tilde { K }$ . Alternatively, we construct the approximation by a pessimistic trajectory truncation strategy, minimizing the worst case of all selections of $\{ \mathbf { y } _ { k } ( \mathbf { x } , \bar { \mathbf { z } } ) \}$ , i.e., $\varphi _ { K } ( \mathbf { x } , \mathbf { z } )$ . By further solving the approximated problems $\operatorname* { m i n } { \varphi _ { K } ( \mathbf { x } , \mathbf { z } ) }$ , we shall provide a lower bound estimation for the optimal value of the BLO problem in Eq. (5).
+
+Lemma 3.2 Let $( { \bf x } _ { K } , { \bf z } _ { K } ) \in \mathrm { ~ a r g m i n ~ } \varphi _ { K } ( { \bf x } , { \bf z } )$ , then x∈X ,z∈Y
+
+$$
+\varphi _ { K } ( \mathbf x _ { K } , \mathbf z _ { K } ) \leq \operatorname* { i n f } _ { \mathbf y \in \hat { \mathcal S } ( \mathbf x ) } F ( \mathbf x , \mathbf y ) , \quad \forall \mathbf x \in \mathcal X .
+$$
+
+Upon together with the uniform convergence result in Lemma 3.1, the gap between the lower bound provided by $\operatorname* { m i n } { \varphi _ { K } ( \mathbf { x } , \mathbf { z } ) }$ and the true optimal value of the BLO problem in Eq. (5) eventually vanishes. To fill in this gap and present the main convergence result of our proposed IAPTT-GM,
+
+we need the continuity of $\mathcal { R } _ { \alpha } ( \mathbf { x } , \mathbf { y } )$ . Indeed, it follows from [43, Theorem 6.42] that $\mathsf { P r o j } _ { \mathcal { Y } }$ is continuous. Combined with the assumed continuity of $\nabla _ { \mathbf y } { f } ( \mathbf x , \mathbf y )$ , we get the desired continuity of $\mathcal { R } _ { \alpha } ( \mathbf { x } , \mathbf { y } )$ immediately.
+
+Theorem 3.1 Let $\{ \mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) \}$ be the sequence generated by Eq. (4) with $\begin{array} { r } { \boldsymbol { \alpha } _ { \mathbf { y } } ^ { k } \in [ \underline { { \boldsymbol { \alpha } } } _ { \mathbf { y } } , \overline { { \boldsymbol { \alpha } } } _ { \mathbf { y } } ] \subset ( 0 , \frac { 2 } { L _ { f } } ) , } \end{array}$ ,   
+and $( { \bf x } _ { K } , { \bf z } _ { K } ) \in \underset { -- } { \operatorname { a r g m i n } } \varphi _ { K } ( { \bf x } , { \bf z } )$ , then we have: $\mathbf { x } { \in } \mathcal { X } , \mathbf { z } { \in } \mathcal { Y }$
+
+(1) Any limit point $\bar { \bf x }$ of the sequence $\left\{ { \bf x } _ { K } \right\}$ is the solution to BLO in Eq. (1), that is $\bar { \textbf { x } } \in$ $\underset { \mathbf { x } \in \mathcal { X } } { \mathrm { a r g m i n } } \varphi ( \mathbf { x } )$ .
+
+$$
+\operatorname* { i n f } _ { \mathbf { x } \in \mathcal { X } , \mathbf { z } \in \mathcal { V } } \varphi _ { K } ( \mathbf { x } , \mathbf { z } ) \to \operatorname* { i n f } _ { \mathbf { x } \in \mathcal { X } } \varphi ( \mathbf { x } ) a s K \to \infty .
+$$
+
+In the above theorem, we justify the global solutions convergence of the approximated problems. We next derive a convergence characterization regarding the local minimums of the approximated problems. In particular, the next theorem shows that any limit point of the local minimums of approximated problems is in some sense a local minimum of the bilevel problem in Eq. (1).
+
+Theorem 3.2 Let $\{ \mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) \}$ be the sequence generated by Eq. (4) with $\begin{array} { r } { \boldsymbol { \alpha } _ { \mathbf { y } } ^ { k } \in [ \underline { { \boldsymbol { \alpha } } } _ { \mathbf { y } } , \overline { { \boldsymbol { \alpha } } } _ { \mathbf { y } } ] \subset ( 0 , \frac { 2 } { L _ { f } } ) , } \end{array}$ , and $\left( { { \bf { x } } _ { K } } , { { \bf { z } } _ { K } } \right)$ be a local minimum of $\varphi _ { K } ( \mathbf { x } , \mathbf { z } )$ with uniform neighborhood modulus $\delta > 0$ , i.e.,
+
+$$
+\varphi _ { K } ( \mathbf { x } _ { K } , \mathbf { z } _ { K } ) \leq \varphi _ { K } ( \mathbf { x } , \mathbf { z } ) , \quad \forall ( \mathbf { x } , \mathbf { z } ) \in \mathbb { B } _ { \delta } ( \mathbf { x } _ { K } , \mathbf { z } _ { K } ) \cap \mathcal { X } \times \mathcal { Y } .
+$$
+
+Then we have that for any limit point $( \bar { \bf x } , \bar { \bf z } )$ of the sequence $\{ ( { \bf x } _ { K } , { \bf z } _ { K } ) \}$ , there exists a limit point $\bar { \mathbf { y } }$ of the sequence $\left\{ { \bf y } _ { K } ( { \bf x } _ { K } , { \bf z } _ { K } ) \right\}$ such that $\bar { \mathbf { y } } \in \hat { S } ( \bar { \mathbf { x } } )$ and $( { \bar { \mathbf { x } } } , { \bar { \mathbf { y } } } )$ satisfies that there exists $\tilde { \delta } > 0$ such that
+
+$$
+\begin{array} { r } { F ( \bar { \mathbf { x } } , \bar { \mathbf { y } } ) \leq F ( \mathbf { x } , \mathbf { z } ) , \quad \forall ( \mathbf { x } , \mathbf { z } ) \in \mathbb { B } _ { \widetilde { \delta } } ( \bar { \mathbf { x } } , \bar { \mathbf { z } } ) \cap \{ \mathbf { x } \in \mathcal { X } , \mathbf { z } \in \mathcal { V } \mid \mathbf { z } \in \hat { S } ( \mathbf { x } ) \} . } \end{array}
+$$
+
+# 3.2 Theoretical Findings of IA-GM (A) for BLO with LLC
+
+A byproduct of our study, which has its own interest, is that thanks to the involved initialization auxiliary, our theory can improve those existing results for classical gradient methods with accelerated gradient descent dynamical iterations under LLC. Nesterov’s acceleration technique [44] has been used widely for solving convex optimization problem and it greatly improves the convergence rate of gradient descent. To illustrate our result, we will take the Nesterov’s acceleration proximal gradient method [45] as the embedded optimization dynamics in classical gradient method for example.
+
+Thanks to the LLC setting, $\mathcal { R } _ { \alpha } ( { \bf x } , { \bf y } _ { K } ( { \bf x } , { \bf z } ) )$ may uniformly converge to zero w.r.t. UL variable $x$ and auxiliary variable $\mathbf { z }$ , thus the pessimistic trajectory truncation operation can be removed. Subsequently, we slightly simplify our algorithm that $\bar { k }$ is simply taken as $K$ , thus the approximation objective admits a succinct form, i.e., $F ( \mathbf { x } , \mathbf { y } _ { K } ( \mathbf { x } , \mathbf { z } ) )$ .
+
+In summary, by constructing ${ \bf y } _ { k } ( { \bf x } , { \bf z } )$ through following Nesterov’s acceleration dynamics,
+
+$$
+\begin{array} { r l } & { \mathbf { y } _ { 0 } ( \mathbf { x } , \mathbf { z } ) = \mathbf { z } , \qquad t _ { 0 } = 1 , \quad t _ { k + 1 } = \frac { 1 + \sqrt { 1 + t _ { k } ^ { 2 } } } { 2 } , k = 0 , \cdots , K - 1 , } \\ & { \mathbf { u } ^ { k + 1 } ( \mathbf { x } , \mathbf { z } ) = \mathbf { y } ^ { k + 1 } ( \mathbf { x } , \mathbf { z } ) + \left( \frac { t _ { k } - 1 } { t _ { k + 1 } } \right) ( \mathbf { y } ^ { k + 1 } ( \mathbf { x } , \mathbf { z } ) - \mathbf { y } ^ { k } ( \mathbf { x } , \mathbf { z } ) ) , k = 0 , \cdots , K - 1 , } \\ & { \mathbf { y } _ { k + 1 } ( \mathbf { x } , \mathbf { z } ) = \mathrm { P r o j } _ { \mathcal { Y } } \left( \mathbf { u } ^ { k } ( \mathbf { x } , \mathbf { z } ) - \alpha \nabla _ { \mathbf { y } } f ( \mathbf { x } , \mathbf { u } ^ { k } ( \mathbf { x } , \mathbf { z } ) ) \right) , k = 0 , \cdots , K - 1 , } \end{array}
+$$
+
+where $\alpha > 0$ is the step size, we propose an accelerated Gradient-based Method with Initialization Auxiliary, named IA-GM(A), via minimizing the approximation objective function,
+
+$$
+\operatorname* { m i n } _ { \mathbf { x } \in \mathcal { X } , \mathbf { z } \in \mathcal { V } } \phi _ { K } ( \mathbf { x } , \mathbf { z } ) : = F ( \mathbf { x } , \mathbf { y } _ { K } ( \mathbf { x } , \mathbf { z } ) ) .
+$$
+
+The detailed description of the proposed IA-GM(A) is stated in the Supplemental Material.
+
+We suppose Assumption 3.1(1)-(4) are satisfied throughout this subsection. The convergence result of our proposed IA-GM(A) with LLC assumption is given as below.
+
+Theorem 3.3 Assume that the generated sequence $\{ \mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) \}$ satisfies that $\mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) \in \mathcal { Y } _ { \mathrm { : } }$ , and ${ \bf y } _ { k } ( { \bf x } , { \bf z } ) = { \bf z }$ for any $\mathbf { z } \in S ( \mathbf { x } )$ , $\mathbf { x } \in \mathcal { X }$ , and either
+
+(a) for any $\epsilon > 0$ , there exists $k ( \epsilon ) > 0$ such that whenever $K > k ( \epsilon )$
+
+whenever $K > k ( \epsilon )$ , or
+
+(b) there exists $\alpha > 0 _ { : }$ , for any $\epsilon > 0$ , there exists $k ( \epsilon ) > 0$ such that,
+
+Let $( \mathbf { x } _ { K } , \mathbf { z } _ { K } ) \in \operatorname { a r g m i n } _ { \mathbf { x } \in \mathcal { X } , \mathbf { z } \in \mathcal { Y } } \phi _ { K } ( \mathbf { x } , \mathbf { z } ) : = F ( \mathbf { x } , \mathbf { y } _ { K } ( \mathbf { x } , \mathbf { z } ) )$ , then we have
+
+(1) any limit point $\bar { x }$ of the sequence $\left\{ { \bf x } _ { K } \right\}$ satisfies that $\bar { \mathbf { x } } \in \underset { \mathbf { x } \in \mathcal { X } } { \mathrm { a r g m i n } } \varphi ( \mathbf { x } ) ,$ , i.e., $\bar { x }$ is the solution
+
+to BLO (1).
+
+$$
+\operatorname* { i n f } _ { \mathbf { x } \in \mathcal { X } , \mathbf { z } \in \mathcal { V } } \phi _ { K } ( \mathbf { x } , \mathbf { z } ) \to \operatorname* { i n f } _ { \mathbf { x } \in \mathcal { X } } \varphi ( \mathbf { x } ) a s K \to \infty .
+$$
+
+Next, we show that the Nesterov’s acceleration dynamics satisfy all the assumptions required in the above convergence theorem.
+
+Theorem 3.4 Let $\{ \mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) \}$ be the sequence generated by Nesterov’s acceleration dynamics in Eq. (9) with $\begin{array} { r } { \alpha = \frac { 1 } { L _ { f } } } \end{array}$ . Then $\{ \mathbf { y } _ { k } ( \mathbf { x } , \mathbf { z } ) \}$ satisfies all the assumptions required by Theorem 3.3.
+
+Remark 3.1 Our convergence result Theorem 3.3 is not only for $\mathbf { y } _ { k }$ generated by Nesterov’s acceleration dynamics. It is a general convergence result that is applicable for the case where $\mathbf { y } _ { k }$ is generated by other dynamics. And the assumptions required in Theorem 3.3 is weak enough to be satisfied by the dynamics introduced by many first-order methods on convex LL problem in Eq. 2.
+
+# 4 Experimental Results
+
+In this section, we first verify the theoretical convergence results on non-convex numerical problems compared with existing EG methods and IG methods. Then we test the performance of IAPTT-GM and demonstrate its generalizability to real-world BLO problems with non-convex followers, which are caused by non-convex regularization and neural network structures. In addition, we further validate the performance of the accelerated version (i.e., IA-GM (A)) under LLC with numerical examples and data hyper-cleaning tasks 2.
+
+# 4.1 Numerical Verification
+
+To verify the convergence property under assumptions provided in Section 3, we consider the following non-convex BLO problem:
+
+$$
+\operatorname* { m i n } _ { x \in \mathcal { X } , y \in \mathbb { R } } x + x y , \quad s . t . \quad y \in \underset { y \in \mathcal { Y } } { \operatorname { a r g m i n } } - \sin ( x y ) ,
+$$
+
+where $\mathcal { X } \ : = \ : [ 1 , 1 0 ]$ and $\mathcal { V } = [ - 2 , 2 ]$ . Given any $x \in \mathcal { X }$ , it satisfies $\begin{array} { r } { \mathrm { a r g m i n } _ { y \in \mathcal { y } } - \mathrm { s i n } ( x y ) \ = } \end{array}$ $\{ ( 2 k \pi + \pi / 2 ) / x \mid k \in \mathbb { Z } \} \cap \mathcal { Y }$ and $\begin{array} { r } { \operatorname* { m i n } _ { y \in \mathcal { V } } - \sin ( x y ) = - 1 } \end{array}$ . The unique solution is $( x ^ { * } , y ^ { * } ) =$ $( 1 1 \pi / 4 , - 2 )$ . It should be noted that the LL problem of Eq. (10) has multiple global minima, which can significantly show the advantage of initialization auxiliary technique. It can be easily verified that the above toy example satisfies Assumption 3.1.
+
+In Figure 1, we separately compared IAPTT-GM with EG methods such as RHG [3], BDA [23], IG methods such as LS [28], NS [7] and IA-GM. From Figure 1.(a) to Figure 1.(b), we can observe that different initialization points only slightly affect the convergence speed of IAPTT-GM. With initialization points distant from $( x ^ { * } , y ^ { * } )$ , IAPTT-GM can still achieve optimal solution of UL variables and optimal objective value, while other methods fail to converge to the true solution. In Figure 1.(g) and Figure 1.( h), we compare the performances of IAPTT-GM with IA-GM, which has no convergence guarantee without LLC assumption. As shown, IA-GM fails to converge to the true solution eventually, which validates the necessity of PTT technique and the effectiveness of IAPTT-GM.
+
+![](images/c127978dc735f90a7baeb2a430c84d4f2d7ad25a940e852e57c541daeb62a434.jpg)  
+Figure 1: Illustrating the convergence behavior of $\| F - F ^ { * } \| / \| F ^ { * } \|$ and $\| x - x ^ { * } \| / \| x ^ { * } \|$ as the training proceeds. $F _ { \mathrm { I A P T T - G M } }$ and $F _ { \mathrm { I A - G M } }$ denote the UL objectives of IAPTT-GM and IA-GM, respectively. Three representative initialization points for UL and LL variables are $( x _ { 0 } , y _ { 0 } ) = ( 1 , 2 )$ , $( x _ { 0 } , y _ { 0 } ) = \mathbf { \bar { ( 5 , 1 ) } }$ , $( x _ { 0 } , \overset { \cdot } { y } _ { 0 } ) = ( 7 , - 1 ) .$ .
+
+Runtime and Memory Analysis. In Figure 2, we report the average steps $\bar { k }$ of IAPTT-GM for PTT and the iterative speed as the UL iteration increases. In comparison with IA-GM, which uses default $K$ for the LL optimization loop, the changing $\bar { k }$ for IAPTT-GM leads to less iterations for the backward recurrent propagation and thus faster iterative updates during optimization. Although IA introduces additional variables and iterations, the PTT technique can choose a small $\bar { k }$ , thus shortens the back-propagation trajectory for computing the UL gradient (see Figure 2). As can be seen in Table 1, the memory required by our IAPTT-GM is less than NS, LS, and BDA and the same as that for RHG. As for the runtime, IAPTT-GM is a bit slower than RHG and NS, and faster than BDA. But please notice that the performance and the theoretical properties of IAPTT-GM are better than these existing approaches.
+
+![](images/e00c87cd413be5c13a123c8f9646faf7864d31c9f7a3a818764051a283f0610c.jpg)  
+Figure 2: Illustrating average steps for PTT technique and average running speed of the numerical example. Note that we conduct the experiments using more LL iterations so as to reduce the measurement error.
+
+Table 1: Memory and runtime of existing methods for solving the above BLO problem. We conduct the experiments using the same parameter settings in Section C of the supplementary materials.   
+
+<table><tr><td rowspan=1 colspan=1>Metrics</td><td rowspan=1 colspan=1>LS</td><td rowspan=1 colspan=1>NS</td><td rowspan=1 colspan=1>RHG</td><td rowspan=1 colspan=1>BDA</td><td rowspan=1 colspan=1>IAPTT-GM</td></tr><tr><td rowspan=1 colspan=1>Memory (GB)</td><td rowspan=1 colspan=1>10.426</td><td rowspan=1 colspan=1>10.387</td><td rowspan=1 colspan=1>10.153</td><td rowspan=1 colspan=1>10.154</td><td rowspan=1 colspan=1>10.153</td></tr><tr><td rowspan=1 colspan=1>Runtime (Sec)</td><td rowspan=1 colspan=1>5.120</td><td rowspan=1 colspan=1>10.815</td><td rowspan=1 colspan=1>9.990</td><td rowspan=1 colspan=1>16.800</td><td rowspan=1 colspan=1>10.835</td></tr></table>
+
+# 4.2 BLO with Non-convex Followers in Different Application Scenarios
+
+To cover various real-world BLO application scenarios, we consider two categories of non-convex LL problems caused by non-convex regularization term and neural network architectures, which refer to few-shot classification and data hyper-cleaning tasks, respectively. Please note that the set constraint $\mathcal { V }$ is only used to guarantee the completeness of our theoretical analysis in Section 3. In application scenarios, we can just consider the constraint as an extra large set, so that all the variables are automatically in this feasible set. In this way, it is natural to ignore the projection operation in practical computations.
+
+# 4.2.1 Few-Shot Classification: Non-convex LL Objective
+
+In few-shot learning, to be more specific, N-way M-shot classification tasks [46], provided with M samples from each class, we train models to take advantage of prior data from similar tasks to quickly classify unseen instances from these $_ \mathrm { N }$ classes. Following the experimental protocol [47], the model parameters are separated into two parts: the hyper representation module (parameterized by $\mathbf { x }$ ) shared by all the tasks and the last classifier (parameterized by $\mathbf { y } ^ { j }$ ) for $j$ -th task. Define the meta training dataset as $\mathcal { D } = \{ \mathcal { D } ^ { j } \}$ , where $\mathcal { D } ^ { j } = \mathcal { D } _ { \mathtt { t r } } ^ { j } \bigcup \mathcal { D } _ { \mathtt { v a l } } ^ { j }$ corresponds to the $j$ -th task.
+
+The cross-entropy loss function is widely used for UL and LL objectives. Referring to Section 3, IAPTT-GM covers the convergence results with non-convex LL model, thus allowing flexible design of the LL objective function. For instance, while non-convex regularization terms, e.g., $\ell _ { q }$ regularization with $0 < q < 1$ , have shown effectiveness to help the LL model converge and avoid over-fitting, existing methods can only guarantee the convergence when $q \geq 1$ , thus almost provide no support for non-convex objectives. We consider the LL subproblem with non-convex loss functions by adding $\ell _ { q }$ regularization 3. Then the UL and LL subproblems can be written as
+
+$$
+F \left( \mathbf { x } , \left\{ \mathbf { y } ^ { j } \right\} \right) = \sum _ { j } \ell \left( \mathbf { x } , \mathbf { y } ^ { j } ; \mathcal { D } _ { \mathrm { v a l } } ^ { j } \right) , \quad f \left( \mathbf { x } , \left\{ \mathbf { y } ^ { j } \right\} \right) = \sum _ { j } \ell \left( \mathbf { x } , \mathbf { y } ^ { j } ; \mathcal { D } _ { \mathrm { t r } } ^ { j } \right) + \Vert \mathbf { y } ^ { j } \Vert _ { q } .
+$$
+
+Table 2: Mean test accuracy of 5-way classification on tieredImageNet and miniImageNet, and the $\pm$ represents $9 5 \%$ confidence intervals over tasks.   
+
+<table><tr><td rowspan=2 colspan=1>Methods</td><td rowspan=2 colspan=1>Backbone</td><td rowspan=1 colspan=2>MiniImagenet</td><td rowspan=1 colspan=2>TieredImagenet</td></tr><tr><td rowspan=1 colspan=1>5-way 1-shot</td><td rowspan=1 colspan=1> 5-way 5-shot</td><td rowspan=1 colspan=1> 5-way 1-shot</td><td rowspan=1 colspan=1>t5-way 5-shot</td></tr><tr><td rowspan=1 colspan=1>Proto Net</td><td rowspan=1 colspan=1>ConvNet-4</td><td rowspan=1 colspan=1>49.42 ± 1.84</td><td rowspan=1 colspan=1>68.20± 0.66</td><td rowspan=1 colspan=1>53.31 ± 0.89</td><td rowspan=1 colspan=1>72.69±0.74</td></tr><tr><td rowspan=1 colspan=1>Relation Net</td><td rowspan=1 colspan=1>ConvNet-4</td><td rowspan=1 colspan=1>50.44 ± 0.82</td><td rowspan=1 colspan=1>65.32 ± 0.70</td><td rowspan=1 colspan=1>54.48 ± 0.93</td><td rowspan=1 colspan=1>65.32 ± 0.70</td></tr><tr><td rowspan=1 colspan=1>MAML</td><td rowspan=1 colspan=1>ConvNet-4</td><td rowspan=1 colspan=1>48.70±0.75</td><td rowspan=1 colspan=1>63.11 ± 0.11</td><td rowspan=1 colspan=1>49.06 ± 0.50</td><td rowspan=1 colspan=1>67.48 ± 0.47</td></tr><tr><td rowspan=1 colspan=1>RHG</td><td rowspan=1 colspan=1>ConvNet-4</td><td rowspan=1 colspan=1>48.89 ±0.81</td><td rowspan=1 colspan=1>63.02 ± 0.70</td><td rowspan=1 colspan=1>49.63 ± 0.67</td><td rowspan=1 colspan=1>66.14 ± 0.57</td></tr><tr><td rowspan=1 colspan=1>T-RHG</td><td rowspan=1 colspan=1>ConvNet-4</td><td rowspan=1 colspan=1>47.67± 0.82</td><td rowspan=1 colspan=1>63.70 ± 0.76</td><td rowspan=1 colspan=1>50.79 ± 0.69</td><td rowspan=1 colspan=1>67.39 ± 0.60</td></tr><tr><td rowspan=1 colspan=1>BDA</td><td rowspan=1 colspan=1>ConvNet-4</td><td rowspan=1 colspan=1>49.08±0.82</td><td rowspan=1 colspan=1>62.17 ± 0.70</td><td rowspan=1 colspan=1>51.56 ± 0.68</td><td rowspan=1 colspan=1>68.21 ±0.58</td></tr><tr><td rowspan=1 colspan=1>MAML</td><td rowspan=1 colspan=1>ResNet-12</td><td rowspan=1 colspan=1>51.03 ± 0.50</td><td rowspan=1 colspan=1>68.26 ± 0.47</td><td rowspan=1 colspan=1>58.58± 0.49</td><td rowspan=1 colspan=1>71.24 ± 0.43</td></tr><tr><td rowspan=1 colspan=1>RHG</td><td rowspan=1 colspan=1>ResNet-12</td><td rowspan=1 colspan=1>50.54±0.85</td><td rowspan=1 colspan=1>64.53 ± 0.68</td><td rowspan=1 colspan=1>58.19 ±0.76</td><td rowspan=1 colspan=1>75.20±0.60</td></tr><tr><td rowspan=1 colspan=1>IAPTT-GM</td><td rowspan=1 colspan=1>ResNet-12</td><td rowspan=1 colspan=1>56.69±0.66</td><td rowspan=1 colspan=1>70.21 ± 0.55</td><td rowspan=1 colspan=1>60.71±0.77</td><td rowspan=1 colspan=1>75.85 ± 0.59</td></tr></table>
+
+Detailed information about the datasets and network architectures can be found in the supplementary materials. We report results of IAPTT-GM and various mainstream methods, e.g., Prototypical Network [48], Relation Net [49] and T-RHG [16] on miniImageNet [47] and tieredImageNet [50] datasets with two different backbones [6, 51] in Table 2. As it is shown, our proposed method outperforms state-of-the-art methods on both 5-way 1-shot and 5-way 5-shot tasks.
+
+# 4.2.2 Data Hyper-Cleaning: Non-convex LL Architecture Structure
+
+Data hyper-cleaning [3] aims to cleanup the corrupted data with noise label. According to [16], the dataset is randomly split to three disjoint subsets: $\mathcal { D } _ { \mathtt { t r } }$ for training, $\mathcal { D } _ { \mathtt { v a l } }$ for validation and $\mathcal { D } _ { \mathrm { t e s t } }$ for testing, then a fixed proportion of the training samples in $\mathcal { D } _ { \mathtt { t r } }$ is randomly corrupted.
+
+Following the classical experimental protocol [3], we choose cross-entroy as the loss function $\ell$ , and the UL and LL subproblem take the form of
+
+$$
+F ( { \mathbf x } , { \mathbf y } ) = \sum _ { ( \mathbf u _ { i } , \mathbf v _ { i } ) \in \mathcal { D } _ { v a } } \ell \left( { \mathbf y } ( { \mathbf x } ) ; \mathbf u _ { i } , { \mathbf v } _ { i } \right) , f ( { \mathbf x } , { \mathbf y } ) = \sum _ { ( \mathbf u _ { i } , \mathbf v _ { i } ) \in \mathcal { D } _ { v a } } [ \sigma ( { \mathbf x } ) ] _ { i } \ell \left( { \mathbf y } ; { \mathbf u } _ { i } , { \mathbf v } _ { i } \right) ,
+$$
+
+where $\left( \mathbf { u } _ { i } , \mathbf { v } _ { i } \right)$ denotes the data pair and $\sigma ( \mathbf { x } )$ represents the element-wise sigmoid function on $\mathbf { x }$ . We define the hyperparameter $\mathbf { x }$ as a vector being trained to label the noisy data, of which the dimension equals to the number of training samples. The LL variables parameterized by $\mathbf { y }$ contain the weights and bias of fully connected layers.
+
+Note that existing methods consider convex a single fully connected layer as the LL model, while more complex neural network structure is not applicable. Under our assumption without LLC, we employ two fully connected layers as the LL network architecture.
+
+In Table 3, we compare IAPTT-GM with IG methods (e.g., LS, NS) and EG methods (e.g., RHG, T-RHG [16]). As it is shown, IAPTT-GM achieves better test performance of both accuracy and F1 score on two datasets, including MNIST [46] and FashionMNIST [52]. Our theoretical results also show that the performance improvement comes from PTT and IA techniques to overcome non-convex LL subproblems.
+
+Table 3: Reporting results of existing methods for solving data hyper-cleaning tasks. Acc. and F1 score denote the test accuracy and the harmonic mean of the precision and recall, respectively.   
+
+<table><tr><td rowspan=2 colspan=1>Method</td><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>FashionMNIST</td></tr><tr><td rowspan=1 colspan=1>Acc.F1 score</td><td rowspan=1 colspan=1>Acc.F1 score</td></tr><tr><td rowspan=1 colspan=1>LS</td><td rowspan=1 colspan=1>89.19 85.96</td><td rowspan=1 colspan=1>83.15  85.13</td></tr><tr><td rowspan=1 colspan=1>NS</td><td rowspan=1 colspan=1>87.54 89.58</td><td rowspan=1 colspan=1>81.37  87.28</td></tr><tr><td rowspan=1 colspan=1>RHG</td><td rowspan=1 colspan=1>87.90 89.36</td><td rowspan=1 colspan=1>81.91  87.12</td></tr><tr><td rowspan=1 colspan=1>T-RHG</td><td rowspan=1 colspan=1>88.57 89.77</td><td rowspan=1 colspan=1>81.85  86.76</td></tr><tr><td rowspan=1 colspan=1>BDA</td><td rowspan=1 colspan=1>87.15 90.38</td><td rowspan=1 colspan=1>79.97 88.24</td></tr><tr><td rowspan=1 colspan=1>IAPTT-GM</td><td rowspan=1 colspan=1>90.88 91.57</td><td rowspan=1 colspan=1>83.67 90.37</td></tr></table>
+
+# 4.3 Evaluations of IA-GM (A) for BLOs under LLC
+
+In addition to non-convex BLO problems, we also raise concerns about the acceleration strategy of our method with LLC condition. We first consider the following BLO with LLC condition [23]:
+
+$$
+\operatorname* { m i n } _ { \mathbf { x } \in \mathcal { X } } \| \mathbf { x } - \mathbf { y } _ { 2 } \| ^ { 4 } + \| \mathbf { y } _ { 1 } - \mathbf { e } \| ^ { 4 } , \quad s . t . \quad ( \mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } ) \in \arg \operatorname* { m i n } _ { \mathbf { y } _ { 1 } \in \mathbb { R } ^ { n } , \mathbf { y } _ { 2 } \in \mathbb { R } ^ { n } } \frac { 1 } { 2 } \| \mathbf { y } _ { 1 } \| ^ { 2 } - \mathbf { x } ^ { \top } \mathbf { y } _ { 1 } ,
+$$
+
+where $n = 5 0$ , $\mathcal { X } = [ - 1 0 0 , 1 0 0 ] \times \cdot \cdot \cdot [ - 1 0 0 , 1 0 0 ] \subset \mathbb { R } ^ { n }$ , and e represents the vector whose elements are all equal to 1. The optimal solution for this problem is $\mathbf { x } ^ { * } = \mathbf { e } , \mathbf { y } _ { 1 } ^ { * } = \mathbf { e } , \mathbf { y } _ { 2 } ^ { * } = \mathbf { e }$ .
+
+As shown in Section 3.2, our method IA-GM (A) incorporates Nesterov’s acceleration strategy for solving Eq. (2). Note that with LLC assumption, IA-GM maintains the convergence property.
+
+![](images/1c2daf47ed3e8975be7489fb933f9bbb8a0401e79da1c16050ce4df719b9a324.jpg)  
+Figure 3: The left two subfigures report the curves of $\| \mathbf { y } - \mathbf { y } ^ { * } \|$ and $\| \mathbf { x } - \mathbf { x } ^ { * } \|$ for IA-GM and IA-GM (A). The figures on the right illustrate the results of mean UL loss and accuracy on the convex data hyper-cleaning problems.
+
+As illustrated in the first two figures in Figure 3, IA-GM (A) shows significant improvement of convergence speed on UL and LL variables, which verifies the convergence results of Theorem 3.4 under LLC. We further study the convex data hyper-cleaning problem, which simply implements single fully connected layer as the network structure. From the right half of Figure 3, we can easily find that IA-GM (A) also performs better than IA-GM on real-world applications.
+
+# 5 Conclusion
+
+This paper presents a generic first-order algorithmic framework named IAPTT-GM to solve BLO problems with non-convex follower. We introduce two features, initialization auxiliary and pessimistic trajectory truncation operation to guarantee the convergence without the LLC hypothesis and achieves better performance on various applications. Meanwhile, we also validate the performance and speed improvement for BLO with LLC condition.
+
+# Acknowledgments and Disclosure of Funding
+
+This work is partially supported by the National Natural Science Foundation of China (Nos. 61922019, 11971220), the National Key R&D Program of China (2020YFB1313503), LiaoNing Revitalization Talents Program (XLYC1807088), the Shenzhen Science and Technology Program (No. RCYX20200714114700072), the Fundamental Research Funds for the Central Universities, the Pacific Institute for the Mathematical Sciences (PIMS), the Stable Support Plan Program of Shenzhen Natural Science Fund (No. 20200925152128002) and the Guangdong Basic and Applied Basic Research Foundation 2019A1515011152.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] Please refer to Section 1.1 for detailed description corresponding to the abstract.   
+(b) Did you describe the limitations of your work? [No]   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] Please refer to Section 3.   
+(b) Did you include complete proofs of all theoretical results? [Yes] We put most of the detailed calculation process in the supplementary materials.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Instruction for the code, datasets are provided in the supplemental material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Information about the dataset and hyperparameters for numerical experiments can be found in the supplemental material.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We specify the detailed configuration of computing devices, GPUs in the supplementary materials.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] We conduct all the experiments based on public data.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/b83ibRX55T/b83ibRX55T_content_list.json

@@ -0,0 +1,1582 @@
+[
+    {
+        "type": "text",
+        "text": "Towards Gradient-based Bilevel Optimization with Non-convex Followers and Beyond ",
+        "text_level": 1,
+        "bbox": [
+            191,
+            122,
+            808,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Risheng Liu1,2 Yaohua Liu1 Shangzhi Zeng3 Jin Zhang ∗4,5 ",
+        "bbox": [
+            261,
+            224,
+            730,
+            241
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1International School of Information Science & Engineering, DUT 2Pazhou Lab, Guangzhou 3Department of Mathematics and Statistics, UVic   \n4Department of Mathematics, SUSTech 5National Center for Applied Mathematics Shenzhen rsliu@dlut.edu.cn liuyaohua_918@163.com zengshangzhi@gmail.com zhangj9@sustech.edu.cn ",
+        "bbox": [
+            186,
+            241,
+            813,
+            311
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            361,
+            535,
+            376
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In recent years, Bi-Level Optimization (BLO) techniques have received extensive attentions from both learning and vision communities. A variety of BLO models in complex and practical tasks are of non-convex follower structure in nature (a.k.a., without Lower-Level Convexity, LLC for short). However, this challenging class of BLOs is lack of developments on both efficient solution strategies and solid theoretical guarantees. In this work, we propose a new algorithmic framework, named Initialization Auxiliary and Pessimistic Trajectory Truncated Gradient Method (IAPTT-GM), to partially address the above issues. In particular, by introducing an auxiliary as initialization to guide the optimization dynamics and designing a pessimistic trajectory truncation operation, we construct a reliable approximate version of the original BLO in the absence of LLC hypothesis. Our theoretical investigations establish the convergence of solutions returned by IAPTT-GM towards those of the original BLO without LLC. As an additional bonus, we also theoretically justify the quality of our IAPTT-GM embedded with Nesterov’s accelerated dynamics under LLC. The experimental results confirm both the convergence of our algorithm without LLC, and the theoretical findings under LLC. ",
+        "bbox": [
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+            391,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            637,
+            310,
+            655
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Bi-Level Optimization (BLO) has been widely used to formulate problems in the field of deep learning [1, 2], especially for hyperparameter optimization [3, 4, 5], meta learning [6, 7, 8, 9], neural architecture search [10, 11, 12], adversarial learning [13], and reinforcement learning [14], etc. BLO aims to tackle nested optimization structures appearing in applications, which has emerged as a prevailing optimization technique for modern machine learning tasks with underlying hierarchy. In the last decade, a large number of BLO methods have been proposed to address different machine learning tasks. In fact, Gradient Methods (GMs), which can effectively handle BLO problems of large scale, thus gain popularity. According to different types of strategies for gradient calculations, existing GMs can be divided into two categories, i.e., the explicit approaches which aim to replace the Lower-Level (LL) problem with dynamic iterations and implicit schemes that apply the implicit function theorem to formulate the first-order optimality condition of the LL problem. ",
+        "bbox": [
+            174,
+            669,
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+            821
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Explicit Gradient Methods for BLOs. In this type, solving the LL problem is regarded as the evolution path of the dynamic system starting from a given initial point of the LL variable. The gradient of the Upper-Level (UL) variable can be directly calculated by automatic differentiation based on the trajectory of LL variable. This class of methods can be further divided into three types, namely, recurrence-based EG (e.g., [3, 15, 6, 16, 10]), initialization-based EG (e.g., [17, 18]) and proxy-based EG methods (e.g., [19, 20, 21, 22]), differing from each other in the way of accessing the gradient of constructed dynamic trajectory. While most of this type of works assume the LLC and Lower-Level Singleton (LLS) to simplify their optimization process and theoretical analysis, cases where the LLS assumption does not hold have been tackled in the recent work [23, 24]. In particular, to eliminate the LLS assumption which is too restrctive to be satisfied in real-world complex tasks, [23] first considers incorporating UL objective information into the dynamic iterations, but the more general cases where LLC does not hold remain unsolved. On the other hand, while most of the mentioned works focus on the asymptotic convergence, the progress on nonasymptotic convergence analysis has been recently witnessed see, e.g., [25, 26, 27]. ",
+        "bbox": [
+            174,
+            835,
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+            878
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Implicit Gradient Methods for BLOs. This type, also known as implicit differentiation [28, 7, 29], replaces the LL problem with its first-order optimality condition and uses the implicit function theorem to calculate the gradient of the UL problem by solving a linear system. This method decouples the calculation of UL gradient from the dynamic system of LL, resulting in a significant speed increase when the dynamic system iterates many times. However, because of the burden originated from computing a Hessian matrix and its inverse, IG methods are usually computationally expensive when linear systems are ill-conditioned. To alleviate this computational issue, there are mainly two kinds of techniques, i.e., IG based on Linear System (LS) [28, 7] and Neumann Series (NS) [29]. On the theoretical side, IG methods rely on the strong convexity of LL problems heavily, which is even more restrictive than the LLC and LLS together. ",
+        "bbox": [
+            174,
+            257,
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+            397
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Initialization Optimization for Learning. In deep learning, the selection of initialization scheme has a great influence on the training speed and performance [30]. As the most representative work in recent years, Model-Agnostic Meta-Learning (MAML) [31] applies the same initialization to all tasks, and is optimized by a loss function common to the task that evaluates the effect of the initial value, resulting in an initialization that achieves good generalization performance with only a few gradient steps on new tasks. Due to its simple form this method has been widely studied and applied [32, 33, 34, 35]. [36] noticed that not all network parameters are suitable for the same initialization, and therefore proposed a strategy to apply co-initialization only on a part of parameters. In theory, [37, 38, 39] give comprehensive study on the convergence and convergence rate of MAML and some MAML-type approaches based on the meta objective function. However, the convergence theory of these existing results are given based on the loss function for evaluating initial values, and the convergence analysis of such type of methods from the perspective of each task is still lacked. ",
+        "bbox": [
+            174,
+            411,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Value-Function Approach. The value function based methods have also emerged as a promising branch to solve BLO problems [40]. Under the special case where the LL is jointly convex with respect to both the UL and LL variables, the BLO problem can be equivalently reformulated into a difference-of-convex program [41], which is numerically solvable. Typically, by reformulating the BLO into an Inner Single Bi-level (ISB) optimization problem with value-function approach, a gradient-based interior-point method name BVFIM [42] is proposed to solve the BLO tasks, which effectively avoids the expensive Hessian-vector and Jacobian-vector products. Generally speaking, the value-function does not admit an explicit form, and is always nonsmooth, non-convex and with jumps. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1.1 Our Motivations and Contributions ",
+        "text_level": 1,
+        "bbox": [
+            178,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As mentioned above, some theoretical progresses have long been witnessed in diversified learning areas, but for most existing BLO methods, extra restrictive assumptions (e.g., LLS, LLC and LL strong convexity) have to be enforced. Their algorithm design and associated theoretical analysis actually are only valid for optimization with a simplified problem structure. Unfortunately, it has been well recognized that LL non-convexity frequently appears in a variety of applications, e.g., sparse $\\ell _ { q }$ regularization ( $0 < q < 1$ ) for avoiding over-fitting, and learning parameters of coupled multi-layer neural networks, etc. Therefore, in challenging real-world scenarios, we are usually required to consider BLO problems where these assumptions (e.g., LLS, LLC and even LL strong convexity) are naturally violated. These fundamental theoretical issues motivate us to propose a series of new techniques to address BLO with non-convex LL problems, which have been frequently appeared in various learning applications. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In particular, by introducing an Initialization Auxiliary (IA) to the LL optimization dynamics and operating a Pessimistic Trajectory Truncation (PTT) strategy during the UL approximation, we construct a Gradient-based Method (GM), named IAPTT-GM, to address BLO in challenging optimization scenarios (i.e., with non-convex follower tasks). We analyze the convergence behaviors of IAPTT-GM on BLOs without LLC and also investigate theoretical properties of the accelerated version of our algorithm on BLOs under LLC. Extensive experiments verify our theoretical results and demonstrate the effectiveness of IAPTT-GM on different learning applications. The main contributions of our IAPTT-GM are summarized as follows: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• We propose IA and PTT, two new mechanisms to efficiently handle complex BLOs where the follower is facing with a non-convex task (i.e., without LLS and even LLC). IA actually paves the way for jointly optimizing both the UL variables and the dynamical initialization, while PTT adaptively reduces the complexity of backward recurrent propagation. To our best knowledge, we establish the first strict convergence guarantee for gradientbased method on BLOs with non-convex follower tasks. We also justify the quality of our IAPTT-GM embedded with Nesterov’s accelerated dynamics under LLC. • We conduct a series of experiments to verify our theoretical findings and evaluate IAPTT-GM on various challenging BLOs, in which the follower tasks are either with non-convex loss functions (e.g., few-shot learning) or coupled network structures (e.g., data hyper-cleaning). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 The Proposed Algorithmic Framework ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this work, we consider the BLO problem in the form: ",
+        "bbox": [
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+            415,
+            540,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/861631915756c6638c2121ee236aaf24517b8d1215ebcb46c616b87bbc65a523.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { x } \\in \\mathcal { X } , \\mathbf { y } } F ( \\mathbf { x } , \\mathbf { y } ) , \\quad s . t . \\quad \\mathbf { y } \\in S ( \\mathbf { x } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            438,
+            614,
+            463
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { x } \\in \\mathbb { R } ^ { n } , \\mathbf { y } \\in \\mathbb { R } ^ { m }$ are UL and LL variables respectively, and $\\boldsymbol { S } ( \\mathbf { x } )$ denotes the set of solutions of the LL problem, i.e., ",
+        "bbox": [
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+            469,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a0753305ab31a68de928f1b612e34c8428132bf195c4f030176d6f884325196e.jpg",
+        "text": "$$\nS ( \\mathbf x ) : = \\arg \\operatorname* { m i n } _ { \\mathbf y \\in \\mathcal y } f ( \\mathbf x , \\mathbf y ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            497,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $f$ is differentiable w.r.t. y. To ensure the BLO model in Eq. (1) is well-defined, we assume that $\\scriptstyle { \\mathcal { S } } ( \\mathbf { x } )$ is nonempty for all $\\mathbf { x } \\in \\mathcal { X }$ . Observe further that the above BLO is structurally different from those in existing literature in the sense that no convexity assumption is required in the LL problem. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the following, we describe the proposed IAPTT-GM to solve the class of BLOs defined in Eq. (1). The mechanism of a classical dynamics-embedded gradient method, approximates the LL solution via a dynamical system drawn from optimization iterations. Choosing gradient descent as the optimization dynamics for example, the approximation ${ \\bf y } _ { K } ( { \\bf x } )$ is accessed by operations repeatedly performed by $K - 1$ steps parameterized by UL variable $\\mathbf { x }$ ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/de21d2710352a92344cb37d1a8b764bd7decaeac4db96f364cc1ccb3b975499f.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { y } _ { k + 1 } ( \\mathbf { x } ) = \\mathbf { y } _ { k } ( \\mathbf { x } ) - s \\nabla _ { \\mathbf { y } } f ( \\mathbf { x } , \\mathbf { y } _ { k } ( \\mathbf { x } ) ) , k = 0 , \\cdots , K - 1 , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            651,
+            692,
+            670
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $s$ is a step size, and $\\mathbf { y } _ { 0 }$ is a fixed initial value. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 Initialization Auxiliary ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            707,
+            372,
+            723
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Embedding the dynamical iterations into the UL problem returns the approximate version $F ( \\mathbf { x } , \\mathbf { y } _ { K } ( \\mathbf { x } ) )$ . As long as $\\nabla _ { \\mathbf { y } } f ( \\mathbf { x } , \\mathbf { y } _ { K } ( \\mathbf { x } ) )$ uniformly converges to zero w.r.t. UL variable $\\mathbf { x }$ varying in $\\mathcal { X }$ , we call this a good approximation. To this end, usually restrictive LL strong convexity assumptions are imposed, thus the desired convergence of solutions of approximation problems towards those of the original BLO follows. By drawing inspiration from the classic dynamics-embedded gradient method which replaces the LL problem with certain optimization dynamics, hence resulting in an approximation of the bi-level problem, we propose a new gradient scheme to solve BLO in Eq. (1) without LLC restriction. Specifically, we let $K$ be a prescribed positive integer and construct the following approximation ${ \\bf y } _ { K } ( { \\bf x } , { \\bf z } )$ of the LL solution drawn from projected gradient descent iterations ",
+        "bbox": [
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+            732,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4285456e369bf45215c7ac3d9a0499049613eeb6e4642d3dcf098db124f7f045.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbf { y } _ { 0 } ( \\mathbf { x } , \\mathbf { z } ) = \\mathbf { z } , } \\\\ & { \\mathbf { y } _ { k + 1 } ( \\mathbf { x } , \\mathbf { z } ) = \\operatorname* { P r o j } _ { \\mathcal { V } } ( \\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) - \\alpha _ { \\mathbf { y } } ^ { k } \\nabla _ { \\mathbf { y } } f ( \\mathbf { x } , \\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) ) ) , k = 0 , \\cdots , K - 1 , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            876,
+            751,
+            916
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\{ \\alpha _ { \\mathbf { y } } ^ { k } \\}$ is a sequence of steps sizes. We next embed the dynamical iterations ${ \\bf y } _ { k } ( { \\bf x } , { \\bf z } )$ into $\\mathrm { m a x } _ { 1 \\le k \\le K } \\{ F ( { \\bf x } , { \\bf y } _ { k } ( { \\bf x } , { \\bf z } ) ) \\}$ , which can be regarded as a pessimistic trajectory truncation of the UL objective. The mechanism of the above scheme, in comparison, accesses the approximation parameterized by UL variable $\\mathbf { x }$ and LL initial point $\\mathbf { y } _ { 0 }$ . Our motivation for the initialization auxiliary variable $\\mathbf { z }$ comes from convergence theory [42] of non-convex first-order optimization methods. In fact, when the non-convex LL problem admits multiple solutions, the gradient descent steps with a “bad” initial point $\\mathbf { y } _ { 0 }$ cannot return a desired point in the LL solution set, simultaneously optimizing the UL objective. To overcome such a difficulty, instead of using a fixed initial value, we introduce an initialization auxiliary variable $\\mathbf { z }$ . Therefore, when it comes to solving the UL approximation problems, together with the UL variable $\\mathbf { x }$ , the auxiliary variable $\\mathbf { z }$ is also updated and hence optimized. As a consequence, we may search for the “best” initial value, starting from which the gradient descent steps approach a solution to the BLO in Eq. (1), i.e., a point in the LL solution set, simultaneously minimizing the UL objective. ",
+        "bbox": [
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+            90,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 Pessimistic Trajectory Truncation ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            296,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This is a striking feature of our algorithm that significantly differs from existing methods and leads to some new convergence results without LLC. The motivation for the design of pessimistic trajectory truncation comes again from convergence theory of non-convex firstorder optimization methods. It is understood that when LL is non-convex, $\\mathcal { R } _ { \\alpha } ( { \\bf x } , { \\bf y } _ { K } ( { \\bf x } , { \\bf z } ) )$ may not uniformly converge w.r.t. $\\mathbf { x }$ and $\\mathbf { z }$ , where $\\mathcal { R } _ { \\alpha } ( \\mathbf { x } , \\mathbf { y } )$ is the proximal gradient residual mapping defined as $\\mathcal { R } _ { \\alpha } ( \\mathbf { x } , \\mathbf { y } ) : = \\textbf { y } -$ $\\mathrm { P r o j } _ { \\mathcal { V } } \\left( \\mathbf { \\bar { y } } - \\alpha \\nabla _ { \\mathbf { y } } f ( \\mathbf { x } , \\mathbf { y } ) \\right) ^ { 1 }$ , which can be used as a measurement of the optimality of LL problem in Eq. (2). Thus a direct embedding of ${ \\bf y } _ { K } ( { \\bf x } , { \\bf z } )$ into UL objective $F ( \\mathbf { x } , \\mathbf { y } )$ may not necessarily provide an appropriate approximation. Fortunately, it is also understood that, for each $\\mathbf { x } \\in \\mathcal { X }$ , $\\mathbf { z } \\in \\mathcal { V }$ and $K > 0$ , there exists at ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 The Proposed IAPTT-GM ",
+        "text_level": 1,
+        "bbox": [
+            498,
+            335,
+            756,
+            351
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1: Initialize $\\mathbf { x } ^ { 0 }$ and $\\mathbf { z } ^ { 0 }$ .   \n2: for $t = 0  T - 1$ do   \n3: $\\mathbf { y } _ { 0 } = \\mathbf { z } ^ { t }$ .   \n4: for $k = 0  K - 1$ do   \n5: $\\%$ LL Updating with $\\mathbf { x } ^ { t }$ and $\\mathbf { z } ^ { t }$   \n6: $\\mathbf { y } _ { k + 1 } = \\mathbf { \\widetilde { P } r o j } _ { \\mathcal { V } } ^ { - } ( \\mathbf { y } _ { k } - \\alpha _ { \\mathbf { y } } ^ { k } \\nabla _ { \\mathbf { y } } f ( \\mathbf { x } ^ { t } , \\mathbf { y } _ { k } ) )$   \n7: end for   \n8: 9: $\\%$ $\\bar { k } = \\arg \\operatorname* { m a x } _ { k } \\{ \\bar { F ( \\mathbf { x } , \\mathbf { y } _ { k } ) } \\} _ { k = 1 } ^ { K } .$ cation.   \n10: $\\%$ UL Updating with ${ \\bf y } _ { \\bar { k } } ( { \\bf x } , { \\bf z } )$   \n11: $\\mathbf { x } ^ { t + 1 } = \\operatorname { P r o j } _ { \\mathcal { X } } ( \\mathbf { x } ^ { t } - \\alpha _ { \\mathbf { x } } \\nabla _ { \\mathbf { x } } F ( \\mathbf { x } ^ { t } , \\mathbf { y } _ { \\bar { k } } ) )$ .   \n12: $\\%$ Initialization Updating with ${ \\bf y } _ { \\bar { k } } ( { \\bf x } , { \\bf z } )$   \n13: $\\mathbf { z } ^ { t + 1 } = \\operatorname { P r o j } _ { \\mathcal { V } } ( \\mathbf { z } ^ { t ^ { \\top } } - \\alpha _ { \\mathbf { z } } \\bar { \\nabla } _ { \\mathbf { z } } F ( \\mathbf { x } ^ { t } , \\mathbf { y } _ { \\bar { k } } ) )$ .   \n14: end for ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "least a $\\tilde { K }$ such that along the selection ${ \\mathbf { y } } _ { \\tilde { K } } ( { \\mathbf { x } } , { \\mathbf { z } } )$ , $\\mathcal { R } _ { \\alpha } ( \\mathbf { x } , \\mathbf { y } _ { \\tilde { K } } ( \\mathbf { x } , \\mathbf { z } ) )$ uniformly converges to zero w.r.t.   \n$\\mathbf { x }$ and $\\mathbf { z }$ , as $K$ tending infinity. ",
+        "bbox": [
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+            574,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "However, in general, it is too ambitious to expect an explicit identification of the exact selection ${ \\bf y } _ { \\tilde { K } } ( { \\bf x } , { \\bf z } )$ . Alternatively, we consider a pessimistic strategy, minimizing the worst case of all selections of $\\{ \\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) \\}$ , i.e., $\\begin{array} { r } { \\operatorname* { m a x } _ { 1 \\leq k \\leq K } \\left\\{ F ( \\mathbf { x } , \\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) ) \\right\\} } \\end{array}$ . By doing so, we successfully reach a good approximation. In addition to the theoretical convergence, we also benefit from this pessimistic strategy in a numerical sense. The pessimistic max operation always results in a favorable trajectory truncation smaller than $K$ . Consequently, this technique offers inexpensive computational cost for computing the hyper-gradient through back propagation, as shown in the numerical experiments. ",
+        "bbox": [
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+            608,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To conclude this section, we state the complete IAPTT-GM in Algorithm 1. Note that $K$ and $T$ represent the numbers of inner and outer iterations, respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 Theoretical Investigations ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            767,
+            419,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "With the purpose of studying the convergence of dynamics-embedded gradient method for BLO without LLC, we involve two signature features in our algorithmic design, i.e., initialization auxiliary and pessimistic trajectory truncated. This section is devoted to the convergence analysis of our proposed algorithm with and without LLC assumption. Please notice that all the proofs of our theoretical results are stated in the Supplemental Material. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 Convergence Analysis of IAPTT-GM for BLO with Non-convex Followers ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this part, we conduct the convergence analysis of the IAPTT-GM for solving BLO in Eq. (1) without LLC. Before presenting our main convergence results, we introduce some notations related to BLO. With introduced function $\\varphi ( \\mathbf x ) : = \\operatorname* { i n f } _ { \\mathbf y \\in { \\cal S } ( \\mathbf x ) } { \\cal F } ( \\mathbf x , \\mathbf y )$ , the BLO in Eq. (1) can be rewritten as ",
+        "bbox": [
+            173,
+            116,
+            825,
+            160
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/024e28096679f21f6e68b4594aee124ab76a39523e4ce37b3bc37e670f0bb011.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { x } \\in \\mathcal { X } } \\varphi ( \\mathbf { x } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            165,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "With given $K \\geq 1$ and defining $\\varphi _ { K } ( \\mathbf x , \\mathbf z ) : = \\mathrm { m a x } _ { k } \\left\\{ F ( \\mathbf x , \\mathbf y _ { k } ( \\mathbf x , \\mathbf z ) ) \\right\\}$ with $\\{ \\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) \\}$ defined in Eq. (4), our proposed IAPTT-GM generates sequence $\\{ ( \\mathbf { x } ^ { t } , \\mathbf { z } ^ { t } ) \\}$ for solving following approximation problem to BLO in Eq. (5), ",
+        "bbox": [
+            173,
+            194,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/798a9a50db87e6130d59274326dbf2882e2e6b8a618f1dca7ac179d80c473744.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { { \\mathbf { x } } \\in { \\mathcal { X } } , { \\mathbf { z } } \\in { \\mathcal { Y } } } { \\varphi _ { K } ( \\mathbf { x } , \\mathbf { z } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            434,
+            236,
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+            260
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This section is mainly devoted to the convergence of solutions of approximation problems in Eq. (6) towards those of the original BLO in Eq. (5). ",
+        "bbox": [
+            176,
+            262,
+            826,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3.1 We make following standing assumptions throughout this section. ",
+        "bbox": [
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+            301,
+            717,
+            318
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "(1) $F , f : \\mathbb { R } ^ { n } \\times \\mathbb { R } ^ { m }  \\mathbb { R }$ are continuous functions.   \n(2) $\\nabla f$ is continuous and $\\nabla _ { \\mathbf y } f$ is $L _ { f }$ Lipschitz continuous with respect to y for any $\\mathbf { x } \\in \\mathcal { X }$ .   \n(3) $\\mathcal { X }$ and $\\mathcal { V }$ are convex compact sets.   \n(4) $\\boldsymbol { S } ( \\mathbf { x } )$ is nonempty for any $\\mathbf { x } \\in \\mathcal { X }$ .   \n(5) For any $( { \\bar { \\mathbf { x } } } , { \\bar { \\mathbf { y } } } )$ minimizing $F ( \\mathbf { x } , \\mathbf { y } )$ over constraints $\\mathbf { x } \\in \\mathcal { X } , \\mathbf { y } \\in \\mathcal { Y }$ and $\\mathbf { y } \\in \\hat { S } ( \\mathbf { x } )$ , it holds that $\\bar { \\mathbf { y } } \\in S ( \\mathbf { x } )$ . ",
+        "bbox": [
+            200,
+            325,
+            823,
+            452
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that $\\hat { S } ( \\mathbf { x } )$ denotes the set of LL stationary points, i.e., $\\hat { \\mathcal { S } } ( \\mathbf { x } ) = \\{ \\mathbf { y } \\in \\mathcal { Y } | 0 = \\nabla \\mathbf { y } f ( \\mathbf { x } , \\mathbf { y } ) +$ $\\mathcal { N } _ { \\mathcal { Y } } ( \\mathbf { y } ) \\}$ . It should be noticed that $\\mathbf { y } \\in \\hat { S } ( \\mathbf { x } )$ if and only if $\\begin{array} { r } { \\mathcal { R } _ { \\alpha } ( \\mathbf { x } , \\mathbf { y } ) = 0 } \\end{array}$ . Assumption 3.1 is standard in bi-level optimziation related literature, which will be shown to be satisfied for the numerical example given in Section 4.1. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As discussed in the preceding section, for each $\\mathbf { x } \\in \\mathcal { X }$ , $\\mathbf { z } \\in \\mathcal { V }$ and $K > 0$ , there exists at least a $\\tilde { K }$ such that along the selection ${ \\mathbf { y } } _ { \\tilde { K } } ( { \\mathbf { x } } , { \\mathbf { z } } )$ , $\\mathcal { R } _ { \\alpha } ( \\mathbf { x } , \\mathbf { y } _ { \\tilde { K } } ( \\mathbf { x } , \\mathbf { z } ) )$ uniformly converges to zero w.r.t. $\\mathbf { x }$ and $\\mathbf { z }$ , as $K$ tending infinity. We next specifically show the existence of such index $\\tilde { K }$ . To this end, $\\tilde { K }$ can be chosen by optimizing $\\| \\mathcal { R } _ { \\alpha } ( \\mathbf { x } , \\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) ) \\|$ among the indices $k = 0 , 1 , . . . , K$ . In particular, as stated in the following lemma, $\\| \\mathcal { R } _ { \\alpha } ( \\mathbf { x } , \\mathbf { y } _ { \\tilde { K } } ( \\mathbf { x } , \\mathbf { z } ) ) \\|$ uniformly decreases with a $\\textstyle { \\frac { 1 } { \\sqrt { K } } }$ rate on $\\mathcal { X } \\times \\mathcal { V }$ as $K$ increases. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.1 Let $\\{ \\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) \\}$ be the sequence defined in Eq. (4) with $\\begin{array} { r } { \\alpha _ { \\mathbf { y } } ^ { k } \\in [ \\underline { { \\alpha } } _ { \\mathbf { y } } , \\overline { { \\alpha } } _ { \\mathbf { y } } ] \\subset ( 0 , \\frac { 2 } { L _ { f } } ) } \\end{array}$ , there exists $C _ { f } > 0$ such that ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e9f145d865985712112ae917a263a769824698c49c66ad68baaaefb8c3be6164.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { 0 \\leq k \\leq K } \\| \\mathcal { R } _ { \\underline { { \\alpha } } _ { \\mathbf { y } } } ( \\mathbf { x } , \\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) ) \\| \\leq \\frac { C _ { f } } { \\sqrt { K + 1 } } , \\quad \\forall \\mathbf { x } \\in \\boldsymbol { \\mathcal { X } } , \\mathbf { z } \\in \\mathcal { Y } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            670,
+            699,
+            704
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As shown in the Appendix, the proof of Lemma 3.1 for the existence cannot offer us an explicit identification of the exact selection of $\\tilde { K }$ . Alternatively, we construct the approximation by a pessimistic trajectory truncation strategy, minimizing the worst case of all selections of $\\{ \\mathbf { y } _ { k } ( \\mathbf { x } , \\bar { \\mathbf { z } } ) \\}$ , i.e., $\\varphi _ { K } ( \\mathbf { x } , \\mathbf { z } )$ . By further solving the approximated problems $\\operatorname* { m i n } { \\varphi _ { K } ( \\mathbf { x } , \\mathbf { z } ) }$ , we shall provide a lower bound estimation for the optimal value of the BLO problem in Eq. (5). ",
+        "bbox": [
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+            714,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.2 Let $( { \\bf x } _ { K } , { \\bf z } _ { K } ) \\in \\mathrm { ~ a r g m i n ~ } \\varphi _ { K } ( { \\bf x } , { \\bf z } )$ , then x∈X ,z∈Y ",
+        "bbox": [
+            173,
+            797,
+            526,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b0e91fa7c0133c4ce283558d7132240e1ee65583af121e41282d510ff14321af.jpg",
+        "text": "$$\n\\varphi _ { K } ( \\mathbf x _ { K } , \\mathbf z _ { K } ) \\leq \\operatorname* { i n f } _ { \\mathbf y \\in \\hat { \\mathcal S } ( \\mathbf x ) } F ( \\mathbf x , \\mathbf y ) , \\quad \\forall \\mathbf x \\in \\mathcal X .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            829,
+            645,
+            857
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Upon together with the uniform convergence result in Lemma 3.1, the gap between the lower bound provided by $\\operatorname* { m i n } { \\varphi _ { K } ( \\mathbf { x } , \\mathbf { z } ) }$ and the true optimal value of the BLO problem in Eq. (5) eventually vanishes. To fill in this gap and present the main convergence result of our proposed IAPTT-GM, ",
+        "bbox": [
+            173,
+            868,
+            825,
+            912
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "we need the continuity of $\\mathcal { R } _ { \\alpha } ( \\mathbf { x } , \\mathbf { y } )$ . Indeed, it follows from [43, Theorem 6.42] that $\\mathsf { P r o j } _ { \\mathcal { Y } }$ is continuous. Combined with the assumed continuity of $\\nabla _ { \\mathbf y } { f } ( \\mathbf x , \\mathbf y )$ , we get the desired continuity of $\\mathcal { R } _ { \\alpha } ( \\mathbf { x } , \\mathbf { y } )$ immediately. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.1 Let $\\{ \\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) \\}$ be the sequence generated by Eq. (4) with $\\begin{array} { r } { \\boldsymbol { \\alpha } _ { \\mathbf { y } } ^ { k } \\in [ \\underline { { \\boldsymbol { \\alpha } } } _ { \\mathbf { y } } , \\overline { { \\boldsymbol { \\alpha } } } _ { \\mathbf { y } } ] \\subset ( 0 , \\frac { 2 } { L _ { f } } ) , } \\end{array}$ ,   \nand $( { \\bf x } _ { K } , { \\bf z } _ { K } ) \\in \\underset { -- } { \\operatorname { a r g m i n } } \\varphi _ { K } ( { \\bf x } , { \\bf z } )$ , then we have: $\\mathbf { x } { \\in } \\mathcal { X } , \\mathbf { z } { \\in } \\mathcal { Y }$ ",
+        "bbox": [
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+            143,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(1) Any limit point $\\bar { \\bf x }$ of the sequence $\\left\\{ { \\bf x } _ { K } \\right\\}$ is the solution to BLO in Eq. (1), that is $\\bar { \\textbf { x } } \\in$ $\\underset { \\mathbf { x } \\in \\mathcal { X } } { \\mathrm { a r g m i n } } \\varphi ( \\mathbf { x } )$ . ",
+        "bbox": [
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+            196,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9216bd2bdf840e52116eb551c31c15ca0b1b7eca1c431f93e0cea7dfdf87f4a6.jpg",
+        "text": "$$\n\\operatorname* { i n f } _ { \\mathbf { x } \\in \\mathcal { X } , \\mathbf { z } \\in \\mathcal { V } } \\varphi _ { K } ( \\mathbf { x } , \\mathbf { z } ) \\to \\operatorname* { i n f } _ { \\mathbf { x } \\in \\mathcal { X } } \\varphi ( \\mathbf { x } ) a s K \\to \\infty .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            242,
+            522,
+            266
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In the above theorem, we justify the global solutions convergence of the approximated problems. We next derive a convergence characterization regarding the local minimums of the approximated problems. In particular, the next theorem shows that any limit point of the local minimums of approximated problems is in some sense a local minimum of the bilevel problem in Eq. (1). ",
+        "bbox": [
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+            275,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2 Let $\\{ \\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) \\}$ be the sequence generated by Eq. (4) with $\\begin{array} { r } { \\boldsymbol { \\alpha } _ { \\mathbf { y } } ^ { k } \\in [ \\underline { { \\boldsymbol { \\alpha } } } _ { \\mathbf { y } } , \\overline { { \\boldsymbol { \\alpha } } } _ { \\mathbf { y } } ] \\subset ( 0 , \\frac { 2 } { L _ { f } } ) , } \\end{array}$ , and $\\left( { { \\bf { x } } _ { K } } , { { \\bf { z } } _ { K } } \\right)$ be a local minimum of $\\varphi _ { K } ( \\mathbf { x } , \\mathbf { z } )$ with uniform neighborhood modulus $\\delta > 0$ , i.e., ",
+        "bbox": [
+            171,
+            340,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/925ff1838349de33f47ef4d0e3959ffab49fe8fdbbbc72fa695352720f571d9a.jpg",
+        "text": "$$\n\\varphi _ { K } ( \\mathbf { x } _ { K } , \\mathbf { z } _ { K } ) \\leq \\varphi _ { K } ( \\mathbf { x } , \\mathbf { z } ) , \\quad \\forall ( \\mathbf { x } , \\mathbf { z } ) \\in \\mathbb { B } _ { \\delta } ( \\mathbf { x } _ { K } , \\mathbf { z } _ { K } ) \\cap \\mathcal { X } \\times \\mathcal { Y } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            378,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Then we have that for any limit point $( \\bar { \\bf x } , \\bar { \\bf z } )$ of the sequence $\\{ ( { \\bf x } _ { K } , { \\bf z } _ { K } ) \\}$ , there exists a limit point $\\bar { \\mathbf { y } }$ of the sequence $\\left\\{ { \\bf y } _ { K } ( { \\bf x } _ { K } , { \\bf z } _ { K } ) \\right\\}$ such that $\\bar { \\mathbf { y } } \\in \\hat { S } ( \\bar { \\mathbf { x } } )$ and $( { \\bar { \\mathbf { x } } } , { \\bar { \\mathbf { y } } } )$ satisfies that there exists $\\tilde { \\delta } > 0$ such that ",
+        "bbox": [
+            176,
+            400,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7893cba9d199b14892a6421f80b7ac9f159186bd046c7c9d31fd34dd7b5adf53.jpg",
+        "text": "$$\n\\begin{array} { r } { F ( \\bar { \\mathbf { x } } , \\bar { \\mathbf { y } } ) \\leq F ( \\mathbf { x } , \\mathbf { z } ) , \\quad \\forall ( \\mathbf { x } , \\mathbf { z } ) \\in \\mathbb { B } _ { \\widetilde { \\delta } } ( \\bar { \\mathbf { x } } , \\bar { \\mathbf { z } } ) \\cap \\{ \\mathbf { x } \\in \\mathcal { X } , \\mathbf { z } \\in \\mathcal { V } \\mid \\mathbf { z } \\in \\hat { S } ( \\mathbf { x } ) \\} . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            258,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 Theoretical Findings of IA-GM (A) for BLO with LLC ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            473,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "A byproduct of our study, which has its own interest, is that thanks to the involved initialization auxiliary, our theory can improve those existing results for classical gradient methods with accelerated gradient descent dynamical iterations under LLC. Nesterov’s acceleration technique [44] has been used widely for solving convex optimization problem and it greatly improves the convergence rate of gradient descent. To illustrate our result, we will take the Nesterov’s acceleration proximal gradient method [45] as the embedded optimization dynamics in classical gradient method for example. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Thanks to the LLC setting, $\\mathcal { R } _ { \\alpha } ( { \\bf x } , { \\bf y } _ { K } ( { \\bf x } , { \\bf z } ) )$ may uniformly converge to zero w.r.t. UL variable $x$ and auxiliary variable $\\mathbf { z }$ , thus the pessimistic trajectory truncation operation can be removed. Subsequently, we slightly simplify our algorithm that $\\bar { k }$ is simply taken as $K$ , thus the approximation objective admits a succinct form, i.e., $F ( \\mathbf { x } , \\mathbf { y } _ { K } ( \\mathbf { x } , \\mathbf { z } ) )$ . ",
+        "bbox": [
+            174,
+            588,
+            825,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In summary, by constructing ${ \\bf y } _ { k } ( { \\bf x } , { \\bf z } )$ through following Nesterov’s acceleration dynamics, ",
+        "bbox": [
+            171,
+            650,
+            771,
+            666
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cd68fd55bedf56b5bd88fe1adebddf6653d9cd75bc00568340ba8d0ae710a413.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbf { y } _ { 0 } ( \\mathbf { x } , \\mathbf { z } ) = \\mathbf { z } , \\qquad t _ { 0 } = 1 , \\quad t _ { k + 1 } = \\frac { 1 + \\sqrt { 1 + t _ { k } ^ { 2 } } } { 2 } , k = 0 , \\cdots , K - 1 , } \\\\ & { \\mathbf { u } ^ { k + 1 } ( \\mathbf { x } , \\mathbf { z } ) = \\mathbf { y } ^ { k + 1 } ( \\mathbf { x } , \\mathbf { z } ) + \\left( \\frac { t _ { k } - 1 } { t _ { k + 1 } } \\right) ( \\mathbf { y } ^ { k + 1 } ( \\mathbf { x } , \\mathbf { z } ) - \\mathbf { y } ^ { k } ( \\mathbf { x } , \\mathbf { z } ) ) , k = 0 , \\cdots , K - 1 , } \\\\ & { \\mathbf { y } _ { k + 1 } ( \\mathbf { x } , \\mathbf { z } ) = \\mathrm { P r o j } _ { \\mathcal { Y } } \\left( \\mathbf { u } ^ { k } ( \\mathbf { x } , \\mathbf { z } ) - \\alpha \\nabla _ { \\mathbf { y } } f ( \\mathbf { x } , \\mathbf { u } ^ { k } ( \\mathbf { x } , \\mathbf { z } ) ) \\right) , k = 0 , \\cdots , K - 1 , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            670,
+            779,
+            761
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha > 0$ is the step size, we propose an accelerated Gradient-based Method with Initialization Auxiliary, named IA-GM(A), via minimizing the approximation objective function, ",
+        "bbox": [
+            174,
+            763,
+            825,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/822a2070773e0bcda23e859c8da2138da4aca7cce53cd38ecb4f6ec307498a2c.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { x } \\in \\mathcal { X } , \\mathbf { z } \\in \\mathcal { V } } \\phi _ { K } ( \\mathbf { x } , \\mathbf { z } ) : = F ( \\mathbf { x } , \\mathbf { y } _ { K } ( \\mathbf { x } , \\mathbf { z } ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            795,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The detailed description of the proposed IA-GM(A) is stated in the Supplemental Material. ",
+        "bbox": [
+            173,
+            824,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We suppose Assumption 3.1(1)-(4) are satisfied throughout this subsection. The convergence result of our proposed IA-GM(A) with LLC assumption is given as below. ",
+        "bbox": [
+            173,
+            844,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.3 Assume that the generated sequence $\\{ \\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) \\}$ satisfies that $\\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) \\in \\mathcal { Y } _ { \\mathrm { : } }$ , and ${ \\bf y } _ { k } ( { \\bf x } , { \\bf z } ) = { \\bf z }$ for any $\\mathbf { z } \\in S ( \\mathbf { x } )$ , $\\mathbf { x } \\in \\mathcal { X }$ , and either ",
+        "bbox": [
+            173,
+            882,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(a) for any $\\epsilon > 0$ , there exists $k ( \\epsilon ) > 0$ such that whenever $K > k ( \\epsilon )$ ",
+        "bbox": [
+            204,
+            90,
+            666,
+            106
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "whenever $K > k ( \\epsilon )$ , or ",
+        "bbox": [
+            232,
+            136,
+            390,
+            150
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(b) there exists $\\alpha > 0 _ { : }$ , for any $\\epsilon > 0$ , there exists $k ( \\epsilon ) > 0$ such that, ",
+        "bbox": [
+            205,
+            157,
+            658,
+            172
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Let $( \\mathbf { x } _ { K } , \\mathbf { z } _ { K } ) \\in \\operatorname { a r g m i n } _ { \\mathbf { x } \\in \\mathcal { X } , \\mathbf { z } \\in \\mathcal { Y } } \\phi _ { K } ( \\mathbf { x } , \\mathbf { z } ) : = F ( \\mathbf { x } , \\mathbf { y } _ { K } ( \\mathbf { x } , \\mathbf { z } ) )$ , then we have ",
+        "bbox": [
+            173,
+            208,
+            624,
+            233
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(1) any limit point $\\bar { x }$ of the sequence $\\left\\{ { \\bf x } _ { K } \\right\\}$ satisfies that $\\bar { \\mathbf { x } } \\in \\underset { \\mathbf { x } \\in \\mathcal { X } } { \\mathrm { a r g m i n } } \\varphi ( \\mathbf { x } ) ,$ , i.e., $\\bar { x }$ is the solution ",
+        "bbox": [
+            202,
+            242,
+            825,
+            267
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "to BLO (1). ",
+        "bbox": [
+            232,
+            265,
+            308,
+            280
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/64aa0e76ee99c26cef8c3bece6bfa16173325ef6c0442cc2b9f171c8fb153241.jpg",
+        "text": "$$\n\\operatorname* { i n f } _ { \\mathbf { x } \\in \\mathcal { X } , \\mathbf { z } \\in \\mathcal { V } } \\phi _ { K } ( \\mathbf { x } , \\mathbf { z } ) \\to \\operatorname* { i n f } _ { \\mathbf { x } \\in \\mathcal { X } } \\varphi ( \\mathbf { x } ) a s K \\to \\infty .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            230,
+            286,
+            521,
+            310
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Next, we show that the Nesterov’s acceleration dynamics satisfy all the assumptions required in the above convergence theorem. ",
+        "bbox": [
+            173,
+            318,
+            825,
+            347
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.4 Let $\\{ \\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) \\}$ be the sequence generated by Nesterov’s acceleration dynamics in Eq. (9) with $\\begin{array} { r } { \\alpha = \\frac { 1 } { L _ { f } } } \\end{array}$ . Then $\\{ \\mathbf { y } _ { k } ( \\mathbf { x } , \\mathbf { z } ) \\}$ satisfies all the assumptions required by Theorem 3.3. ",
+        "bbox": [
+            173,
+            354,
+            821,
+            386
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Remark 3.1 Our convergence result Theorem 3.3 is not only for $\\mathbf { y } _ { k }$ generated by Nesterov’s acceleration dynamics. It is a general convergence result that is applicable for the case where $\\mathbf { y } _ { k }$ is generated by other dynamics. And the assumptions required in Theorem 3.3 is weak enough to be satisfied by the dynamics introduced by many first-order methods on convex LL problem in Eq. 2. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Experimental Results ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            469,
+            387,
+            488
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first verify the theoretical convergence results on non-convex numerical problems compared with existing EG methods and IG methods. Then we test the performance of IAPTT-GM and demonstrate its generalizability to real-world BLO problems with non-convex followers, which are caused by non-convex regularization and neural network structures. In addition, we further validate the performance of the accelerated version (i.e., IA-GM (A)) under LLC with numerical examples and data hyper-cleaning tasks 2. ",
+        "bbox": [
+            173,
+            501,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 Numerical Verification ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            599,
+            370,
+            614
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To verify the convergence property under assumptions provided in Section 3, we consider the following non-convex BLO problem: ",
+        "bbox": [
+            173,
+            625,
+            823,
+            654
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/946c4d728ecc2bf1780065fa070b5928b27f577a5f4e5cb6660294053b26fa00.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathcal { X } , y \\in \\mathbb { R } } x + x y , \\quad s . t . \\quad y \\in \\underset { y \\in \\mathcal { Y } } { \\operatorname { a r g m i n } } - \\sin ( x y ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            655,
+            651,
+            680
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { X } \\ : = \\ : [ 1 , 1 0 ]$ and $\\mathcal { V } = [ - 2 , 2 ]$ . Given any $x \\in \\mathcal { X }$ , it satisfies $\\begin{array} { r } { \\mathrm { a r g m i n } _ { y \\in \\mathcal { y } } - \\mathrm { s i n } ( x y ) \\ = } \\end{array}$ $\\{ ( 2 k \\pi + \\pi / 2 ) / x \\mid k \\in \\mathbb { Z } \\} \\cap \\mathcal { Y }$ and $\\begin{array} { r } { \\operatorname* { m i n } _ { y \\in \\mathcal { V } } - \\sin ( x y ) = - 1 } \\end{array}$ . The unique solution is $( x ^ { * } , y ^ { * } ) =$ $( 1 1 \\pi / 4 , - 2 )$ . It should be noted that the LL problem of Eq. (10) has multiple global minima, which can significantly show the advantage of initialization auxiliary technique. It can be easily verified that the above toy example satisfies Assumption 3.1. ",
+        "bbox": [
+            173,
+            681,
+            825,
+            753
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In Figure 1, we separately compared IAPTT-GM with EG methods such as RHG [3], BDA [23], IG methods such as LS [28], NS [7] and IA-GM. From Figure 1.(a) to Figure 1.(b), we can observe that different initialization points only slightly affect the convergence speed of IAPTT-GM. With initialization points distant from $( x ^ { * } , y ^ { * } )$ , IAPTT-GM can still achieve optimal solution of UL variables and optimal objective value, while other methods fail to converge to the true solution. In Figure 1.(g) and Figure 1.( h), we compare the performances of IAPTT-GM with IA-GM, which has no convergence guarantee without LLC assumption. As shown, IA-GM fails to converge to the true solution eventually, which validates the necessity of PTT technique and the effectiveness of IAPTT-GM. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/c127978dc735f90a7baeb2a430c84d4f2d7ad25a940e852e57c541daeb62a434.jpg",
+        "image_caption": [
+            "Figure 1: Illustrating the convergence behavior of $\\| F - F ^ { * } \\| / \\| F ^ { * } \\|$ and $\\| x - x ^ { * } \\| / \\| x ^ { * } \\|$ as the training proceeds. $F _ { \\mathrm { I A P T T - G M } }$ and $F _ { \\mathrm { I A - G M } }$ denote the UL objectives of IAPTT-GM and IA-GM, respectively. Three representative initialization points for UL and LL variables are $( x _ { 0 } , y _ { 0 } ) = ( 1 , 2 )$ , $( x _ { 0 } , y _ { 0 } ) = \\mathbf { \\bar { ( 5 , 1 ) } }$ , $( x _ { 0 } , \\overset { \\cdot } { y } _ { 0 } ) = ( 7 , - 1 ) .$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Runtime and Memory Analysis. In Figure 2, we report the average steps $\\bar { k }$ of IAPTT-GM for PTT and the iterative speed as the UL iteration increases. In comparison with IA-GM, which uses default $K$ for the LL optimization loop, the changing $\\bar { k }$ for IAPTT-GM leads to less iterations for the backward recurrent propagation and thus faster iterative updates during optimization. Although IA introduces additional variables and iterations, the PTT technique can choose a small $\\bar { k }$ , thus shortens the back-propagation trajectory for computing the UL gradient (see Figure 2). As can be seen in Table 1, the memory required by our IAPTT-GM is less than NS, LS, and BDA and the same as that for RHG. As for the runtime, IAPTT-GM is a bit slower than RHG and NS, and faster than BDA. But please notice that the performance and the theoretical properties of IAPTT-GM are better than these existing approaches. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/e00c87cd413be5c13a123c8f9646faf7864d31c9f7a3a818764051a283f0610c.jpg",
+        "image_caption": [
+            "Figure 2: Illustrating average steps for PTT technique and average running speed of the numerical example. Note that we conduct the experiments using more LL iterations so as to reduce the measurement error. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            552,
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f31269f4047bed220b0a9a9fb6ad063ce5289e4d6a59cae54669be28d902d850.jpg",
+        "table_caption": [
+            "Table 1: Memory and runtime of existing methods for solving the above BLO problem. We conduct the experiments using the same parameter settings in Section C of the supplementary materials. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Metrics</td><td rowspan=1 colspan=1>LS</td><td rowspan=1 colspan=1>NS</td><td rowspan=1 colspan=1>RHG</td><td rowspan=1 colspan=1>BDA</td><td rowspan=1 colspan=1>IAPTT-GM</td></tr><tr><td rowspan=1 colspan=1>Memory (GB)</td><td rowspan=1 colspan=1>10.426</td><td rowspan=1 colspan=1>10.387</td><td rowspan=1 colspan=1>10.153</td><td rowspan=1 colspan=1>10.154</td><td rowspan=1 colspan=1>10.153</td></tr><tr><td rowspan=1 colspan=1>Runtime (Sec)</td><td rowspan=1 colspan=1>5.120</td><td rowspan=1 colspan=1>10.815</td><td rowspan=1 colspan=1>9.990</td><td rowspan=1 colspan=1>16.800</td><td rowspan=1 colspan=1>10.835</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 BLO with Non-convex Followers in Different Application Scenarios ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To cover various real-world BLO application scenarios, we consider two categories of non-convex LL problems caused by non-convex regularization term and neural network architectures, which refer to few-shot classification and data hyper-cleaning tasks, respectively. Please note that the set constraint $\\mathcal { V }$ is only used to guarantee the completeness of our theoretical analysis in Section 3. In application scenarios, we can just consider the constraint as an extra large set, so that all the variables are automatically in this feasible set. In this way, it is natural to ignore the projection operation in practical computations. ",
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+    {
+        "type": "text",
+        "text": "4.2.1 Few-Shot Classification: Non-convex LL Objective ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In few-shot learning, to be more specific, N-way M-shot classification tasks [46], provided with M samples from each class, we train models to take advantage of prior data from similar tasks to quickly classify unseen instances from these $_ \\mathrm { N }$ classes. Following the experimental protocol [47], the model parameters are separated into two parts: the hyper representation module (parameterized by $\\mathbf { x }$ ) shared by all the tasks and the last classifier (parameterized by $\\mathbf { y } ^ { j }$ ) for $j$ -th task. Define the meta training dataset as $\\mathcal { D } = \\{ \\mathcal { D } ^ { j } \\}$ , where $\\mathcal { D } ^ { j } = \\mathcal { D } _ { \\mathtt { t r } } ^ { j } \\bigcup \\mathcal { D } _ { \\mathtt { v a l } } ^ { j }$ corresponds to the $j$ -th task. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The cross-entropy loss function is widely used for UL and LL objectives. Referring to Section 3, IAPTT-GM covers the convergence results with non-convex LL model, thus allowing flexible design of the LL objective function. For instance, while non-convex regularization terms, e.g., $\\ell _ { q }$ regularization with $0 < q < 1$ , have shown effectiveness to help the LL model converge and avoid over-fitting, existing methods can only guarantee the convergence when $q \\geq 1$ , thus almost provide no support for non-convex objectives. We consider the LL subproblem with non-convex loss functions by adding $\\ell _ { q }$ regularization 3. Then the UL and LL subproblems can be written as ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/0a6e221a79cc78d4f2f102918e974ada388f2f8704d4499ed4b7784b1e574f7c.jpg",
+        "text": "$$\nF \\left( \\mathbf { x } , \\left\\{ \\mathbf { y } ^ { j } \\right\\} \\right) = \\sum _ { j } \\ell \\left( \\mathbf { x } , \\mathbf { y } ^ { j } ; \\mathcal { D } _ { \\mathrm { v a l } } ^ { j } \\right) , \\quad f \\left( \\mathbf { x } , \\left\\{ \\mathbf { y } ^ { j } \\right\\} \\right) = \\sum _ { j } \\ell \\left( \\mathbf { x } , \\mathbf { y } ^ { j } ; \\mathcal { D } _ { \\mathrm { t r } } ^ { j } \\right) + \\Vert \\mathbf { y } ^ { j } \\Vert _ { q } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/ca04572e4a05f2bc61214104a6cdc9a1ed29e43e796ba96f1cd1b69e6881f1f5.jpg",
+        "table_caption": [
+            "Table 2: Mean test accuracy of 5-way classification on tieredImageNet and miniImageNet, and the $\\pm$ represents $9 5 \\%$ confidence intervals over tasks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=2 colspan=1>Methods</td><td rowspan=2 colspan=1>Backbone</td><td rowspan=1 colspan=2>MiniImagenet</td><td rowspan=1 colspan=2>TieredImagenet</td></tr><tr><td rowspan=1 colspan=1>5-way 1-shot</td><td rowspan=1 colspan=1> 5-way 5-shot</td><td rowspan=1 colspan=1> 5-way 1-shot</td><td rowspan=1 colspan=1>t5-way 5-shot</td></tr><tr><td rowspan=1 colspan=1>Proto Net</td><td rowspan=1 colspan=1>ConvNet-4</td><td rowspan=1 colspan=1>49.42 ± 1.84</td><td rowspan=1 colspan=1>68.20± 0.66</td><td rowspan=1 colspan=1>53.31 ± 0.89</td><td rowspan=1 colspan=1>72.69±0.74</td></tr><tr><td rowspan=1 colspan=1>Relation Net</td><td rowspan=1 colspan=1>ConvNet-4</td><td rowspan=1 colspan=1>50.44 ± 0.82</td><td rowspan=1 colspan=1>65.32 ± 0.70</td><td rowspan=1 colspan=1>54.48 ± 0.93</td><td rowspan=1 colspan=1>65.32 ± 0.70</td></tr><tr><td rowspan=1 colspan=1>MAML</td><td rowspan=1 colspan=1>ConvNet-4</td><td rowspan=1 colspan=1>48.70±0.75</td><td rowspan=1 colspan=1>63.11 ± 0.11</td><td rowspan=1 colspan=1>49.06 ± 0.50</td><td rowspan=1 colspan=1>67.48 ± 0.47</td></tr><tr><td rowspan=1 colspan=1>RHG</td><td rowspan=1 colspan=1>ConvNet-4</td><td rowspan=1 colspan=1>48.89 ±0.81</td><td rowspan=1 colspan=1>63.02 ± 0.70</td><td rowspan=1 colspan=1>49.63 ± 0.67</td><td rowspan=1 colspan=1>66.14 ± 0.57</td></tr><tr><td rowspan=1 colspan=1>T-RHG</td><td rowspan=1 colspan=1>ConvNet-4</td><td rowspan=1 colspan=1>47.67± 0.82</td><td rowspan=1 colspan=1>63.70 ± 0.76</td><td rowspan=1 colspan=1>50.79 ± 0.69</td><td rowspan=1 colspan=1>67.39 ± 0.60</td></tr><tr><td rowspan=1 colspan=1>BDA</td><td rowspan=1 colspan=1>ConvNet-4</td><td rowspan=1 colspan=1>49.08±0.82</td><td rowspan=1 colspan=1>62.17 ± 0.70</td><td rowspan=1 colspan=1>51.56 ± 0.68</td><td rowspan=1 colspan=1>68.21 ±0.58</td></tr><tr><td rowspan=1 colspan=1>MAML</td><td rowspan=1 colspan=1>ResNet-12</td><td rowspan=1 colspan=1>51.03 ± 0.50</td><td rowspan=1 colspan=1>68.26 ± 0.47</td><td rowspan=1 colspan=1>58.58± 0.49</td><td rowspan=1 colspan=1>71.24 ± 0.43</td></tr><tr><td rowspan=1 colspan=1>RHG</td><td rowspan=1 colspan=1>ResNet-12</td><td rowspan=1 colspan=1>50.54±0.85</td><td rowspan=1 colspan=1>64.53 ± 0.68</td><td rowspan=1 colspan=1>58.19 ±0.76</td><td rowspan=1 colspan=1>75.20±0.60</td></tr><tr><td rowspan=1 colspan=1>IAPTT-GM</td><td rowspan=1 colspan=1>ResNet-12</td><td rowspan=1 colspan=1>56.69±0.66</td><td rowspan=1 colspan=1>70.21 ± 0.55</td><td rowspan=1 colspan=1>60.71±0.77</td><td rowspan=1 colspan=1>75.85 ± 0.59</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Detailed information about the datasets and network architectures can be found in the supplementary materials. We report results of IAPTT-GM and various mainstream methods, e.g., Prototypical Network [48], Relation Net [49] and T-RHG [16] on miniImageNet [47] and tieredImageNet [50] datasets with two different backbones [6, 51] in Table 2. As it is shown, our proposed method outperforms state-of-the-art methods on both 5-way 1-shot and 5-way 5-shot tasks. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.2.2 Data Hyper-Cleaning: Non-convex LL Architecture Structure ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Data hyper-cleaning [3] aims to cleanup the corrupted data with noise label. According to [16], the dataset is randomly split to three disjoint subsets: $\\mathcal { D } _ { \\mathtt { t r } }$ for training, $\\mathcal { D } _ { \\mathtt { v a l } }$ for validation and $\\mathcal { D } _ { \\mathrm { t e s t } }$ for testing, then a fixed proportion of the training samples in $\\mathcal { D } _ { \\mathtt { t r } }$ is randomly corrupted. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Following the classical experimental protocol [3], we choose cross-entroy as the loss function $\\ell$ , and the UL and LL subproblem take the form of ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/26ec85e05191c935b88bf582a79a2bb85c418b69d8a8465af29d91c293eb2aa3.jpg",
+        "text": "$$\nF ( { \\mathbf x } , { \\mathbf y } ) = \\sum _ { ( \\mathbf u _ { i } , \\mathbf v _ { i } ) \\in \\mathcal { D } _ { v a } } \\ell \\left( { \\mathbf y } ( { \\mathbf x } ) ; \\mathbf u _ { i } , { \\mathbf v } _ { i } \\right) , f ( { \\mathbf x } , { \\mathbf y } ) = \\sum _ { ( \\mathbf u _ { i } , \\mathbf v _ { i } ) \\in \\mathcal { D } _ { v a } } [ \\sigma ( { \\mathbf x } ) ] _ { i } \\ell \\left( { \\mathbf y } ; { \\mathbf u } _ { i } , { \\mathbf v } _ { i } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            202,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $\\left( \\mathbf { u } _ { i } , \\mathbf { v } _ { i } \\right)$ denotes the data pair and $\\sigma ( \\mathbf { x } )$ represents the element-wise sigmoid function on $\\mathbf { x }$ . We define the hyperparameter $\\mathbf { x }$ as a vector being trained to label the noisy data, of which the dimension equals to the number of training samples. The LL variables parameterized by $\\mathbf { y }$ contain the weights and bias of fully connected layers. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Note that existing methods consider convex a single fully connected layer as the LL model, while more complex neural network structure is not applicable. Under our assumption without LLC, we employ two fully connected layers as the LL network architecture. ",
+        "bbox": [
+            173,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In Table 3, we compare IAPTT-GM with IG methods (e.g., LS, NS) and EG methods (e.g., RHG, T-RHG [16]). As it is shown, IAPTT-GM achieves better test performance of both accuracy and F1 score on two datasets, including MNIST [46] and FashionMNIST [52]. Our theoretical results also show that the performance improvement comes from PTT and IA techniques to overcome non-convex LL subproblems. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/983c2d9210a50a7f2b35fed3bfb87b45d1e3a6109344962f994ccd3877678059.jpg",
+        "table_caption": [
+            "Table 3: Reporting results of existing methods for solving data hyper-cleaning tasks. Acc. and F1 score denote the test accuracy and the harmonic mean of the precision and recall, respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=2 colspan=1>Method</td><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>FashionMNIST</td></tr><tr><td rowspan=1 colspan=1>Acc.F1 score</td><td rowspan=1 colspan=1>Acc.F1 score</td></tr><tr><td rowspan=1 colspan=1>LS</td><td rowspan=1 colspan=1>89.19 85.96</td><td rowspan=1 colspan=1>83.15  85.13</td></tr><tr><td rowspan=1 colspan=1>NS</td><td rowspan=1 colspan=1>87.54 89.58</td><td rowspan=1 colspan=1>81.37  87.28</td></tr><tr><td rowspan=1 colspan=1>RHG</td><td rowspan=1 colspan=1>87.90 89.36</td><td rowspan=1 colspan=1>81.91  87.12</td></tr><tr><td rowspan=1 colspan=1>T-RHG</td><td rowspan=1 colspan=1>88.57 89.77</td><td rowspan=1 colspan=1>81.85  86.76</td></tr><tr><td rowspan=1 colspan=1>BDA</td><td rowspan=1 colspan=1>87.15 90.38</td><td rowspan=1 colspan=1>79.97 88.24</td></tr><tr><td rowspan=1 colspan=1>IAPTT-GM</td><td rowspan=1 colspan=1>90.88 91.57</td><td rowspan=1 colspan=1>83.67 90.37</td></tr></table>",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "4.3 Evaluations of IA-GM (A) for BLOs under LLC ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In addition to non-convex BLO problems, we also raise concerns about the acceleration strategy of our method with LLC condition. We first consider the following BLO with LLC condition [23]: ",
+        "bbox": [
+            169,
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+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2fb0b62ed02fb26a8e8dcb0d852f43226dca54aa90087617d0e9e96e68ed2229.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { x } \\in \\mathcal { X } } \\| \\mathbf { x } - \\mathbf { y } _ { 2 } \\| ^ { 4 } + \\| \\mathbf { y } _ { 1 } - \\mathbf { e } \\| ^ { 4 } , \\quad s . t . \\quad ( \\mathbf { y } _ { 1 } , \\mathbf { y } _ { 2 } ) \\in \\arg \\operatorname* { m i n } _ { \\mathbf { y } _ { 1 } \\in \\mathbb { R } ^ { n } , \\mathbf { y } _ { 2 } \\in \\mathbb { R } ^ { n } } \\frac { 1 } { 2 } \\| \\mathbf { y } _ { 1 } \\| ^ { 2 } - \\mathbf { x } ^ { \\top } \\mathbf { y } _ { 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            204,
+            393,
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+            426
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where $n = 5 0$ , $\\mathcal { X } = [ - 1 0 0 , 1 0 0 ] \\times \\cdot \\cdot \\cdot [ - 1 0 0 , 1 0 0 ] \\subset \\mathbb { R } ^ { n }$ , and e represents the vector whose elements are all equal to 1. The optimal solution for this problem is $\\mathbf { x } ^ { * } = \\mathbf { e } , \\mathbf { y } _ { 1 } ^ { * } = \\mathbf { e } , \\mathbf { y } _ { 2 } ^ { * } = \\mathbf { e }$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "As shown in Section 3.2, our method IA-GM (A) incorporates Nesterov’s acceleration strategy for solving Eq. (2). Note that with LLC assumption, IA-GM maintains the convergence property. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/1c2daf47ed3e8975be7489fb933f9bbb8a0401e79da1c16050ce4df719b9a324.jpg",
+        "image_caption": [
+            "Figure 3: The left two subfigures report the curves of $\\| \\mathbf { y } - \\mathbf { y } ^ { * } \\|$ and $\\| \\mathbf { x } - \\mathbf { x } ^ { * } \\|$ for IA-GM and IA-GM (A). The figures on the right illustrate the results of mean UL loss and accuracy on the convex data hyper-cleaning problems. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "As illustrated in the first two figures in Figure 3, IA-GM (A) shows significant improvement of convergence speed on UL and LL variables, which verifies the convergence results of Theorem 3.4 under LLC. We further study the convex data hyper-cleaning problem, which simply implements single fully connected layer as the network structure. From the right half of Figure 3, we can easily find that IA-GM (A) also performs better than IA-GM on real-world applications. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This paper presents a generic first-order algorithmic framework named IAPTT-GM to solve BLO problems with non-convex follower. We introduce two features, initialization auxiliary and pessimistic trajectory truncation operation to guarantee the convergence without the LLC hypothesis and achieves better performance on various applications. Meanwhile, we also validate the performance and speed improvement for BLO with LLC condition. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This work is partially supported by the National Natural Science Foundation of China (Nos. 61922019, 11971220), the National Key R&D Program of China (2020YFB1313503), LiaoNing Revitalization Talents Program (XLYC1807088), the Shenzhen Science and Technology Program (No. RCYX20200714114700072), the Fundamental Research Funds for the Central Universities, the Pacific Institute for the Mathematical Sciences (PIMS), the Stable Support Plan Program of Shenzhen Natural Science Fund (No. 20200925152128002) and the Guangdong Basic and Applied Basic Research Foundation 2019A1515011152. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            237,
+            266,
+            253
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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A method for solving the convex programming problem with convergence rate o $( 1 / \\mathrm { k } ^ { \\sim } 2 )$ . In Dokl. akad. nauk Sssr, volume 269, pages 543–547, 1983.   \n[45] Amir Beck and Marc Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. 2009.   \n[46] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.   \n[47] Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Daan Wierstra, et al. Matching networks for one shot learning. In NeurIPS, 2016.   \n[48] Jake Snell, Kevin Swersky, and Richard S Zemel. Prototypical networks for few-shot learning. arXiv:1703.05175, 2017.   \n[49] Flood Sung, Yongxin Yang, Li Zhang, Tao Xiang, Philip HS Torr, and Timothy M Hospedales. Learning to compare: Relation network for few-shot learning. 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parse/train/bajTv_cvkI/bajTv_cvkI.md

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+# Exploiting Domain-Specific Features to Enhance Domain Generalization
+
+Manh-Ha Bui1 Toan Tran1 Anh Tuan Tran1 Dinh Phung1,2 1 VinAI Research, Vietnam 2 Monash University, Australia {v.habm1, v.toantm3, v.anhtt152, v.dinhpq2}@vinai.io ∗
+
+# Abstract
+
+Domain Generalization (DG) aims to train a model, from multiple observed source domains, in order to perform well on unseen target domains. To obtain the generalization capability, prior DG approaches have focused on extracting domaininvariant information across sources to generalize on target domains, while useful domain-specific information which strongly correlates with labels in individual domains and the generalization to target domains is usually ignored. In this paper, we propose meta-Domain Specific-Domain Invariant (mDSDI) - a novel theoretically sound framework that extends beyond the invariance view to further capture the usefulness of domain-specific information. Our key insight is to disentangle features in the latent space while jointly learning both domain-invariant and domainspecific features in a unified framework. The domain-specific representation is optimized through the meta-learning framework to adapt from source domains, targeting a robust generalization on unseen domains. We empirically show that mDSDI provides competitive results with state-of-the-art techniques in DG. A further ablation study with our generated dataset, Background-Colored-MNIST, confirms the hypothesis that domain-specific is essential, leading to better results when compared with only using domain-invariant.
+
+# 1 Introduction and Related work
+
+Domain Generalization (DG) has recently become an important research topic in machine learning due to its real-world applicability and its close connection to the way humans generalize to learn in a new domain. In a DG framework, the learner is trained on multiple datasets collected under different environments without any access to any data on the target domain (1). One of the most notable approaches to this problem is to learn the “domain-invariant” features across these training datasets, with the assumption that these invariant representations are also held in unseen target domains (2; 3; 4; 5; 6). While this has been shown to work well in practice, its key drawback is completely ignoring “domain-specific” information that could aid the generalization performance, especially when the number of source domains increases (7).
+
+For instance, consider the problem of classifying dog or fish images from two source domains: sketch and photo. While the sketch contains a conceptual drawing of the animal, the photo includes their taken picture within a background. In this case, sketch domain-invariant is kept across domains, while domain-specific, e.g., a dog in a house or fish in the ocean, will be discarded due to only existing in the photo domain. However, this background information, when present, could lead to an improvement of the classification performance in target domains due to common association between the objects of its background, and when negligent sketches are hard to distinguish. From a theoretical standpoint, there has also been strong recent evidence to indicate the insufficiency of learning domaininvariant representation for successful adaptation in domain adaptation problems (8; 9). For example,
+
+Zhao et al. (8) has pointed out the degradation in target predictive performance if domain-invariant representations are forced while the marginal label distributions on the source and target domains are overly different.
+
+Utilizing domain-specific features in DG has been widely studied in recent works (e.g., (10; 7)). Ding and Fu (10) introduce multiple domain-specific networks for each domain, then use the structured low-rank constraints to align them with domain-invariant. While this encourages the better transfer of knowledge, its main problem is the requirement of too many domain-specific networks. More recently, Chattopadhyay et al. (7) proposed a masking strategy to disentangle domain-invariant and domain-specific to further boost domain-specific learning, but its key drawback is that domaininvariant/domain-specific representations might not be disentangled since the learning and inferring procedures are performed implicitly (i.e., without any theoretical guarantee) through a mask generalization process. That means it lacks a clear motivation as well as theoretical justifications.
+
+Regarding meta-learning related work, a typical approach involving meta-learning in DG is MLDG (11) that is based on gradient update which simulates train/test domain shift within each mini-batch, mainly to learn transferable weight representations from meta-source domains to quickly adapt to the meta-target domain, and so improve generalization ability. However, their task objective adapts for all representation features which include domain-invariant, since low effectiveness because domain-invariant is stable across domains, pushing to adapt those features might affect the stability of those domain-invariant, leading to a lower generalization performance on the target domain.
+
+To handle these domain-invariant shortcomings, in this paper, we propose a novel theoretically sound DG approach that aims to extract label-informative domain-specific and then explicitly disentangles the domain-invariant and domain-specific representations in an efficient way without training multiple networks for domain-specific. Following the meta-learning idea and mitigating previous work’s drawbacks, we apply a meta-learning technique specifically to exploit domain-specific quality which should need to be adapted to unseen domains from source domains. Our contributions in this work are summarized as follows:
+
+• We provide a theoretical analysis based on the information bottleneck principle to point out the limitation of only learning invariant and the importance of domain-specific representation by a certainly plausible assumption.   
+• We then develop a rigorous framework to formulate elements of domain-invariant/domainspecific representations, in which our key insight is to introduce an effective metaoptimization training framework (11) to learn domain-specific representation from multiple training domains. Without accessing any data from unseen target domains, the meta-training procedure provides a suitable mechanism to self-learn domain-specific representation. We term our approach meta-Domain Specific-Domain Invariant (mDSDI) and provide necessary theoretical verifications for it.   
+• To demonstrate the merit of the proposed mDSDI framework, we extensively evaluate mDSDI on several state-of-the-art DG benchmark datasets, including Colored-MNIST, Rotated-MNIST, VLCS, PACS, Office-Home, Terra Incognita, DomainNet in addition to our newly created Background-Colored-MNIST for the ablation study to examine the behavior of our mDSDI.
+
+# 2 Methodology
+
+# 2.1 Problem setting and Definitions
+
+Let $\boldsymbol { \mathcal { X } } \subset \mathbb { R } ^ { D }$ be the sample space and $\mathcal { V } \subset \mathbb { R }$ the label space. Denote the set of joint probability distributions on $\mathcal { X } \times \mathcal { V }$ by $\mathcal { P } _ { \mathcal { X } \times \mathcal { Y } }$ , and the set of probability marginal distributions on $\mathcal { X }$ by $\mathcal { P } _ { \mathcal { X } }$ . A domain is defined by a joint distribution $P ( x , y ) \in \mathcal P _ { \mathcal { X } \times \mathcal { Y } }$ , and let $\mathcal { P }$ be a measure on $\mathcal { P } _ { \mathcal { X } \times \mathcal { Y } }$ , i.e., whose realizations are distributions on $\mathcal { X } \times \mathcal { V }$ .
+
+Denote $N$ source domains by $S ^ { ( i ) } = \{ ( x _ { j } ^ { ( i ) } , y _ { j } ^ { ( i ) } ) \} _ { j = 1 } ^ { n _ { i } }$ , $i = 1 , \ldots , N$ , where $n _ { i }$ is the number of data points in $S ^ { ( i ) }$ , i.e., $( x _ { j } ^ { ( i ) } , y _ { j } ^ { ( i ) } ) \stackrel { i i d } { \sim } P ^ { ( i ) } ( x , y )$ where $P ^ { ( i ) } ( x , y ) \sim \mathcal { P }$ ; and $x _ { j } ^ { ( i ) } \sim P _ { \mathcal { X } } ^ { ( i ) }$ , in which $P _ { \mathcal { X } } ^ { ( i ) } \sim P _ { \mathcal { X } }$ . In a typical DG framework, a learning model which is only trained on the set of source domains $\{ S ^ { ( i ) } \} _ { i = 1 } ^ { N }$ without any access to the (unlabeled) data points in the target domain, arrives at a good generalization performance on the test dataset $S ^ { T } = \{ ( x _ { j } ^ { T } , y _ { j } ^ { T } ) \} _ { j = 1 } ^ { n _ { T } }$ , where $( x _ { j } ^ { T } , y _ { j } ^ { T } ) \overset { i i d } { \sim } P ^ { T } ( x , y )$ and $P ^ { T } ( x , y ) \sim \mathcal { P }$ .
+
+First, we present the definition of domain-invariant representation in a latent space $\mathcal { Z }$ under covariate shift assumption (i.e., the conditional distribution $P ( \boldsymbol { y } | \boldsymbol { x } )$ is unchanged across the source domains):
+
+Definition 1. A feature extraction mapping $Q : \mathcal { X }  \mathcal { Z }$ is said to be domain-invariant if the distribution $P _ { Q } ( Q ( X ) )$ is unchanged across the source domains, i.e., $\forall i , j = 1 , \ldots , N , i \neq j w e$ have $P _ { Q } ^ { ( i ) } ( Q ( X ) ) \equiv P _ { Q } ^ { ( j ) } ( Q ( X ) )$ , where $P _ { Q } ^ { ( i ) } ( Q ( X ) ) = P _ { Q } ( Q ( X ) | X \sim P _ { \ X } ^ { ( i ) } )$ , $i = 1 , \ldots , N .$ . In this case, the corresponding latent representation $Z _ { I } = Q ( X )$ is then called the domain-invariant representation (see (3) also).
+
+As mentioned in the example in the introduction part, the definition 1 reveals that the extracted domain-invariant latent $Z _ { I }$ could be the conceptual drawing of the animal which is shared in both sketch and photo domains. However, when existing background information is taken by a picture such as a house or ocean, it is crucial to take these backgrounds into account because the domain-invariant feature extraction $Q$ might ignore them by only existing in the photo domain. Therefore, we next introduce the definition of domain-specific in latent space as follows:
+
+Definition 2. A feature extraction mapping $R : \mathcal { X }  \mathcal { Z }$ is said to be domain-specific $i f \ \forall i , j =$ $1 , \dots , N , \ i \ \neq \ j$ such that $P _ { R } ^ { ( i ) } ( R ( X ) ) \neq P _ { R } ^ { ( j ) } ( R ( X ) )$ , where $P _ { R } ^ { ( i ) } ( R ( X ) ) = P _ { R } ( R ( X ) | X \sim$ $P _ { \mathcal { X } } ^ { ( i ) } )$ ), $i = 1 , \ldots , N$ . In this case, given $X \sim P _ { \mathcal { X } } ^ { ( i ) }$ the corresponding latent representation $Z _ { S } ^ { ( i ) } = $ $R ( X )$ is then called the domain-specific representation w.r.t. the domain .
+
+Definition 2 states that for any domain $i$ and $j$ , the distributions of the domain-specific latent $P _ { R } ^ { ( i ) } ( R ( X ) )$ and ven t $P _ { R } ^ { ( j ) } ( R ( X ) )$ must bes from comand letely different. For instance, following our menthat are sketch and photo domain, the mapping $i$ $j$ $R ( X )$ should extract specific information that only belongs to the domain including the shadow of the fish drawing in the sketch and ocean background information in the photo domain.
+
+To this end, this paper aims to show that only learning domain-invariant will limit the prediction performance and generalization ability. Hence, we next provide a formal explanation for the motivation of learning domain-specific, by showing the potential drawback of only learning domain-invariance in terms of predicting class labels.
+
+![](images/3edf5c6eda874cebbcf7af6b99450ae6f38b1d6ec3646b7fb1831c1c5f553958.jpg)  
+2.2 A theoretical analysis under the Information bottleneck method   
+Figure 1: Venn diagram showing relationships between source domains represented by $X ^ { 1 }$ , $X ^ { 2 }$ , target domain represented by $X ^ { T }$ , and label $Y$ . (a) The learning procedure of minimal and sufficient label-related representation in definition 3. (b) Explaining the theorem 1 where the domain-invariant based method provides an inferior prediction performance to our proposed method that incorporates both domain-invariant $I ( Z _ { I ^ { * } } ; Y )$ and labelrelated domain-specific values $\epsilon$ made by our assumption 1. (c) A case when the unseen (target) domain has different $X ^ { T }$ and $Y ^ { T }$ , while domain-invariant information is still stable across domains, some domain-specific in source domains become redundant information.
+
+Notations. Given three arbitrary random variables $A , B$ , and $C$ , let us use $I ( A ; B )$ to represent mutual information between $A$ and $B$ ; $I ( A ; B | C )$ to represent conditional mutual information of $A$ and $B$ given $C$ ; $H ( A )$ to represent entropy of $A$ ; and $H ( A | B )$ to represent conditional entropy for random variables $A$ given $B$ . For simplicity, we consider the case with two source domains $S ^ { 1 } , S ^ { 2 }$ (the results with multiple source domains can be naturally extended from there). We also define two corresponding random variables $X ^ { i } \sim P _ { \mathcal { X } } ^ { ( i ) }$ , that are sampled from the marginal distribution $P _ { \mathcal { X } }$ in the domain $S ^ { i } , i = 1 , 2$ .
+
+Figure 1 illustrates all the definitions and assumptions above used for our theoretical verification (in Theorem 1). In particular, in that figure, each of the four colored rectangles represents an individual entropy: $H ( X ^ { 1 } )$ for domain $S ^ { 1 }$ is in blue, $H ( X ^ { 2 } )$ for domain $S ^ { 2 }$ is in red, $\overset { \vartriangle } { \boldsymbol { H } } ( X ^ { T } )$ for the target domain is in black, and $H ( Y )$ for the class label is in green border rectangle.
+
+We first show the ineffectiveness of only learning domain-invariant information when compared with incorporating domain-specific in source domain $S ^ { 1 }$ (and similarly with domain $S ^ { 2 }$ ). Our justification partly relies on the following assumption about the correlation between the domainspecific representation and the class label:
+
+Assumption 1. (Label-correlated domain-specificity) Assuming that there exists a domain-specific representation $Z _ { S } ^ { ( 1 ) }$ extracted by the deterministic mapping $Z _ { S } ^ { ( 1 ) } = R ( X ^ { 1 } )$ in definition 2, which correlates with label in domain $S ^ { 1 }$ such that $I ( Z _ { S } ^ { ( 1 ) } ; Y | X ^ { 2 } ) = I ( X ^ { 1 } ; Y | X ^ { 2 } ) = \varepsilon _ { 1 }$ , where $\varepsilon _ { 1 } > 0$ is a constant.
+
+Assumption 1 indicates that, for the source domain $S ^ { 1 }$ , we can learn $Z _ { S } ^ { ( 1 ) } = R ( X ^ { 1 } )$ such that $I ( Z _ { S } ^ { ( 1 ) } ; Y | X ^ { 2 } )$ is strictly positive and equals to $I ( X ^ { 1 } ; Y | X ^ { 2 } )$ , where $I ( X ^ { 1 } ; Y | X ^ { 2 } )$ is the specific information that correlates with the label in the domain $S ^ { 1 }$ , but not in the domain $S ^ { 2 }$ (12). For instance, in the example mentioned in the introduction, if domain $S ^ { 1 }$ is “photo” while $S ^ { 2 }$ is “sketch”, the value of $\epsilon _ { 1 }$ should be positive because the background information such as a house, the ocean also provides information to predict whether the object is a dog or fish without considering its conceptual drawing. This assumption is particularly valid and practically plausible and is demonstrated by several examples observed in our experiments. For instance, for the DomainNet benchmark dataset, in the real-world domain, many bed pictures contain a bed in the room or bike pictures that have bicycles parked on the street. Other examples are in PACS such as dogs in the yard or guitars lying on a table in photo and art domains. These examples are strongly related to assumption 1, in which specific information correlates with labels in a particular domain.
+
+We next present supervised learning frameworks under the umbrella of the information theory (13; 14) and the information bottleneck method (13; 15) that generalizes minimal sufficient statistics to the minimal (i.e., less complexity) and sufficient (i.e, better fidelity) representations. The learning process of such representations is equivalent to solving the following objectives:
+
+Definition 3. (Minimal and sufficient representations with label $( l 4 ) ,$ ). Let $Z _ { X ^ { 1 } } = G ( X ^ { 1 } )$ is the output of a deterministic latent mapping $G$ . A representation $Z _ { s u p }$ is said to be the sufficient label-related representation and $Z _ { \mathrm { s u p } ^ { * } }$ is said to be the minimal and sufficient representation $i f$ :
+
+$$
+Z _ { s u p } = \mathop { \mathrm { a r g m a x } } _ { G } I ( Z _ { X ^ { 1 } } ; Y ) ~ a n d ~ Z _ { \mathrm { s u p } ^ { * } } = \mathop { \mathrm { a r g m i n } } _ { Z _ { \mathrm { s u p } } } I ( Z _ { \mathrm { s u p } } ; X ^ { 1 } ) ~ s . t . ~ I ( Z _ { \mathrm { s u p } } ; Y )
+$$
+
+The learning procedure for definition 3 is illustrated in Figure 1: (a). The method is equivalent to employ compressed representations to reduce the complexity (redundant information) of ${ \bar { I } } ( Z _ { X ^ { 1 } } ; X ^ { 1 } )$ by minimizing and providing sufficient representation to class label $Y$ by maximizing $I ( Z _ { X ^ { 1 } } ; Y )$ . Similarly and motivated by multi-view information bottleneck settings (16), we present the objective of learning sufficient (and minimal) representations with domain-invariant information in the below definition:
+
+Definition 4. (Minimal and sufficient representations with domain-invariance $( I 6 )$ ). Let $Z _ { X ^ { 1 } } =$ $Q ( X ^ { 1 } )$ is the output of a deterministic domain invariant mapping $Q$ in the definition $^ { l }$ . Then $Z _ { I }$ is said to be the sufficient domain-invariant representation and $Z _ { I ^ { * } }$ is said to be the minimal and sufficient representation $i f$ :
+
+$$
+Z _ { I } = \underset { Q } { \operatorname { a r g m a x } } I ( Z _ { X ^ { 1 } } ; X ^ { 2 } ) \mathrm { ~ } a n d ~ Z _ { I ^ { * } } = \underset { Z _ { I } } { \operatorname { a r g m i n } } I ( Z _ { I } ; X ^ { 1 } ) \mathrm { ~ } s . t . \mathrm { ~ } I ( Z _ { I } ; X ^ { 2 } ) \mathrm { ~ } i s \mathrm { ~ } m a ^ { \dag }
+$$
+
+Definition 4 introduces a learning strategy for domain-invariance across domains (or views) that preserves shared information across two domains by maximizing $I ( Z _ { X ^ { 1 } } ; X ^ { 2 } )$ ; and also reduces specificity (redundant information) of the domain $S ^ { 1 }$ by minimizing $I ( Z _ { X ^ { 1 } } ; X ^ { 1 } )$ . We next present a lemma about the conditional independence between the latent representation $Z _ { X ^ { 1 } }$ and both the label $Y$ and the random variable $X ^ { 2 }$ when $Q , R ,$ , and $G$ are deterministic functions of the random variable $X ^ { 1 }$ :
+
+Lemma 1. (Determinism $( l 4 ) ,$ ) If $P ( Z _ { X ^ { 1 } } | X ^ { 1 } )$ is a Dirac delta function, then the following conditional independence holds: $Y$ ⊥⊥ $Z _ { X ^ { 1 } } | X ^ { 1 }$ and $X ^ { 2 }$ ⊥⊥ $Z _ { X ^ { 1 } } | X ^ { 1 }$ , inducing a Markov chain $X ^ { 2 }  Y  \mathbf { \bar { } } X ^ { 1 }  Z _ { X ^ { 1 } }$ .
+
+The proof of Lemma 1 is provided in Appendix A.1.
+
+Lemma 1 simply states that $Z _ { X ^ { 1 } }$ contains no more information than $X ^ { 1 }$ . Now, we show the ineffectiveness of only learning domain-invariant approach, based on the existence of the label-related domain-specific in the following theorem:
+
+Theorem 1. (Label-related information with domain-specificity) Assuming that there exists a domainspecific value $\varepsilon _ { 1 } > 0$ in domain $S ^ { 1 }$ (see Assumption $I$ ), the label-related representation - based learning approach (i.e., using $Z _ { s u p }$ and $Z _ { s u p ^ { * } }$ ) provides better prediction performance than the domain-invariant representation - based method (i.e., using $Z _ { I }$ and $Z _ { I ^ { * } }$ ). Formally,
+
+$$
+I ( X ^ { 1 } ; Y ) = I ( Z _ { s u p } ; Y ) = I ( Z _ { s u p ^ { * } } ; Y ) = I ( Z _ { I ^ { * } } ; Y ) + \varepsilon _ { 1 } > I ( Z _ { I ^ { * } } ; Y ) .
+$$
+
+The proof of Theorem 1 is mainly based on the result of Lemma 1, and is provided in Appendix A.2.
+
+The visualization of Theorem 1 is depicted by Figure 1: (b), where the domain-specific value for domain $S ^ { 2 }$ , $\varepsilon _ { 2 }$ is obtained in the same way as $\varepsilon _ { 1 }$ . It indicates that if an existing domain-specific representation has the positive corresponding information value $\varepsilon$ , the domain-invariance-based learning method provides an inferior prediction performance to our proposed method that incorporates both domain-invariant and label-related domain-specific.
+
+Now, Theorem 1 suggests that besides optimizing a domain-invariant mapping $Q$ as usual, we should jointly optimize domain-specific mapping $R$ to achieve a better generalization performance. However, in domain generalization, we are not allowed to access the target domain for training and must use $Q$ and $R$ from source domains. As pointed out in (10; 17; 7), although domain-invariant might be the same because it is unchanged across source domains, there is no guarantee whether this domain-specific information on the source domain is relevant to the target domain while making the prediction. Figure 1: (c) illustrates this case when the target domain has different $X ^ { T }$ and $Y ^ { T }$ , then some extracted domain-specific from source domains become redundant information. Therefore, the next raising question is how to learn domain-invariant and domain-specific effectively.
+
+To handle these shortcomings, we next propose a unified framework that jointly optimizes both $Q$ and $R$ by disentangling their feature representation. In particular, the deterministic mapping $Q$ is optimized by adversarial learning to extract useful domain-invariant features across domains. Meanwhile, by leveraging the transfer weight representations from meta-source domains to adapt to a meta-target domain, we apply meta-learning to deterministic mapping $R$ to force it to extract relevant domain-specific features of the target domain to improve generalization ability.
+
+# 2.3 Algorithm: meta-Domain Specific-Domain Invariant (mDSDI)
+
+So far, we have discussed the main ideas of our proposed method. Here we discuss implementation details for our proposed mDSDI approach.
+
+Figure 2 shows the graphical model and overview of our mDSDI framework. In particular, our unified network consists of the following components: a domain-invariant representation $Z _ { I } = Q _ { \theta _ { Q } } ( X )$ ; a domain-specific representation $Z _ { S } = R _ { \theta _ { R } } ( X )$ ; a domain discriminator $D _ { \theta _ { D _ { I } } } : Z _ { I } \to \overline { { { 1 , N } } }$ ; a domain classifier $D _ { \theta _ { D _ { S } } } : Z _ { S }  \overline { { { 1 , N } } }$ and a classifier $F _ { \theta _ { F } } : Z _ { I } \oplus Z _ { S } \to \mathcal { V } .$ . We also denote domain random variable by $D$ , sample space by $\mathcal { D }$ , and outcome by $d$ .
+
+Domain-Invariant and Domain-Specific Extraction. The domain-invariant representation $Z _ { I }$ defined in Definition 1, is obtained by using an adversarial training framework (3), in which the domain discriminator $D _ { \theta _ { D _ { I } } }$ tries to maximize the prediction probability of the domain label from the latent $Z _ { I }$ I, while the goal of the encoder $Q _ { \theta _ { Q } }$ is to map the sample $X$ to the latent $Z _ { I }$ , such that $D _ { \theta _ { D _ { I } } }$ cannot discriminate the domain of $X$ . This task can be performed by solving the following min-max game:
+
+$$
+\operatorname* { m i n } _ { \theta _ { Q } } \operatorname* { m a x } _ { \theta _ { D _ { I } } } \left\{ L _ { Z _ { I } } : = - \mathbb { E } _ { x , d \sim X , D } \left[ d \log D _ { I } ( Q ( x ) ) \right] \right\} .
+$$
+
+To extract the domain-specific $Z _ { S }$ defined in Definition 2, we propose the use of the domain classifier $D _ { \theta _ { D _ { S } } }$ , that is trained to predict the domain label from $Z _ { S }$ . The corresponding parameters $\theta _ { D _ { S } }$ and
+
+![](images/2fb1d00477803a4293ca60c05c0e62a26e208e119e7c3924bd0a57369cfabaad.jpg)  
+Figure 2: The graphical model (a) and overall architecture (b) for our proposed mDSDI, including: domaininvariant $Z _ { I }$ is optimized via adversarial training with domain discriminator $D _ { I }$ , domain-specific $Z _ { S }$ is optimized via domain classifier $D _ { S }$ , these latent $Z _ { I }$ and $\bar { Z } _ { S }$ are disentangled by using covariance matrix. To push them to contain label information, these latents are integrated into a classifier $F$ which is optimized via cross-entropy with the label $Y$ . To make the model able to adapt specific information from source to unseen domain while still remaining domain-invariance information across domains, we additionally push $Z _ { S }$ through a meta-learning procedure.
+
+$\theta _ { R }$ are, therefore, optimized with the objective function below:
+
+$$
+\operatorname* { m i n } _ { \theta _ { D _ { S } } , \theta _ { R } } \left\{ L _ { Z _ { S } } : = - \mathbb { E } _ { x , d \sim X , D } \left[ d \log D _ { S } ( R ( x ) ) \right] \right\} .
+$$
+
+Disentanglement between Domain-Invariant and Domain-Specific. The disentanglement condition between two random vectors $Z _ { I }$ and $Z _ { S }$ can be solved by forcing their covariance matrix, denoted by $\mathrm { C o v } ( Z _ { I } , Z _ { S } )$ close to 0. A detailed discussion of disentangled two representations is provided in Appendix B.3. The related parameters $( \theta _ { Q } , \theta _ { R } )$ are then updated in the following optimization problem:
+
+$$
+\operatorname* { m i n } _ { \theta _ { Q } , \theta _ { R } } \left\{ L _ { D } : = \mathbb { E } _ { x \sim X } \left[ \| \mathrm { C o v } ( Q ( x ) , R ( x ) ) \| _ { 2 } \right] \right\} ,
+$$
+
+where $\| \cdot \| _ { 2 }$ is the $L _ { 2 }$ norm.
+
+Sufficiency of domain-specific and domain-invariant w.r.t. the classification task. The goal of the classifier $F$ parameterized by $\theta _ { F }$ is to predict the label of the original sample $X$ based on the domain-invariant $Z _ { I }$ and domain-specific $Z _ { S }$ , i.e.,
+
+$$
+\hat { Y } = F _ { \theta _ { F } } ( Z _ { I } \oplus Z _ { S } ) ,
+$$
+
+where $\oplus$ denotes the concatenation operation. Then, the training process of $F$ is then performed by solving
+
+$$
+\operatorname* { m i n } _ { \theta _ { Q } , \theta _ { R } , \theta _ { F } } \left\{ L _ { T } : = - \mathbb { E } _ { x , y \sim X , Y } \left[ y \log F ( Q ( x ) , R ( x ) ) \right] \right\} .
+$$
+
+Meta-Training for Domain-Specific Information. To encourage the domain-specific representation $Z _ { S }$ to adapt information learned from the source domains to the unseen target domain, we introduce the use of meta-learning framework (11), targeting a robust generalization. Note that the domaininvariant feature $Z _ { I }$ remains during the meta-learning procedure. In particular, each source domain $S _ { m } , \ m \in \overline { { 1 , N } }$ is split into two sub-domains, namely meta-train $S _ { m r }$ and meta-test $S _ { m e }$ . The domain-specific parameters $\theta _ { R }$ and the classifier parameters $\theta _ { F }$ are then jointly optimized as follows:
+
+$$
+\operatorname* { m i n } _ { w } \left\{ L _ { T _ { m } } : = f \left( w - \nabla f \left( w , S _ { m r } \right) , S _ { m e } \right) \right\} ,
+$$
+
+where $w = ( \theta _ { R } , \theta _ { F } )$ and
+
+$$
+f \left( \boldsymbol { w } , \boldsymbol { S _ { m } } \right) = - \mathbb { E } _ { \boldsymbol { x } , \boldsymbol { y } \sim \boldsymbol { X } , \boldsymbol { Y } } \left[ \boldsymbol { y } \log \boldsymbol { F } ( \boldsymbol { Z } _ { I } , \boldsymbol { R } ( \boldsymbol { x } ) ) \right] .
+$$
+
+Training and Inference. The pseudo-code for training and inference processes of our proposed mDSDI framework is presented in Algorithm 1. Each iteration of the training process consists of two steps:
+
+i) First, we integrate the objective functions (1), (2), (3) and (5) to construct an objective function $L _ { A }$ defined as follows:
+
+$$
+\operatorname* { m i n } _ { \theta _ { Q } , \theta _ { D _ { S } } , \theta _ { R } , \theta _ { F } } \operatorname* { m a x } _ { \theta _ { D _ { I } } } \left\{ L _ { A } : = \lambda _ { Z _ { I } } L _ { Z _ { I } } + \lambda _ { Z _ { S } } L _ { Z _ { S } } + \lambda _ { D } L _ { D } + L _ { T } \right\} ,
+$$
+
+where $\lambda _ { Z _ { I } } , \lambda _ { Z _ { S } }$ and $\lambda _ { D }$ are selected as the balanced parameters.
+
+ii) The second step is to employ meta-training to adapt task-related domain-specific from source domains to unseen domains. In each mini-batch, the meta-train and meta-test are split, then the gradient transformation step from meta-train domains to the meta-test domain is performed by solving the optimization problem (6).
+
+Algorithm 1: Training and Inference processes of mDSDI
+
+Training Input: Source domain $S ^ { ( i ) }$ , encoder $Q _ { \theta _ { Q } }$ , $R _ { \theta _ { R } }$ , domain classifier $D _ { \theta _ { D _ { I } } }$ , $D _ { \theta _ { D _ { S } } }$ for $Z _ { I }$ ,   
+$Z _ { S }$ , task classifier $F _ { \theta _ { F } }$ , batch size $B$ , learning rate $\eta$ . Output: The optimal: $Q _ { \theta _ { Q } } ^ { * } , R _ { \theta _ { R } } ^ { * } , F _ { \theta _ { F } } ^ { * } ;$   
+for $i t e = 1 $ iterations do Sample $S _ { B }$ with a mini-batch $B$ for each domain $S ^ { ( i ) }$ ; Compute $L _ { A }$ using Eq. (8) and perform gradient update $\nabla _ { \theta _ { Q } , \theta _ { R } , \theta _ { D _ { I } } , \theta _ { D _ { S } } , \theta _ { F } } L _ { A }$ with $\eta$ .; for $j = 1  N$ (number of source domains) do Split Meta-train $S _ { B / j }$ , Meta-test $S _ { j }$ ; Meta-train: Perform gradient update $\nabla _ { { \boldsymbol { \theta } } _ { R } , { \boldsymbol { \theta } } _ { F } }$ by minimizing Eq. (7) with $S _ { B / j }$ and $\eta$ ; Meta-test: Compute $L _ { T _ { m } }$ using Eq. (6) with $S _ { j }$ and updated gradient from Meta-train; Meta-optimization: Perform gradient update $\dot { \nabla } _ { \theta _ { R } , \theta _ { F } } L _ { T _ { m } }$ with $\eta$ ; end   
+end   
+Inference Input: Target domain $S ^ { T }$ , optimal: $Q _ { \theta _ { Q } } ^ { * }$ , $R _ { \theta _ { R } } ^ { * }$ , $F _ { \theta _ { F } } ^ { * }$ . Output: $Y ^ { T }$ using Eq. (4);
+
+# 3 Experiments
+
+# 3.1 Experimental settings
+
+Dataset. To evaluate the effectiveness of the proposed method, we utilize 7 commonly used datasets including: Colored-MNIST (18): includes 70000 samples of dimension (2, 28, 28) in binary classification problem with noisy label, from MNIST over 3 domains with noisy rate $d \in \{ 0 . 1 , 0 . 3$ , $0 . 9 \}$ , Rotated-MNIST (19): contains 70000 samples of dimension $( 1 , 2 8 , 2 8 )$ and 10 classes, rotated from MNIST over 6 domains $d \in \{ 0 , 1 5 , 3 0 , 4 5 , 6 0 , 7 5 \}$ , VLCS (20): includes 10729 samples of dimension $( 3 , 2 2 4 , 2 2 4 )$ and 5 classes, over 4 photographic domains $d \in \{ \mathrm { C a l t e c h } 1 0 1$ , LabelMe, SUN09, $\mathrm { V O C } 2 0 0 7 \}$ , PACS (2): contains 9991 images of dimension (3, 224, 224) and 7 classes, over 4 domains $d \in \{$ artpaint, cartoon, sketches, photo $\}$ , Office-Home (21): has 15500 daily images of dimension $( 3 , 2 2 4 , 2 2 4 )$ and 65 categories, over 4 domains $d \in \{ \mathrm { a r t } .$ , clipart, product, real $\}$ , Terra Incognita (22): includes 24778 wild photographs of dimension $( 3 , 2 2 4 , 2 2 4 )$ and 10 animals, over 4 camera-trap domains $d \in \{ \mathrm { L 1 0 0 , L 3 8 , L 4 3 , L 4 6 } \}$ , and DomainNet (23): contains 586575 images of dimension (3, 224, 224) and 345 classes, over 6 domains $d \in \{$ clipart, infograph, painting, quickdraw, real, sketch $\}$ . The detail of each dataset is provided in Appendix C.1.
+
+Baseline. Following DomainBed (24) settings, we compare our model with 14 related methods in DG which are divided by 5 common techniques, including: Standard Empirical Risk Minimization: Empirical Risk Minimization (ERM (25)); domain-specific-learning: Group Distributionally Robust Optimization (GroupDRO (26)), Marginal Transfer Learning (MTL (1; 27)), Adaptive Risk Minimization (ARM (28)); Meta-learning: Meta-Learning for DG (MLDG (11)); domain-invariantlearning: Invariant Risk Minimization (IRM (18)), Deep CORrelation ALignment (CORAL (29)), Maximum Mean Discrepancy (MMD (30)), Domain Adversarial Neural Networks (DANN (31)), Class-conditional DANN (CDANN (32)), Risk Extrapolation (VREx (33)); Augmenting data: Interdomain Mixup (Mixup (34; 35; 36)), Style-Agnostic Networks (SagNets (37)), Representation Self Challenging (RSC (38)). The detail of each method is provided in Appendix C.2.
+
+We use the training-domain validation set technique as proposed in DomainBed (24) for model selection. In particular, for all datasets, we first merge the raw training and validation, then, we run the test three times with three different seeds. For each random seed, we randomly split training and validation and choose the model maximizing the accuracy on the validation set, then compute performance on the given test sets. The mean and standard deviation of classification accuracy from these three runs are reported. We evaluate generalization performance based on backbones MNIST-ConvNet (24) for MNIST datasets and ResNet-50 (39) for non-MNIST datasets to compare with the mentioned methods. Data-processing techniques, model architectures, hyper-parameters, and changes of objective functions during training are presented in detail in Appendix C.3 C.5. All source code to reproduce results are available at https://github.com/VinAIResearch/mDSDI.
+
+# 3.2 Results
+
+Table 1: Classification accuracy $( \% )$ for all algorithms and datasets summarization. Our mDSDI method achieves highest accuracy on average when comparing 14 popular DG algorithms across 7 benchmark datasets.   
+
+<table><tr><td>Method</td><td>CMNIST</td><td>RMNIST</td><td>VLCS</td><td>PACS</td><td>OfficeHome</td><td>TerraInc</td><td>DomainNet</td><td>Average</td></tr><tr><td>ERM (25)</td><td>51.5±0.1</td><td>98.0±0.0</td><td>77.5±0.4</td><td>85.5±0.2</td><td>66.5±0.3</td><td>46.1±1.8</td><td>40.9±0.1</td><td>66.6</td></tr><tr><td>IRM (18)</td><td>52.0±0.1</td><td>97.7±0.1</td><td>78.5±0.5</td><td>83.5±0.8</td><td>64.3±2.2</td><td>47.6±0.8</td><td>33.9±2.8</td><td>65.4</td></tr><tr><td>GroupDRO (26)</td><td>52.1±0.0</td><td>98.0±0.0</td><td>76.7±0.6</td><td>84.4±0.8</td><td>66.0±0.7</td><td>43.2±1.1</td><td>33.3±0.2</td><td>64.8</td></tr><tr><td>Mixup (34; 35;36)</td><td>52.1±0.2</td><td>98.0±0.1</td><td>77.4±0.6</td><td>84.6±0.6</td><td>68.1±0.3</td><td>47.9±0.8</td><td>39.2±0.1</td><td>66.7</td></tr><tr><td>MLDG (11)</td><td>51.5±0.1</td><td>97.9±0.0</td><td>77.2±0.4</td><td>84.9±1.0</td><td>66.8±0.6</td><td>47.7±0.9</td><td>41.2±0.1</td><td>66.7</td></tr><tr><td>CORAL (29)</td><td>51.5±0.1</td><td>98.0±0.1</td><td>78.8±0.6</td><td>86.2±0.3</td><td>68.7±0.3</td><td>47.6±1.0</td><td>41.5±0.1</td><td>67.5</td></tr><tr><td>MMD (30)</td><td>51.5±0.2</td><td>97.9±0.0</td><td>77.5±0.9</td><td>84.6±0.5</td><td>66.3±0.1</td><td>42.2±1.6</td><td>23.4±9.5</td><td>63.3</td></tr><tr><td>DANN (31)</td><td>51.5±0.3</td><td>97.8±0.1</td><td>78.6±0.4</td><td>83.6±0.4</td><td>65.9±0.6</td><td>46.7±0.5</td><td>38.3±0.1</td><td>66.1</td></tr><tr><td>CDANN (32)</td><td>51.7±0.1</td><td>97.9±0.1</td><td>77.5±0.1</td><td>82.6±0.9</td><td>65.8±1.3</td><td>45.8±1.6</td><td>38.3±0.3</td><td>65.6</td></tr><tr><td>MTL (1; 27)</td><td>51.4±0.1</td><td>97.9±0.0</td><td>77.2±0.4</td><td>84.6±0.5</td><td>66.4±0.5</td><td>45.6±1.2</td><td>40.6±0.1</td><td>66.2</td></tr><tr><td>SagNets (37)</td><td>51.7±0.0</td><td>98.0±0.0</td><td>77.8±0.5</td><td>86.3±0.2</td><td>68.1±0.1</td><td>48.6±1.0</td><td>40.3±0.1</td><td>67.2</td></tr><tr><td>ARM (28)</td><td>56.2±0.2</td><td>98.2±0.1</td><td>77.6±0.3</td><td>85.1±0.4</td><td>64.8±0.3</td><td>45.5±0.3</td><td>35.5±0.2</td><td>66.1</td></tr><tr><td>VREx (33)</td><td>51.8±0.1</td><td>97.9±0.1</td><td>78.3±0.2</td><td>84.9±0.6</td><td>66.4±0.6</td><td>46.4±0.6</td><td>33.6±2.9</td><td>65.6</td></tr><tr><td>RSC (38)</td><td>51.7±0.2</td><td>97.6±0.1</td><td>77.1±0.5</td><td>85.2±0.9</td><td>65.5±0.9</td><td>46.6±1.0</td><td>38.9±0.5</td><td>66.1</td></tr><tr><td>mDSDI(Ours)</td><td>52.2±0.2</td><td>98.0±0.1</td><td>79.0±0.3</td><td>86.2±0.2</td><td>69.2±0.4</td><td>48.1±1.4</td><td>42.8±0.1</td><td>67.9</td></tr></table>
+
+Table 1 summarizes the results of our experiments on 7 benchmark datasets when compared with mentioned methods. The full result per dataset and domain is provided in Appendix C.4. From these results, we draw three conclusions about our mDSDI model:
+
+Our mDSDI still preserves domain-invariant information. We observe in some target domains which have background-less images and assume only contain domain-invariant information such as Colored-MNIST, Rotated-MNIST, or Terra Incognita (similar observation in cartoon or sketch in PACS, clip-art or product in OfficeHome, and quickdraw in DomainNet. Full results in Appendix C.4), our mDSDI model still achieves competitive results with other baselines (e.g., $5 2 . 2 \%$ in ColoredMNIST, $9 8 . 0 \%$ in Rotated-MNIST, and $4 8 . 1 \%$ in Terra Incognita) which are based on domaininvariant-learning techniques such as DANN, C-DANN, CORAL, MMD, IRM, and VREx. Those results demonstrate the effectiveness of our adversarial training technique for extracting domaininvariant features. Furthermore, due to considering disentangled domain-invariant and domainspecific latent, even in situations where the samples do not have domain-specific, our model still performs well by retaining informative domain-invariance.
+
+Our mDSDI could capture the usefulness of domain-specific information. In contrast, we observe that in some target domains that have relevant domain-specific with source domains such as landscape background of the object class from photographic pictures in VLCS (similar observation in PACS such as dogs in the yard or guitars lying on a table in photo and art domain, bed in the room or bike parked on the street in the Art and Real-world domain of OfficeHome. Full results in Appendix C.4), mDSDI achieves significantly higher results than other methods (e.g., $7 9 . 0 \%$ in VLCS, $8 6 . 2 \%$ in PACS). This means that our domain-invariant features not only support generalization better but also our domain-specific ones cover helpful information in special scenarios such as backgrounds and colors related to objects in the classification task. Moreover, when comparing with other domain-specific based techniques such as GroupDRO, MTL, and ARM, the results showed that domain-specific features learned by meta-training from our model are more helpful than theirs and have captured useful domain-specific features from those object-background relations.
+
+Extending beyond the invariance view to usefulness domain-specific information is important. As shown in Table 1, our mDSDI has the highest average number with $6 7 . 9 \%$ (highlighted with statistically significant according to a $t$ -test at a significance level $\alpha = 0 . 0 5$ ). The reason why our model outperforms other baselines could be explained by the fact that their domain-invariant methods are not able to capture domain-specific information, and so have poor performance. Meanwhile, when comparing with other domain-specific based methods, their models only concentrate on domainspecific techniques, and so provide inferior domain-invariant information to our techniques in some background-less images. In contrast, due to considering disentangled domain-invariant and domainspecific features, and having the right strategy to learn each latent, our model captures both this useful information, hence, outperforms their results. Not only has the highest average number, but our method also dominates other methods on a known large-scale dataset such as $6 9 . 2 \%$ in Office-Home or $4 2 . 8 \%$ in DomainNet. This implies that besides the essential combination between domain-invariant and domain-specific, when the number of datasets increases, our method can extract more relevant information for complex tasks, such as classifying 345 classes in DomainNet. These results also mean that our model has a balance between informative domain-invariant and domainspecific features to adapt better to different environments than others, therefore showing the highest average in all settings.
+
+# 3.3 How does mDSDI work?
+
+![](images/fa9fdc3f39d19694c628748adf77167091b1d3b2559a70bb240ea60e48b62849.jpg)  
+Figure 3: Feature visualization for domain-invariant: (a): different colors represent different classes; (b): different colors indicate different domains. Feature visualization for domain-specific: (c): different colors represent different classes; (d): different colors indicate different domains. Source domain includes: art (red), cartoon (green), sketch (blue) while target domain is photo (black) in the domain plots. Best viewed in color (Zoom in for details).
+
+To better understand our framework, we visualize the distribution of the learned features with tSNE (40) to analyze the feature space of domain-invariant and domain-specific. As shown in Figure 3 on PACS Dataset, our domain-invariant extractor can minimize the distance between the distribution of the domains (see Figure 3: (b)). However, these domain-invariant features still make mistakes on the classification task, indicated by a mixture of points from different class labels in the middle (see Figure 3: (a)), and many of these points are from the target domain (black color in the Figure 3: (b)). Meanwhile, the domain-specific representation better distinguishes points by class label (see Figure 3: (c)). More importantly, the photo domain’s specific features (black) are close to the art domain (red) (see Figure 3: (d)). This is reasonable because only these two domains include backgrounds related to the object class. It implies that meta-training in our model well learns specific features that can be adapted to the new unseen domain.
+
+# 3.4 Ablation study: Important of mDSDI on the Background-Colored-MNIST dataset
+
+This section examines our system design by checking its performance under different settings in the real scenario with our generated dataset (a similar experiment with PACS benchmark dataset is in Appendix C.6).
+
+Background-Colored-MNIST dataset. Figure 4 illustrates our Background-Colored-MNIST, generated from the original MNIST. We assume the domain-invariant is the digit’s sketch and design the dataset so that the background color is domain-specific. As a result, on the unseen domain, domain-specific will be useful for the classification task. Specifically, the dataset includes three source domains $d _ { t r }$ , different by digit’s color red, green, blue	, generated from a subset with 1000 training images for MNIST per each domain. In each source domain, the background color is the same for intra-class images but different across classes. In the target domain, 10000 testing images of MNIST are colored for one target domain $d _ { t e }$ with digit color $\left\{ \mathrm { o r a n g e } \right\}$ . In this domain, each class’s background color is similar to the same class’s background color in one of three source domains.
+
+Importance of mDSDI. We aim to prove the combination of learning disentangled representation domain-specific, domain-invariant, and meta-training on domain-specific are important in this scenario. To do so, we compare our model under nine settings: learning domain-invariant only (DI), learning domain-specific only (DS), meta-training on domain-invariant (DI-Meta), meta-training on domain-specific (DS-Meta), a combination of domain-invariant and domain-specific without disentanglement loss $L _ { D }$ (DSDI-Without $L _ { D }$ ), a combination of domain-invariant and domain-specific without meta-training (DSDI-Without Meta), meta-training on both representation $Z _ { I }$ and $Z _ { S }$ (DSDIMeta), meta-training on domain-invariant without domain-specific (DSDI-Meta DI) and our proposed framework (mDSDI-Meta DS), which is meta-training on domain-specific without domain-invariant. Table 2 shows that our model is the best setting with $8 9 . 7 \%$ . It proves that combining domain-invariant and domain-specific is crucial by dominating the settings with only domain-invariant or domainspecific (DI, DI-Meta, DS, DS-Meta). Regarding disentangling two representations $Z _ { I }$ and $Z _ { S }$ , it is worth noting that without disentanglement loss $L _ { D }$ , the model only achieves $8 1 . 4 \%$ , which is lower than mDSDI. It reveals adding disentanglement loss $L _ { D }$ is essential to boost our model performance. Compared with meta-training on both, which only reaches $8 2 . 1 \%$ and on domain-invariant are $7 9 . 0 \%$ , it implies that meta-training is necessary but only for the domain-specific, and our arguments are reasonable. It also shows that our combination with domain-invariant and domain-specific is not easy: a deep ensemble between two neural networks, but existing a deep-down insight in our framework in DG, when compared with mDSDI-Without Meta which can be seen as a type of ensemble, is only around $8 0 . 4 \%$ .
+
+![](images/f6f4e395e2fdc76b3ae09c624dbf54bffba0a3cfc77dc72a48647677c428f3ba.jpg)  
+Figure 4: Background-Colored-MNIST Dataset, where source domains include $\{$ red, green, blue $\}$ digit colors and target domain has $\left\{ { \mathrm { o r a n g e } } \right\}$ color.
+
+Table 2: Classification accuracy $( \% )$ on BackgroundColored-MNIST. Ablation study shows impact of domain-invariant when combined with meta-training on domain-specific in our method.   
+
+<table><tr><td>Method</td><td>Accuracy</td></tr><tr><td>DI</td><td>65.7±4.6</td></tr><tr><td>DI-Meta</td><td>63.6±5.1</td></tr><tr><td>DS</td><td>70.7±4.8</td></tr><tr><td>DS-Meta</td><td>75.3±3.4</td></tr><tr><td>DSDI-Without L D</td><td>81.4±2.6</td></tr><tr><td>DSDI-WithoutMeta</td><td>80.4±1.7</td></tr><tr><td>DSDI-Meta</td><td>82.1±1.4</td></tr><tr><td>DSDI-Meta DI</td><td>79.0±2.3</td></tr><tr><td>mDSDI-Meta DS (Ours)</td><td>89.7±0.8</td></tr></table>
+
+# 4 Conclusion and Discussion
+
+Despite being aware of the importance of domain-specific information, little investigation into the theory and a rigorous algorithm to explore its representation. To the best of our knowledge, our work provides the first theoretical analysis to understand and realize the efficiency of domain-specific information in domain generalization. The domain-specific contains unique characteristics and when combined with domain-invariant information can significantly aid performance on unseen domains. Following our theoretical insights based on the information bottleneck principle, we propose a mDSDI algorithm which disentangles these features. We next introduce the use of the meta-training scheme to support domain-specific to adapt information from source domains to unseen domains. Our experimental results demonstrate mDSDI brings out competitive results with related approaches in domain generalization. In addition, the ablation study with our Background-Colored-MNIST further illustrated and demonstrated the efficiency of combining domain-invariant and domain-specific via our proposed mDSDI. Our theoretical analysis and proposed mDSDI framework can facilitate fundamental progress in understanding the behavior of both domain-invariant and domain-specific representation in domain generalization.
+
+Toward a robustness algorithm that can effectively learn both domain-invariant and domain-specific features, there are certainly many challenges that remain in our paper, for example, a theorem to explain when domain-specific may hurt performance in the unseen domain, a stronger connection between theory in implementation, a method to make two representations to be non-linearly independent as well as a lower computational cost of the covariance matrix. In the future, we plan to continue tackling these challenges to provide a better understanding and learning framework in domain generalization then extending to broader settings of transfer learning. A detailed clarification, discussion, and plausible methods in the future work are provided in Appendix B.
+
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parse/train/bajTv_cvkI/bajTv_cvkI_content_list.json

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+        "type": "text",
+        "text": "Exploiting Domain-Specific Features to Enhance Domain Generalization ",
+        "text_level": 1,
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+        "text": "Manh-Ha Bui1 Toan Tran1 Anh Tuan Tran1 Dinh Phung1,2 1 VinAI Research, Vietnam 2 Monash University, Australia {v.habm1, v.toantm3, v.anhtt152, v.dinhpq2}@vinai.io ∗ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        ],
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+    },
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+        "type": "text",
+        "text": "Domain Generalization (DG) aims to train a model, from multiple observed source domains, in order to perform well on unseen target domains. To obtain the generalization capability, prior DG approaches have focused on extracting domaininvariant information across sources to generalize on target domains, while useful domain-specific information which strongly correlates with labels in individual domains and the generalization to target domains is usually ignored. In this paper, we propose meta-Domain Specific-Domain Invariant (mDSDI) - a novel theoretically sound framework that extends beyond the invariance view to further capture the usefulness of domain-specific information. Our key insight is to disentangle features in the latent space while jointly learning both domain-invariant and domainspecific features in a unified framework. The domain-specific representation is optimized through the meta-learning framework to adapt from source domains, targeting a robust generalization on unseen domains. We empirically show that mDSDI provides competitive results with state-of-the-art techniques in DG. A further ablation study with our generated dataset, Background-Colored-MNIST, confirms the hypothesis that domain-specific is essential, leading to better results when compared with only using domain-invariant. ",
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+        "text": "1 Introduction and Related work ",
+        "text_level": 1,
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+        "text": "Domain Generalization (DG) has recently become an important research topic in machine learning due to its real-world applicability and its close connection to the way humans generalize to learn in a new domain. In a DG framework, the learner is trained on multiple datasets collected under different environments without any access to any data on the target domain (1). One of the most notable approaches to this problem is to learn the “domain-invariant” features across these training datasets, with the assumption that these invariant representations are also held in unseen target domains (2; 3; 4; 5; 6). While this has been shown to work well in practice, its key drawback is completely ignoring “domain-specific” information that could aid the generalization performance, especially when the number of source domains increases (7). ",
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+        "text": "For instance, consider the problem of classifying dog or fish images from two source domains: sketch and photo. While the sketch contains a conceptual drawing of the animal, the photo includes their taken picture within a background. In this case, sketch domain-invariant is kept across domains, while domain-specific, e.g., a dog in a house or fish in the ocean, will be discarded due to only existing in the photo domain. However, this background information, when present, could lead to an improvement of the classification performance in target domains due to common association between the objects of its background, and when negligent sketches are hard to distinguish. From a theoretical standpoint, there has also been strong recent evidence to indicate the insufficiency of learning domaininvariant representation for successful adaptation in domain adaptation problems (8; 9). For example, ",
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+        "text": "Zhao et al. (8) has pointed out the degradation in target predictive performance if domain-invariant representations are forced while the marginal label distributions on the source and target domains are overly different. ",
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+        "text": "Utilizing domain-specific features in DG has been widely studied in recent works (e.g., (10; 7)). Ding and Fu (10) introduce multiple domain-specific networks for each domain, then use the structured low-rank constraints to align them with domain-invariant. While this encourages the better transfer of knowledge, its main problem is the requirement of too many domain-specific networks. More recently, Chattopadhyay et al. (7) proposed a masking strategy to disentangle domain-invariant and domain-specific to further boost domain-specific learning, but its key drawback is that domaininvariant/domain-specific representations might not be disentangled since the learning and inferring procedures are performed implicitly (i.e., without any theoretical guarantee) through a mask generalization process. That means it lacks a clear motivation as well as theoretical justifications. ",
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+        "text": "Regarding meta-learning related work, a typical approach involving meta-learning in DG is MLDG (11) that is based on gradient update which simulates train/test domain shift within each mini-batch, mainly to learn transferable weight representations from meta-source domains to quickly adapt to the meta-target domain, and so improve generalization ability. However, their task objective adapts for all representation features which include domain-invariant, since low effectiveness because domain-invariant is stable across domains, pushing to adapt those features might affect the stability of those domain-invariant, leading to a lower generalization performance on the target domain. ",
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+    {
+        "type": "text",
+        "text": "To handle these domain-invariant shortcomings, in this paper, we propose a novel theoretically sound DG approach that aims to extract label-informative domain-specific and then explicitly disentangles the domain-invariant and domain-specific representations in an efficient way without training multiple networks for domain-specific. Following the meta-learning idea and mitigating previous work’s drawbacks, we apply a meta-learning technique specifically to exploit domain-specific quality which should need to be adapted to unseen domains from source domains. Our contributions in this work are summarized as follows: ",
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+        "text": "• We provide a theoretical analysis based on the information bottleneck principle to point out the limitation of only learning invariant and the importance of domain-specific representation by a certainly plausible assumption.   \n• We then develop a rigorous framework to formulate elements of domain-invariant/domainspecific representations, in which our key insight is to introduce an effective metaoptimization training framework (11) to learn domain-specific representation from multiple training domains. Without accessing any data from unseen target domains, the meta-training procedure provides a suitable mechanism to self-learn domain-specific representation. We term our approach meta-Domain Specific-Domain Invariant (mDSDI) and provide necessary theoretical verifications for it.   \n• To demonstrate the merit of the proposed mDSDI framework, we extensively evaluate mDSDI on several state-of-the-art DG benchmark datasets, including Colored-MNIST, Rotated-MNIST, VLCS, PACS, Office-Home, Terra Incognita, DomainNet in addition to our newly created Background-Colored-MNIST for the ablation study to examine the behavior of our mDSDI. ",
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+    {
+        "type": "text",
+        "text": "2 Methodology ",
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+        "text": "2.1 Problem setting and Definitions ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Let $\\boldsymbol { \\mathcal { X } } \\subset \\mathbb { R } ^ { D }$ be the sample space and $\\mathcal { V } \\subset \\mathbb { R }$ the label space. Denote the set of joint probability distributions on $\\mathcal { X } \\times \\mathcal { V }$ by $\\mathcal { P } _ { \\mathcal { X } \\times \\mathcal { Y } }$ , and the set of probability marginal distributions on $\\mathcal { X }$ by $\\mathcal { P } _ { \\mathcal { X } }$ . A domain is defined by a joint distribution $P ( x , y ) \\in \\mathcal P _ { \\mathcal { X } \\times \\mathcal { Y } }$ , and let $\\mathcal { P }$ be a measure on $\\mathcal { P } _ { \\mathcal { X } \\times \\mathcal { Y } }$ , i.e., whose realizations are distributions on $\\mathcal { X } \\times \\mathcal { V }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Denote $N$ source domains by $S ^ { ( i ) } = \\{ ( x _ { j } ^ { ( i ) } , y _ { j } ^ { ( i ) } ) \\} _ { j = 1 } ^ { n _ { i } }$ , $i = 1 , \\ldots , N$ , where $n _ { i }$ is the number of data points in $S ^ { ( i ) }$ , i.e., $( x _ { j } ^ { ( i ) } , y _ { j } ^ { ( i ) } ) \\stackrel { i i d } { \\sim } P ^ { ( i ) } ( x , y )$ where $P ^ { ( i ) } ( x , y ) \\sim \\mathcal { P }$ ; and $x _ { j } ^ { ( i ) } \\sim P _ { \\mathcal { X } } ^ { ( i ) }$ , in which $P _ { \\mathcal { X } } ^ { ( i ) } \\sim P _ { \\mathcal { X } }$ . In a typical DG framework, a learning model which is only trained on the set of source domains $\\{ S ^ { ( i ) } \\} _ { i = 1 } ^ { N }$ without any access to the (unlabeled) data points in the target domain, arrives at a good generalization performance on the test dataset $S ^ { T } = \\{ ( x _ { j } ^ { T } , y _ { j } ^ { T } ) \\} _ { j = 1 } ^ { n _ { T } }$ , where $( x _ { j } ^ { T } , y _ { j } ^ { T } ) \\overset { i i d } { \\sim } P ^ { T } ( x , y )$ and $P ^ { T } ( x , y ) \\sim \\mathcal { P }$ . ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "First, we present the definition of domain-invariant representation in a latent space $\\mathcal { Z }$ under covariate shift assumption (i.e., the conditional distribution $P ( \\boldsymbol { y } | \\boldsymbol { x } )$ is unchanged across the source domains): ",
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+        "type": "text",
+        "text": "Definition 1. A feature extraction mapping $Q : \\mathcal { X }  \\mathcal { Z }$ is said to be domain-invariant if the distribution $P _ { Q } ( Q ( X ) )$ is unchanged across the source domains, i.e., $\\forall i , j = 1 , \\ldots , N , i \\neq j w e$ have $P _ { Q } ^ { ( i ) } ( Q ( X ) ) \\equiv P _ { Q } ^ { ( j ) } ( Q ( X ) )$ , where $P _ { Q } ^ { ( i ) } ( Q ( X ) ) = P _ { Q } ( Q ( X ) | X \\sim P _ { \\ X } ^ { ( i ) } )$ , $i = 1 , \\ldots , N .$ . In this case, the corresponding latent representation $Z _ { I } = Q ( X )$ is then called the domain-invariant representation (see (3) also). ",
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+        "type": "text",
+        "text": "As mentioned in the example in the introduction part, the definition 1 reveals that the extracted domain-invariant latent $Z _ { I }$ could be the conceptual drawing of the animal which is shared in both sketch and photo domains. However, when existing background information is taken by a picture such as a house or ocean, it is crucial to take these backgrounds into account because the domain-invariant feature extraction $Q$ might ignore them by only existing in the photo domain. Therefore, we next introduce the definition of domain-specific in latent space as follows: ",
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+        "type": "text",
+        "text": "Definition 2. A feature extraction mapping $R : \\mathcal { X }  \\mathcal { Z }$ is said to be domain-specific $i f \\ \\forall i , j =$ $1 , \\dots , N , \\ i \\ \\neq \\ j$ such that $P _ { R } ^ { ( i ) } ( R ( X ) ) \\neq P _ { R } ^ { ( j ) } ( R ( X ) )$ , where $P _ { R } ^ { ( i ) } ( R ( X ) ) = P _ { R } ( R ( X ) | X \\sim$ $P _ { \\mathcal { X } } ^ { ( i ) } )$ ), $i = 1 , \\ldots , N$ . In this case, given $X \\sim P _ { \\mathcal { X } } ^ { ( i ) }$ the corresponding latent representation $Z _ { S } ^ { ( i ) } = $ $R ( X )$ is then called the domain-specific representation w.r.t. the domain . ",
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+        "type": "text",
+        "text": "Definition 2 states that for any domain $i$ and $j$ , the distributions of the domain-specific latent $P _ { R } ^ { ( i ) } ( R ( X ) )$ and ven t $P _ { R } ^ { ( j ) } ( R ( X ) )$ must bes from comand letely different. For instance, following our menthat are sketch and photo domain, the mapping $i$ $j$ $R ( X )$ should extract specific information that only belongs to the domain including the shadow of the fish drawing in the sketch and ocean background information in the photo domain. ",
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+        "type": "text",
+        "text": "To this end, this paper aims to show that only learning domain-invariant will limit the prediction performance and generalization ability. Hence, we next provide a formal explanation for the motivation of learning domain-specific, by showing the potential drawback of only learning domain-invariance in terms of predicting class labels. ",
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+            "2.2 A theoretical analysis under the Information bottleneck method ",
+            "Figure 1: Venn diagram showing relationships between source domains represented by $X ^ { 1 }$ , $X ^ { 2 }$ , target domain represented by $X ^ { T }$ , and label $Y$ . (a) The learning procedure of minimal and sufficient label-related representation in definition 3. (b) Explaining the theorem 1 where the domain-invariant based method provides an inferior prediction performance to our proposed method that incorporates both domain-invariant $I ( Z _ { I ^ { * } } ; Y )$ and labelrelated domain-specific values $\\epsilon$ made by our assumption 1. (c) A case when the unseen (target) domain has different $X ^ { T }$ and $Y ^ { T }$ , while domain-invariant information is still stable across domains, some domain-specific in source domains become redundant information. "
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+    {
+        "type": "text",
+        "text": "Notations. Given three arbitrary random variables $A , B$ , and $C$ , let us use $I ( A ; B )$ to represent mutual information between $A$ and $B$ ; $I ( A ; B | C )$ to represent conditional mutual information of $A$ and $B$ given $C$ ; $H ( A )$ to represent entropy of $A$ ; and $H ( A | B )$ to represent conditional entropy for random variables $A$ given $B$ . For simplicity, we consider the case with two source domains $S ^ { 1 } , S ^ { 2 }$ (the results with multiple source domains can be naturally extended from there). We also define two corresponding random variables $X ^ { i } \\sim P _ { \\mathcal { X } } ^ { ( i ) }$ , that are sampled from the marginal distribution $P _ { \\mathcal { X } }$ in the domain $S ^ { i } , i = 1 , 2$ . ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Figure 1 illustrates all the definitions and assumptions above used for our theoretical verification (in Theorem 1). In particular, in that figure, each of the four colored rectangles represents an individual entropy: $H ( X ^ { 1 } )$ for domain $S ^ { 1 }$ is in blue, $H ( X ^ { 2 } )$ for domain $S ^ { 2 }$ is in red, $\\overset { \\vartriangle } { \\boldsymbol { H } } ( X ^ { T } )$ for the target domain is in black, and $H ( Y )$ for the class label is in green border rectangle. ",
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+        "type": "text",
+        "text": "We first show the ineffectiveness of only learning domain-invariant information when compared with incorporating domain-specific in source domain $S ^ { 1 }$ (and similarly with domain $S ^ { 2 }$ ). Our justification partly relies on the following assumption about the correlation between the domainspecific representation and the class label: ",
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+        "type": "text",
+        "text": "Assumption 1. (Label-correlated domain-specificity) Assuming that there exists a domain-specific representation $Z _ { S } ^ { ( 1 ) }$ extracted by the deterministic mapping $Z _ { S } ^ { ( 1 ) } = R ( X ^ { 1 } )$ in definition 2, which correlates with label in domain $S ^ { 1 }$ such that $I ( Z _ { S } ^ { ( 1 ) } ; Y | X ^ { 2 } ) = I ( X ^ { 1 } ; Y | X ^ { 2 } ) = \\varepsilon _ { 1 }$ , where $\\varepsilon _ { 1 } > 0$ is a constant. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Assumption 1 indicates that, for the source domain $S ^ { 1 }$ , we can learn $Z _ { S } ^ { ( 1 ) } = R ( X ^ { 1 } )$ such that $I ( Z _ { S } ^ { ( 1 ) } ; Y | X ^ { 2 } )$ is strictly positive and equals to $I ( X ^ { 1 } ; Y | X ^ { 2 } )$ , where $I ( X ^ { 1 } ; Y | X ^ { 2 } )$ is the specific information that correlates with the label in the domain $S ^ { 1 }$ , but not in the domain $S ^ { 2 }$ (12). For instance, in the example mentioned in the introduction, if domain $S ^ { 1 }$ is “photo” while $S ^ { 2 }$ is “sketch”, the value of $\\epsilon _ { 1 }$ should be positive because the background information such as a house, the ocean also provides information to predict whether the object is a dog or fish without considering its conceptual drawing. This assumption is particularly valid and practically plausible and is demonstrated by several examples observed in our experiments. For instance, for the DomainNet benchmark dataset, in the real-world domain, many bed pictures contain a bed in the room or bike pictures that have bicycles parked on the street. Other examples are in PACS such as dogs in the yard or guitars lying on a table in photo and art domains. These examples are strongly related to assumption 1, in which specific information correlates with labels in a particular domain. ",
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+        "type": "text",
+        "text": "We next present supervised learning frameworks under the umbrella of the information theory (13; 14) and the information bottleneck method (13; 15) that generalizes minimal sufficient statistics to the minimal (i.e., less complexity) and sufficient (i.e, better fidelity) representations. The learning process of such representations is equivalent to solving the following objectives: ",
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+        "type": "text",
+        "text": "Definition 3. (Minimal and sufficient representations with label $( l 4 ) ,$ ). Let $Z _ { X ^ { 1 } } = G ( X ^ { 1 } )$ is the output of a deterministic latent mapping $G$ . A representation $Z _ { s u p }$ is said to be the sufficient label-related representation and $Z _ { \\mathrm { s u p } ^ { * } }$ is said to be the minimal and sufficient representation $i f$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/63288722cadfc752178bc22f9618858ec4de8c3aeacecdb3894f1e1b28e2d13e.jpg",
+        "text": "$$\nZ _ { s u p } = \\mathop { \\mathrm { a r g m a x } } _ { G } I ( Z _ { X ^ { 1 } } ; Y ) ~ a n d ~ Z _ { \\mathrm { s u p } ^ { * } } = \\mathop { \\mathrm { a r g m i n } } _ { Z _ { \\mathrm { s u p } } } I ( Z _ { \\mathrm { s u p } } ; X ^ { 1 } ) ~ s . t . ~ I ( Z _ { \\mathrm { s u p } } ; Y )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The learning procedure for definition 3 is illustrated in Figure 1: (a). The method is equivalent to employ compressed representations to reduce the complexity (redundant information) of ${ \\bar { I } } ( Z _ { X ^ { 1 } } ; X ^ { 1 } )$ by minimizing and providing sufficient representation to class label $Y$ by maximizing $I ( Z _ { X ^ { 1 } } ; Y )$ . Similarly and motivated by multi-view information bottleneck settings (16), we present the objective of learning sufficient (and minimal) representations with domain-invariant information in the below definition: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Definition 4. (Minimal and sufficient representations with domain-invariance $( I 6 )$ ). Let $Z _ { X ^ { 1 } } =$ $Q ( X ^ { 1 } )$ is the output of a deterministic domain invariant mapping $Q$ in the definition $^ { l }$ . Then $Z _ { I }$ is said to be the sufficient domain-invariant representation and $Z _ { I ^ { * } }$ is said to be the minimal and sufficient representation $i f$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/51060c7a439f2fef09882ae59bfc266026d91824d92149e8b9b985d9364eb8c5.jpg",
+        "text": "$$\nZ _ { I } = \\underset { Q } { \\operatorname { a r g m a x } } I ( Z _ { X ^ { 1 } } ; X ^ { 2 } ) \\mathrm { ~ } a n d ~ Z _ { I ^ { * } } = \\underset { Z _ { I } } { \\operatorname { a r g m i n } } I ( Z _ { I } ; X ^ { 1 } ) \\mathrm { ~ } s . t . \\mathrm { ~ } I ( Z _ { I } ; X ^ { 2 } ) \\mathrm { ~ } i s \\mathrm { ~ } m a ^ { \\dag }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Definition 4 introduces a learning strategy for domain-invariance across domains (or views) that preserves shared information across two domains by maximizing $I ( Z _ { X ^ { 1 } } ; X ^ { 2 } )$ ; and also reduces specificity (redundant information) of the domain $S ^ { 1 }$ by minimizing $I ( Z _ { X ^ { 1 } } ; X ^ { 1 } )$ . We next present a lemma about the conditional independence between the latent representation $Z _ { X ^ { 1 } }$ and both the label $Y$ and the random variable $X ^ { 2 }$ when $Q , R ,$ , and $G$ are deterministic functions of the random variable $X ^ { 1 }$ : ",
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+    {
+        "type": "text",
+        "text": "Lemma 1. (Determinism $( l 4 ) ,$ ) If $P ( Z _ { X ^ { 1 } } | X ^ { 1 } )$ is a Dirac delta function, then the following conditional independence holds: $Y$ ⊥⊥ $Z _ { X ^ { 1 } } | X ^ { 1 }$ and $X ^ { 2 }$ ⊥⊥ $Z _ { X ^ { 1 } } | X ^ { 1 }$ , inducing a Markov chain $X ^ { 2 }  Y  \\mathbf { \\bar { } } X ^ { 1 }  Z _ { X ^ { 1 } }$ . ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "The proof of Lemma 1 is provided in Appendix A.1. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1 simply states that $Z _ { X ^ { 1 } }$ contains no more information than $X ^ { 1 }$ . Now, we show the ineffectiveness of only learning domain-invariant approach, based on the existence of the label-related domain-specific in the following theorem: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 1. (Label-related information with domain-specificity) Assuming that there exists a domainspecific value $\\varepsilon _ { 1 } > 0$ in domain $S ^ { 1 }$ (see Assumption $I$ ), the label-related representation - based learning approach (i.e., using $Z _ { s u p }$ and $Z _ { s u p ^ { * } }$ ) provides better prediction performance than the domain-invariant representation - based method (i.e., using $Z _ { I }$ and $Z _ { I ^ { * } }$ ). Formally, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/e5fc9955e1f37c4f35b059a061a60ccc327f9b1fb76a4b56a44432b9f3432e25.jpg",
+        "text": "$$\nI ( X ^ { 1 } ; Y ) = I ( Z _ { s u p } ; Y ) = I ( Z _ { s u p ^ { * } } ; Y ) = I ( Z _ { I ^ { * } } ; Y ) + \\varepsilon _ { 1 } > I ( Z _ { I ^ { * } } ; Y ) .\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "The proof of Theorem 1 is mainly based on the result of Lemma 1, and is provided in Appendix A.2. ",
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+    {
+        "type": "text",
+        "text": "The visualization of Theorem 1 is depicted by Figure 1: (b), where the domain-specific value for domain $S ^ { 2 }$ , $\\varepsilon _ { 2 }$ is obtained in the same way as $\\varepsilon _ { 1 }$ . It indicates that if an existing domain-specific representation has the positive corresponding information value $\\varepsilon$ , the domain-invariance-based learning method provides an inferior prediction performance to our proposed method that incorporates both domain-invariant and label-related domain-specific. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Now, Theorem 1 suggests that besides optimizing a domain-invariant mapping $Q$ as usual, we should jointly optimize domain-specific mapping $R$ to achieve a better generalization performance. However, in domain generalization, we are not allowed to access the target domain for training and must use $Q$ and $R$ from source domains. As pointed out in (10; 17; 7), although domain-invariant might be the same because it is unchanged across source domains, there is no guarantee whether this domain-specific information on the source domain is relevant to the target domain while making the prediction. Figure 1: (c) illustrates this case when the target domain has different $X ^ { T }$ and $Y ^ { T }$ , then some extracted domain-specific from source domains become redundant information. Therefore, the next raising question is how to learn domain-invariant and domain-specific effectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To handle these shortcomings, we next propose a unified framework that jointly optimizes both $Q$ and $R$ by disentangling their feature representation. In particular, the deterministic mapping $Q$ is optimized by adversarial learning to extract useful domain-invariant features across domains. Meanwhile, by leveraging the transfer weight representations from meta-source domains to adapt to a meta-target domain, we apply meta-learning to deterministic mapping $R$ to force it to extract relevant domain-specific features of the target domain to improve generalization ability. ",
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+    {
+        "type": "text",
+        "text": "2.3 Algorithm: meta-Domain Specific-Domain Invariant (mDSDI) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "So far, we have discussed the main ideas of our proposed method. Here we discuss implementation details for our proposed mDSDI approach. ",
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+    {
+        "type": "text",
+        "text": "Figure 2 shows the graphical model and overview of our mDSDI framework. In particular, our unified network consists of the following components: a domain-invariant representation $Z _ { I } = Q _ { \\theta _ { Q } } ( X )$ ; a domain-specific representation $Z _ { S } = R _ { \\theta _ { R } } ( X )$ ; a domain discriminator $D _ { \\theta _ { D _ { I } } } : Z _ { I } \\to \\overline { { { 1 , N } } }$ ; a domain classifier $D _ { \\theta _ { D _ { S } } } : Z _ { S }  \\overline { { { 1 , N } } }$ and a classifier $F _ { \\theta _ { F } } : Z _ { I } \\oplus Z _ { S } \\to \\mathcal { V } .$ . We also denote domain random variable by $D$ , sample space by $\\mathcal { D }$ , and outcome by $d$ . ",
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+        "type": "text",
+        "text": "Domain-Invariant and Domain-Specific Extraction. The domain-invariant representation $Z _ { I }$ defined in Definition 1, is obtained by using an adversarial training framework (3), in which the domain discriminator $D _ { \\theta _ { D _ { I } } }$ tries to maximize the prediction probability of the domain label from the latent $Z _ { I }$ I, while the goal of the encoder $Q _ { \\theta _ { Q } }$ is to map the sample $X$ to the latent $Z _ { I }$ , such that $D _ { \\theta _ { D _ { I } } }$ cannot discriminate the domain of $X$ . This task can be performed by solving the following min-max game: ",
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+    {
+        "type": "equation",
+        "img_path": "images/38d18a9f73e28c22b1315f6c0ad1de8c093d64dd338392ba66a48e304ba402dd.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta _ { Q } } \\operatorname* { m a x } _ { \\theta _ { D _ { I } } } \\left\\{ L _ { Z _ { I } } : = - \\mathbb { E } _ { x , d \\sim X , D } \\left[ d \\log D _ { I } ( Q ( x ) ) \\right] \\right\\} .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "To extract the domain-specific $Z _ { S }$ defined in Definition 2, we propose the use of the domain classifier $D _ { \\theta _ { D _ { S } } }$ , that is trained to predict the domain label from $Z _ { S }$ . The corresponding parameters $\\theta _ { D _ { S } }$ and ",
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+    {
+        "type": "image",
+        "img_path": "images/2fb1d00477803a4293ca60c05c0e62a26e208e119e7c3924bd0a57369cfabaad.jpg",
+        "image_caption": [
+            "Figure 2: The graphical model (a) and overall architecture (b) for our proposed mDSDI, including: domaininvariant $Z _ { I }$ is optimized via adversarial training with domain discriminator $D _ { I }$ , domain-specific $Z _ { S }$ is optimized via domain classifier $D _ { S }$ , these latent $Z _ { I }$ and $\\bar { Z } _ { S }$ are disentangled by using covariance matrix. To push them to contain label information, these latents are integrated into a classifier $F$ which is optimized via cross-entropy with the label $Y$ . To make the model able to adapt specific information from source to unseen domain while still remaining domain-invariance information across domains, we additionally push $Z _ { S }$ through a meta-learning procedure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "$\\theta _ { R }$ are, therefore, optimized with the objective function below: ",
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+        "page_idx": 5
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+    {
+        "type": "equation",
+        "img_path": "images/088092ebc2a521af33fa77c578f34cee9b016eee6e6d50d9937733bd12511afe.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta _ { D _ { S } } , \\theta _ { R } } \\left\\{ L _ { Z _ { S } } : = - \\mathbb { E } _ { x , d \\sim X , D } \\left[ d \\log D _ { S } ( R ( x ) ) \\right] \\right\\} .\n$$",
+        "text_format": "latex",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Disentanglement between Domain-Invariant and Domain-Specific. The disentanglement condition between two random vectors $Z _ { I }$ and $Z _ { S }$ can be solved by forcing their covariance matrix, denoted by $\\mathrm { C o v } ( Z _ { I } , Z _ { S } )$ close to 0. A detailed discussion of disentangled two representations is provided in Appendix B.3. The related parameters $( \\theta _ { Q } , \\theta _ { R } )$ are then updated in the following optimization problem: ",
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+        "img_path": "images/10be3e4b3c412759b4c85fdcf7b2ac028adf205bca79a811187fd0acb83dc4e9.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta _ { Q } , \\theta _ { R } } \\left\\{ L _ { D } : = \\mathbb { E } _ { x \\sim X } \\left[ \\| \\mathrm { C o v } ( Q ( x ) , R ( x ) ) \\| _ { 2 } \\right] \\right\\} ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "where $\\| \\cdot \\| _ { 2 }$ is the $L _ { 2 }$ norm. ",
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+        "type": "text",
+        "text": "Sufficiency of domain-specific and domain-invariant w.r.t. the classification task. The goal of the classifier $F$ parameterized by $\\theta _ { F }$ is to predict the label of the original sample $X$ based on the domain-invariant $Z _ { I }$ and domain-specific $Z _ { S }$ , i.e., ",
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+        "type": "equation",
+        "img_path": "images/fde077f58a681b76ff41c06320b8a5ee38343efa8f20849aa3fccf9a955206b9.jpg",
+        "text": "$$\n\\hat { Y } = F _ { \\theta _ { F } } ( Z _ { I } \\oplus Z _ { S } ) ,\n$$",
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+        "type": "text",
+        "text": "where $\\oplus$ denotes the concatenation operation. Then, the training process of $F$ is then performed by solving ",
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+        "type": "equation",
+        "img_path": "images/6ffd71e3f2796faf8b4c6fe72c6ad753ab951e72d452ba79e1a0937796e10435.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta _ { Q } , \\theta _ { R } , \\theta _ { F } } \\left\\{ L _ { T } : = - \\mathbb { E } _ { x , y \\sim X , Y } \\left[ y \\log F ( Q ( x ) , R ( x ) ) \\right] \\right\\} .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Meta-Training for Domain-Specific Information. To encourage the domain-specific representation $Z _ { S }$ to adapt information learned from the source domains to the unseen target domain, we introduce the use of meta-learning framework (11), targeting a robust generalization. Note that the domaininvariant feature $Z _ { I }$ remains during the meta-learning procedure. In particular, each source domain $S _ { m } , \\ m \\in \\overline { { 1 , N } }$ is split into two sub-domains, namely meta-train $S _ { m r }$ and meta-test $S _ { m e }$ . The domain-specific parameters $\\theta _ { R }$ and the classifier parameters $\\theta _ { F }$ are then jointly optimized as follows: ",
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+        "type": "equation",
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+        "text": "$$\n\\operatorname* { m i n } _ { w } \\left\\{ L _ { T _ { m } } : = f \\left( w - \\nabla f \\left( w , S _ { m r } \\right) , S _ { m e } \\right) \\right\\} ,\n$$",
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+        "type": "text",
+        "text": "where $w = ( \\theta _ { R } , \\theta _ { F } )$ and ",
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+        "type": "equation",
+        "img_path": "images/bde9058b646d02887dda31b54f7ee2d93739a9d685e564308e546f2d6e26e7be.jpg",
+        "text": "$$\nf \\left( \\boldsymbol { w } , \\boldsymbol { S _ { m } } \\right) = - \\mathbb { E } _ { \\boldsymbol { x } , \\boldsymbol { y } \\sim \\boldsymbol { X } , \\boldsymbol { Y } } \\left[ \\boldsymbol { y } \\log \\boldsymbol { F } ( \\boldsymbol { Z } _ { I } , \\boldsymbol { R } ( \\boldsymbol { x } ) ) \\right] .\n$$",
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+        "type": "text",
+        "text": "Training and Inference. The pseudo-code for training and inference processes of our proposed mDSDI framework is presented in Algorithm 1. Each iteration of the training process consists of two steps: ",
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+        "type": "text",
+        "text": "i) First, we integrate the objective functions (1), (2), (3) and (5) to construct an objective function $L _ { A }$ defined as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/99dc3deb3bf2a4ae2d293b92acbc92f32480f6e097c023dc976351bc9c575d6f.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta _ { Q } , \\theta _ { D _ { S } } , \\theta _ { R } , \\theta _ { F } } \\operatorname* { m a x } _ { \\theta _ { D _ { I } } } \\left\\{ L _ { A } : = \\lambda _ { Z _ { I } } L _ { Z _ { I } } + \\lambda _ { Z _ { S } } L _ { Z _ { S } } + \\lambda _ { D } L _ { D } + L _ { T } \\right\\} ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\lambda _ { Z _ { I } } , \\lambda _ { Z _ { S } }$ and $\\lambda _ { D }$ are selected as the balanced parameters. ",
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+    {
+        "type": "text",
+        "text": "ii) The second step is to employ meta-training to adapt task-related domain-specific from source domains to unseen domains. In each mini-batch, the meta-train and meta-test are split, then the gradient transformation step from meta-train domains to the meta-test domain is performed by solving the optimization problem (6). ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1: Training and Inference processes of mDSDI ",
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+    },
+    {
+        "type": "text",
+        "text": "Training Input: Source domain $S ^ { ( i ) }$ , encoder $Q _ { \\theta _ { Q } }$ , $R _ { \\theta _ { R } }$ , domain classifier $D _ { \\theta _ { D _ { I } } }$ , $D _ { \\theta _ { D _ { S } } }$ for $Z _ { I }$ ,   \n$Z _ { S }$ , task classifier $F _ { \\theta _ { F } }$ , batch size $B$ , learning rate $\\eta$ . Output: The optimal: $Q _ { \\theta _ { Q } } ^ { * } , R _ { \\theta _ { R } } ^ { * } , F _ { \\theta _ { F } } ^ { * } ;$   \nfor $i t e = 1 $ iterations do Sample $S _ { B }$ with a mini-batch $B$ for each domain $S ^ { ( i ) }$ ; Compute $L _ { A }$ using Eq. (8) and perform gradient update $\\nabla _ { \\theta _ { Q } , \\theta _ { R } , \\theta _ { D _ { I } } , \\theta _ { D _ { S } } , \\theta _ { F } } L _ { A }$ with $\\eta$ .; for $j = 1  N$ (number of source domains) do Split Meta-train $S _ { B / j }$ , Meta-test $S _ { j }$ ; Meta-train: Perform gradient update $\\nabla _ { { \\boldsymbol { \\theta } } _ { R } , { \\boldsymbol { \\theta } } _ { F } }$ by minimizing Eq. (7) with $S _ { B / j }$ and $\\eta$ ; Meta-test: Compute $L _ { T _ { m } }$ using Eq. (6) with $S _ { j }$ and updated gradient from Meta-train; Meta-optimization: Perform gradient update $\\dot { \\nabla } _ { \\theta _ { R } , \\theta _ { F } } L _ { T _ { m } }$ with $\\eta$ ; end   \nend   \nInference Input: Target domain $S ^ { T }$ , optimal: $Q _ { \\theta _ { Q } } ^ { * }$ , $R _ { \\theta _ { R } } ^ { * }$ , $F _ { \\theta _ { F } } ^ { * }$ . Output: $Y ^ { T }$ using Eq. (4); ",
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+    {
+        "type": "text",
+        "text": "3 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 Experimental settings ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Dataset. To evaluate the effectiveness of the proposed method, we utilize 7 commonly used datasets including: Colored-MNIST (18): includes 70000 samples of dimension (2, 28, 28) in binary classification problem with noisy label, from MNIST over 3 domains with noisy rate $d \\in \\{ 0 . 1 , 0 . 3$ , $0 . 9 \\}$ , Rotated-MNIST (19): contains 70000 samples of dimension $( 1 , 2 8 , 2 8 )$ and 10 classes, rotated from MNIST over 6 domains $d \\in \\{ 0 , 1 5 , 3 0 , 4 5 , 6 0 , 7 5 \\}$ , VLCS (20): includes 10729 samples of dimension $( 3 , 2 2 4 , 2 2 4 )$ and 5 classes, over 4 photographic domains $d \\in \\{ \\mathrm { C a l t e c h } 1 0 1$ , LabelMe, SUN09, $\\mathrm { V O C } 2 0 0 7 \\}$ , PACS (2): contains 9991 images of dimension (3, 224, 224) and 7 classes, over 4 domains $d \\in \\{$ artpaint, cartoon, sketches, photo $\\}$ , Office-Home (21): has 15500 daily images of dimension $( 3 , 2 2 4 , 2 2 4 )$ and 65 categories, over 4 domains $d \\in \\{ \\mathrm { a r t } .$ , clipart, product, real $\\}$ , Terra Incognita (22): includes 24778 wild photographs of dimension $( 3 , 2 2 4 , 2 2 4 )$ and 10 animals, over 4 camera-trap domains $d \\in \\{ \\mathrm { L 1 0 0 , L 3 8 , L 4 3 , L 4 6 } \\}$ , and DomainNet (23): contains 586575 images of dimension (3, 224, 224) and 345 classes, over 6 domains $d \\in \\{$ clipart, infograph, painting, quickdraw, real, sketch $\\}$ . The detail of each dataset is provided in Appendix C.1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Baseline. Following DomainBed (24) settings, we compare our model with 14 related methods in DG which are divided by 5 common techniques, including: Standard Empirical Risk Minimization: Empirical Risk Minimization (ERM (25)); domain-specific-learning: Group Distributionally Robust Optimization (GroupDRO (26)), Marginal Transfer Learning (MTL (1; 27)), Adaptive Risk Minimization (ARM (28)); Meta-learning: Meta-Learning for DG (MLDG (11)); domain-invariantlearning: Invariant Risk Minimization (IRM (18)), Deep CORrelation ALignment (CORAL (29)), Maximum Mean Discrepancy (MMD (30)), Domain Adversarial Neural Networks (DANN (31)), Class-conditional DANN (CDANN (32)), Risk Extrapolation (VREx (33)); Augmenting data: Interdomain Mixup (Mixup (34; 35; 36)), Style-Agnostic Networks (SagNets (37)), Representation Self Challenging (RSC (38)). The detail of each method is provided in Appendix C.2. ",
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+    {
+        "type": "text",
+        "text": "We use the training-domain validation set technique as proposed in DomainBed (24) for model selection. In particular, for all datasets, we first merge the raw training and validation, then, we run the test three times with three different seeds. For each random seed, we randomly split training and validation and choose the model maximizing the accuracy on the validation set, then compute performance on the given test sets. The mean and standard deviation of classification accuracy from these three runs are reported. We evaluate generalization performance based on backbones MNIST-ConvNet (24) for MNIST datasets and ResNet-50 (39) for non-MNIST datasets to compare with the mentioned methods. Data-processing techniques, model architectures, hyper-parameters, and changes of objective functions during training are presented in detail in Appendix C.3 C.5. All source code to reproduce results are available at https://github.com/VinAIResearch/mDSDI. ",
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+    {
+        "type": "text",
+        "text": "3.2 Results ",
+        "text_level": 1,
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+    {
+        "type": "table",
+        "img_path": "images/7284a03e6d02688dd889df1962ca7ed6ead709a05448297e75af766f4446c265.jpg",
+        "table_caption": [
+            "Table 1: Classification accuracy $( \\% )$ for all algorithms and datasets summarization. Our mDSDI method achieves highest accuracy on average when comparing 14 popular DG algorithms across 7 benchmark datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>CMNIST</td><td>RMNIST</td><td>VLCS</td><td>PACS</td><td>OfficeHome</td><td>TerraInc</td><td>DomainNet</td><td>Average</td></tr><tr><td>ERM (25)</td><td>51.5±0.1</td><td>98.0±0.0</td><td>77.5±0.4</td><td>85.5±0.2</td><td>66.5±0.3</td><td>46.1±1.8</td><td>40.9±0.1</td><td>66.6</td></tr><tr><td>IRM (18)</td><td>52.0±0.1</td><td>97.7±0.1</td><td>78.5±0.5</td><td>83.5±0.8</td><td>64.3±2.2</td><td>47.6±0.8</td><td>33.9±2.8</td><td>65.4</td></tr><tr><td>GroupDRO (26)</td><td>52.1±0.0</td><td>98.0±0.0</td><td>76.7±0.6</td><td>84.4±0.8</td><td>66.0±0.7</td><td>43.2±1.1</td><td>33.3±0.2</td><td>64.8</td></tr><tr><td>Mixup (34; 35;36)</td><td>52.1±0.2</td><td>98.0±0.1</td><td>77.4±0.6</td><td>84.6±0.6</td><td>68.1±0.3</td><td>47.9±0.8</td><td>39.2±0.1</td><td>66.7</td></tr><tr><td>MLDG (11)</td><td>51.5±0.1</td><td>97.9±0.0</td><td>77.2±0.4</td><td>84.9±1.0</td><td>66.8±0.6</td><td>47.7±0.9</td><td>41.2±0.1</td><td>66.7</td></tr><tr><td>CORAL (29)</td><td>51.5±0.1</td><td>98.0±0.1</td><td>78.8±0.6</td><td>86.2±0.3</td><td>68.7±0.3</td><td>47.6±1.0</td><td>41.5±0.1</td><td>67.5</td></tr><tr><td>MMD (30)</td><td>51.5±0.2</td><td>97.9±0.0</td><td>77.5±0.9</td><td>84.6±0.5</td><td>66.3±0.1</td><td>42.2±1.6</td><td>23.4±9.5</td><td>63.3</td></tr><tr><td>DANN (31)</td><td>51.5±0.3</td><td>97.8±0.1</td><td>78.6±0.4</td><td>83.6±0.4</td><td>65.9±0.6</td><td>46.7±0.5</td><td>38.3±0.1</td><td>66.1</td></tr><tr><td>CDANN (32)</td><td>51.7±0.1</td><td>97.9±0.1</td><td>77.5±0.1</td><td>82.6±0.9</td><td>65.8±1.3</td><td>45.8±1.6</td><td>38.3±0.3</td><td>65.6</td></tr><tr><td>MTL (1; 27)</td><td>51.4±0.1</td><td>97.9±0.0</td><td>77.2±0.4</td><td>84.6±0.5</td><td>66.4±0.5</td><td>45.6±1.2</td><td>40.6±0.1</td><td>66.2</td></tr><tr><td>SagNets (37)</td><td>51.7±0.0</td><td>98.0±0.0</td><td>77.8±0.5</td><td>86.3±0.2</td><td>68.1±0.1</td><td>48.6±1.0</td><td>40.3±0.1</td><td>67.2</td></tr><tr><td>ARM (28)</td><td>56.2±0.2</td><td>98.2±0.1</td><td>77.6±0.3</td><td>85.1±0.4</td><td>64.8±0.3</td><td>45.5±0.3</td><td>35.5±0.2</td><td>66.1</td></tr><tr><td>VREx (33)</td><td>51.8±0.1</td><td>97.9±0.1</td><td>78.3±0.2</td><td>84.9±0.6</td><td>66.4±0.6</td><td>46.4±0.6</td><td>33.6±2.9</td><td>65.6</td></tr><tr><td>RSC (38)</td><td>51.7±0.2</td><td>97.6±0.1</td><td>77.1±0.5</td><td>85.2±0.9</td><td>65.5±0.9</td><td>46.6±1.0</td><td>38.9±0.5</td><td>66.1</td></tr><tr><td>mDSDI(Ours)</td><td>52.2±0.2</td><td>98.0±0.1</td><td>79.0±0.3</td><td>86.2±0.2</td><td>69.2±0.4</td><td>48.1±1.4</td><td>42.8±0.1</td><td>67.9</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1 summarizes the results of our experiments on 7 benchmark datasets when compared with mentioned methods. The full result per dataset and domain is provided in Appendix C.4. From these results, we draw three conclusions about our mDSDI model: ",
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+        "type": "text",
+        "text": "Our mDSDI still preserves domain-invariant information. We observe in some target domains which have background-less images and assume only contain domain-invariant information such as Colored-MNIST, Rotated-MNIST, or Terra Incognita (similar observation in cartoon or sketch in PACS, clip-art or product in OfficeHome, and quickdraw in DomainNet. Full results in Appendix C.4), our mDSDI model still achieves competitive results with other baselines (e.g., $5 2 . 2 \\%$ in ColoredMNIST, $9 8 . 0 \\%$ in Rotated-MNIST, and $4 8 . 1 \\%$ in Terra Incognita) which are based on domaininvariant-learning techniques such as DANN, C-DANN, CORAL, MMD, IRM, and VREx. Those results demonstrate the effectiveness of our adversarial training technique for extracting domaininvariant features. Furthermore, due to considering disentangled domain-invariant and domainspecific latent, even in situations where the samples do not have domain-specific, our model still performs well by retaining informative domain-invariance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our mDSDI could capture the usefulness of domain-specific information. In contrast, we observe that in some target domains that have relevant domain-specific with source domains such as landscape background of the object class from photographic pictures in VLCS (similar observation in PACS such as dogs in the yard or guitars lying on a table in photo and art domain, bed in the room or bike parked on the street in the Art and Real-world domain of OfficeHome. Full results in Appendix C.4), mDSDI achieves significantly higher results than other methods (e.g., $7 9 . 0 \\%$ in VLCS, $8 6 . 2 \\%$ in PACS). This means that our domain-invariant features not only support generalization better but also our domain-specific ones cover helpful information in special scenarios such as backgrounds and colors related to objects in the classification task. Moreover, when comparing with other domain-specific based techniques such as GroupDRO, MTL, and ARM, the results showed that domain-specific features learned by meta-training from our model are more helpful than theirs and have captured useful domain-specific features from those object-background relations. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Extending beyond the invariance view to usefulness domain-specific information is important. As shown in Table 1, our mDSDI has the highest average number with $6 7 . 9 \\%$ (highlighted with statistically significant according to a $t$ -test at a significance level $\\alpha = 0 . 0 5$ ). The reason why our model outperforms other baselines could be explained by the fact that their domain-invariant methods are not able to capture domain-specific information, and so have poor performance. Meanwhile, when comparing with other domain-specific based methods, their models only concentrate on domainspecific techniques, and so provide inferior domain-invariant information to our techniques in some background-less images. In contrast, due to considering disentangled domain-invariant and domainspecific features, and having the right strategy to learn each latent, our model captures both this useful information, hence, outperforms their results. Not only has the highest average number, but our method also dominates other methods on a known large-scale dataset such as $6 9 . 2 \\%$ in Office-Home or $4 2 . 8 \\%$ in DomainNet. This implies that besides the essential combination between domain-invariant and domain-specific, when the number of datasets increases, our method can extract more relevant information for complex tasks, such as classifying 345 classes in DomainNet. These results also mean that our model has a balance between informative domain-invariant and domainspecific features to adapt better to different environments than others, therefore showing the highest average in all settings. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.3 How does mDSDI work? ",
+        "text_level": 1,
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+    },
+    {
+        "type": "image",
+        "img_path": "images/fa9fdc3f39d19694c628748adf77167091b1d3b2559a70bb240ea60e48b62849.jpg",
+        "image_caption": [
+            "Figure 3: Feature visualization for domain-invariant: (a): different colors represent different classes; (b): different colors indicate different domains. Feature visualization for domain-specific: (c): different colors represent different classes; (d): different colors indicate different domains. Source domain includes: art (red), cartoon (green), sketch (blue) while target domain is photo (black) in the domain plots. Best viewed in color (Zoom in for details). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To better understand our framework, we visualize the distribution of the learned features with tSNE (40) to analyze the feature space of domain-invariant and domain-specific. As shown in Figure 3 on PACS Dataset, our domain-invariant extractor can minimize the distance between the distribution of the domains (see Figure 3: (b)). However, these domain-invariant features still make mistakes on the classification task, indicated by a mixture of points from different class labels in the middle (see Figure 3: (a)), and many of these points are from the target domain (black color in the Figure 3: (b)). Meanwhile, the domain-specific representation better distinguishes points by class label (see Figure 3: (c)). More importantly, the photo domain’s specific features (black) are close to the art domain (red) (see Figure 3: (d)). This is reasonable because only these two domains include backgrounds related to the object class. It implies that meta-training in our model well learns specific features that can be adapted to the new unseen domain. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "3.4 Ablation study: Important of mDSDI on the Background-Colored-MNIST dataset ",
+        "text_level": 1,
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+        "text": "This section examines our system design by checking its performance under different settings in the real scenario with our generated dataset (a similar experiment with PACS benchmark dataset is in Appendix C.6). ",
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+        "text": "Background-Colored-MNIST dataset. Figure 4 illustrates our Background-Colored-MNIST, generated from the original MNIST. We assume the domain-invariant is the digit’s sketch and design the dataset so that the background color is domain-specific. As a result, on the unseen domain, domain-specific will be useful for the classification task. Specifically, the dataset includes three source domains $d _ { t r }$ , different by digit’s color \bred, green, blue\t, generated from a subset with 1000 training images for MNIST per each domain. In each source domain, the background color is the same for intra-class images but different across classes. In the target domain, 10000 testing images of MNIST are colored for one target domain $d _ { t e }$ with digit color $\\left\\{ \\mathrm { o r a n g e } \\right\\}$ . In this domain, each class’s background color is similar to the same class’s background color in one of three source domains. ",
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+        "text": "Importance of mDSDI. We aim to prove the combination of learning disentangled representation domain-specific, domain-invariant, and meta-training on domain-specific are important in this scenario. To do so, we compare our model under nine settings: learning domain-invariant only (DI), learning domain-specific only (DS), meta-training on domain-invariant (DI-Meta), meta-training on domain-specific (DS-Meta), a combination of domain-invariant and domain-specific without disentanglement loss $L _ { D }$ (DSDI-Without $L _ { D }$ ), a combination of domain-invariant and domain-specific without meta-training (DSDI-Without Meta), meta-training on both representation $Z _ { I }$ and $Z _ { S }$ (DSDIMeta), meta-training on domain-invariant without domain-specific (DSDI-Meta DI) and our proposed framework (mDSDI-Meta DS), which is meta-training on domain-specific without domain-invariant. Table 2 shows that our model is the best setting with $8 9 . 7 \\%$ . It proves that combining domain-invariant and domain-specific is crucial by dominating the settings with only domain-invariant or domainspecific (DI, DI-Meta, DS, DS-Meta). Regarding disentangling two representations $Z _ { I }$ and $Z _ { S }$ , it is worth noting that without disentanglement loss $L _ { D }$ , the model only achieves $8 1 . 4 \\%$ , which is lower than mDSDI. It reveals adding disentanglement loss $L _ { D }$ is essential to boost our model performance. Compared with meta-training on both, which only reaches $8 2 . 1 \\%$ and on domain-invariant are $7 9 . 0 \\%$ , it implies that meta-training is necessary but only for the domain-specific, and our arguments are reasonable. It also shows that our combination with domain-invariant and domain-specific is not easy: a deep ensemble between two neural networks, but existing a deep-down insight in our framework in DG, when compared with mDSDI-Without Meta which can be seen as a type of ensemble, is only around $8 0 . 4 \\%$ . ",
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+        "img_path": "images/f6f4e395e2fdc76b3ae09c624dbf54bffba0a3cfc77dc72a48647677c428f3ba.jpg",
+        "image_caption": [
+            "Figure 4: Background-Colored-MNIST Dataset, where source domains include $\\{$ red, green, blue $\\}$ digit colors and target domain has $\\left\\{ { \\mathrm { o r a n g e } } \\right\\}$ color. "
+        ],
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+        "table_caption": [
+            "Table 2: Classification accuracy $( \\% )$ on BackgroundColored-MNIST. Ablation study shows impact of domain-invariant when combined with meta-training on domain-specific in our method. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Accuracy</td></tr><tr><td>DI</td><td>65.7±4.6</td></tr><tr><td>DI-Meta</td><td>63.6±5.1</td></tr><tr><td>DS</td><td>70.7±4.8</td></tr><tr><td>DS-Meta</td><td>75.3±3.4</td></tr><tr><td>DSDI-Without L D</td><td>81.4±2.6</td></tr><tr><td>DSDI-WithoutMeta</td><td>80.4±1.7</td></tr><tr><td>DSDI-Meta</td><td>82.1±1.4</td></tr><tr><td>DSDI-Meta DI</td><td>79.0±2.3</td></tr><tr><td>mDSDI-Meta DS (Ours)</td><td>89.7±0.8</td></tr></table>",
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+        "text": "4 Conclusion and Discussion ",
+        "text_level": 1,
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+        "text": "Despite being aware of the importance of domain-specific information, little investigation into the theory and a rigorous algorithm to explore its representation. To the best of our knowledge, our work provides the first theoretical analysis to understand and realize the efficiency of domain-specific information in domain generalization. The domain-specific contains unique characteristics and when combined with domain-invariant information can significantly aid performance on unseen domains. Following our theoretical insights based on the information bottleneck principle, we propose a mDSDI algorithm which disentangles these features. We next introduce the use of the meta-training scheme to support domain-specific to adapt information from source domains to unseen domains. Our experimental results demonstrate mDSDI brings out competitive results with related approaches in domain generalization. In addition, the ablation study with our Background-Colored-MNIST further illustrated and demonstrated the efficiency of combining domain-invariant and domain-specific via our proposed mDSDI. Our theoretical analysis and proposed mDSDI framework can facilitate fundamental progress in understanding the behavior of both domain-invariant and domain-specific representation in domain generalization. ",
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+        "text": "Toward a robustness algorithm that can effectively learn both domain-invariant and domain-specific features, there are certainly many challenges that remain in our paper, for example, a theorem to explain when domain-specific may hurt performance in the unseen domain, a stronger connection between theory in implementation, a method to make two representations to be non-linearly independent as well as a lower computational cost of the covariance matrix. In the future, we plan to continue tackling these challenges to provide a better understanding and learning framework in domain generalization then extending to broader settings of transfer learning. A detailed clarification, discussion, and plausible methods in the future work are provided in Appendix B. ",
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