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+# UNSUPERVISED DATA AUGMENTATION FOR CONSISTENCY TRAINING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Semi-supervised learning lately has shown much promise in improving deep learning models when labeled data is scarce. Common among recent approaches is the use of consistency training on a large amount of unlabeled data to constrain model predictions to be invariant to input noise. In this work, we present a new perspective on how to effectively noise unlabeled examples and argue that the quality of noising, specifically those produced by advanced data augmentation methods, plays a crucial role in semi-supervised learning. By substituting simple noising operations with advanced data augmentation methods, our method brings substantial improvements across six language and three vision tasks under the same consistency training framework. On the IMDb text classification dataset, with only 20 labeled examples, our method achieves an error rate of 4.20, outperforming the state-of-the-art model trained on 25,000 labeled examples. On a standard semi-supervised learning benchmark, CIFAR-10, our method outperforms all previous approaches and achieves an error rate of $2 . 7 \%$ with only 4,000 examples, nearly matching the performance of models trained on 50,000 labeled examples. Our method also combines well with transfer learning, e.g., when finetuning from BERT, and yields improvements in high-data regime, such as ImageNet, whether when there is only $10 \%$ labeled data or when a full labeled set with 1.3M extra unlabeled examples is used. 1
+
+# 1 INTRODUCTION
+
+A fundamental weakness of deep learning is that it typically requires a lot of labeled data to work well. Semi-supervised learning (SSL) (Chapelle et al., 2009) is one of the most promising methods of leveraging unlabeled data to address this weakness. The recent works in SSL are diverse but those that are based on consistency training (Bachman et al., 2014; Rasmus et al., 2015; Laine & Aila, 2016; Tarvainen & Valpola, 2017) have shown to work well on many benchmarks.
+
+In a nutshell, consistency training methods simply regularize model predictions to be invariant to small noise applied to either input examples (Miyato et al., 2018; Sajjadi et al., 2016; Clark et al., 2018) or hidden states (Bachman et al., 2014; Laine & Aila, 2016). This framework makes sense intuitively because a good model should be robust to any small change in an input example or hidden states. Under this framework, different methods in this category differ mostly in how and where the noise injection is applied. Typical noise injection methods are additive Gaussian noise, dropout noise or adversarial noise.
+
+In this work, we investigate the role of noise injection in consistency training and observe that advanced data augmentation methods, specifically those work best in supervised learning (Simard et al., 1998; Krizhevsky et al., 2012; Cubuk et al., 2018; Yu et al., 2018), also perform well in semisupervised learning. There is indeed a strong correlation between the performance of data augmentation operations in supervised learning and their performance in consistency training. We, hence, propose to substitute the traditional noise injection methods with high quality data augmentation methods in order to improve consistency training. To emphasize the use of better data augmentation in consistency training, we name our method Unsupervised Data Augmentation or UDA.
+
+We evaluate UDA on a wide variety of language and vision tasks. On six text classification tasks, our method achieves significant improvements over state-of-the-art models. Notably, on IMDb, UDA with 20 labeled examples outperforms the state-of-the-art model trained on $1 2 5 0 \mathrm { x }$ more labeled data. We also evaluate UDA on standard semi-supervised learning benchmarks for vision such as CIFAR-10 and SVHN. UDA outperforms all existing semi-supervised learning methods by significant margins. On CIFAR-10 with 4,000 labeled examples, UDA achieves an error rate of 5.29, nearly matching the performance of the fully supervised model that uses 50,000 labeled examples. Furthermore, with a better architecture, PyramidNet+ShakeDrop, UDA achieves a new state-ofthe-art error rate of 2.7. On SVHN, UDA achieves an error rate of 2.55 with only 1,000 labeled examples. Finally, we also find UDA to be beneficial when there is a large amount of supervised data. For instance, on ImageNet, UDA leads to improvements of top-1 accuracy from 58.84 to 68.78 with $1 0 \%$ of the labeled set and from 78.43 to 79.05 when we use the full labeled set and an external dataset with 1.3M unlabeled examples.
+
+Our key contributions and findings can be summarized as follows:
+
+• First, we show that state-of-the-art data augmentations found in supervised learning can also serve as a superior source of perturbation under the consistency enforcing semi-supervised framework. See results in Table 1 and Table 2.   
+• Second, we show that UDA can match and even outperform purely supervised learning that uses orders of magnitude more labeled data. State-of-the-art results for both vision and language tasks are reported in Table 3 and 4. The effectiveness of UDA across different training data sizes are highlighted in Figure 4 and 5.   
+• Finally, we show that UDA combines well with transfer learning, e.g., when fine-tuning from BERT (see Table 4), and is effective at high-data regime, e.g. on ImageNet (see Table 5).
+
+# 2 UNSUPERVISED DATA AUGMENTATION (UDA)
+
+In this section, we first formulate our task and then present the key method and insights behind UDA. Throughout this paper, we focus on classification problems and will use $x$ to denote the input and $y ^ { * }$ to denote its ground-truth prediction target. We are interested in learning a model $p _ { \theta } ( y \mid x )$ to predict $y ^ { * }$ based on the input $x$ , where $\theta$ denotes the model parameters. Finally, we will use $L$ and $U$ to denote the sets of labeled and unlabeled examples respectively.
+
+# 2.1 BACKGROUND: SUPERVISED DATA AUGMENTATION
+
+Data augmentation aims at creating novel and realistic-looking training data by applying a transformation to an example, without changing its label. Formally, let $q ( \bar { x } \mid x )$ be the augmentation transformation from which one can draw augmented examples $\hat { x }$ based on an original example $x$ . For an augmentation transformation to be valid, it is required that any example ${ \hat { x } } \sim q ( { \hat { x } } \mid x )$ drawn from the distribution shares the same ground-truth label as $x$ . Given a valid augmentation transformation, we can simply minimize the negative log-likelihood on augmented examples.
+
+Supervised data augmentation can be equivalently seen as constructing an augmented labeled set from the original supervised set and then training the model on the augmented set. Therefore, the augmented set needs to provide additional inductive biases to be more effective. How to design the augmentation transformation has, thus, become critical.
+
+In recent years, there have been significant advancements on the design of data augmentations for NLP (Yu et al., 2018), vision (Krizhevsky et al., 2012; Cubuk et al., 2018) and speech (Hannun et al., 2014; Park et al., 2019) in supervised settings. Despite the promising results, data augmentation is mostly regarded as the “cherry on the cake” which provides a steady but limited performance boost because these augmentations has so far only been applied to a set of labeled examples which is usually of a small size. Motivated by this limitation, via the consistency training framework, we extend the advancement in supervised data augmentation to semi-supervised learning where abundant unlabeled data is available.
+
+![](images/87dbf12e693e4fa1ebe2e7d799c65864b5d966ed735b831120d0fb1e1c6877ca.jpg)  
+Figure 1: Training objective for UDA, where M is a model that predicts a distribution of $y$ given $x$ .
+
+# 2.2 UNSUPERVISED DATA AUGMENTATION
+
+As discussed in the introduction, a recent line of work in semi-supervised learning has been utilizing unlabeled examples to enforce smoothness of the model. The general form of these works can be summarized as follows:
+
+• Given an input $x$ , compute the output distribution $p _ { \theta } ( y \mid x )$ given $x$ and a noised version $p _ { \theta } ( y ~ |$ $x , \epsilon )$ by injecting a small noise $\epsilon$ . The noise can be applied to $x$ or hidden states. • Minimize a divergence metric between the two distributions $\mathcal { D } \left( p _ { \theta } ( y \mid x ) \parallel p _ { \theta } ( y \mid x , \epsilon ) \right)$ .
+
+This procedure enforces the model to be insensitive to the noise $\epsilon$ and hence smoother with respect to changes in the input (or hidden) space. From another perspective, minimizing the consistency loss gradually propagates label information from labeled examples to unlabeled ones.
+
+In this work, we are interested in a particular setting where the noise is injected to the input $x$ , i.e., $\hat { x } = q ( x , \epsilon )$ , as considered by prior works (Sajjadi et al., 2016; Laine & Aila, 2016; Miyato et al., 2018). But different from existing work, we focus on the unattended question of how the form or “quality” of the noising operation $q$ can influence the performance of this consistency training framework. Specifically, to enforce consistency, prior methods generally employ simple noise injection methods such as adding Gaussian noise, simple input augmentations to noise unlabeled examples. In contrast, we hypothesize that stronger data augmentations in supervised learning can also lead to superior performance when used to noise unlabeled examples in the semi-supervised consistency training framework, since it has been shown that more advanced data augmentations that are more diverse and natural can lead to significant performance gain in the supervised setting.
+
+Following this idea, we propose to use a rich set of state-of-the-art data augmentations verified in various supervised settings to inject noise and optimize the same consistency training objective on unlabeled examples. When jointly trained with labeled examples, we utilize a weighting factor $\lambda$ to balance the supervised cross entropy and the unsupervised consistency training loss, which is illustrated in Figure 1. Formally, the full objective can be written as follows:
+
+$$
+\operatorname* { m i n } _ { \theta } \mathcal { I } ( \theta ) = \mathbb { E } _ { x , y ^ { * } \in L } \left[ - \log p _ { \theta } ( y ^ { * } \mid x ) \right] + \lambda \mathbb { E } _ { x \in U } \mathbb { E } _ { \hat { x } \sim q ( \hat { x } | x ) } \left[ \mathcal { D } _ { \mathrm { K L } } \left( p _ { \tilde { \theta } } ( y \mid x ) \ \lVert \ p _ { \theta } ( y \mid \hat { x } ) ) \right) \right] .
+$$
+
+where $q ( { \hat { x } } \mid x )$ is a data augmentation transformation and $\tilde { \theta }$ is a fixed copy of the current parameters $\theta$ indicating that the gradient is not propagated through $\tilde { \theta }$ , as suggested by Miyato et al. (2018). We also follow VAT (Miyato et al., 2018) to use the KL divergence. We set $\lambda$ to 1 for most of our experiments and use different batch sizes for the supervised data and the unsupervised data. In the vision domain, simple augmentations including cropping and flipping are applied to labeled examples. To minimize the discrepancy between supervised training and prediction on unlabeled examples, we apply the same simple augmentations to unlabeled examples for computing $p _ { \tilde { \theta } } ( y \mid x )$ .
+
+Discussion. Before detailing the augmentation operations used in this work, we first provide some intuitions on how more advanced data augmentations can provide extra advantages over simple ones used in earlier works from three aspects:
+
+• Valid noise: Advanced data augmentation methods that achieve great performance in supervised learning usually generate realistic augmented examples that share the same ground-truth labels with the original example. Thus, it is safe to encourage the consistency between predictions on the original unlabeled example and the augmented unlabeled examples.
+
+• Diverse noise: Advanced data augmentation can generate a diverse set of examples since it can make large modifications to the input example without changing its label, while simple Gaussian noise only make local changes. Encouraging consistency on a diverse set of augmented examples can significantly improve the sample efficiency.
+
+• Targeted inductive biases: Different tasks require different inductive biases. Data augmentation operations that work well in supervised training essentially provides the missing or most wanted inductive biases in an original labeled set.
+
+# 2.3 AUGMENTATION STRATEGIES FOR DIFFERENT TASKS
+
+We now detail the augmentation methods, tailored for different tasks, that we use in this work.
+
+RandAugment for Image Classification. We make use of a data augmentation method called RandAugment, which is inspired by AutoAugment (Cubuk et al., 2018). AutoAugment uses a search method to combine all image processing transformations in the Python Image Library (PIL) to find a good augmentation strategy. In RandAugment, we do not use search, but instead uniformly sample from the same set of augmentation transformations in PIL. In other words, RandAugment is simpler and requires no labeled data as there is no need to search for optimal policies.
+
+Back-translation for Text Classification. When used as an augmentation method, backtranslation (Sennrich et al., 2015; Edunov et al., 2018) refers to the procedure of translating an existing example $x$ in language $A$ into another language $B$ and then translating it back into $A$ to obtain an augmented example $\hat { x }$ . As observed by Yu et al. (2018), back-translation can generate diverse paraphrases while preserving the semantics of the original sentences, leading to significant performance improvements in question answering. In our case, we use back-translation to paraphrase the training data of our text classification tasks.2
+
+We find that the diversity of the paraphrases is more important than the quality or the validity. Hence, we employ random sampling with a tunable temperature instead of beam search for the generation. As shown in Figure 2, the paraphrases generated by back-translation sentence are diverse and have similar semantic meanings. More specifically, we use WMT’14 English-French translation models (in both directions) to perform back-translation on each sentence. To facilitate future research, we have open-sourced our back-translation system together with the translation checkpoints.
+
+![](images/4cdd12835430eea86a7d9a619a2a3a2afe8958713c4ab32bb4b8dff477e76dc8.jpg)  
+Figure 2: Augmented examples using back-translation and RandAugment.
+
+Word replacing with TF-IDF for Text Classification. While back-translation is good at maintaining the global semantics of a sentence, there is little control over which words will be retained. This requirement is important for topic classification tasks, such as DBPedia, in which some keywords are more informative than other words in determining the topic. We, therefore, propose an augmentation method that replaces uninformative words with low TF-IDF scores while keeping those with high TF-IDF values. We refer readers to Appendix C for a detailed description.
+
+# 2.4 TRAINING SIGNAL ANNEALING FOR LOW-DATA REGIME
+
+In semi-supervised learning, we often encounter a situation where there is a huge gap between the amount of unlabeled data and that of labeled data. Hence, the model often quickly overfits the limited amount of labeled data while still underfitting the unlabeled data. To tackle this difficulty, we introduce a new training technique, called Training Signal Annealing (TSA), which gradually releases the “training signals” of the labeled examples as training progresses. Intuitively, we only utilize a labeled example if the model’s confidence on that example is lower than a predefined threshold which increases according to a schedule. Specifically, at training step $t$ , if the model’s predicted probability for the correct category $p _ { \theta } ( y ^ { \ast } \mid x )$ is higher than a threshold $\eta _ { t }$ , we remove that example from the loss function. Suppose $K$ is the number of categories, by gradually increase $\eta _ { t }$ from $\textstyle { \frac { 1 } { K } } $ to 1, the threshold $\eta _ { t }$ serves as a ceiling to prevent over-training on easy labeled examples.
+
+We consider three increasing schedules of $\eta _ { t }$ with different application scenarios. Let $T$ be the total number of training steps, the three schedules are shown in Figure 3. Intuitively, when the model is prone to overfit, e.g., when the problem is relatively easy or the number of labeled examples is very limited, the exp-schedule is most suitable as the supervised signal is mostly released at the end of training. In contrast, when the model is less likely to overfit (e.g., when we have abundant labeled examples or when the model employs effective regularization), the log-schedule can serve well.
+
+![](images/113c6c44647a30d9d7e78b133896c1a245e30dd854fa25c70b7fed58bcd93e81.jpg)  
+Figure 3: Three schedules of TSA. We set $\begin{array} { r } { \eta _ { t } = \alpha _ { t } * \left( 1 - \frac { 1 } { K } \right) + \frac { 1 } { K } } \end{array}$ . $\alpha _ { t }$ is set to $\textstyle 1 - \exp \bigl ( - \frac { t } { T } * 5 \bigr )$ $\textstyle { \frac { t } { T } }$ and $\textstyle \exp \bigl ( \bigl ( \frac { t } { T } - 1 \bigr ) * 5 \bigr )$ for the log, linear and exp schedules.
+
+# 3 EXPERIMENTS
+
+In this section, we evaluate UDA on a variety of language and vision tasks. For language, we rely on six text classification benchmark datasets, including IMDb, Yelp-2, Yelp-5, Amazon-2 and Amazon5 sentiment classification and DBPedia topic classification (Maas et al., 2011; Zhang et al., 2015). For vision, we employ two smaller datasets CIFAR-10 (Krizhevsky & Hinton, 2009), SVHN (Netzer et al., 2011), which are often used to compare semi-supervised algorithms, as well as ImageNet (Deng et al., 2009) of a larger scale to test the scalability of UDA. For details of the labeled and unlabeled data and experiment details, we refer readers to Appendix E.
+
+# 3.1 CORRELATION BETWEEN SUPERVISED AND SEMI-SUPERVISED PERFORMANCES
+
+As the first step, we try to verify the fundamental idea of UDA, i.e., there is a positive correlation of data augmentation’s effectiveness in supervised learning and semi-supervised learning. Based on Yelp-5 (a language task) and CIFAR-10 (a vision task), we compare the performance of different data augmentation methods in either fully supervised or semi-supervised settings. For Yelp-5, apart from back-translation, we include a simpler method Switchout (Wang et al., 2018) which replaces a token with a random token uniformly sampled from the vocabulary. For CIFAR-10, we compare RandAugment with two simpler methods: (1) cropping & flipping augmentation and (2) Cutout.
+
+Based on this setting, Table 1 and Table 2 exhibit a strong correlation of an augmentation’s effectiveness between supervised and semi-supervised settings. This validates our idea of stronger data augmentations found in supervised learning can always lead to more gains when applied to the semi-supervised learning settings.
+
+<table><tr><td>Augmentation (# Sup examples)</td><td>Sup (50k)</td><td>Semi-Sup (4k)</td></tr><tr><td>Crop&amp;flip</td><td>5.36</td><td>16.17</td></tr><tr><td>Cutout</td><td>4.42</td><td>6.42</td></tr><tr><td>RandAugment</td><td>4.23</td><td>5.29</td></tr></table>
+
+Table 1: Error rates on CIFAR-10.
+
+Table 2: Error rate on Yelp-5.   
+
+<table><tr><td>Augmentation (# Sup examples)</td><td>Sup (650k)</td><td>Semi-sup (2.5k)</td></tr><tr><td>X</td><td>38.36</td><td>50.80</td></tr><tr><td>Switchout</td><td>37.24</td><td>43.38</td></tr><tr><td>Back-translation</td><td>36.71</td><td>41.35</td></tr></table>
+
+# 3.2 ALGORITHM COMPARISON ON VISION SEMI-SUPERVISED LEARNING BENCHMARKS
+
+With the correlation established above, the next question we ask is how well UDA performs compared to existing semi-supervised learning algorithms. To answer the question, we focus on the most commonly used semi-supervised learning benchmarks CIFAR-10 and SVHN.
+
+Vary the size of labeled data. Firstly, we follow the settings in (Oliver et al., 2018) and employ Wide-ResNet-28-2 (Zagoruyko & Komodakis, 2016; He et al., 2016) as the backbone model and evaluate UDA with varied supervised data sizes. Specifically, we compare UDA with two highly competitive baselines: (1) Virtual adversarial training (VAT) (Miyato et al., 2018), an algorithm that generates adversarial Gaussian noise on input, and (2) MixMatch (Berthelot et al., 2019), a parallel work that combines previous advancements in semi-supervised learning. The comparison is shown in Figure 4 with two key observations.3
+
+• First, UDA consistently outperforms the two baselines with a clear margin given different sizes of labeled data.   
+• Moreover, the performance difference between UDA and VAT shows the superiority of data augmentation based noise. The difference of UDA and VAT is essentially the noise process. While the noise produced by VAT often contain high-frequency artifacts that do not exist in real images, data augmentation mostly generates diverse and realistic images.
+
+![](images/f8f181f11a4ad184e5b17ad7505a41bb3ae73b6be1c54c39cfd10ad05474ac35.jpg)  
+Figure 4: Comparison with two semi-supervised learning methods on CIFAR-10 and SVHN with varied number of labeled examples.
+
+Comparisons with published results Next, we directly compare UDA with previously published results under different model architectures. Following previous work, 4k and 1k labeled examples are used for CIFAR-10 and SVHN respectively. As shown in Table 3, given the same architecture, UDA outperforms all published results by significant margins. This shows the huge potential of state-of-the-art data augmentations under the consistency training framework in the vision domain.
+
+Table 3: Comparison between methods using different models where PyramidNet is used with ShakeDrop regularization. Fully supervised Wide-ResNet-28-2 and PyramidNet+ShakeDrop have an error rate of 5.4 and 2.7 when trained on 50,000 examples without RandAugment. On CIFAR10, with only 4,000 labeled examples, UDA matches the performance of the two fully supervised models. On SVHN, UDA also matches the performance of our fully supervised model trained on 73,257 examples without RandAugment, which has an error rate of 2.84.   
+
+<table><tr><td>Method</td><td>Model</td><td>#Param</td><td>CIFAR-10 (4k)</td><td>SVHN (1k)</td></tr><tr><td>II-Model (Laine &amp; Aila,2016)</td><td>Conv-Large</td><td>3.1M</td><td>12.36 ± 0.31</td><td>4.82 ± 0.17</td></tr><tr><td>Mean Teacher (Tarvainen &amp; Valpola,2017)</td><td>Conv-Large</td><td>3.1M</td><td>12.31 ± 0.28</td><td>3.95 ± 0.19</td></tr><tr><td>VAT+ EntMin (Miyato et al., 2018)</td><td>Conv-Large</td><td>3.1M</td><td>10.55 ± 0.05</td><td>3.86 ± 0.11</td></tr><tr><td>SNTG (Luo et al., 2018)</td><td>Conv-Large</td><td>3.1M</td><td>10.93 ± 0.14</td><td>3.86 ± 0.27</td></tr><tr><td>VAdD (Park et al., 2018)</td><td>Conv-Large</td><td>3.1M</td><td>11.32 ± 0.11</td><td>4.16 ± 0.08</td></tr><tr><td>Fast-SWA (Athiwaratkun et al., 2018)</td><td>Conv-Large</td><td>3.1M</td><td>9.05</td><td></td></tr><tr><td>ICT (Verma et al., 2019)</td><td>Conv-Large</td><td>3.1M</td><td>7.29 ± 0.02</td><td>3.89 ±0.04</td></tr><tr><td>Pseudo-Label (Lee,2013)</td><td>WRN-28-2</td><td>1.5M</td><td>16.21 ± 0.11</td><td>7.62 ± 0.29</td></tr><tr><td>LGA + VAT (Jackson &amp; Schulman, 2019)</td><td>WRN-28-2</td><td>1.5M</td><td>12.06 ± 0.19</td><td>6.58 ± 0.36</td></tr><tr><td>mixmixup (Hataya &amp; Nakayama, 2019)</td><td>WRN-28-2</td><td>1.5M</td><td>10</td><td>=</td></tr><tr><td>ICT (Verma et al., 2019)</td><td>WRN-28-2</td><td>1.5M</td><td>7.66 ± 0.17</td><td>3.53 ± 0.07</td></tr><tr><td>MixMatch (Berthelot et al.,2019)</td><td>WRN-28-2</td><td>1.5M</td><td>6.24 ± 0.06</td><td>2.89 ± 0.06</td></tr><tr><td>Mean Teacher (Tarvainen&amp; Valpola,2017)</td><td>Shake-Shake</td><td>26M</td><td>6.28 ± 0.15</td><td>-</td></tr><tr><td>Fast-SWA (Athiwaratkun et al., 2018)</td><td>Shake-Shake</td><td>26M</td><td>5.0</td><td></td></tr><tr><td>MixMatch (Berthelot et al.,2019)</td><td>WRN</td><td>26M</td><td>4.95 ± 0.08</td><td>=</td></tr><tr><td>UDA (RandAugment)</td><td>WRN-28-2</td><td>1.5M</td><td>5.29 ± 0.25</td><td>2.55 ± 0.09</td></tr><tr><td>UDA (RandAugment)</td><td>Shake-Shake</td><td>26M</td><td>3.7</td><td></td></tr><tr><td>UDA (RandAugment)</td><td>PyramidNet</td><td>26M</td><td>2.7</td><td></td></tr></table>
+
+# 3.3 EVALUATION ON TEXT CLASSIFICATION DATASETS
+
+Next, we further evaluate UDA in the language domain. Moreover, in order to test whether UDA can be combined with the success of unsupervised representation learning, such as BERT (Devlin et al., 2018), we further consider four initialization schemes: (a) random Transformer; (b) BERTBASE; (c) BERTLARGE; (d) BERTFINETUNE: BERTLARGE fine-tuned on in-domain unlabeled data4. Under each of these four initialization schemes, we compare the performances with and without UDA.
+
+The results are presented in Table 4 where we would like to emphasize three observations:
+
+• First, even with very few labeled examples, UDA can offer decent or even competitive performances compared to the SOTA model trained with full supervised data. Particularly, on binary sentiment analysis tasks, with only 20 supervised examples, UDA outperforms the previous SOTA trained with full supervised data on IMDb and is competitive on Yelp-2 and Amazon-2. • Second, UDA is complementary to transfer learning / representation learning. As we can see, when initialized with BERT and further finetuned on in-domain data, UDA can still significantly reduce the error rate from 6.50 to 4.20 on IMDb. • Finally, we also note that for five-category sentiment classification tasks, there still exists a clear gap between UDA with 500 labeled examples per class and BERT trained on the entire supervised set. Intuitively, five-category sentiment classifications are much more difficult than their binary counterparts. This suggests a room for further improvement in the future.
+
+Results with different labeled set sizes. We also show in Figure 5 that UDA leads to consistent improvements across all labeled data sizes on IMDb and Yelp-2.
+
+<table><tr><td colspan="8">Fully supervised baseline</td></tr><tr><td>Datasets (# Sup examples)</td><td>IMDb (25k)</td><td>Yelp-2 (560k)</td><td>Yelp-5 (650k)</td><td>Amazon-2 (3.6m)</td><td>Amazon-5 (3m)</td><td>DBpedia (560k)</td></tr><tr><td>Pre-BERT SOTA BERTLARGE</td><td>4.32 4.51</td><td>2.16 1.89</td><td>29.98 29.32</td><td>3.32 2.63</td><td>34.81 34.17</td><td>0.70 0.64</td></tr><tr><td colspan="7">Semi-supervised setting</td></tr><tr><td>Initialization UDA</td><td>IMDb (20)</td><td>Yelp-2 (20)</td><td>Yelp-5 (2.5k)</td><td>Amazon-2 (20)</td><td>Amazon-5 (2.5k)</td><td>DBpedia (140)</td></tr><tr><td>X Random</td><td>43.27 25.23</td><td>40.25 8.33</td><td>50.80 41.35</td><td>45.39 16.16</td><td>55.70 44.19</td><td>41.14 7.24</td></tr><tr><td>X BERTBASE</td><td>18.40 5.45</td><td>13.60 2.61</td><td>41.00 33.80</td><td>26.75 3.96</td><td>44.09 38.40</td><td>2.58 1.33</td></tr><tr><td>X BERTLARGE</td><td>11.72 4.78</td><td>10.55 2.50</td><td>38.90 33.54</td><td>15.54 3.93</td><td>42.30 37.80</td><td>1.68 1.09</td></tr><tr><td>BERTFINETUNE</td><td>X 6.50 4.20</td><td>2.94 2.05</td><td>32.39 32.08</td><td>12.17 3.50</td><td>37.32 37.12</td><td>- -</td></tr></table>
+
+Table 4: Error rates on text classification datasets. In the fully supervised settings, the pre-BERT SOTAs include ULMFiT (Howard & Ruder, 2018) for Yelp-2 and Yelp-5, DPCNN (Johnson & Zhang, 2017) for Amazon-2 and Amazon-5, Mixed VAT (Sachan et al., 2018) for IMDb and DBPedia. All of our experiments use a sequence length of 512.
+
+![](images/731ae10a632f4f3d0e6c2394811c6a9f4a6ed503003af707fcf41fde324f664e.jpg)  
+Figure 5: Accuracy on IMDb and Yelp-2 with different number of labeled examples. In the largedata regime, with the full training set of IMDb, UDA also provides robust gains.
+
+# 3.4 SCALABILITY TEST ON THE IMAGENET DATASET
+
+Then, to evaluate whether UDA can scale to problems with a large scale and a higher difficulty, we now turn to the ImageNet dataset with ResNet-50 being the underlying architecture. Specifically, we consider two experiment settings with different natures:
+
+• We use $10 \%$ of the supervised data of ImageNet while using all other data as unlabeled data. As a result, the unlabeled exmaples are entirely in-domain. • In the second setting, we keep all images in ImageNet as supervised data. Then, we use the domain-relevance data filtering method (See Appendix B for details) to filter out 1.3M images from an anonymous dataset. Hence, the unlabeled set is not necessarily in-domain.
+
+The results are summarized in Table 5. In both $10 \%$ and the full data settings, UDA consistently brings significant gains compared to the supervised baseline. This shows UDA is not only able to scale but also able to utilize out-of-domain unlabeled examples to improve model performance. In parallel to our work, S4L (Zhai et al., 2019b) and CPC (Henaff et al., 2019) also show significant ´ improvements on ImageNet.
+
+Table 5: Top- $^ { . 1 / }$ top-5 accuracy on ImageNet with $10 \%$ and $100 \%$ of the labeled set. We use image size 224 and 331 for the $10 \%$ and $100 \%$ experiments respectively.   
+
+<table><tr><td>Methods</td><td>SSL</td><td>10%</td><td>100%</td></tr><tr><td>ResNet-50</td><td>×</td><td>55.09 /77.26</td><td>77.28 /93.73</td></tr><tr><td> w. RandAugment</td><td></td><td>58.84 / 80.56</td><td>78.43 /94.37</td></tr><tr><td>UDA (RandAugment)|√</td><td></td><td>68.78 / 88.80</td><td>79.05 / 94.49</td></tr></table>
+
+# 3.5 ABLATION STUDIES FOR TSA
+
+Lastly, we study the effect of TSA on two tasks with different amounts of unlabeled data: (a) Yelp-5 where we have only $2 . 5 \mathrm { k }$ labeled examples and $6 \mathrm { m }$ unlabeled examples. (b) CIFAR-10 where we have 4k labeled examples and $5 0 \mathrm { k }$ unlabeled examples. For Yelp-5, we use a randomly initialized transformer in this study to rule out factors of having a pre-trained representation.
+
+As shown in Table 6, on Yelp-5, where there is a lot more unlabeled data than labeled data, TSA reduces the error rate from 50.81 to 41.35 when compared to the baseline without TSA. More specifically, the best performance is achieved when we choose to postpone releasing the supervised training signal to the end of the training, i.e, exp-schedule leads to the best performance. On the other hand, linear-schedule is the sweet spot on CIFAR-10 in terms of the speed of releasing supervised training signals, where the amount of unlabeled data is comparable to that of supervised data.
+
+Table 6: Ablation study for Training Signal Annealing (TSA) on Yelp-5 and CIFAR-10. The shown numbers are error rates.   
+
+<table><tr><td>TSA schedule</td><td>Yelp-5</td><td>CIFAR-10</td></tr><tr><td>X</td><td>50.81</td><td>5.67</td></tr><tr><td>log-schedule</td><td>49.06</td><td>5.67</td></tr><tr><td>linear-schedule</td><td>45.41</td><td>5.29</td></tr><tr><td>exp-schedule</td><td>41.35</td><td>7.81</td></tr></table>
+
+# 4 RELATED WORK
+
+Existing works in consistency training does make use of data augmentation (Laine & Aila, 2016; Sajjadi et al., 2016); however, they only apply weak augmentation methods such as random translations and cropping. In parallel to our work, ICT (Verma et al., 2019) and MixMatch (Berthelot et al., 2019) also show improvements for semi-supervised learning. These methods employ mixup (Zhang et al., 2017) on top of simple augmentations such as flipping and cropping; instead, UDA emphasizes on the use of state-of-the-art data augmentations, leading to significantly better results on CIFAR-10 and SVHN. In addition, UDA is also applicable to language domain and can also scale well to more challenging vision datasets, such as ImageNet.
+
+Other works in the consistency training family mostly differ in how the noise is defined: Pseudoensemble (Bachman et al., 2014) directly applies Gaussian noise and Dropout noise; VAT (Miyato et al., 2018; 2016) defines the noise by approximating the direction of change in the input space that the model is most sensitive to; Cross-view training (Clark et al., 2018) masks out part of the input data. Apart from enforcing consistency on the input examples and the hidden representations, another line of research enforces consistency on the model parameter space. Works in this category include Mean Teacher (Tarvainen & Valpola, 2017), fast-Stochastic Weight Averaging (Athiwaratkun et al., 2018) and Smooth Neighbors on Teacher Graphs (Luo et al., 2018). For a complete version of related work, see Appendix D.
+
+# 5 CONCLUSION
+
+In this paper, we show that data augmentation and semi-supervised learning are well connected: better data augmentation can lead to significantly better semi-supervised learning. Our method,
+
+UDA, employs state-of-the-art data augmentation found in supervised learning to generate diverse and realistic noise and enforces the model to be consistent with respect to these noise. For text, UDA combines well with representation learning, e.g., BERT, and is very effective in low-data regime where state-of-the-art performance is achieved on IMDb with only 20 examples. For vision, UDA outperforms prior works by a clear margin and nearly matches the performance of the fully supervised models trained on the full labeled sets which are one order of magnitude larger. Lastly, UDA can effectively leverage out-of-domain unlabeled data and achieve improved performances on ImageNet where we have a large amount of supervised data. We hope that UDA will encourage future research to transfer advanced supervised augmentation to semi-supervised setting for different tasks.
+
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+
+# A MORE EXPERIMENTS
+
+A.1 ABLATIONS STUDIES ON RANDAUGMENT
+
+We hypothesize that the success of RandAugment should be credited to the diversity of the augmentation transformations, since RandAugment works very well for multiple different datasets while does not require a search algorithm to find out the most effective policies. To verify this hypothesis, we test UDA’s performance when we restrict the number of possible transformations used in RandAugment. As shown in Figure 6, the performance gradually improves as we use more augmentation transformations.
+
+![](images/376a93ef160eb5d0c57caf29b3b056eb5656528130872dbb9e9309f4bd847d3a.jpg)  
+Figure 6: Error rate of UDA on CIFAR-10 with different numbers of possible transformations in RandAugment. UDA achieves lower error rate when we increase the number of possible transformations, which demonstrates the importance of a rich set of augmentation transformations.
+
+# A.2 RESULTS ON CIFAR-10 AND SVHN WITH VARIED LABEL SET SIZES
+
+CIFAR-10 In Table 7, we show results for compared methods of Figure 4a and results of PseudoLabel (Lee, 2013), Π-Model (Laine & Aila, 2016), Mean Teacher (Tarvainen & Valpola, 2017). Fully supervised learning using 50,000 examples achieves an error rate of 5.36 and 4.23 with or without RandAugment. The performance of the baseline models are reported by MixMatch (Berthelot et al., 2019).
+
+To make sure that the performance reported by MixMatch and our results are comparable, we reimplement MixMatch in our codebase and find that the results in the original paper is comparable but slightly higher than our reimplementation, which results in a more competitive comparison for UDA. For example, our reimplementation of MixMatch achieves an error rate of $7 . 0 0 \pm 0 . 5 9$ and $7 . 3 9 \pm 0 . 1 1$ with 4,000 and 2,000 examples. MixMatch uses a different model implementation and employs exponential moving average (EMA) on the model parameters, while we do not use EMA for our implementations.
+
+Table 7: Error rate $( \% )$ for CIFAR-10.   
+
+<table><tr><td>Methods /# Sup</td><td>250</td><td>500</td><td>1,000</td><td>2,000</td><td>4,000</td></tr><tr><td>Pseudo-Label</td><td>49.98 ± 1.17</td><td>40.55 ± 1.70</td><td>30.91 ± 1.73</td><td>21.96 ± 0.42</td><td>16.21 ± 0.11</td></tr><tr><td>II-Model</td><td>53.02 ± 2.05</td><td>41.82 ± 1.52</td><td>31.53 ± 0.98</td><td>23.07 ± 0.66</td><td>17.41 ± 0.37</td></tr><tr><td>Mean Teacher</td><td>47.32 ± 4.71</td><td>42.01 ± 5.86</td><td>17.32 ± 4.00</td><td>12.17 ± 0.22</td><td>10.36 ± 0.25</td></tr><tr><td>VAT</td><td>36.03 ± 2.82</td><td>26.11 ± 1.52</td><td>18.68 ± 0.40</td><td>14.40 ± 0.15</td><td>11.05 ± 0.31</td></tr><tr><td>MixMatch</td><td>11.08 ± 0.87</td><td>9.65 ± 0.94</td><td>7.75 ± 0.32</td><td>7.03 ± 0.15</td><td>6.24 ± 0.06</td></tr><tr><td>UDA (RandAugment)</td><td>8.76 ± 0.90</td><td>6.68 ± 0.24</td><td>5.87 ± 0.13</td><td>5.51 ± 0.21</td><td>5.29 ± 0.25</td></tr></table>
+
+SVHN In Table 8, we similarly show results for compared methods of Figure 4b and results of methods mentioned above. Fully supervised learning using 73,257 examples achieves an error rate of 2.84 and 2.28 with or without RandAugment. The performance of the baseline models are reported by MixMatch (Berthelot et al., 2019). Our reimplementation of MixMatch also resulted in comparable but higher error rates than the reported ones.
+
+<table><tr><td>Methods /# Sup</td><td>250</td><td>500</td><td>1,000</td><td>2,000</td><td>4,000</td></tr><tr><td>Pseudo-Label</td><td>21.16 ± 0.88</td><td>14.35 ± 0.37</td><td>10.19 ± 0.41</td><td>7.54 ± 0.27</td><td>5.71 ± 0.07</td></tr><tr><td>II-Model</td><td>17.65 ± 0.27</td><td>11.44 ± 0.39</td><td>8.60 ± 0.18</td><td>6.94 ± 0.27</td><td>5.57 ± 0.14</td></tr><tr><td>Mean Teacher</td><td>6.45 ± 2.43</td><td>3.82 ± 0.17</td><td>3.75 ± 0.10</td><td>3.51 ± 0.09</td><td>3.39 ± 0.11</td></tr><tr><td>VAT</td><td>8.41 ± 1.01</td><td>7.44 ± 0.79</td><td>5.98 ± 0.21</td><td>4.85 ± 0.23</td><td>4.20 ± 0.15</td></tr><tr><td>MixMatch</td><td>3.78 ± 0.26</td><td>3.64 ± 0.46</td><td>3.27 ± 0.31</td><td>3.04 ± 0.13</td><td>2.89 ± 0.06</td></tr><tr><td>UDA (RandAugment)</td><td>2.76 ± 0.17</td><td>2.70 ± 0.09</td><td>2.55 ± 0.09</td><td>2.57 ± 0.09</td><td>2.47 ± 0.15</td></tr></table>
+
+Table 8: Error rate $( \% )$ for SVHN.
+
+# B ADDITIONAL TRAINING TECHNIQUES
+
+While UDA generally works well, there are several practical issues in consistency training that may dampen the performance gain if not carefully dealt with. This section presents additional techniques targeting at some commonly encountered problems.
+
+Sharpening Predictions. Entropy minimization (Grandvalet & Bengio, 2005) has been shown to be effective in semi-supervised learning methods such as VAT (Miyato et al., 2018). To apply entropy minimization in our method, we simply add a loss term to the objective to regularize the predicted distributions on unlabeled examples to have a low entropy. Alternatively, when the number of labeled examples are extremely small, we find it helpful to mask out examples that the current model is not confident about and use a low Softmax temperature when computing the target distribution on unlabeled examples. Specifically, in each minibatch, the consistency loss term is computed only on examples whose highest probability among classification categories is greater than a threshold.
+
+Domain-relevance Data Filtering. Ideally, we would like to make use of out-of-domain unlabeled data since it is usually much easier to collect, but the class distributions of out-of-domain data are mismatched with those of in-domain data, which can result in performance loss if directly used (Oliver et al., 2018). To obtain data relevant to the domain for the task at hand, we adopt a common technique for detecting out-of-domain data. We use our baseline model trained on the in-domain data to infer the labels of data in a large out-of-domain dataset and pick out examples that the model is most confident about. Specifically, for each category, we sort all examples based on the classified probabilities of being in that category and select the examples with the highest probabilities.
+
+# C EXTENDED AUGMENTATION STRATEGIES FOR DIFFERENT TASKS
+
+Discussion on Trade-off Between Diversity and Validity for Data Augmentation. Despite that state-of-the-art data augmentation methods can generate diverse and valid augmented examples as discussed in section 2.2, there is a trade-off between diversity and validity since diversity is achieved by changing a part of the original example, naturally leading to the risk of altering the ground-truth label. We find it beneficial to tune the trade-off between diversity and validity for data augmentation methods. For text classification, we tune the temperature of random sampling. On the one hand, when we use a temperature of 0, decoding by random sampling degenerates into greedy decoding and generates perfectly valid but identical paraphrases. On the other hand, when we use a temperature of 1, random sampling generates very diverse but barely readable paraphrases. We find that setting the Softmax temperature to 0.7, 0.8 or 0.9 leads to the best performances.
+
+RandAugment Details. In our implementation of RandAugment, each sub-policy is composed of two operations, where each operation is represented by the transformation name, probability, and magnitude that is specific to that operation. For example, a sub-policy can be [(Sharpness, 0.6, 2), (Posterize, 0.3, 9)].
+
+For each operation, we randomly sample a transformation from 15 possible transformations, a magnitude in [1, 10) and fix the probability to 0.5. Specifically, we sample from the following 15 transformations: Invert, Cutout, Sharpness, AutoContrast, Posterize, ShearX, TranslateX, TranslateY, ShearY, Rotate, Equalize, Contrast, Color, Solarize, Brightness. We find this setting to work well in our first try and did not tune the magnitude range and the probability. Tuning these hyperparameters might result in further gains in accuracy.
+
+TF-IDF based word replacing Details. We describe the TF-IDF based word replacing data augmentation method in this section. Ideally, we would like the augmentation method to generate both diverse and valid examples. Hence, the augmentation is designed to retain keywords and replace uninformative words with other uninformative words. We use BERT’s word tokenizer since BERT first tokenizes sentences into a sequence of words and then tokenize words into subwords although the model uses subwords as input.
+
+Specifically, Suppose $\mathrm { I D F } ( w )$ is the IDF score for word $w$ computed on the whole corpus, and $\mathrm { T F } ( w )$ is the TF score for word $w$ in a sentence. We compute the TF-IDF score as $\mathrm { T F I D \bar { F } } ( w ) =$ $\mathrm { T F } ( w ) \mathrm { I D F } ( w )$ . Suppose the maximum TF-IDF score in a sentence $x$ is $C = \mathrm { m a x } _ { i } \mathrm { T F I D F } ( x _ { i } )$ . To make the probability of having a word replaced to negatively correlate with its TF-IDF score, we set the probability to $\mathrm { \dot { m i n } } ( p ( C - \mathrm { T F I D F } ( \bar { x _ { i } } ) ) / Z , 1 )$ , where $p$ is a hyperparameter that controls the magnitude of the augmentation and $\begin{array} { r } { Z = \sum _ { i } ( C - \mathrm { T F I D F } ( x _ { i } ) ) / | x | } \end{array}$ is the average score. $p$ is set to 0.7 for experiments on DBPedia.
+
+When a word is replaced, we sample another word from the whole vocabulary for the replacement. Intuitively, the sampled words should not be keywords to prevent changing the ground-truth labels of the sentence. To measure if a word is keyword, we compute a score of each word on the whole corpus. Specifically, we compute the score as $S ( w ) = \mathrm { f r e q } ( w ) \mathrm { I D F } ( w )$ where $\operatorname { f r e q } ( w )$ is the frequency of word $w$ on the whole corpus. We set the probability of sampling word $w$ as $( \operatorname* { m a x } _ { w ^ { \prime } } \bar { S ( w ^ { \prime } ) } - \bar { S ( w ) } ) / Z ^ { \prime }$ where $\begin{array} { r } { Z ^ { \prime } = \sum _ { w } \operatorname* { i n a x } _ { w ^ { \prime } } S ( w ^ { \prime } ) - \bar { S ( w ) } } \end{array}$ is a normalization term.
+
+# D EXTENDED RELATED WORK
+
+Semi-supervised Learning. Due to the long history of semi-supervised learning (SSL), we refer readers to (Chapelle et al., 2009) for a general review. More recently, many efforts have been made to renovate classic ideas into deep neural instantiations. For example, graph-based label propagation (Zhu et al., 2003) has been extended to neural methods via graph embeddings (Weston et al., 2012; Yang et al., 2016) and later graph convolutions (Kipf & Welling, 2016). Similarly, with the variational auto-encoding framework and reinforce algorithm, classic graphical models based SSL methods with target variable being latent can also take advantage of deep architectures (Kingma et al., 2014; Maaløe et al., 2016; Yang et al., 2017). Besides the direct extensions, it was found that training neural classifiers to classify out-of-domain examples into an additional class (Salimans et al., 2016) works very well in practice. Later, Dai et al. (2017) shows that this can be seen as an instantiation of low-density separation.
+
+Apart from enforcing consistency on the noised input examples and the hidden representations, another line of research enforces consistency under different model parameters, which is complementary to our method. For example, Mean Teacher (Tarvainen & Valpola, 2017) maintains a teacher model with parameters being the ensemble of a student model’s parameters and enforces the consistency between the predictions of the two models. Recently, Athiwaratkun et al. (2018) propose fast-SWA that improves Mean Teacher by encouraging the model to explore a diverse set of plausible parameters. In addition to parameter-level consistency, SNTG (Luo et al., 2018) also enforces input-level consistency by constructing a similarity graph between unlabeled examples.
+
+Data Augmentation. Also related to our work is the field of data augmentation research. Besides the conventional approaches and two data augmentation methods mentioned in Section 2.1, a recent approach MixUp (Zhang et al., 2017) goes beyond data augmentation from a single data point and performs interpolation of data pairs to achieve augmentation. Recently, Hernandez-Garc ´ ´ıa & Konig (2018) have shown that data augmentation can be regarded as a kind of explicit regularization ¨ methods similar to Dropout.
+
+Diverse Back Translation. Diverse paraphrases generated by back-translation has been a key component in the significant performance improvements in our text classification experiments. We use random sampling instead of beam search for decoding similar to the work by Edunov et al. (2018). There are also recent works on generating diverse translations (He et al., 2018; Shen et al., 2019; Kool et al., 2019) that might lead to further improvements when used as data augmentations.
+
+Unsupervised Representation Learning. Apart from semi-supervised learning, unsupervised representation learning offers another way to utilize unsupervised data. Collobert & Weston (2008) demonstrated that word embeddings learned by language modeling can improve the performance significantly on semantic role labeling. Later, the pre-training of word embeddings was simplified and substantially scaled in Word2Vec (Mikolov et al., 2013) and Glove (Pennington et al., 2014). More recently, Dai & Le (2015); Peters et al. (2018); Radford et al. (2018); Howard & Ruder (2018); Devlin et al. (2018) have shown that pre-training using language modeling and denoising auto-encoding leads to significant improvements on many tasks in the language domain. There is also a growing interest in self-supervised learning for vision (Zhai et al., 2019b; Henaff et al., 2019; ´ Trinh et al., 2019).
+
+Consistency Training in Other Domains. Similar ideas of consistency training has also been applied in other domains. For example, recently, enforcing adversarial consistency on unsupervised data has also been shown to be helpful in adversarial robustness (Stanforth et al., 2019; Zhai et al., 2019a; Carmon et al., 2019). Enforcing consistency w.r.t data augmentation has also been shown to work well for representation learning (Hu et al., 2017; Ye et al., 2019). Invariant representation learning (Liang et al., 2018; Salazar et al., 2018) applies the consistency loss not only to the predicted distributions but also to representations and has been shown significant improvements on speech recognition.
+
+# E EXPERIMENT DETAILS
+
+In this section, we provide experiment details for the performed experiments.
+
+# E.1 TEXT CLASSIFICATIONS
+
+Datasets. In our semi-supervised setting, we randomly sampled labeled examples from the full supervised set5 and use the same number of examples for each category. For unlabeled data, we use the whole training set for DBPedia, the concatenation of the training set and the unlabeled set for IMDb and external data for Yelp-2, Yelp-5, Amazon-2 and Amazon-5 (McAuley et al., 2015)6. Note that for Yelp and Amazon based datasets, the label distribution of the unlabeled set might not match with that of labeled datasets since there are different number of examples in different categories. Nevertheless, we find it works well to use all the unlabeled data.
+
+Preprocessing. We find the sequence length to be an important factor in achieving good performance. For all text classification datasets, we truncate the input to 512 subwords since BERT is pretrained with a maximum sequence length of 512. Further, when the length of an example is greater than 512, we keep the last 512 subwords instead of the first 512 subwords as keeping the latter part of the sentence lead to better performances on IMDb.
+
+Fine-tuning BERT on in-domain unsupervised data. We fine-tune the BERT model on in-domain unsupervised data using the code released by BERT. We try learning rate of 2e-5, 5e-5 and 1e-4, batch size of 32, 64 and 128 and number of training steps of $3 0 \mathrm { k }$ , $1 0 0 \mathrm { k }$ and $3 0 0 \mathrm { k }$ . We pick the fine-tuned models by the BERT loss on a held-out set instead of the performance on a downstream task.
+
+Random initialized Transformer. For the experiments with randomly initialized Transformer, we adopt hyperparameters for BERT base except that we only use 6 hidden layers and 8 attention heads. We also increase the dropout rate on the attention and the hidden states to 0.2, When we train UDA with randomly initialized architectures, we train UDA for $5 0 0 \mathrm { k }$ or 1M steps on Amazon-5 and Yelp5 where we have abundant unlabeled data.
+
+BERT hyperparameters. Following the common BERT fine-tuning procedure, we keep a dropout rate of 0.1, and try learning rate of 1e-5, 2e-5 and 5e-5 and batch size of 32 and 128. We also tune the number of steps ranging from 30 to $1 0 0 \mathrm { k }$ for various data sizes.
+
+UDA hyperparameters. We set the weight on the unsupervised objective $\lambda$ to 1 in all of our experiments. We use a batch size of 32 for the supervised objective since 32 is the smallest batch size on v3-32 Cloud TPU Pod. We use a batch size of 224 for the unsupervised objective when the Transformer is initialized with BERT so that the model can be trained on more unlabeled data. We find that generating one augmented example for each unlabeled example is enough for BERTFINETUNE.
+
+All experiments in this part are performed on a v3-32 Cloud TPU Pod.
+
+# E.2 SEMI-SUPERVISED LEARNING BENCHMARKS CIFAR-10 AND SVHN
+
+Hyperparameters for Wide-ResNet-28-2. For hyperparameter tuning, for simplicity, we performed a random sampling search over hyperparameters and choose the best one based on validation sets ( $20 \%$ of the training sets with different sizes). We use the averaged results of multiple experiments to reduce the performance variance measured on the small validation sets. Specifically, we tried the following ranges:
+
+• training steps: 50k, 100k;   
+• learning rate: 0.03, 0.05, 0.1;   
+• TSA schedules: log-schedule, linear-schedule, exp-schedule, not using TSA; entropy minimization loss weight: 0, 0.1, 0.3; consistency loss weight: 1, 3, 6; unlabeled data batch size: 960, 1280;   
+• weight decay rate: 5e-4, 7e-4, 1e-3;   
+• softmax temperature: 1, 0.9;   
+• confidence threshold: 0, 0.8.
+
+where the values in bold black text are our default hyperparameters. We found that given a reasonably large labeled set, our method is robust to hyper-parameters. Therefore, we use the same hyper-parameters in these cases. Specifically, we use the above default hyperparameters for CIFAR10 with 4,000, 2,000, 1,000 and 500 examples. For SVHN with 4,000, 2,000, 1,000, 500, 250 examples, we additionally set the learning rate to 0.05 and unlabeled batch size to 1280. For the case of 250 examples on CIFAR-10, we use a different set of hyper-parameters: training steps: 50k; TSA schedule: log-schedule; consistency loss coefficient: 6; weight decay: 7e-4; unlabeled data batch size: 1280; softmax temperature: 0.9; consistency threshold: 0.8.
+
+Other hyperparameters not mentioned above are the same to the the original paper of WideResNet (Zagoruyko & Komodakis, 2016). In order to reduce training time, we generate augmented examples before training and dump them to disk. For CIFAR-10, we generate 100 augmented examples for each unlabeled example. Note that generating augmented examples in an online fashion is always better or as good as using dumped augmented examples since the model can see different augmented examples in different epochs, leading to more diverse samples. We report the average performance and the standard deviation for 10 runs.
+
+Hyperparameters for Shake-Shake and PyramidNet. For the experiments with Shake-Shake, we train UDA for 300k steps and use a batch size of 128 for the supervised objective and use a batch size of 512 for the unsuperivsed objective. For the experiments with PyramidNet+ShakeDrop, we train UDA for $7 0 0 \mathrm { k }$ steps and use a batch size of 64 for the supervised objective and a batch size of 128 for the unsupervised objective. For both models, we use a learning rate of 0.03 and use a cosine learning decay with one annealing cycle following AutoAugment.
+
+All experiments in this part are performed on a v3-32 Cloud TPU v3 Pod.
+
+# E.3 IMAGENET
+
+$10 \%$ Labeled Set Setting. Unless otherwise stated, we follow the standard hyperparameters used in an open-source implementation of ResNet.7 For the $10 \%$ labeled set setting, we use a batch size of 512 for the supervised objective and a batch size of 15,360 for the unsupervised objective. We use a base learning rate of 0.3 that is decayed by 10 for four times and set the weight on the unsupervised objective $\lambda$ to 20. We mask out unlabeled examples whose highest probabilities across categories are less than 0.5 and set the Softmax temperature to 0.4. The model is trained for 40k steps. Experiments in this part are performed on a v3-64 Cloud TPU v3 Pod.
+
+Full Labeled Set Setting. For experiments on the full ImageNet, we use a batch size of 8,192 for the supervised objective and a batch size of 16,384 for the unsupervised objective. The weight on the unsupervised objective $\lambda$ is set to 1. We use entropy minimization to sharpen the prediction. We use a base learning rate of 1.6 and decay it by 10 for four times. Experiments in this part are performed on a v3-128 Cloud TPU v3 Pod.

parse/train/ee6W5UgQLa/ee6W5UgQLa.md

@@ -0,0 +1,269 @@
+# MULTIMODALQA: COMPLEX QUESTION ANSWERING OVER TEXT, TABLES AND IMAGES
+
+Alon Talmor∗,1,2 Ori Yoran∗,1,2 Amnon Catav∗,2 Dan Lahav∗,2 Yizhong Wang3   
+Akari Asai3 Gabriel Ilharco3 Hannaneh Hajishirzi2,3 Jonathan Berant1,2
+
+1The Allen Institute for AI, 2Tel-Aviv University, 3University of Washington {alont,oriy,jonathan}@allenai.org   
+{amnoncatav,lahav}@mail.tau.ac.il   
+{yizhongw,akari,gamaga,hannaneh}@cs.washington.edu
+
+# ABSTRACT
+
+When answering complex questions, people can seamlessly combine information from visual, textual and tabular sources. While interest in models that reason over multiple pieces of evidence has surged in recent years, there has been relatively little work on question answering models that reason across multiple modalities. In this paper, we present MULTIMODALQA (MMQA): a challenging question answering dataset that requires joint reasoning over text, tables and images. We create MMQA using a new framework for generating complex multi-modal questions at scale, harvesting tables from Wikipedia, and attaching images and text paragraphs using entities that appear in each table. We then define a formal language that allows us to take questions that can be answered from a single modality, and combine them to generate cross-modal questions. Last, crowdsourcing workers take these automatically generated questions and rephrase them into more fluent language. We create 29,918 questions through this procedure, and empirically demonstrate the necessity of a multi-modal multi-hop approach to solve our task: our multi-hop model, ImplicitDecomp, achieves an average $\mathrm { F _ { 1 } }$ of 51.7 over cross-modal questions, substantially outperforming a strong baseline that achieves $3 8 . 2 \mathrm { F _ { 1 } }$ , but still lags significantly behind human performance, which is at $9 0 . 1 \mathrm { F _ { 1 } }$ .
+
+# 1 INTRODUCTION
+
+When presented with complex questions, people often do not know in advance what source(s) of information are relevant for answering it. In general scenarios, these sources can encompass multiple modalities, be it paragraphs of text, structured tables, images or combinations of those. For instance, a user might ponder “When was the famous painting with two touching fingers completed?”, if she cannot remember the exact name of the painting. Answering this question is made possible by integrating information across both the textual and visual modalities.
+
+Recently, there has been substantial interest in question answering (QA) models that reason over multiple pieces of evidence (multi-hop questions (Yang et al., 2018; Talmor & Berant, 2018; Welbl et al., 2017)). In most prior work, the question is phrased in natural language and the answer is in a context, which may be a paragraph (Rajpurkar, 2016), a table (Pasupat & Liang, 2015), or an image (Antol et al., 2015). However, there has been relatively little work on answering questions that require integrating information across modalities. Hannan et al. (2020) created MANYMODALQA: a dataset where the context for each question includes information from multiple modalities. However, the answer to each question can be derived from a single modality only, and no cross-modality reasoning is needed. Thus, the task is focused on identifying the relevant modality. Recently, Chen et al. (2020b) presented HYBRIDQA, a dataset that requires reasoning over tabular and textual data. While HYBRIDQA requires cross-modal reasoning, it does not require visual inference, limiting the types of questions that can be represented (See Table 1 for a comparison between the datasets).
+
+# Multimodal Context
+
+![](images/5c1613ac705899207e7e983dc4f8aad672769668f3aa396af2992ec3ecdfc5b5.jpg)  
+Q: Which B.Piazza titlecame earlier: the movie S.Stalone'sson staredinor the movie with halfofalady'sface on the poster? A:Tell Me That You Love Me, Junie Moon   
+Figure 1: Example of a MMQA question, answer and context. In green are the text modality question and answer, and in red the image modality. The table is used to perform the year comparison between the answers of the text and image question parts.
+
+In this work, we present MMQA, the first large-scale (29,918 examples) QA dataset that requires integrating information across free text, semi-structured tables, and images, where $3 5 . 7 \%$ of the questions require cross-modality reasoning. Figure 1 shows an example question: “Which B.Piazza title came earlier: the movie S. Stallon’s son starred in, or the movie with half of a lady’s face on the poster?”. Answering this question entails (i) decomposing the question into a sequence of simpler questions, (ii) determining the modalities for the simpler questions and answering them, i.e., information on the poster is in an image, the information on ${ } ^ { \mathfrak { a } } S .$ Stallon’s son” is in free text, and the years of the movies are in the table, (iii) combining the information from the simpler questions to compute the answer: “Tell Me that you love me, Junie Moon”.
+
+Our methodology for creating MMQA involves three high-level steps. (a) Context construction: we harvest tables from Wikipedia, and connect each table to images and paragraphs that appear in existing Reading Comprehension (RC) datasets (Kwiatkowski et al., 2019; Clark et al., 2019; Yang et al., 2018); (b) Question generation: Following past work (Talmor & Berant, 2018), we use the linked structure of the context to automatically generate questions that require multiple reasoning operations (composition, conjunction, comparison) across modalities in pseudo-language ; (c) Paraphrasing: we use crowdsourcing workers to paraphrase the pseudo-language questions into more fluent English.
+
+To tackle MMQA, we introduce ImplicitDecomp, a new model that predicts a program that specifies the required reasoning steps over different modalities, and executes the program with dedicated text, table, and image models. ImplicitDecomp performs multi-hop multimodal reasoning without the need for an explicit decomposition of the question.
+
+We empirically evaluate MMQA by comparing ImplicitDecomp to strong baselines that do not perform cross-modal reasoning and to human performance. We find that on multimodal questions, ImplicitDecomp improves $\mathrm { F _ { 1 } }$ from
+
+Table 1: A comparison of MULTIMODALQA to MANYMODALQA and HYBRIDQA. We compare dataset size, use of images, and whether the dataset supports multihop questions and an open-domain full-wiki setup.   
+
+<table><tr><td>Dataset</td><td>Size</td><td>Full- wiki</td><td>Uses images</td><td>Multi- hop</td></tr><tr><td>MANYMODALQA</td><td>10K</td><td>X</td><td>√</td><td>X</td></tr><tr><td>HYBRIDQA</td><td>70K</td><td>X</td><td>×</td><td>√</td></tr><tr><td>MULTIMODALQA</td><td>30K</td><td>√</td><td>√</td><td>√</td></tr></table>
+
+$3 8 . 2  5 1 . 7 $ over a single-hop approach. Humans are able to reach $9 0 . 1 \mathrm { F _ { 1 } }$ , significantly outperforming our best model. Because automatic evaluation is non-trivial, we also manually analyze human performance and find humans correctly answer $9 4 . 5 \%$ of the questions in MMQA. Finally, our dataset can be used in an open-domain setup over all of Wikipedia. In this setup, the $\mathrm { F _ { 1 } }$ of humans is 84.8.
+
+To summarize, our key contributions are:
+
+• MMQA: a dataset with 29,918 questions and answers, $3 5 . 7 \%$ of which require cross-modal reasoning. • A methodology for generating multimodal questions over text, tables and images at scale. • ImplicitDecomp, A model for implicitly decomposing multimodal questions, which improves on a single-hop model by 13.5 absolute $\mathrm { F _ { 1 } }$ points on questions requiring cross-modal reasoning. • Our dataset and code are available at https://allenai.github.io/multimodalqa.
+
+# 2 DATASET GENERATION
+
+Our goal is to develop a method that allows generating complex questions over multiple modalities at scale. An overview of the methodology is captured in Figure 2. We first select a Wikipedia table as an anchor, to which we add images and texts paragraphs and obtain a context. Single modality questions are generated based on these contexts, and used to automatically create multimodal, multihop questions. AMT workers rephrase the questions into natural language, and finally distractor paragraphs and images are selected for each question. We now elaborate on the 6 steps of the process.
+
+![](images/c677a9a1b1216fe181547eefdaed522f58fff6f5ef071ef7e788cd277e95cbb4.jpg)  
+Figure 2: An overview of MMQA dataset generation process.
+
+2.1 Wikipedia tables as anchors The 01-01-2020 English Wikipedia dump contains roughly $3 M$ tables. We extracted all tables and selected those that meet the following criteria: (a) The tables contain 10-25 rows (b) At least 3 images are associated with the table. This results in a total of $7 0 0 \mathrm { k }$ tables. (see supp. material for more information). These tables are the anchors of our contexts, which we enrich with images and text for multimodal question generation. A key element of the tables are Wikipedia Entities (WikiEntities) that appear in them, i.e., concepts linked to other Wikipedia entries. We use them to connect different modalities, bridge questions, and solve ambiguities (details below).
+
+2.2 Connecting Images and Text to Tables Images. We consider two cases: (a) in-table images and (b) images from pages of linked WikiEntities. In the former, the images are featured inside the table cells. In the latter, the table contains a column of WikiEntities that potentially have images, e.g. a table describing the filmography of an actor often contains a column of film names, which may have posters in their respective pages. To associate entities with their representative image, we map entities and their profile images in their Wikipedia pages. Overall, we obtain 57,713 images, with 889 in-table images and 56,824 WikiEntities images. Text. We build on texts from contexts appearing in existing reading comprehension datasets. We elaborate on this process next.
+
+2.3 Generating Single-Modality Questions Tables. We generate pseudo-language table questions in the following form “In [table title] of [Wikipedia page title] which cells in [column X] have the [value Y] in [column Z]?”. We additionally support numeric computations over columns classified as dates or numbers, such as min and max values, e.g., “In [Doubles] of [WCT Tournament of Champions], what was the MOST RECENT [Year](s) where the [Location] was [Forest Hills]”.
+
+Images. We use crowdsourcing to generate single-modality questions about images. We generated two types of image questions, based on the images we retrieved from the previous step: (i) questions over a single image, (ii) questions over a list of images.
+
+When generating single-image questions, we show Amazon Mechanical Turk (AMT) crowd workers an image alongside its WikiEntity, and ask them to phrase a question about the image with the entity being the focus of the question. E.g, if the entity is “Roger Federer”, a potential question is “What’s the hair color of Roger Federer?”. For questions to have meaning in an open-domain setting, we primed AMT workers to ask questions that correspond to “stable” features, i.e., features that are unlikely to change in different images and are thus appropriate in an open-domain setting.
+
+Table 2: All 16 compositional templates in MMQA with an example and their relative frequency.   
+
+<table><tr><td rowspan=1 colspan=1>Type</td><td rowspan=1 colspan=1>Q&amp;A</td><td rowspan=1 colspan=1>%</td></tr><tr><td rowspan=1 colspan=1>TextQ</td><td rowspan=1 colspan=1>What wasthe territorial capitalof the territory opposingOhioin theToledoWar?Detroit</td><td rowspan=1 colspan=1>31.0</td></tr><tr><td rowspan=1 colspan=1>TableQ</td><td rowspan=1 colspan=1>DoestheGerman state Baden-Wurttemberg orThuringia have moreresidents?Baden-Wurttemberg</td><td rowspan=1 colspan=1>18.3</td></tr><tr><td rowspan=1 colspan=1>ImageQ</td><td rowspan=1 colspan=1>WhatweaponisthestatueinNottinghamholding?bow</td><td rowspan=1 colspan=1>8.9</td></tr><tr><td rowspan=1 colspan=1>Compose(TextQ,TableQ)</td><td rowspan=1 colspan=1>Atwhatagedid theCleveland Cavaliersplayer with 6190rebounds enter the NBA?19</td><td rowspan=1 colspan=1>7.8</td></tr><tr><td rowspan=1 colspan=1>ImageListQ</td><td rowspan=1 colspan=1>Whatisthecommon nameof thebushwarblerinThailand that hasanorange stripe above itseye?Chestnut-crowned bush warbler</td><td rowspan=1 colspan=1>6.1</td></tr><tr><td rowspan=1 colspan=1>Compose(TableQ,ImageListQ)</td><td rowspan=1 colspan=1>Thefilm that starredChrisEllisonwhereamanwasholding anewspaper on the poster, was released what year?1988</td><td rowspan=1 colspan=1>5.4</td></tr><tr><td rowspan=1 colspan=1>Compose(ImageQ,TableQ)</td><td rowspan=1 colspan=1>OntheposterfortheTVshowinwhich TomMisonplayedDorianCrane,what kind of structure can be seen behind the two men? castle</td><td rowspan=1 colspan=1>4.5</td></tr><tr><td rowspan=1 colspan=1>Compare(Compose(TableQ,ImageQ),TableQ)</td><td rowspan=1 colspan=1>Which manufacturerhas fewerwinsattheFirst Data 5Oo:Buick or thebrand with across for a logo?Buick</td><td rowspan=1 colspan=1>3.5</td></tr><tr><td rowspan=1 colspan=1>Compose(TableQ,TextQ)</td><td rowspan=1 colspan=1>Onwhat date did the original artistwho sangSweetChildof Mine haveaconcert atUSBank Stadium?July 30,2017</td><td rowspan=1 colspan=1>3.2</td></tr><tr><td rowspan=1 colspan=1>Intersect(TableQ,TextQ)</td><td rowspan=1 colspan=1>Who was the artistfor DamonFox in20o6who also sings &quot;You gotthemoves like Jagger&quot;?Christina Aguilera</td><td rowspan=1 colspan=1>2.6</td></tr><tr><td rowspan=1 colspan=1>Compose(TextQ,ImageListQ)</td><td rowspan=1 colspan=1>Ontheposterfor themoviebasedonthebook &quot;ActlikeaLady,ThinkLikea Man,&quot;how many people are there in total? nine</td><td rowspan=1 colspan=1>2.4</td></tr><tr><td rowspan=1 colspan=1>Intersect(ImageListQ,TableQ)</td><td rowspan=1 colspan=1>WhatcoversoftheChandlerCanterburyfilmsfrom2oo9hasmorethanoneperson?PowderBlueBallsOut,Gary theTennisCoach,After.Life</td><td rowspan=1 colspan=1>2.3</td></tr><tr><td rowspan=1 colspan=1>Compare(TableQ,Compose(TableQ,TextQ))</td><td rowspan=1 colspan=1>DidChelseaorclubthat singsYou&#x27;llNeverWalkAlonerank higherinDeloitteFootball MoneyLeague20o7?Chelsea</td><td rowspan=1 colspan=1>2.1</td></tr><tr><td rowspan=1 colspan=1>Compose(ImageQ,TextQ)</td><td rowspan=1 colspan=1>DidGary Oldmantakepart inthemoviewhoseposterfeaturestwomenholding handguns,and which had MarkL.Smith as a writer? no</td><td rowspan=1 colspan=1>1.0</td></tr><tr><td rowspan=1 colspan=1>Compare(Compose(TableQ,ImageQ),Compose(TableQ,TextQ))</td><td rowspan=1 colspan=1>Wasthefilmthat featuresa gianteyeonitsposter or the firstWolverinemovie theearlierfilm thatScott Silverworkedon? Requiem fora Dream</td><td rowspan=1 colspan=1>0.8</td></tr><tr><td rowspan=1 colspan=1>Intersect(ImageListQ,TextQ)</td><td rowspan=1 colspan=1>Whatcommonlaw statewith aneagleontheflag hasan institutionin the North region of DivisionIIof theNCCAA?Iowa</td><td rowspan=1 colspan=1>0.2</td></tr></table>
+
+For questions with a list of images, we use images that appear in the same column of a table. To generate these questions, AMT workers were given the images and asked to phrase a binary question about a distinctive feature of the entities that a subset of the images share. E.g., given a list of statues, the worker could ask “Which of the statues features a horse?” This process results in 2,764 single image questions and 7,773 list image questions that are later used to create multimodal questions.
+
+Text. To obtain questions answerable over text paragraphs we build on existing reading comprehension datasets: Natural Questions (NQ) (Kwiatkowski et al., 2019) consists of about $3 0 0 K$ questions issued to the Google search engine. This dataset mostly contains simple questions where a single paragraph suffices to answer each question. BoolQ (Clark et al., 2019) contains 15, 942 yes/no questions, gathered using the same pipeline as NQ. HotpotQA (Yang et al., 2018) contains $1 1 2 K$ training questions, where crowd workers were shown pairs of related Wikipedia paragraphs and were asked to author questions that require multi-hop reasoning over the paragraphs.
+
+To use questions from the above datasets as building blocks for multi-hop multimodal questions, we unified them into a corpus that consists of triples of (i) a text question, (ii) an answer and (iii) 1-2 gold paragraphs from Wikipedia. We link a question to a table, by matching WikiEntities in the table to entities in the text of the question (see supplementary material for further details). Overall, we retrieved 6,644 questions from NQ, 1,246 from BoolQ and 4,733 from HotpotQA.
+
+2.4 Generating multimodal complex questions We present an automatic method for creating at scale multimodal compositional questions (i,e., questions that require answering a sequence of subquestions to conclude the final answer). Our first step is to introduce a formal language that allows to combine questions answerable from a single modality. Below we introduce the logical operations that allow to generate such pseudo-language (PL) questions, while keeping a formal representation of how they were constructed. In Table 2, we illustrate this process with all 16 different compositional templates used for question generation. We now describe our logical operations.
+
+Logical Operations Functions in our formal language take arguments and return a PL partial question, as well as answers that can be a list of one or more strings, or a list of one or more WikiEntities. All operations have access to the full context. In addition, we prepend a prefix containing the Wikipedia table name and page title—e.g. “In the Filmography of Brad Pitt,”—to all our PL questions to support an open-domain QA setup. Our set of logical operations are:
+
+1. TABLEQ: Returns a question from the table questions generated in $\ S 2 . 3$ , as well as a list of WikiEntities or a list of strings as answers.
+
+2. TEXTQ: Returns a text corpus question (see $\ S 2 . 3$ ) and a list of WikiEntities or strings as answers.
+
+3. IMAGEQ: Returns a question about a single image associated with a WikiEntity and a single token answer from a fixed vocabulary (see $\ S 2 . 3 \AA$ ).
+
+4. IMAGELISTQ: Returns a question about a list of images and a list of WikiEntities corresponding to the images that answer the question (see $\ S 2 . 3$ ).
+
+5. COMPOSE $( \cdot , \cdot )$ : Takes a PL question containing a single WikiEntity as a first argument, and a PL question that produces that WikiEntity as the output answer as its second argument. E.g., COMPOSE(“Where was Barack Obama born?”,“Who was the 44th president of the USA?”). The function replaces the WikiEntity in the first-argument PL question with the second-argument PL question and returns the resulting PL question (“Where was the 44th president of the USA born?”).
+
+6. INTERSECT $( \cdot , \cdot )$ : Takes two PL questions that return lists of more than one WikiEntity, and returns their intersection as the answer. The resulting $\mathrm { P L }$ question is of the form ${ } ^ { \cdot } P L _ { 1 }$ and $P L _ { 2 }$ ” omitting $\mathrm { P L _ { 2 } }$ ’s first word (“Who was born in Hawaii and is the parent of Sasha Obama?”).
+
+7. COMPARE $( \cdot , \cdot )$ : Takes two PL questions each returning one WikiEntity that can be linked to one cell in the table, denoted by $\mathrm { { A n s } _ { 1 } }$ , $\mathrm { { A n s } _ { 2 } }$ . We first choose a numeric or date column in the table, if such exists. We then compare the values of this column corresponding to the rows of $\mathbf { A n s } _ { 1 }$ and Ans2. Depending on the comparison outcome, output one of $( \mathrm { A n s _ { 1 } }$ , Ans2) as the operation answer. The PL question created is of the form “What has compare-op numeric-column-name, $P L _ { 1 }$ or $P L _ { 2 }$ ?” omitting $\mathrm { P L _ { 1 } }$ and $\mathrm { P L _ { 2 } }$ ’s first word. E.g. “What has most recent creation year, the rocket of Appolo program, or the rocket of Gemini program?”
+
+2.5 Paraphrasing using AMT We used English-speaking AMT workers to paraphrase automaticallygenerated PL questions into natural language (NL). Each question was paraphrased by 1 worker and validated by 1-3 other workers. To avoid annotator bias (Geva et al., 2019), the number of annotators who worked on both the training and evaluation set was kept to a minimum. We also deployed a feedback mechanism, where workers receive a bonus if a baseline model correctly answered the question after their first paraphrasing attempt, but incorrectly after they refined the paraphrase. See supp. material for print-screens of the AMT annotator interface.
+
+To generate diversity, workers got a bonus if the normalized edit distance of a paraphrase compared to the PL question was higher than 0.7. A total of 971 workers were involved, and 29,918 examples were produced with an average cost of $0 . 3 3 \ S$ per question. We split the dataset into 23,817 training, 2,441 development (dev.), and 3,660 test set examples. Context components in the dev. and test sets are disjoint, and were constructed from a disjoint set of single-modality questions.
+
+A shortcoming of our method for automatically generating examples is that the question distribution does not come from a “natural” source. We argue that developing models that are capable of performing reasoning over multiple modalities is an important direction and MMQA provides an opportunity to develop and evaluate such models. Moreover, this method allows to control the compositional questions created, proving effective in creating a cheap and scalable dataset.
+
+2.6 Adding distractors to the context Images. Questions from the IMAGELISTQ operator require reasoning over a list of images from the same column, and hence do not require additional distractors. For IMAGEQ questions (single-image), we randomly add images that are associated with the WikiEntities that appear in the table, setting a maximum of 15 distractors per question.
+
+Text. We used DPR (Karpukhin et al., 2020), a neural information retrieval model, to retrieve distractors for all questions. Each context includes exactly 10 paragraphs, where 1-2 are gold paragraphs and the rest are distractors. Specifically, we encode the first 2 paragraphs of each Wikipedia article with the DPR encoder, and use as distractors the paragraphs with the highest dot product between their encoding and the question encoding. We do not allow: (a) an overlap between the distractors in the training and evaluation sets, (b) distractors originating from the gold article, (c) distractors containing an exact match to the gold answer.
+
+To summarize, each of our examples contains a question, an answer, the formal representation of the PL question (ignored by our models), and all distractors and gold context for all modalities. This renders MMQA useful for both open-domain multimodal QA, as well as context-dependant QA.
+
+![](images/53d9aa241337ef2392714bb6dfb2adc55db8678aa5705c36b68b58c902fb16dd.jpg)  
+Figure 3: Domain diversity in MMQA. The area of each color corresponds to the topic frequency in the dataset.
+
+# 3 DATASET ANALYSIS
+
+To highlight the diversity of MMQA we analyze its key statistics, domains, and lexical richness.
+
+Key Statistics MMQA contains 29, 918 questions, and their main statistics are in Table 3. Since we focus on multimodality, we upsample the number of multimodal questions in the dev. and test sets compared to the training set. Also, about $60 \%$ of the questions in MMQA are compositional. Questions are relatively long (18.2 words), but answers tend to be short (2.1 words). The answer for each question can be a single answer or a list of answers. While list answers comprise only $7 . 4 \%$ of the data, when considering compositional questions that contain an intermediate question within them, the proportion of list answers in intermediate questions is higher $( 1 8 . 9 \% )$ ).
+
+Table 3: Key statistics for MMQA.   
+
+<table><tr><td>Measurement</td><td>Value</td></tr><tr><td># Distinct Questions</td><td>29,918</td></tr><tr><td>Train multimodal questions</td><td>34.6%</td></tr><tr><td>Dev.+test multimodal questions</td><td>40.1%</td></tr><tr><td>Train compositional questions</td><td>58.8%</td></tr><tr><td>Dev.+test compositional questions</td><td>62.3%</td></tr><tr><td>Average question length (words)</td><td>18.2</td></tr><tr><td>Average # of answers per question</td><td>1.16</td></tr><tr><td>List answers</td><td>7.4%</td></tr><tr><td>List answers per intermediate question</td><td>18.9%</td></tr><tr><td>Average answer length (words)</td><td>2.1</td></tr><tr><td># of distinct words in questions</td><td>49,649</td></tr><tr><td># of distinct words in answers</td><td>20,820</td></tr><tr><td># of distinct context tables</td><td>11,022</td></tr></table>
+
+Domain Diversity Figure 3 shows a sample of questions from MMQA categorized to different domains. While entertainment categories occupy a large portion of our dataset (Films $36 \%$ , TV $19 \%$ ), we observe questions represent a wide variety of topics.
+
+Lexical Richness Workers received a bonus when substantially modifying the PL questions. We observe that the average normalized edit distance between the NL questions and the PL questions is high (0.7), that NL questions are shorter (avg. length of 20.02 vs. 22.16 words for PL questions), and use a richer vocabulary (#unique words 39, 319 vs 37, 108).
+
+# 4 MODELS
+
+Here we present our baseline models. We first train models that interact with a single modality given a question (§4.1), and use those as building blocks in our multimodal approaches (§4.2). We denote the question by $Q$ , context paragraphs by $\mathcal { P }$ , Table by $T$ and context images by $\mathcal { T }$ .
+
+# 4.1 SINGLE-MODALITY QA MODULES
+
+Text QA Module Following prior work (Min et al., $2 0 1 9 \mathrm { a }$ ; Asai et al., 2020), our text QA module takes as input a question $Q$ and a paragraph $p \in \mathcal P$ and answers $Q$ by selecting a span in each paragraph $p$ independently, predicting the start and end positions (Devlin et al., 2019). Additionally, the model returns four scores for for every paragraph $p$ corresponding to: if the answer is $( i )$ a span in $p$ ; $( i i )$ “yes”; $( i i i )$ “no”; or $( i \nu )$ not in $p$ . At inference time, the model selects the paragraph that has the lowest score for $i \nu$ – the answer is not the paragraph. Our model is based on a pre-trained RoBERTa-large model (Liu et al., 2019), fine-tuned on MMQA.
+
+Table QA Module Following prior work (Herzig et al., 2020), our table QA module takes as input the question $Q$ and the table $T$ , and selects a subset of the table cells and an aggregation operation to compute the final answer. Specifically, we linearize the table $T$ by rows, with column names prepended to the corresponding cells (Chen et al., 2019). For example, this converts the table in Figure 1 to the following text: “Row 1: year is 1957; title is a dangerous age; role is David. Row 2...”. Next, we concatenate the question to the linearized table, and encode them using RoBERTa-large. We then pass the contextualized representation of every token in the table cell to a linear classifier that computes the probability of the token being selected. The score for a cell is the average of its tokens. Cells with probability $> 0 . 5$ are selected. Finally, another linear multi-class classifier predicts an aggregation operation from SUM, MEAN, COUNT, YES, NO, and NONE. Aggregation operations are applied on the selected cells, YES and NO operations output “yes” or “no”, and the NONE operation outputs all selected cells.
+
+Image QA Module Questions with visual information are handled by a multimodal transformer that processes the text question and pre-computed image features. For a question $Q$ and a set of images $\mathcal { T }$ , we feed the model the question and the visual features $\Phi ( i )$ extracted from each image $i \in \mathcal { T }$ , along with the name of the WikiEntity associated with the image. For each image and question, the model predicts an answer from a fixed vocabulary determined by the answers in the training set and 3 special tokens: $y _ { \mathrm { d t r } }$ , $y _ { \mathrm { p } }$ and $y _ { \mathrm { n } }$ . In questions where the expected answer is a phrase (e.g., ImageQ), we return the answer from the image where $p ( y _ { \mathrm { d t r } } )$ is lowest (similar to text QA). In questions where the expected answer is a subset of the images (e.g., Compose(TableQ,ImageQ)), we return all images where $p ( y _ { \mathrm { p } } ) > p ( y _ { \mathrm { n } } )$ . Our model is based on the pre-trained model VILBERT-MT1 (Lu et al., 2020). Visual features are extracted by a vision network $\Phi$ , comprised of a Faster R-CNN (Ren et al., 2015) pre-trained on Visual Genome (Krishna et al., 2017).
+
+# 4.2 MULTIMODALITY QA MODELS
+
+We turn to models that interact with multiple modalities.
+
+Multi-Hop Implicit Decomposition (ImplicitDecomp) Our dataset is designed to test reasoning across modalities. As a first attempt towards this goal, we introduce a 2-hop implicit decomposition baseline, capable of combining information scattered across modalities (illustrated in Figure 4).
+
+We first train a question-type classifier, based on RoBERTa-large, that takes a question $Q$ as input, and predicts one of the 16 possible question types (Table 2). The question type can be viewed as a program, specifying the relevant modalities, their order, and the logical operations. For example, if the question type is Compose(TextQ,TableQ), the first hop should be conducted on the table $T$ , and the second hop on the paragraphs $\mathcal { P }$ . In each hop, we feed the model with the question $Q$ , the question type, the hop number, and the context of the corresponding modality. The model automatically identifies which part of the question is relevant at the current hop and does not explicitly decompose the question into sub-questions (hence the name implicit decomposition). In the second hop, answers from the first hop are also given as input so that the model can leverage this information and conduct cross-modal reasoning to output the final answer. For all single-modality question types (such as TextQ and TableQ), the model uses only the first hop to get the answer.
+
+Single-Hop Routing (AutoRouting) A simple approach for answering questions without cross-modal reasoning is to first determine the modality where the answer is expected to occur, and then run the corresponding single-modality module. We use the aforementioned question type classifier to determine the modality where the answer will appear, route the question and the context for the predicted modality into the corresponding module, and use the output as the final answer.
+
+Question-only and Context-only baselines We run the question-only and context-only baselines, suggested by Kaushik & Lipton (2018). Our question-only baseline is BART-large (Lewis et al., 2019): a sequence-to-sequence model that directly generates the answer given the question. For the context-only baseline, we first predict the question type using the classifier described above to pick a target module. We then feed the relevant context to the target module, replacing the question with an empty string.
+
+# 4.3 TRAINING AND SUPERVISION
+
+Our dataset provides rich supervision including not only the final answer but also question types and intermediate results. Therefore, we can train the pipeline modules in a supervised fashion. Specifically, we train the question type classifier using a cross entropy loss w.r.t the gold question type. For AutoRouting, each QA module is trained with the subset of samples whose final answer can be extracted from the corresponding modality. For ImplicitDecomp, only one model is trained per modality, which is used to answer both the first-hop and second-hop questions. The question-only and context-only baselines are trained in the corresponding format.
+
+![](images/838228a9af05a2952896cfe89472ebe9701295d9d6234c351935f7aed3b0739e.jpg)  
+Figure 4: ImplicitDecomp: Modules with the same color share parameters. In this example, the text QA module is activated to produce the 1st-hop answer, and this intermediate answer is fed into the Image QA model to produce the final answer. Question type ([Q Type]) is determined by a separate classifier.
+
+# 5 EXPERIMENTS
+
+We evaluate models in three different setups: (1) questions that require a single modality to answer (Single Modality); (2) questions that require reasoning over multiple modalities (Multi Modality); (3) and all questions $( A l l )$ . Our evaluation metrics need to support lists of answers, and thus we use average $\mathrm { F _ { 1 } }$ and Exact Match (EM), as described in Dua et al. (2019), where answers on the gold and predicted lists are aligned. Human performance is estimated with 9 expert annotators, who answered 145 questions. Test results are are reported using a single run (one random seed).
+
+We show results in Table 4.2 ImplicitDecomp achieves significantly higher performance $( 5 5 . 9 \mathrm { F _ { 1 , } } )$ compared to the other baselines, but lower than human performance $( 9 1 . 2 \mathrm { F _ { 1 } }$ with provided context, and $8 4 . 8 \mathrm { F _ { 1 } }$ in the open-domain setting over all of Wikipedia), suggesting ample room for improvement. On the Multi Modality subset, ImplicitDecomp
+
+Table 4: Test set results   
+
+<table><tr><td></td><td>Single Modality</td><td colspan="2">Mutli Modality</td><td colspan="2">All</td></tr><tr><td></td><td>EM</td><td>F1 EM</td><td>F1</td><td>EM</td><td>F1</td></tr><tr><td>Question-only2 Context-only</td><td>14.2 8.0</td><td>17.0 10.2</td><td>16.9 19.5 6.6 8.5</td><td>15.3 7.4</td><td>18.0 9.5</td></tr><tr><td>AutoRouting ImplicitDecomp</td><td>48.9 51.1</td><td>57.1 58.8</td><td>32.0 38.2 46.5 51.7</td><td>42.1 49.3</td><td>49.5 55.9</td></tr><tr><td>Human</td><td>87.9</td><td>92.5</td><td>84.8 90.1</td><td>86.2</td><td>91.2</td></tr></table>
+
+substantially improves performance compared to AutoRouting $\mathrm { 3 8 . 2 }  \mathrm { 5 1 . 7 } $ , emphasizing the superiority of our approach on multi-hop questions, while on single-hop questions this gap is smaller.
+
+Since automatic evaluation of performance is non-trivial in our setup, we also manually evaluate human performance. In $9 4 . 5 \%$ of the cases, answers are either identical or semantically equivalent to the gold answer, $0 . 7 \%$ have an error in the question, and $4 . 8 \%$ are human errors. Human errors are owing to the length of the context, resulting in human fatigue (which models do not suffer from).
+
+Analysis To demonstrate that ImplicitDecomp indeed performs multi-hop reasoning, successfully answering intermediate questions, we analyze ImplicitDecomp predictions for multimodal questions generated using the Compose, Compare and Intersect operations (Table 5). For these questions, we find that when the 1st-hop answer is correct, the model achieves an $\mathrm { F _ { 1 } }$ of 63.9, whereas when the 1st-hop prediction is incorrect, the $\mathrm { F _ { 1 } }$ drops to 37.4. This suggests that the model relies on the 1st-hop answer, effectively performing multi-hop reasoning. Last, our question type classifier obtains a high accuracy of $9 1 . 5 \%$ on the test set.
+
+Table 5: Examples where ImplicitDecomp correctly answers both the intermediate and the entire question, and a breakdown of the 1st-hop $\mathrm { F _ { 1 } }$ and final $\mathrm { F _ { 1 } }$ for the three logical operations: Compose, Compare, and Intersect.   
+
+<table><tr><td rowspan=1 colspan=1>Type</td><td rowspan=1 colspan=1>Question</td><td rowspan=1 colspan=1>1st-hop prediction</td><td rowspan=1 colspan=1>Final prediction</td><td rowspan=1 colspan=1>1st-hop F1</td><td rowspan=1 colspan=1>F1</td></tr><tr><td rowspan=1 colspan=1>Compose</td><td rowspan=1 colspan=1>WhatpartdidKymKarathplayintheTVshowwhose poster featuresa dog?</td><td rowspan=1 colspan=1>Lassie</td><td rowspan=1 colspan=1>Kathy Vaughn</td><td rowspan=1 colspan=1>62.3</td><td rowspan=1 colspan=1>50.8</td></tr><tr><td rowspan=1 colspan=1>Compare</td><td rowspan=1 colspan=1>Whichvideo gamewasWes Johnson involved in earlier:Fallout4 or the gamewhose cover shows agun-wielding man?</td><td rowspan=1 colspan=1>Hammer&amp;Sickle</td><td rowspan=1 colspan=1>Hammer&amp;Sickle</td><td rowspan=1 colspan=1>55.7</td><td rowspan=1 colspan=1>61.1</td></tr><tr><td rowspan=1 colspan=1>Intersect</td><td rowspan=1 colspan=1>Whichalbum,released inDecemberof201l,hasamanwearingsunglasses on its cover, and was released under the RCA label?</td><td rowspan=1 colspan=1>TY.O,Back to Love</td><td rowspan=1 colspan=1>Back to Love</td><td rowspan=1 colspan=1>33.5</td><td rowspan=1 colspan=1>55.1</td></tr></table>
+
+To test whether the compositional questions created are indeed multi-hop, we conducted a qualitative analysis over 50 questions as suggested by (Min et al., 2019b). We find that $6 \%$ are of the Weak Distractors category, that is, questions such as “What year... ” when there is only one year appearing in the context, making the question easy. $2 \%$ have Redundant evidence, that is, questions such as “Which Donald Trump TV show has ...” where there is only one TV show starring Donald Trump, making the rest of the question redundant. The remaining $92 \%$ indeed require multi-hop reasoning.
+
+# 6 RELATED WORK
+
+Visual question answering—i.e., the task of answering questions about images—has been widely explored in previous work (Antol et al., 2015; Zhang et al., 2016; Goyal et al., 2017; Johnson et al., 2017; Hudson & Manning, 2019; Zellers et al., 2019; Singh et al., 2019; Methani et al., 2020), ranging from synthetic images to scientific plots. Our work differs significantly from those, by including more complex, multi-hop questions that require reasoning over text, tables and images. Currently, the most successful paradigm in VQA is fine-tuning models pre-trained on large amounts of image captioning data (Tan & Bansal, 2019; Lu et al., 2019; 2020; Su et al., 2020; Chen et al., 2020c; Li et al., 2020), an approach we follow for answering image-related questions.
+
+MANYMODALQA (Hannan et al., 2020) move beyond directing the question to an image-only context, to choosing between an image, a text, and a table. Their work focuses on routing the question to the correct context modality. Our question-type classifier, based on RoBERTa-large reaches an accuracy of $9 1 . 4 \%$ on our 16 possible question types, showing that the main challenge in MMQA is reasoning over the context rather than identifying the question type.
+
+HYBRIDQA (Chen et al., 2020b) presents a cross-modality reasoning challenge over tabular and textual data. A fundamental difference is that our setup offers cross-modality reasoning over images as well. In addition, our approach is cheaper to annotate since it requires only paraphrasing, and the question type distribution is more controllable (we offer 16 major question types vs. 6 in HYBRIDQA). Moreover, our text passages are chosen using the question, answer and table, while in HYBRIDQA only WikiEntities from the table are used to find text passages.
+
+The model proposed in HYBRIDQA introduces a heuristic for linking the text passage to the table cells, which may lead to performance degradation. Conversely, our model uses (automaticallyannotated) intermediate multi-hop answers, to perform reasoning and linking implicitly over the full table and text, which should lead to more robust reasoning, in particular when reasoning over multiple table cells, as well as for narrative tracking and co-reference over the full text. In parallel to this work, a new open-domain variant of HYBRIDQA has been released by Chen et al. (2020a).
+
+# 7 CONCLUSION
+
+We present MMQA, a new QA dataset that contains 29,918 examples, $3 5 . 7 \%$ of which require crossmodality reasoning. We describe a novel framework for generating complex multimodal questions at scale, and showcase the diversity and multimodal properties of the resulting dataset. We evaluate MMQA using a variety of models, and confirm that the best model exploits the multimodality of the dataset and takes into account multi-hop reasoning via implicit decomposition. However, human performance substantially exceeds the best model, establishing the need for further research involving multiple modalities in question answering systems, which we hope that our work will drive.
+
+# 8 ACKNOWLEDGMENTS
+
+We thank our colleagues at The Allen Institute of AI, James Ferguson and Amir Globerson. This research was partially supported by The Blavatnik Computer Science Research Fund and The Yandex Initiative for Machine Learning, and the European Union’s Seventh Framework Programme (FP7) under grant agreement no. 802800-DELPHI. Special thanks to Carrot Search which allowed use to use their FoamTree visualization.
+
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+
+# A APPENDIX
+
+Please see separate file for supplementary material.

parse/train/ee6W5UgQLa/ee6W5UgQLa_content_list.json

@@ -0,0 +1,1490 @@
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+    {
+        "type": "text",
+        "text": "MULTIMODALQA: COMPLEX QUESTION ANSWERING OVER TEXT, TABLES AND IMAGES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Alon Talmor∗,1,2 Ori Yoran∗,1,2 Amnon Catav∗,2 Dan Lahav∗,2 Yizhong Wang3   \nAkari Asai3 Gabriel Ilharco3 Hannaneh Hajishirzi2,3 Jonathan Berant1,2 ",
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+    },
+    {
+        "type": "text",
+        "text": "1The Allen Institute for AI, 2Tel-Aviv University, 3University of Washington {alont,oriy,jonathan}@allenai.org   \n{amnoncatav,lahav}@mail.tau.ac.il   \n{yizhongw,akari,gamaga,hannaneh}@cs.washington.edu ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "When answering complex questions, people can seamlessly combine information from visual, textual and tabular sources. While interest in models that reason over multiple pieces of evidence has surged in recent years, there has been relatively little work on question answering models that reason across multiple modalities. In this paper, we present MULTIMODALQA (MMQA): a challenging question answering dataset that requires joint reasoning over text, tables and images. We create MMQA using a new framework for generating complex multi-modal questions at scale, harvesting tables from Wikipedia, and attaching images and text paragraphs using entities that appear in each table. We then define a formal language that allows us to take questions that can be answered from a single modality, and combine them to generate cross-modal questions. Last, crowdsourcing workers take these automatically generated questions and rephrase them into more fluent language. We create 29,918 questions through this procedure, and empirically demonstrate the necessity of a multi-modal multi-hop approach to solve our task: our multi-hop model, ImplicitDecomp, achieves an average $\\mathrm { F _ { 1 } }$ of 51.7 over cross-modal questions, substantially outperforming a strong baseline that achieves $3 8 . 2 \\mathrm { F _ { 1 } }$ , but still lags significantly behind human performance, which is at $9 0 . 1 \\mathrm { F _ { 1 } }$ . ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            599,
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+            616
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "When presented with complex questions, people often do not know in advance what source(s) of information are relevant for answering it. In general scenarios, these sources can encompass multiple modalities, be it paragraphs of text, structured tables, images or combinations of those. For instance, a user might ponder “When was the famous painting with two touching fingers completed?”, if she cannot remember the exact name of the painting. Answering this question is made possible by integrating information across both the textual and visual modalities. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Recently, there has been substantial interest in question answering (QA) models that reason over multiple pieces of evidence (multi-hop questions (Yang et al., 2018; Talmor & Berant, 2018; Welbl et al., 2017)). In most prior work, the question is phrased in natural language and the answer is in a context, which may be a paragraph (Rajpurkar, 2016), a table (Pasupat & Liang, 2015), or an image (Antol et al., 2015). However, there has been relatively little work on answering questions that require integrating information across modalities. Hannan et al. (2020) created MANYMODALQA: a dataset where the context for each question includes information from multiple modalities. However, the answer to each question can be derived from a single modality only, and no cross-modality reasoning is needed. Thus, the task is focused on identifying the relevant modality. Recently, Chen et al. (2020b) presented HYBRIDQA, a dataset that requires reasoning over tabular and textual data. While HYBRIDQA requires cross-modal reasoning, it does not require visual inference, limiting the types of questions that can be represented (See Table 1 for a comparison between the datasets). ",
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+    },
+    {
+        "type": "text",
+        "text": "Multimodal Context ",
+        "text_level": 1,
+        "bbox": [
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+            106,
+            281,
+            114
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/5c1613ac705899207e7e983dc4f8aad672769668f3aa396af2992ec3ecdfc5b5.jpg",
+        "image_caption": [
+            "Q: Which B.Piazza titlecame earlier: the movie S.Stalone'sson staredinor the movie with halfofalady'sface on the poster? A:Tell Me That You Love Me, Junie Moon ",
+            "Figure 1: Example of a MMQA question, answer and context. In green are the text modality question and answer, and in red the image modality. The table is used to perform the year comparison between the answers of the text and image question parts. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we present MMQA, the first large-scale (29,918 examples) QA dataset that requires integrating information across free text, semi-structured tables, and images, where $3 5 . 7 \\%$ of the questions require cross-modality reasoning. Figure 1 shows an example question: “Which B.Piazza title came earlier: the movie S. Stallon’s son starred in, or the movie with half of a lady’s face on the poster?”. Answering this question entails (i) decomposing the question into a sequence of simpler questions, (ii) determining the modalities for the simpler questions and answering them, i.e., information on the poster is in an image, the information on ${ } ^ { \\mathfrak { a } } S .$ Stallon’s son” is in free text, and the years of the movies are in the table, (iii) combining the information from the simpler questions to compute the answer: “Tell Me that you love me, Junie Moon”. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our methodology for creating MMQA involves three high-level steps. (a) Context construction: we harvest tables from Wikipedia, and connect each table to images and paragraphs that appear in existing Reading Comprehension (RC) datasets (Kwiatkowski et al., 2019; Clark et al., 2019; Yang et al., 2018); (b) Question generation: Following past work (Talmor & Berant, 2018), we use the linked structure of the context to automatically generate questions that require multiple reasoning operations (composition, conjunction, comparison) across modalities in pseudo-language ; (c) Paraphrasing: we use crowdsourcing workers to paraphrase the pseudo-language questions into more fluent English. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To tackle MMQA, we introduce ImplicitDecomp, a new model that predicts a program that specifies the required reasoning steps over different modalities, and executes the program with dedicated text, table, and image models. ImplicitDecomp performs multi-hop multimodal reasoning without the need for an explicit decomposition of the question. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We empirically evaluate MMQA by comparing ImplicitDecomp to strong baselines that do not perform cross-modal reasoning and to human performance. We find that on multimodal questions, ImplicitDecomp improves $\\mathrm { F _ { 1 } }$ from ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/df7b843ae7a3dea4ad4080e4907f23c8c062f6e183b2baacae430a3f1e08dd47.jpg",
+        "table_caption": [
+            "Table 1: A comparison of MULTIMODALQA to MANYMODALQA and HYBRIDQA. We compare dataset size, use of images, and whether the dataset supports multihop questions and an open-domain full-wiki setup. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Size</td><td>Full- wiki</td><td>Uses images</td><td>Multi- hop</td></tr><tr><td>MANYMODALQA</td><td>10K</td><td>X</td><td>√</td><td>X</td></tr><tr><td>HYBRIDQA</td><td>70K</td><td>X</td><td>×</td><td>√</td></tr><tr><td>MULTIMODALQA</td><td>30K</td><td>√</td><td>√</td><td>√</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$3 8 . 2  5 1 . 7 $ over a single-hop approach. Humans are able to reach $9 0 . 1 \\mathrm { F _ { 1 } }$ , significantly outperforming our best model. Because automatic evaluation is non-trivial, we also manually analyze human performance and find humans correctly answer $9 4 . 5 \\%$ of the questions in MMQA. Finally, our dataset can be used in an open-domain setup over all of Wikipedia. In this setup, the $\\mathrm { F _ { 1 } }$ of humans is 84.8. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To summarize, our key contributions are: ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• MMQA: a dataset with 29,918 questions and answers, $3 5 . 7 \\%$ of which require cross-modal reasoning. • A methodology for generating multimodal questions over text, tables and images at scale. • ImplicitDecomp, A model for implicitly decomposing multimodal questions, which improves on a single-hop model by 13.5 absolute $\\mathrm { F _ { 1 } }$ points on questions requiring cross-modal reasoning. • Our dataset and code are available at https://allenai.github.io/multimodalqa. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 DATASET GENERATION ",
+        "text_level": 1,
+        "bbox": [
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+            236,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to develop a method that allows generating complex questions over multiple modalities at scale. An overview of the methodology is captured in Figure 2. We first select a Wikipedia table as an anchor, to which we add images and texts paragraphs and obtain a context. Single modality questions are generated based on these contexts, and used to automatically create multimodal, multihop questions. AMT workers rephrase the questions into natural language, and finally distractor paragraphs and images are selected for each question. We now elaborate on the 6 steps of the process. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/c677a9a1b1216fe181547eefdaed522f58fff6f5ef071ef7e788cd277e95cbb4.jpg",
+        "image_caption": [
+            "Figure 2: An overview of MMQA dataset generation process. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 Wikipedia tables as anchors The 01-01-2020 English Wikipedia dump contains roughly $3 M$ tables. We extracted all tables and selected those that meet the following criteria: (a) The tables contain 10-25 rows (b) At least 3 images are associated with the table. This results in a total of $7 0 0 \\mathrm { k }$ tables. (see supp. material for more information). These tables are the anchors of our contexts, which we enrich with images and text for multimodal question generation. A key element of the tables are Wikipedia Entities (WikiEntities) that appear in them, i.e., concepts linked to other Wikipedia entries. We use them to connect different modalities, bridge questions, and solve ambiguities (details below). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Connecting Images and Text to Tables Images. We consider two cases: (a) in-table images and (b) images from pages of linked WikiEntities. In the former, the images are featured inside the table cells. In the latter, the table contains a column of WikiEntities that potentially have images, e.g. a table describing the filmography of an actor often contains a column of film names, which may have posters in their respective pages. To associate entities with their representative image, we map entities and their profile images in their Wikipedia pages. Overall, we obtain 57,713 images, with 889 in-table images and 56,824 WikiEntities images. Text. We build on texts from contexts appearing in existing reading comprehension datasets. We elaborate on this process next. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 Generating Single-Modality Questions Tables. We generate pseudo-language table questions in the following form “In [table title] of [Wikipedia page title] which cells in [column X] have the [value Y] in [column Z]?”. We additionally support numeric computations over columns classified as dates or numbers, such as min and max values, e.g., “In [Doubles] of [WCT Tournament of Champions], what was the MOST RECENT [Year](s) where the [Location] was [Forest Hills]”. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Images. We use crowdsourcing to generate single-modality questions about images. We generated two types of image questions, based on the images we retrieved from the previous step: (i) questions over a single image, (ii) questions over a list of images. ",
+        "bbox": [
+            176,
+            791,
+            823,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "When generating single-image questions, we show Amazon Mechanical Turk (AMT) crowd workers an image alongside its WikiEntity, and ask them to phrase a question about the image with the entity being the focus of the question. E.g, if the entity is “Roger Federer”, a potential question is “What’s the hair color of Roger Federer?”. For questions to have meaning in an open-domain setting, we primed AMT workers to ask questions that correspond to “stable” features, i.e., features that are unlikely to change in different images and are thus appropriate in an open-domain setting. ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/ea5fba34f94002c7fd1884833c9554b85899f9c26392b4264b740ee060194688.jpg",
+        "table_caption": [
+            "Table 2: All 16 compositional templates in MMQA with an example and their relative frequency. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Type</td><td rowspan=1 colspan=1>Q&amp;A</td><td rowspan=1 colspan=1>%</td></tr><tr><td rowspan=1 colspan=1>TextQ</td><td rowspan=1 colspan=1>What wasthe territorial capitalof the territory opposingOhioin theToledoWar?Detroit</td><td rowspan=1 colspan=1>31.0</td></tr><tr><td rowspan=1 colspan=1>TableQ</td><td rowspan=1 colspan=1>DoestheGerman state Baden-Wurttemberg orThuringia have moreresidents?Baden-Wurttemberg</td><td rowspan=1 colspan=1>18.3</td></tr><tr><td rowspan=1 colspan=1>ImageQ</td><td rowspan=1 colspan=1>WhatweaponisthestatueinNottinghamholding?bow</td><td rowspan=1 colspan=1>8.9</td></tr><tr><td rowspan=1 colspan=1>Compose(TextQ,TableQ)</td><td rowspan=1 colspan=1>Atwhatagedid theCleveland Cavaliersplayer with 6190rebounds enter the NBA?19</td><td rowspan=1 colspan=1>7.8</td></tr><tr><td rowspan=1 colspan=1>ImageListQ</td><td rowspan=1 colspan=1>Whatisthecommon nameof thebushwarblerinThailand that hasanorange stripe above itseye?Chestnut-crowned bush warbler</td><td rowspan=1 colspan=1>6.1</td></tr><tr><td rowspan=1 colspan=1>Compose(TableQ,ImageListQ)</td><td rowspan=1 colspan=1>Thefilm that starredChrisEllisonwhereamanwasholding anewspaper on the poster, was released what year?1988</td><td rowspan=1 colspan=1>5.4</td></tr><tr><td rowspan=1 colspan=1>Compose(ImageQ,TableQ)</td><td rowspan=1 colspan=1>OntheposterfortheTVshowinwhich TomMisonplayedDorianCrane,what kind of structure can be seen behind the two men? castle</td><td rowspan=1 colspan=1>4.5</td></tr><tr><td rowspan=1 colspan=1>Compare(Compose(TableQ,ImageQ),TableQ)</td><td rowspan=1 colspan=1>Which manufacturerhas fewerwinsattheFirst Data 5Oo:Buick or thebrand with across for a logo?Buick</td><td rowspan=1 colspan=1>3.5</td></tr><tr><td rowspan=1 colspan=1>Compose(TableQ,TextQ)</td><td rowspan=1 colspan=1>Onwhat date did the original artistwho sangSweetChildof Mine haveaconcert atUSBank Stadium?July 30,2017</td><td rowspan=1 colspan=1>3.2</td></tr><tr><td rowspan=1 colspan=1>Intersect(TableQ,TextQ)</td><td rowspan=1 colspan=1>Who was the artistfor DamonFox in20o6who also sings &quot;You gotthemoves like Jagger&quot;?Christina Aguilera</td><td rowspan=1 colspan=1>2.6</td></tr><tr><td rowspan=1 colspan=1>Compose(TextQ,ImageListQ)</td><td rowspan=1 colspan=1>Ontheposterfor themoviebasedonthebook &quot;ActlikeaLady,ThinkLikea Man,&quot;how many people are there in total? nine</td><td rowspan=1 colspan=1>2.4</td></tr><tr><td rowspan=1 colspan=1>Intersect(ImageListQ,TableQ)</td><td rowspan=1 colspan=1>WhatcoversoftheChandlerCanterburyfilmsfrom2oo9hasmorethanoneperson?PowderBlueBallsOut,Gary theTennisCoach,After.Life</td><td rowspan=1 colspan=1>2.3</td></tr><tr><td rowspan=1 colspan=1>Compare(TableQ,Compose(TableQ,TextQ))</td><td rowspan=1 colspan=1>DidChelseaorclubthat singsYou&#x27;llNeverWalkAlonerank higherinDeloitteFootball MoneyLeague20o7?Chelsea</td><td rowspan=1 colspan=1>2.1</td></tr><tr><td rowspan=1 colspan=1>Compose(ImageQ,TextQ)</td><td rowspan=1 colspan=1>DidGary Oldmantakepart inthemoviewhoseposterfeaturestwomenholding handguns,and which had MarkL.Smith as a writer? no</td><td rowspan=1 colspan=1>1.0</td></tr><tr><td rowspan=1 colspan=1>Compare(Compose(TableQ,ImageQ),Compose(TableQ,TextQ))</td><td rowspan=1 colspan=1>Wasthefilmthat featuresa gianteyeonitsposter or the firstWolverinemovie theearlierfilm thatScott Silverworkedon? Requiem fora Dream</td><td rowspan=1 colspan=1>0.8</td></tr><tr><td rowspan=1 colspan=1>Intersect(ImageListQ,TextQ)</td><td rowspan=1 colspan=1>Whatcommonlaw statewith aneagleontheflag hasan institutionin the North region of DivisionIIof theNCCAA?Iowa</td><td rowspan=1 colspan=1>0.2</td></tr></table>",
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For questions with a list of images, we use images that appear in the same column of a table. To generate these questions, AMT workers were given the images and asked to phrase a binary question about a distinctive feature of the entities that a subset of the images share. E.g., given a list of statues, the worker could ask “Which of the statues features a horse?” This process results in 2,764 single image questions and 7,773 list image questions that are later used to create multimodal questions. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Text. To obtain questions answerable over text paragraphs we build on existing reading comprehension datasets: Natural Questions (NQ) (Kwiatkowski et al., 2019) consists of about $3 0 0 K$ questions issued to the Google search engine. This dataset mostly contains simple questions where a single paragraph suffices to answer each question. BoolQ (Clark et al., 2019) contains 15, 942 yes/no questions, gathered using the same pipeline as NQ. HotpotQA (Yang et al., 2018) contains $1 1 2 K$ training questions, where crowd workers were shown pairs of related Wikipedia paragraphs and were asked to author questions that require multi-hop reasoning over the paragraphs. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To use questions from the above datasets as building blocks for multi-hop multimodal questions, we unified them into a corpus that consists of triples of (i) a text question, (ii) an answer and (iii) 1-2 gold paragraphs from Wikipedia. We link a question to a table, by matching WikiEntities in the table to entities in the text of the question (see supplementary material for further details). Overall, we retrieved 6,644 questions from NQ, 1,246 from BoolQ and 4,733 from HotpotQA. ",
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+    {
+        "type": "text",
+        "text": "2.4 Generating multimodal complex questions We present an automatic method for creating at scale multimodal compositional questions (i,e., questions that require answering a sequence of subquestions to conclude the final answer). Our first step is to introduce a formal language that allows to combine questions answerable from a single modality. Below we introduce the logical operations that allow to generate such pseudo-language (PL) questions, while keeping a formal representation of how they were constructed. In Table 2, we illustrate this process with all 16 different compositional templates used for question generation. We now describe our logical operations. ",
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+    {
+        "type": "text",
+        "text": "Logical Operations Functions in our formal language take arguments and return a PL partial question, as well as answers that can be a list of one or more strings, or a list of one or more WikiEntities. All operations have access to the full context. In addition, we prepend a prefix containing the Wikipedia table name and page title—e.g. “In the Filmography of Brad Pitt,”—to all our PL questions to support an open-domain QA setup. Our set of logical operations are: ",
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+        "type": "text",
+        "text": "1. TABLEQ: Returns a question from the table questions generated in $\\ S 2 . 3$ , as well as a list of WikiEntities or a list of strings as answers. ",
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+        "type": "text",
+        "text": "2. TEXTQ: Returns a text corpus question (see $\\ S 2 . 3$ ) and a list of WikiEntities or strings as answers. ",
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+        "type": "text",
+        "text": "3. IMAGEQ: Returns a question about a single image associated with a WikiEntity and a single token answer from a fixed vocabulary (see $\\ S 2 . 3 \\AA$ ). ",
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+    {
+        "type": "text",
+        "text": "4. IMAGELISTQ: Returns a question about a list of images and a list of WikiEntities corresponding to the images that answer the question (see $\\ S 2 . 3$ ). ",
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+        "type": "text",
+        "text": "5. COMPOSE $( \\cdot , \\cdot )$ : Takes a PL question containing a single WikiEntity as a first argument, and a PL question that produces that WikiEntity as the output answer as its second argument. E.g., COMPOSE(“Where was Barack Obama born?”,“Who was the 44th president of the USA?”). The function replaces the WikiEntity in the first-argument PL question with the second-argument PL question and returns the resulting PL question (“Where was the 44th president of the USA born?”). ",
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+    {
+        "type": "text",
+        "text": "6. INTERSECT $( \\cdot , \\cdot )$ : Takes two PL questions that return lists of more than one WikiEntity, and returns their intersection as the answer. The resulting $\\mathrm { P L }$ question is of the form ${ } ^ { \\cdot } P L _ { 1 }$ and $P L _ { 2 }$ ” omitting $\\mathrm { P L _ { 2 } }$ ’s first word (“Who was born in Hawaii and is the parent of Sasha Obama?”). ",
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+    {
+        "type": "text",
+        "text": "7. COMPARE $( \\cdot , \\cdot )$ : Takes two PL questions each returning one WikiEntity that can be linked to one cell in the table, denoted by $\\mathrm { { A n s } _ { 1 } }$ , $\\mathrm { { A n s } _ { 2 } }$ . We first choose a numeric or date column in the table, if such exists. We then compare the values of this column corresponding to the rows of $\\mathbf { A n s } _ { 1 }$ and Ans2. Depending on the comparison outcome, output one of $( \\mathrm { A n s _ { 1 } }$ , Ans2) as the operation answer. The PL question created is of the form “What has compare-op numeric-column-name, $P L _ { 1 }$ or $P L _ { 2 }$ ?” omitting $\\mathrm { P L _ { 1 } }$ and $\\mathrm { P L _ { 2 } }$ ’s first word. E.g. “What has most recent creation year, the rocket of Appolo program, or the rocket of Gemini program?” ",
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+        "text": "2.5 Paraphrasing using AMT We used English-speaking AMT workers to paraphrase automaticallygenerated PL questions into natural language (NL). Each question was paraphrased by 1 worker and validated by 1-3 other workers. To avoid annotator bias (Geva et al., 2019), the number of annotators who worked on both the training and evaluation set was kept to a minimum. We also deployed a feedback mechanism, where workers receive a bonus if a baseline model correctly answered the question after their first paraphrasing attempt, but incorrectly after they refined the paraphrase. See supp. material for print-screens of the AMT annotator interface. ",
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+        "type": "text",
+        "text": "To generate diversity, workers got a bonus if the normalized edit distance of a paraphrase compared to the PL question was higher than 0.7. A total of 971 workers were involved, and 29,918 examples were produced with an average cost of $0 . 3 3 \\ S$ per question. We split the dataset into 23,817 training, 2,441 development (dev.), and 3,660 test set examples. Context components in the dev. and test sets are disjoint, and were constructed from a disjoint set of single-modality questions. ",
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+        "type": "text",
+        "text": "A shortcoming of our method for automatically generating examples is that the question distribution does not come from a “natural” source. We argue that developing models that are capable of performing reasoning over multiple modalities is an important direction and MMQA provides an opportunity to develop and evaluate such models. Moreover, this method allows to control the compositional questions created, proving effective in creating a cheap and scalable dataset. ",
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+        "type": "text",
+        "text": "2.6 Adding distractors to the context Images. Questions from the IMAGELISTQ operator require reasoning over a list of images from the same column, and hence do not require additional distractors. For IMAGEQ questions (single-image), we randomly add images that are associated with the WikiEntities that appear in the table, setting a maximum of 15 distractors per question. ",
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+        "type": "text",
+        "text": "Text. We used DPR (Karpukhin et al., 2020), a neural information retrieval model, to retrieve distractors for all questions. Each context includes exactly 10 paragraphs, where 1-2 are gold paragraphs and the rest are distractors. Specifically, we encode the first 2 paragraphs of each Wikipedia article with the DPR encoder, and use as distractors the paragraphs with the highest dot product between their encoding and the question encoding. We do not allow: (a) an overlap between the distractors in the training and evaluation sets, (b) distractors originating from the gold article, (c) distractors containing an exact match to the gold answer. ",
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+        "type": "text",
+        "text": "To summarize, each of our examples contains a question, an answer, the formal representation of the PL question (ignored by our models), and all distractors and gold context for all modalities. This renders MMQA useful for both open-domain multimodal QA, as well as context-dependant QA. ",
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+        "type": "image",
+        "img_path": "images/53d9aa241337ef2392714bb6dfb2adc55db8678aa5705c36b68b58c902fb16dd.jpg",
+        "image_caption": [
+            "Figure 3: Domain diversity in MMQA. The area of each color corresponds to the topic frequency in the dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 DATASET ANALYSIS ",
+        "text_level": 1,
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+        "text": "To highlight the diversity of MMQA we analyze its key statistics, domains, and lexical richness. ",
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+        "text": "Key Statistics MMQA contains 29, 918 questions, and their main statistics are in Table 3. Since we focus on multimodality, we upsample the number of multimodal questions in the dev. and test sets compared to the training set. Also, about $60 \\%$ of the questions in MMQA are compositional. Questions are relatively long (18.2 words), but answers tend to be short (2.1 words). The answer for each question can be a single answer or a list of answers. While list answers comprise only $7 . 4 \\%$ of the data, when considering compositional questions that contain an intermediate question within them, the proportion of list answers in intermediate questions is higher $( 1 8 . 9 \\% )$ ). ",
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+    {
+        "type": "table",
+        "img_path": "images/3a5853eafd1c50bbd2052c8188e70b115b1c647dde2c0ccb502ec140830907c3.jpg",
+        "table_caption": [
+            "Table 3: Key statistics for MMQA. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Measurement</td><td>Value</td></tr><tr><td># Distinct Questions</td><td>29,918</td></tr><tr><td>Train multimodal questions</td><td>34.6%</td></tr><tr><td>Dev.+test multimodal questions</td><td>40.1%</td></tr><tr><td>Train compositional questions</td><td>58.8%</td></tr><tr><td>Dev.+test compositional questions</td><td>62.3%</td></tr><tr><td>Average question length (words)</td><td>18.2</td></tr><tr><td>Average # of answers per question</td><td>1.16</td></tr><tr><td>List answers</td><td>7.4%</td></tr><tr><td>List answers per intermediate question</td><td>18.9%</td></tr><tr><td>Average answer length (words)</td><td>2.1</td></tr><tr><td># of distinct words in questions</td><td>49,649</td></tr><tr><td># of distinct words in answers</td><td>20,820</td></tr><tr><td># of distinct context tables</td><td>11,022</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Domain Diversity Figure 3 shows a sample of questions from MMQA categorized to different domains. While entertainment categories occupy a large portion of our dataset (Films $36 \\%$ , TV $19 \\%$ ), we observe questions represent a wide variety of topics. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Lexical Richness Workers received a bonus when substantially modifying the PL questions. We observe that the average normalized edit distance between the NL questions and the PL questions is high (0.7), that NL questions are shorter (avg. length of 20.02 vs. 22.16 words for PL questions), and use a richer vocabulary (#unique words 39, 319 vs 37, 108). ",
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+    {
+        "type": "text",
+        "text": "4 MODELS ",
+        "text_level": 1,
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+        "text": "Here we present our baseline models. We first train models that interact with a single modality given a question (§4.1), and use those as building blocks in our multimodal approaches (§4.2). We denote the question by $Q$ , context paragraphs by $\\mathcal { P }$ , Table by $T$ and context images by $\\mathcal { T }$ . ",
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+    {
+        "type": "text",
+        "text": "4.1 SINGLE-MODALITY QA MODULES",
+        "text_level": 1,
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+        "text": "Text QA Module Following prior work (Min et al., $2 0 1 9 \\mathrm { a }$ ; Asai et al., 2020), our text QA module takes as input a question $Q$ and a paragraph $p \\in \\mathcal P$ and answers $Q$ by selecting a span in each paragraph $p$ independently, predicting the start and end positions (Devlin et al., 2019). Additionally, the model returns four scores for for every paragraph $p$ corresponding to: if the answer is $( i )$ a span in $p$ ; $( i i )$ “yes”; $( i i i )$ “no”; or $( i \\nu )$ not in $p$ . At inference time, the model selects the paragraph that has the lowest score for $i \\nu$ – the answer is not the paragraph. Our model is based on a pre-trained RoBERTa-large model (Liu et al., 2019), fine-tuned on MMQA. ",
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+    {
+        "type": "text",
+        "text": "Table QA Module Following prior work (Herzig et al., 2020), our table QA module takes as input the question $Q$ and the table $T$ , and selects a subset of the table cells and an aggregation operation to compute the final answer. Specifically, we linearize the table $T$ by rows, with column names prepended to the corresponding cells (Chen et al., 2019). For example, this converts the table in Figure 1 to the following text: “Row 1: year is 1957; title is a dangerous age; role is David. Row 2...”. Next, we concatenate the question to the linearized table, and encode them using RoBERTa-large. We then pass the contextualized representation of every token in the table cell to a linear classifier that computes the probability of the token being selected. The score for a cell is the average of its tokens. Cells with probability $> 0 . 5$ are selected. Finally, another linear multi-class classifier predicts an aggregation operation from SUM, MEAN, COUNT, YES, NO, and NONE. Aggregation operations are applied on the selected cells, YES and NO operations output “yes” or “no”, and the NONE operation outputs all selected cells. ",
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+    {
+        "type": "text",
+        "text": "Image QA Module Questions with visual information are handled by a multimodal transformer that processes the text question and pre-computed image features. For a question $Q$ and a set of images $\\mathcal { T }$ , we feed the model the question and the visual features $\\Phi ( i )$ extracted from each image $i \\in \\mathcal { T }$ , along with the name of the WikiEntity associated with the image. For each image and question, the model predicts an answer from a fixed vocabulary determined by the answers in the training set and 3 special tokens: $y _ { \\mathrm { d t r } }$ , $y _ { \\mathrm { p } }$ and $y _ { \\mathrm { n } }$ . In questions where the expected answer is a phrase (e.g., ImageQ), we return the answer from the image where $p ( y _ { \\mathrm { d t r } } )$ is lowest (similar to text QA). In questions where the expected answer is a subset of the images (e.g., Compose(TableQ,ImageQ)), we return all images where $p ( y _ { \\mathrm { p } } ) > p ( y _ { \\mathrm { n } } )$ . Our model is based on the pre-trained model VILBERT-MT1 (Lu et al., 2020). Visual features are extracted by a vision network $\\Phi$ , comprised of a Faster R-CNN (Ren et al., 2015) pre-trained on Visual Genome (Krishna et al., 2017). ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "4.2 MULTIMODALITY QA MODELS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We turn to models that interact with multiple modalities. ",
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+    {
+        "type": "text",
+        "text": "Multi-Hop Implicit Decomposition (ImplicitDecomp) Our dataset is designed to test reasoning across modalities. As a first attempt towards this goal, we introduce a 2-hop implicit decomposition baseline, capable of combining information scattered across modalities (illustrated in Figure 4). ",
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+        "type": "text",
+        "text": "We first train a question-type classifier, based on RoBERTa-large, that takes a question $Q$ as input, and predicts one of the 16 possible question types (Table 2). The question type can be viewed as a program, specifying the relevant modalities, their order, and the logical operations. For example, if the question type is Compose(TextQ,TableQ), the first hop should be conducted on the table $T$ , and the second hop on the paragraphs $\\mathcal { P }$ . In each hop, we feed the model with the question $Q$ , the question type, the hop number, and the context of the corresponding modality. The model automatically identifies which part of the question is relevant at the current hop and does not explicitly decompose the question into sub-questions (hence the name implicit decomposition). In the second hop, answers from the first hop are also given as input so that the model can leverage this information and conduct cross-modal reasoning to output the final answer. For all single-modality question types (such as TextQ and TableQ), the model uses only the first hop to get the answer. ",
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+    {
+        "type": "text",
+        "text": "Single-Hop Routing (AutoRouting) A simple approach for answering questions without cross-modal reasoning is to first determine the modality where the answer is expected to occur, and then run the corresponding single-modality module. We use the aforementioned question type classifier to determine the modality where the answer will appear, route the question and the context for the predicted modality into the corresponding module, and use the output as the final answer. ",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Question-only and Context-only baselines We run the question-only and context-only baselines, suggested by Kaushik & Lipton (2018). Our question-only baseline is BART-large (Lewis et al., 2019): a sequence-to-sequence model that directly generates the answer given the question. For the context-only baseline, we first predict the question type using the classifier described above to pick a target module. We then feed the relevant context to the target module, replacing the question with an empty string. ",
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+    {
+        "type": "text",
+        "text": "4.3 TRAINING AND SUPERVISION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our dataset provides rich supervision including not only the final answer but also question types and intermediate results. Therefore, we can train the pipeline modules in a supervised fashion. Specifically, we train the question type classifier using a cross entropy loss w.r.t the gold question type. For AutoRouting, each QA module is trained with the subset of samples whose final answer can be extracted from the corresponding modality. For ImplicitDecomp, only one model is trained per modality, which is used to answer both the first-hop and second-hop questions. The question-only and context-only baselines are trained in the corresponding format. ",
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+    {
+        "type": "image",
+        "img_path": "images/838228a9af05a2952896cfe89472ebe9701295d9d6234c351935f7aed3b0739e.jpg",
+        "image_caption": [
+            "Figure 4: ImplicitDecomp: Modules with the same color share parameters. In this example, the text QA module is activated to produce the 1st-hop answer, and this intermediate answer is fed into the Image QA model to produce the final answer. Question type ([Q Type]) is determined by a separate classifier. "
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We evaluate models in three different setups: (1) questions that require a single modality to answer (Single Modality); (2) questions that require reasoning over multiple modalities (Multi Modality); (3) and all questions $( A l l )$ . Our evaluation metrics need to support lists of answers, and thus we use average $\\mathrm { F _ { 1 } }$ and Exact Match (EM), as described in Dua et al. (2019), where answers on the gold and predicted lists are aligned. Human performance is estimated with 9 expert annotators, who answered 145 questions. Test results are are reported using a single run (one random seed). ",
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+    {
+        "type": "text",
+        "text": "We show results in Table 4.2 ImplicitDecomp achieves significantly higher performance $( 5 5 . 9 \\mathrm { F _ { 1 , } } )$ compared to the other baselines, but lower than human performance $( 9 1 . 2 \\mathrm { F _ { 1 } }$ with provided context, and $8 4 . 8 \\mathrm { F _ { 1 } }$ in the open-domain setting over all of Wikipedia), suggesting ample room for improvement. On the Multi Modality subset, ImplicitDecomp ",
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+    {
+        "type": "table",
+        "img_path": "images/203720ab4ba71e2eaebfded2f026770864be7380e4c6d04857e737e6b409aa7a.jpg",
+        "table_caption": [
+            "Table 4: Test set results "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Single Modality</td><td colspan=\"2\">Mutli Modality</td><td colspan=\"2\">All</td></tr><tr><td></td><td>EM</td><td>F1 EM</td><td>F1</td><td>EM</td><td>F1</td></tr><tr><td>Question-only2 Context-only</td><td>14.2 8.0</td><td>17.0 10.2</td><td>16.9 19.5 6.6 8.5</td><td>15.3 7.4</td><td>18.0 9.5</td></tr><tr><td>AutoRouting ImplicitDecomp</td><td>48.9 51.1</td><td>57.1 58.8</td><td>32.0 38.2 46.5 51.7</td><td>42.1 49.3</td><td>49.5 55.9</td></tr><tr><td>Human</td><td>87.9</td><td>92.5</td><td>84.8 90.1</td><td>86.2</td><td>91.2</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "substantially improves performance compared to AutoRouting $\\mathrm { 3 8 . 2 }  \\mathrm { 5 1 . 7 } $ , emphasizing the superiority of our approach on multi-hop questions, while on single-hop questions this gap is smaller. ",
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+        "type": "text",
+        "text": "Since automatic evaluation of performance is non-trivial in our setup, we also manually evaluate human performance. In $9 4 . 5 \\%$ of the cases, answers are either identical or semantically equivalent to the gold answer, $0 . 7 \\%$ have an error in the question, and $4 . 8 \\%$ are human errors. Human errors are owing to the length of the context, resulting in human fatigue (which models do not suffer from). ",
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+    {
+        "type": "text",
+        "text": "Analysis To demonstrate that ImplicitDecomp indeed performs multi-hop reasoning, successfully answering intermediate questions, we analyze ImplicitDecomp predictions for multimodal questions generated using the Compose, Compare and Intersect operations (Table 5). For these questions, we find that when the 1st-hop answer is correct, the model achieves an $\\mathrm { F _ { 1 } }$ of 63.9, whereas when the 1st-hop prediction is incorrect, the $\\mathrm { F _ { 1 } }$ drops to 37.4. This suggests that the model relies on the 1st-hop answer, effectively performing multi-hop reasoning. Last, our question type classifier obtains a high accuracy of $9 1 . 5 \\%$ on the test set. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/1a5ba392a0cad931baf425ee9f07269cb44efa5dd6e309d05ac269a02736c9ae.jpg",
+        "table_caption": [
+            "Table 5: Examples where ImplicitDecomp correctly answers both the intermediate and the entire question, and a breakdown of the 1st-hop $\\mathrm { F _ { 1 } }$ and final $\\mathrm { F _ { 1 } }$ for the three logical operations: Compose, Compare, and Intersect. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Type</td><td rowspan=1 colspan=1>Question</td><td rowspan=1 colspan=1>1st-hop prediction</td><td rowspan=1 colspan=1>Final prediction</td><td rowspan=1 colspan=1>1st-hop F1</td><td rowspan=1 colspan=1>F1</td></tr><tr><td rowspan=1 colspan=1>Compose</td><td rowspan=1 colspan=1>WhatpartdidKymKarathplayintheTVshowwhose poster featuresa dog?</td><td rowspan=1 colspan=1>Lassie</td><td rowspan=1 colspan=1>Kathy Vaughn</td><td rowspan=1 colspan=1>62.3</td><td rowspan=1 colspan=1>50.8</td></tr><tr><td rowspan=1 colspan=1>Compare</td><td rowspan=1 colspan=1>Whichvideo gamewasWes Johnson involved in earlier:Fallout4 or the gamewhose cover shows agun-wielding man?</td><td rowspan=1 colspan=1>Hammer&amp;Sickle</td><td rowspan=1 colspan=1>Hammer&amp;Sickle</td><td rowspan=1 colspan=1>55.7</td><td rowspan=1 colspan=1>61.1</td></tr><tr><td rowspan=1 colspan=1>Intersect</td><td rowspan=1 colspan=1>Whichalbum,released inDecemberof201l,hasamanwearingsunglasses on its cover, and was released under the RCA label?</td><td rowspan=1 colspan=1>TY.O,Back to Love</td><td rowspan=1 colspan=1>Back to Love</td><td rowspan=1 colspan=1>33.5</td><td rowspan=1 colspan=1>55.1</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To test whether the compositional questions created are indeed multi-hop, we conducted a qualitative analysis over 50 questions as suggested by (Min et al., 2019b). We find that $6 \\%$ are of the Weak Distractors category, that is, questions such as “What year... ” when there is only one year appearing in the context, making the question easy. $2 \\%$ have Redundant evidence, that is, questions such as “Which Donald Trump TV show has ...” where there is only one TV show starring Donald Trump, making the rest of the question redundant. The remaining $92 \\%$ indeed require multi-hop reasoning. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            369,
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+            386
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Visual question answering—i.e., the task of answering questions about images—has been widely explored in previous work (Antol et al., 2015; Zhang et al., 2016; Goyal et al., 2017; Johnson et al., 2017; Hudson & Manning, 2019; Zellers et al., 2019; Singh et al., 2019; Methani et al., 2020), ranging from synthetic images to scientific plots. Our work differs significantly from those, by including more complex, multi-hop questions that require reasoning over text, tables and images. Currently, the most successful paradigm in VQA is fine-tuning models pre-trained on large amounts of image captioning data (Tan & Bansal, 2019; Lu et al., 2019; 2020; Su et al., 2020; Chen et al., 2020c; Li et al., 2020), an approach we follow for answering image-related questions. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "MANYMODALQA (Hannan et al., 2020) move beyond directing the question to an image-only context, to choosing between an image, a text, and a table. Their work focuses on routing the question to the correct context modality. Our question-type classifier, based on RoBERTa-large reaches an accuracy of $9 1 . 4 \\%$ on our 16 possible question types, showing that the main challenge in MMQA is reasoning over the context rather than identifying the question type. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "HYBRIDQA (Chen et al., 2020b) presents a cross-modality reasoning challenge over tabular and textual data. A fundamental difference is that our setup offers cross-modality reasoning over images as well. In addition, our approach is cheaper to annotate since it requires only paraphrasing, and the question type distribution is more controllable (we offer 16 major question types vs. 6 in HYBRIDQA). Moreover, our text passages are chosen using the question, answer and table, while in HYBRIDQA only WikiEntities from the table are used to find text passages. ",
+        "bbox": [
+            174,
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+            681
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The model proposed in HYBRIDQA introduces a heuristic for linking the text passage to the table cells, which may lead to performance degradation. Conversely, our model uses (automaticallyannotated) intermediate multi-hop answers, to perform reasoning and linking implicitly over the full table and text, which should lead to more robust reasoning, in particular when reasoning over multiple table cells, as well as for narrative tracking and co-reference over the full text. In parallel to this work, a new open-domain variant of HYBRIDQA has been released by Chen et al. (2020a). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            794,
+            318,
+            809
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We present MMQA, a new QA dataset that contains 29,918 examples, $3 5 . 7 \\%$ of which require crossmodality reasoning. We describe a novel framework for generating complex multimodal questions at scale, and showcase the diversity and multimodal properties of the resulting dataset. We evaluate MMQA using a variety of models, and confirm that the best model exploits the multimodality of the dataset and takes into account multi-hop reasoning via implicit decomposition. However, human performance substantially exceeds the best model, establishing the need for further research involving multiple modalities in question answering systems, which we hope that our work will drive. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "8 ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We thank our colleagues at The Allen Institute of AI, James Ferguson and Amir Globerson. This research was partially supported by The Blavatnik Computer Science Research Fund and The Yandex Initiative for Machine Learning, and the European Union’s Seventh Framework Programme (FP7) under grant agreement no. 802800-DELPHI. Special thanks to Carrot Search which allowed use to use their FoamTree visualization. ",
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+        "page_idx": 9
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+        "text": "A APPENDIX ",
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+]

parse/train/r1DPFCyA-/r1DPFCyA-.md

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+# DISCRIMINATIVE K-SHOT LEARNING USINGPROBABILISTIC MODELS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+This paper introduces a probabilistic framework for $\mathbf { k }$ -shot image classification. The goal is to generalise from an initial large-scale classification task to a separate task comprising new classes and small numbers of examples. The new approach not only leverages the feature-based representation learned by a neural network from the initial task (representational transfer), but also information about the classes (concept transfer). The concept information is encapsulated in a probabilistic model for the final layer weights of the neural network which acts as a prior for probabilistic $\mathbf { k }$ -shot learning. We show that even a simple probabilistic model achieves state-of-the-art on a standard $\mathbf { k }$ -shot learning dataset by a large margin. Moreover, it is able to accurately model uncertainty, leading to well calibrated classifiers, and is easily extensible and flexible, unlike many recent approaches to $\mathbf { k }$ -shot learning.
+
+# 1 INTRODUCTION
+
+A child encountering images of helicopters for the first time is able to generalize to instances with radically different appearance from only a handful of labelled examples. This remarkable feat is supported in part by a high-level feature-representation of images acquired from past experience. However, it is likely that information about previously learned concepts, such as aeroplanes and vehicles, is also leveraged (e.g. that sets of features like tails and rotors or objects like pilots/drivers are likely to appear in new images). The goal of this paper is to build machine systems for performing $\mathbf { k }$ -shot learning, which leverage both existing feature representations of the inputs and existing class information that have both been honed by learning from large amounts of labelled data.
+
+K-shot learning has enjoyed a recent resurgence in the academic community (Lake et al., 2015; Koch et al., 2015; Vinyals et al., 2016; Snell et al., 2017; Srivastava & Salakhutdinov, 2013). Current stateof-the-art methods use complex deep learning architectures and claim that learning good features for $\mathbf { k }$ -shot learning entails training for $k$ -shot specifically via episodic training that simulates many $\mathbf { k }$ -shot tasks. In contrast, this paper proposes a general framework based upon the combination of a deep feature extractor, trained on batch classification, and traditional probabilistic modelling. It subsumes two existing approaches in this vein (Srivastava & Salakhutdinov, 2013; Burgess et al., 2016), and is motivated by similar ideas from multi-task learning (Bakker & Heskes, 2003). The intuition is that deep learning will learn powerful feature representations, whereas probabilistic inference will transfer top-down conceptual information from old classes. Representational learning is driven by the large number of training examples from the original classes making it amenable to standard deep learning. In contrast, the transfer of conceptual information to the new classes relies on a relatively small number of existing classes and $\mathbf { k }$ -shot data points, which means probabilistic inference is appropriate.
+
+While generalisation accuracy is often the key objective when training a classifier, calibration is also a fundamental concern in many applications such as decision making for autonomous driving and medicine. Here, calibration refers to the agreement between a classifier’s uncertainty and the frequency of its mistakes, which has recently received increased attention. For example, Guo et al., 2017 show that the calibration of deep architectures deteriorates as depth and complexity increase. Calibration is closely related to catastrophic forgetting in continual learning. However, to our knowledge, uncertainty has so far been over-looked by the $\mathbf { k }$ -shot community even though it is high in this setting.
+
+Our basic setup mimics that of the motivating example above: a standard deep convolutional neural network (CNN) is trained on a large labelled training set. This learns a rich representation of images at the top hidden layer of the CNN. Accumulated knowledge about classes is embodied in the top layer softmax weights of the network. This information is extracted by training a probabilistic model on these weights. K-shot learning can then 1) use the representation of images provided by the CNN as input to a new softmax function, 2) learn the new softmax weights by combining prior information about their likely form derived from the original dataset with the k-shot likelihood.
+
+# The main contributions of our paper are:
+
+1) We propose a probabilistic framework for k-shot learning. It combines deep convolutional features with a probabilistic model that treats the top-level weights of a neural network as data, which can be used to regularize the weights at k-shot time in a principled Bayesian fashion. We show that the framework recovers $L _ { 2 }$ -regularised logistic regression, with an automatically determined setting of the regularisation parameter, as a special case.
+
+2) We show that our approach achieves state-of-the-art results on the miniImageNet dataset by a wide margin of roughly $6 \%$ for 1- and 5-shot learning. We further show that architectures with better batch classification accuracy also provide features which generalize better at $\mathbf { k }$ -shot time. This finding is contrary to the current belief that episodic training is necessary for good performance and puts the success of recent complex deep learning approaches to $\mathbf { k }$ -shot learning into context.
+
+3) We show on miniImageNet and CIFAR-100 that our framework achieves a good trade-off between classification accuracy and calibration, and it strikes a good balance between learning new classes and forgetting the old ones.
+
+# 2 PROBABILISTIC K-SHOT LEARNING
+
+K-shot learning task. We consider the following discriminative $\mathbf { k }$ -shot learning task: First, we receive a large dataset $\widetilde { \mathcal { D } } = \{ \widetilde { \mathbf { u } } _ { i } , \widetilde { y } _ { i } \} _ { i = 1 } ^ { \widetilde { N } }$ of images $\widetilde { \mathbf { u } } _ { i }$ and labels $\widetilde { y } _ { i } \in \{ 1 , \ldots , \widetilde { C } \}$ that indicate which of the $\widetilde { C }$ classes each image belongs to. Second, we receive a small dataset $\mathcal { D } = \{ \mathbf { u } _ { i } , y _ { i } \} _ { i = 1 } ^ { N }$ of $C$ new classes, $y _ { i } \in \{ \widetilde { C } + 1 , \widetilde { C } + C \}$ , with $k$ images from each new class. Our goal is to construct a model that can leverage the information in $\widetilde { \mathcal { D } }$ and $\mathcal { D }$ to predict well on unseen images $\mathbf { u } ^ { * }$ from the new classes; the performance is evaluated against ground truth labels $y ^ { * }$ .
+
+Summary. In contrast to several recent $\mathbf { k }$ -shot learning approaches that mimic the $\mathbf { k }$ -shot learning task by episodic training on simulated $\mathbf { k }$ -shot tasks, we propose to use the large dataset $\widetilde { \mathcal { D } }$ to train a powerful feature extractor on batch classification, which can then be used in conjunction with a simple probabilistic model to perform k-shot learning. In 2003, Bakker & Heskes introduced a general probabilistic framework for multi-task learning with multi-head models, in which all parameters of a generic feature extractor are shared between a set of tasks, and only the weights of the top linear layer (the “heads”) are task dependent. In the following, we frame k-shot learning in a similar setting and propose a probabilistic framework for $\mathbf { k }$ -shot learning in this vein. Our framework comprises four phases that we refer to as 1) representational learning, 2) concept learning, 3) $k$ -shot learning, and 4) $k$ -shot testing, cf. Fig. 1 (right).
+
+We then show that, for certain modelling assumptions, the obtained method is equivalent/related to regularised logistic regression with a specific choice for the regularisation parameter.
+
+![](images/05e7b76b9d4184116c57ca3c381979d553b04840cc4f3fe43ca24a9ff259946b.jpg)  
+Figure 1: left: Shared feature extractor $\Phi _ { \varphi }$ and separate top linear layers W and $\widetilde { \mathrm { W } }$ with corresponding softmax units on old and new classes. right: Graphical model for probabilistic $\mathbf { k }$ -shot learning.
+
+We provide a high-level description of the probabilistic framework and present a more detailed derivation in Appendix A. While it might appear overly formal, the resulting scheme will be simple and practical, and the probabilistic phrasing will make it extensible and automatic (no free parameters).
+
+Feature extractor and representational learning. We first introduce a convolutional neural network (CNN) $\Phi _ { \varphi }$ as feature extractor whose last hidden layer activations are mapped to two sets of softmax output units corresponding to the $\widetilde { C }$ classes in the large dataset $\widetilde { \mathcal { D } }$ and the $C$ classes in the small dataset $\mathcal { D }$ , respectively. These separate mappings are parametrized by weight matrices $\widetilde { \mathrm { W } }$ for the old classes and $\mathrm { W }$ for the new classes. Denoting the output of the final hidden layer as $\mathbf { x } = \Phi _ { \varphi } ( \mathbf { u } )$ , the first softmax units compute $p ( \widetilde { y } _ { n } | \widetilde { \mathbf { x } } _ { n } , \widetilde { \mathbf { W } } ) = \mathrm { s o f f m a x } ( \widetilde { \mathbf { W } } \widetilde { \mathbf { x } } _ { n } )$ and the second $p ( y _ { n } | \mathbf { x } _ { n } , \mathrm { W } ) = \operatorname { s o f t m a x } ( \mathrm { W } \mathbf { x } _ { n } )$ , cf. Fig. 1 (left).
+
+For representational learning (phase $^ { l }$ ) the large dataset $\widetilde { \mathcal { D } }$ is used to train the CNN $\Phi _ { \varphi }$ using standard deep learning optimisation approaches. This involves learning the parameters $\varphi$ of the feature extractor up to the last hidden layer, as well as the softmax weights $\widetilde { \mathrm { W } }$ . The network parameters $\varphi$ are fixed from this point on and shared across later phases.
+
+Probabilistic modelling. The next goal is to build a probabilistic method for $\mathrm { k }$ -shot prediction that transfers structure from the trained softmax weights $\widetilde { \mathrm { W } }$ to the new $\mathbf { k }$ -shot softmax weights $\mathrm { W }$ and combines it with the $\mathbf { k }$ -shot training examples. Thus, given a test image $\mathbf { u } ^ { * }$ during $k$ -shot testing (phase 4), we compute its feature representation $\mathbf { x } ^ { * } = \Phi ( \mathbf { u } ^ { * } )$ , and the prediction for the new label $y ^ { * }$ is found by averaging the softmax outputs over the posterior distribution of the softmax weights given the two datasets,
+
+$$
+p ( y ^ { * } \mid \mathbf { x } ^ { * } , \mathcal { D } , \widetilde { \mathcal { D } } ) = \int p ( y ^ { * } \mid \mathbf { x } ^ { * } , \mathbf { W } ) p ( \mathbf { W } \mid \mathcal { D } , \widetilde { \mathcal { D } } ) \mathrm { d } \mathbf { W } .
+$$
+
+To this end, we consider a general class of probabilistic models in which the two sets of softmax weights are generated from shared hyperparameters $\theta$ , so that $p ( \widetilde { \mathrm { W } } , \mathrm { W } , \theta ) = p ( \theta ) p ( \widetilde { \mathrm { W } } | \theta ) p ( \mathrm { W } | \theta )$ as indicated in the graphical model in Fig. 1 (right). In this way, the large dataset $\widetilde { \mathcal { D } }$ contains information about $\theta$ that in turn constrains the new softmax weights W. We further assume that there is very little uncertainty in $\widetilde { \mathrm { W } }$ once the large initial training set is observed and so a maximum a posteriori (MAP) estimate, as returned by standard deep learning, suffices. As a consequence of this approximation and the structure of the model, the original data $\widetilde { \mathcal { D } }$ are not required for the $\mathbf { k }$ -shot learning phase. Instead, the weights learned from these data, ${ \widetilde { \mathrm { W } } } ^ { \mathrm { M A P } }$ , can themselves be treated as observed data, which induce a predictive distribution over the $\mathbf { k }$ -shot weights $p ( \mathrm { W } | \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } )$ via Bayes’ rule. This argument is fully explained in Appendix A. We refer to this step as concept learning (phase 2) and note that all probabilistic modelling happens in the definition of $p ( \widetilde { \mathrm { W } } , \mathrm { W } , \theta )$ , (see Secs. 2.2 and 2.3).
+
+During $k$ -shot learning (phase 3) we treat this predictive distribution as our new prior on the weights and again use Bayes’ rule to combine it with the softmax likelihood of the $\mathbf { k }$ -shot training examples $\mathcal { D }$ to obtain a new posterior over the weights that now also incorporates $\mathcal { D }$ ,
+
+$$
+p ( \mathbf { W } \mid \mathcal { D } , \widetilde { \mathcal { D } } ) \approx p ( \mathbf { W } \mid \mathcal { D } , \widetilde { \mathbf { W } } ^ { \mathrm { M A P } } ) \propto p ( \mathbf { W } \mid \widetilde { \mathbf { W } } ^ { \mathrm { M A P } } ) \prod _ { n = 1 } ^ { N } p ( y _ { n } | \mathbf { x } _ { n } , \mathbf { W } ) .
+$$
+
+Finally, we approximate Eq. (2) by its MAP estimate ${ \mathrm { W } } ^ { \mathrm { M A P } }$ , so that the integral in Eq. (1) becomes
+
+$$
+p ( y ^ { * } \mid \mathbf { x } ^ { * } , \mathcal { D } , \widetilde { \mathcal { D } } ) \approx p ( y ^ { * } \mid \mathbf { x } ^ { * } , \mathcal { D } , \widetilde { \mathbf { W } } ^ { \mathrm { M A P } } ) \approx p ( y ^ { * } \mid \mathbf { x } ^ { * } , \mathbf { W } ^ { \mathrm { M A P } } ) .
+$$
+
+# 2.2 CHOOSING A MODEL FOR THE WEIGHTS
+
+The probabilistic model over the weights is key: a good model will transfer useful knowledge that improves performance. However, the usual trade-off between model complexity and learnability is particularly egregious in our setting as the weights $\widetilde { \mathrm { W } }$ are few and high-dimensional and the number of $\mathbf { k }$ -shot samples is small. With an eye on simplicity, we make two simplifying assumptions. First, treating the weights from the hidden layer to the softmax outputs as a vector, we assume independence.
+
+Second, we assume the distribution between the weights of old and new classes to be identical,
+
+$$
+p ( \widetilde { \mathbf { W } } , \mathbf { W } , \theta ) = p ( \theta ) \prod _ { c ^ { \prime } = 1 } ^ { \widetilde { C } } p ( \widetilde { \mathbf { w } } _ { c ^ { \prime } } | \theta ) \prod _ { c = 1 } ^ { C } p ( \mathbf { w } _ { c } | \theta ) \mathrm { w h e r e } p ( \widetilde { \mathbf { w } } _ { c ^ { \prime } } | \theta ) \overset { \mathrm { d i s t } } { = } p ( \mathbf { w } _ { c } | \theta ) .
+$$
+
+After extensive testing, we found that a Gaussian model for the weights strikes the best compromise in the trade-off between complexity and learnability, cf. Sec. 4.2 for a detailed model comparison.
+
+# 2.3 GAUSSIAN MODEL AND ITS RELATION TO LOGISTIC REGRESSION
+
+Our method. We use a simple Gaussian model $p ( \mathbf { w } | \boldsymbol { \theta } ) = \mathcal { N } ( \mathbf { w } | \boldsymbol { \mu } , \boldsymbol { \Sigma } )$ with its conjugate Normalinverse-Wishart prior $p ( \theta ) = p ( \mu , \Sigma ) = \mathcal { N } \mathcal { T } \mathcal { W } ( \mu , \Sigma | \mu _ { 0 } , \kappa _ { 0 } , \Lambda _ { 0 } , \nu _ { 0 } )$ , and estimate MAP solutions for the parameters $\theta ^ { \mathrm { M A P } } = \{ \mu ^ { \mathrm { M A P } } , \Sigma ^ { \mathrm { M A P } } \}$ |. The approximations discussed in Sec. 2.1 lead to $p ( \mathrm { W } \mid \tilde { \mathcal { D } } ) \approx p ( \mathrm { W } \mid \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } ) = \mathcal { N } ( \mathrm { W } \mid \mu ^ { \mathrm { M A P } } , \Sigma ^ { \mathrm { M A P } } )$ , and the posterior at $\mathbf { k }$ -shot time becomes
+
+$$
+p ( \mathrm { W } | \mathcal { D } , \widetilde { \mathcal { D } } ) \propto \mathcal { N } ( \mathrm { W } | \mu ^ { \mathrm { M A P } } , \Sigma ^ { \mathrm { M A P } } ) \prod _ { n = 1 } ^ { N } p ( y _ { n } | \mathbf { x } _ { n } , \mathrm { W } ) .
+$$
+
+For details see Appendix C.1. For $\mathrm { k }$ -shot testing we use the MAP estimates for the weights of the new classes. We found that restricting the covariance matrix to be isotropic, $\Sigma = \sigma ^ { 2 } \mathrm { I }$ , performed best at $\mathbf { k }$ -shot learning, probably due to the small number of data points to learn from as mentioned above.
+
+Relation to logistic regression. Standard logistic regression corresponds to the maximum likelihood (MLE) solution of the softmax likelihood $p ( y _ { n } \mid \mathbf { x } _ { n } , \mathrm { W } ) \ = \ \mathrm { s o f t m a x } ( \mathrm { W } \mathbf { x } _ { n } )$ . Often, $L _ { 2 }$ - regularisation on the weights W with inverse regularisation strength $1 / C _ { \mathrm { r e g } }$ is used; the solution to this regularised optimisation problem corresponds to the MAP solution of a model with isotropic Gaussian prior on the weights with zero mean: This method is analogous to Eq. (5). However, $\begin{array} { r } { p ( \mathrm { W } | \mathcal { D } ) \propto \mathcal { N } ( \mathrm { W } | 0 , \frac { 1 } { 2 } C _ { \mathrm { r e g } } \mathrm { I } ) \prod _ { n = 1 } ^ { N } p ( y _ { n } | \mathbf { x } _ { n } , \mathrm { W } ) } \end{array}$ i) modelling assumptions and approximations are made explicit, ii) it is strictly more general and can incorporate non-zero means $\mu ^ { \mathrm { { \dot { M A P } } } }$ , whereas standard regularised logistic regression assumes zero mean, and iii) the probabilistic interpretation provides a principled way of choosing the regularisation constant using the trained weights $\widetilde { \mathrm { W } }$ : $C _ { \mathrm { r e g } } = 2 \sigma _ { \mathrm { W } } ^ { 2 }$ , where $\sigma _ { \widetilde { \mathrm { W } } } ^ { 2 }$ is the empirical variance of the weights ${ \widetilde { \mathrm { W } } } ^ { \mathrm { M A P } }$ . In $\mathbf { k }$ f f-shot learning, alternative (frequentist) methods such as cross-validation suffer in the face of the small number of $\mathbf { k }$ -shot examples, and are not applicable in 1-shot learning at all.
+
+# 3 RELATED WORK
+
+Embedding methods map the k-shot training and test points into a non-linear space and perform classification by assessing which training points are closest, according to a metric, to the test points. Siamese Networks (Koch et al., 2015) train the embedding using a same/different prediction task derived from the original dataset and use a weighted $L _ { 1 }$ metric for classification. Matching Networks (Vinyals et al., 2016) construct a set of $\mathbf { k }$ -shot learning tasks from the original dataset to train an embedding defined through an attention mechanism that linearly combines training labels weighted by their proximity to test points. More recently, Prototypical Networks (Snell et al., 2017) are a streamlined version of Matching Networks in which embedded classes are summarised by their mean in the embedding space. These embedding methods learn representations for $\mathbf { k }$ -shot learning, but do not directly leverage concept transfer.
+
+Amortised optimisation methods (Ravi & Larochelle, 2017) also simulate related k-shot learning tasks from the initial dataset, but instead train a second network to initialise and optimise a CNN to perform accurate classification on these small datasets. This method can then be applied for new k-shot tasks.
+
+Importantly, both embedding and amortised inference methods improve when the system is trained for a specific $k$ -shot task: to perform well in 5-shot learning, training is carried out with episodes containing 5 examples in each class. The general statement appears to be that training specifically for $\mathbf { k }$ -shot learning is essential for building features which generalise well at $\mathbf { k }$ -shot testing time. The approach proposed in this paper is more flexible; it is not tailored for a specific $k$ and, thus, does not require retraining when switching, e.g., from 5-shot to 10-shot learning. Moreover, Snell et al. (2017)
+
+find that using a larger number of $\mathbf { k }$ -shot classes for the training episodes (e.g., train with $2 0 \mathrm { k }$ -shot classes per episode when testing on only 5 new $\mathbf { k }$ -shot classes) can be beneficial, and they choose this number by cross-validation on a validation-set. This is in alignment with our finding that training with more data and more classes improves performance at $\mathbf { k }$ -shot time.
+
+Deep probabilistic methods include the approach developed in this paper. The methods in this family are not unique to deep learning, and the idea of treating weights as data from which to transfer has been widely applied in multi-task learning (Bakker & Heskes, 2003). The work most closely related to our own is not an approach to $\mathbf { k }$ -shot learning per se, but rather a method for training CNNs with highly imbalanced classes (Srivastava & Salakhutdinov, 2013). It is similar in that it trains a form of Gaussian mixture model over the final layer weights using MAP inference that regularises learning. Burgess et al. (2016) propose an elegant approach to $\mathbf { k } .$ -shot learning that is an instance of the framework described here: a Gaussian model is fit to the weights with MAP inference. The evaluation is promising, but preliminary. One of the goals of this paper is to provide a comprehensive evaluation. While not using a probabilistic approach, Qiao et al., 2017 develop a method for $\mathbf { k }$ -shot learning that trains a recognition model to amortise MAP inference for the softmax weights which can then be used at k-shot learning time. While this method trains the mapping from activation to weights jointly with the classifier, and thus does not learn from the weights per se, it does exploit the structure in the weights for k-shot learning.
+
+# 4 EXPERIMENTS
+
+The code used to produce the following experiments will be made available after review.
+
+Dataset. miniImageNet has become a standard testbed for k-shot learning and is derived from the ImageNet ILSVRC12 dataset (Russakovsky et al., 2015) by extracting 100 out of the 1000 classes. Each class contains 600 images downscaled to $8 4 \times 8 4$ pixels. We use the 100 classes (64 train, 16 validation, 20 test) proposed by Ravi & Larochelle (2017). As our approach does not require a validation set, we use both the training and validation data for the representational learning.
+
+Representational learning. We employ standard CNNs that are inspired by ResNet-34 (He et al., 2016) and VGG (Simonyan & Zisserman, 2014) for the representational learning on the $\widetilde { C }$ base classes, cf. Phase 1 in Sec. 2.1. These trained networks provide both ${ \widetilde { \mathrm { W } } } ^ { \mathrm { M A P } }$ and the fixed feature representation $\Phi _ { \varphi }$ for the $\mathbf { k }$ -shot learning and testing. We employed standard data augmentation from ImageNet for the representational learning but highlight that no data augmentation was used during the $\mathbf { k }$ -shot training and testing. For details on the architecture, training, and data augmentation see Appendix D.4. t-SNE embeddings (Van der Maaten & Hinton, 2008) of the learned last layer weights show sensible clusters, which highlights the structure exploited by the probabilistic model, see Appendix E.1.
+
+Baselines and competing methods. We compare against several baselines as well as recent stateof-the-art methods mentioned in Sec. 3. The baselines are computed on the features $\mathbf { x } = \Phi _ { \phi } ( \mathbf { u } )$ from the last hidden layer of the trained CNN: (i) Nearest Neighbours with cosine distance and (ii) regularized logistic regression with regularisation constant set either by cross-validation or (iii) using the variance of the weights, $C = 2 \sigma _ { W } ^ { 2 }$ , as motivated by our probabilistic framework, cf. Sec. 2.3. We also compare against three recent $\mathbf { k }$ -shot methods: (i) Matching Networks1 (Vinyals et al., 2016), (ii) Prototypical Networks, with numbers reported from Snell et al., 2017 and (iii) Meta-learner LSTM, with numbers reported from Ravi & Larochelle, 2017.
+
+Testing protocol. We evaluate the methods on 600 random k-shot tasks by randomly sampling 5 classes from the 20 test classes and perform 5-way $\mathbf { k }$ -shot learning. Following Snell et al. (2017), we use 15 randomly selected images per class for $\mathbf { k }$ -shot testing to compute accuracies and calibration.
+
+# 4.1 RESULTS ON miniIMAGENET
+
+Overall k-shot performance. We report performance on the miniImageNet dataset in Tab. 1 and Figs. 2 and 3. The best method uses as feature extractor a modified ResNet-34 with 256 features, trained with all 600 examples per training class, and a simple isotropic Gaussian model on the weights for concept learning. Despite its simplicity, our method achieves state-of-the-art and beats prototypical networks by a wide margin of about $6 \%$ . The baseline methods using the same feature extractor are also state-of-the-art compared to prototypical networks and both logistic regressions show comparable accuracy to our methods except for on 1-shot learning. In terms of log-likelihoods, Log Reg $\bar { C } = 2 \sigma _ { \widetilde { W } } ^ { 2 } )$ ) fares slightly better, whereas Log Reg (cv) is much worse.
+
+Table 1: Accuracy on 5-way classification on miniImageNet. Our best method, an isotropic Gaussian model using ResNet-34 features consistently outperforms all competing methods by a wide margin.   
+
+<table><tr><td>Method</td><td>1-shot</td><td>5-shot</td><td>10-shot</td></tr><tr><td>ResNet-34 + Isotropic Gaussian (ours)</td><td>56.3 ± 0.4%</td><td>73.9 ± 0.3%</td><td>78.5 ± 0.3%</td></tr><tr><td>Matching Networks (reimplemented, 1-shot)</td><td>46.8 ± 0.5%</td><td>=</td><td></td></tr><tr><td>Matching Networks (reimplemented, 5-shot)</td><td></td><td>62.7 ± 0.5%</td><td></td></tr><tr><td>Meta-LearnerLSTM(Ravi&amp;Larochelle,2017)</td><td>43.4 ± 0.8%</td><td>60.6 ± 0.7%</td><td></td></tr><tr><td>Prototypical Nets (1-shot) (Snell et al., 2017)</td><td>49.4 ± 0.8%</td><td>65.4 ± 0.7%</td><td></td></tr><tr><td>Prototypical Nets (5-shot) (Snell et al., 2017)</td><td>45.1 ± 0.8%</td><td>68.2 ± 0.7%</td><td></td></tr></table>
+
+![](images/2abc8431807cb98b08f5efed3aa341b8068966734595e8e18b2bc4047ae8316f.jpg)  
+Figure 2: Results for miniImageNet with ResNet-34 style architecture and 600 training images per class. From left to right: accuracy and log likelihood (higher is better) for different k, Expected Calibration Error (ECE, lower is better) vs accuracy for 5-shot learning, and Calibration curve for 5-shot learning. Results on other architectures can be found in Appendix E.2
+
+Deeper features lead to better k-shot learning. We investigate the influence of different feature extractors of increasing complexity on performance in Fig. 3: i) a VGG style network (500 train images per class), ii) a ResNet-34 (500 examples per class), and iii) a ResNet-34 (all 600 examples per class). We find that the complexity of the feature extractor as well as training set size consistently correlate with the accuracy at $\mathbf { k }$ -shot time. For instance, on 5-shot, Gauss (iso) achieves $6 5 \%$ accuracy with a VGG network and $7 4 \%$ with a ResNet trained with all available data, a significant increase of almost $1 0 \%$ . Moreover, Gauss (iso) outperforms Log Reg $( C = 2 \sigma _ { W } ^ { 2 } )$ on 1-shot learning across fmodels, and performs similarly on 5- and 10-shot. We attribute the difference to the former’s ability to also model the mean of the Gaussian, whereas logistic regression assumes a zero mean.
+
+Importantly, this result implies that training specifically for $\mathbf { k }$ -shot learning is not necessary for achieving high generalisation performance on this $\mathbf { k }$ -shot problem. On the contrary, training a powerful deep feature extractor on batch classification using all of the available training data, then building a simple probabilistic model using the learned features and weights achieves state-of-the-art. Recent models that use episodic training cannot leverage such deep feature extractors as for them the depth of the model is limited by the nature of training itself. The reference baseline in the $\mathbf { k }$ -shot learning literature is nearest neighbours, which performs on par with Gauss (iso) on 1-shot learning but is outperformed by all methods on 5- and 10-shot. This is evidence that building a simple classifier on top of the learned features works significantly better for k-shot learning than nearest neighbours.
+
+Calibration. A classifier is said to be calibrated when the probability it predicts for belonging to a given class is on par with the probability of it being the correct prediction. In other words, when examples for which it predicts a probability $p$ of belonging to a given class are correctly classified for a fraction $p$ of the examples. A calibration curve visualises the proportion of examples correctly classified as a function of their predicted probability; a perfectly calibrated classifier should result in a diagonal line. Following Guo et al. (2017), we consider the log likelihood on the $\mathbf { k }$ -shot test examples as well as Expected Calibration Error (ECE) as summary measures of calibration. ECE can be interpreted as the weighted average of the distance of the calibration curve to the diagonal. We find that Log Reg $C = 2 \bar { \sigma } _ { \widetilde { W } } ^ { 2 } ,$ ) and Gauss (iso) provide better accuracy and calibration than Log fReg (cross-validation), cf. Fig. 2. The difference in calibration quality for different regularisations of logistic regression highlights the importance of choosing the right constant, as we discuss now.
+
+![](images/34602529141e976b5411c28bcce3497ad243c5e6887b4a21cb7c66d0c8c625df.jpg)  
+Figure 3: Comparison of different network architectures and training set sizes on the k-shot learning task: VGG style network (trained on 500 images per class) and ResNet-34 style network (trained on 500 and 600 images per class, respectively). Both, deeper networks and larger number of training images, give rise to features that transfer better to k-shot learning.
+
+![](images/1466dac2146a89967aac74d59260f7d2012b5cc9089cd459d7154dc65008ca14.jpg)  
+Figure 4: Choice of regularisation constant for logistic regression for $\mathbf { k }$ -shot learning. Results for $C _ { \mathrm { r e g } } ^ { - } = 2 \sigma _ { \mathrm { W } } ^ { 2 }$ are drawn as black triangles. Dashed lines correspond to logistic regression with crossfvalidated (changing) regularisation constant. Colour brightness of the markers ranges from dark $( C = 1 0 ^ { - 5 }$ ) to bright $C = 1 0 $ ). ECE plots are provided in Appendix E.3.
+
+Choice of the regularisation constant for logistic regression. The results so far suggest that training a simple linear model such as regularised logistic regression might be sufficient to perform well in $\mathbf { k }$ -shot learning. However, while the accuracy at $\mathbf { k }$ -shot time does not vary dramatically as the regularisation constant changes, the calibration does, and jointly maximizing both quantities is not possible, cf. the first two plots of Fig. 4. The standard (frequentist) method to tune this constant is cross validation, which is not applicable in the 1-shot setting, and suffers from lack of data in 5- and 10-shot. Contrary, our probabilistic framework provides a principled way of selecting this regularisation parameter by transfer from the training weights: Log Reg $\dot { ( } C = 2 \dot { \sigma } _ { \widetilde { W } } ^ { 2 }$ ) strikes a good fbalance between accuracy and log-likelihood. The third plot in Fig. 4 reports log-likelihood as a function of accuracy and provides further visualisation of the achieved trade-off between accuracy and calibration for Log Reg $( C = 2 \sigma _ { W } ^ { 2 } )$ ), as well as the failure of Log Reg (cross-validation) to fachieve a good compromise in 5- and 10-shot.
+
+Evaluation in an online setting. We also briefly consider the online setting, in which we jointly test on 80 old and 5 new classes, for which catastrophic forgetting (French, 1999) is a well known problem. During $\mathbf { k }$ -shot learning and testing we employ a softmax which includes both the new and the old weights resulting in a total of 85 weight vectors. We utilise ResNet-34 trained on 500 images per class to retain 100 test images on the old classes. While the k-shot weights were modelled probabilistically, we use the MAP estimate $\widetilde { W } ^ { \mathrm { M A P } }$ for the old weights. Accuracies are reported in
+
+![](images/bfdb92abd1775509a6e5e3559880b374e980dfff29d687fa07c29e8a04a43bb2.jpg)  
+Figure 5: Online learning with ResNet-34 features. Gauss (iso) and Log Reg $( 2 \sigma _ { \widetilde { W } } ^ { 2 } )$ strike a good ftrade-off between learning on new classes and forgetting of old classes. Unregularised Log Reg (MLE) and Log Reg $( 2 \sigma _ { \widetilde { W } } ^ { 2 }$ , only new), which has not been trained in the presence of the old weights, either fcompletely forget the old classes or do not learn anything, respectively.
+
+Fig. 5 for i) all the 85 classes, ii) the old 80 classes only, and iii) the new 5 classes only. For 5- and 10-shot, Gauss (iso) and Log Reg $( 2 \sigma _ { \widetilde { W } } ^ { 2 } )$ only lose a couple of percent on the accuracy of the old fclasses, and perform well on the new classes, striking a good trade-off between forgetting and learning at k-shot time. For unregularised (MLE) logistic regression, the new weights completely dominate the old ones, highlighting that the right regularisation is important. Yet, cross-validation in this setting is often very challenging. When training Logistic Regression without including the old weights (“only new”), the new weights are dominated by the old ones and fail to learn the new classes, making training in the presence of the old weights an essential component for online learning.
+
+# 4.2 MODEL COMPARISON ON CIFAR-100
+
+We performed an extensive comparison between different probabilistic models of the weights using different inference procedures, which we present in Appendix E.4. We report results on the CIFAR-100 dataset on (i) Gaussian, (ii) mixture of Gaussians, and (iii) Laplace, all with either MAP estimation or Hybrid Monte Carlo sampling. We found that the simple Gaussian model is on par with or outperforms other methods at k-shot time, which we attribute to it striking a good balance between choosing a complex model, which may better fit the weights, and statistical efficiency, as the number of weights $\widetilde { C }$ (80 in our case) is often smaller than the dimensionality of the feature representation (256 in our case), cf. Sec. 2. This finding is supported by computing the log-likelihood of held out training weights under such model, with the Gaussian model performing best. Experiments using Hybrid Monte Carlo sampling for $\mathbf { k }$ -shot learning returned very similar performance to MAP estimation and at a much higher computational cost, due to the difficulty of performing sampling in such a high dimensional parameter space. Our recommendation is that practitioners should use simple models and employ simple inference schemes to estimate all free parameters thereby avoiding expending valuable data on validation sets.
+
+# 5 CONCLUSION
+
+We present a probabilistic framework for $\mathbf { k }$ -shot learning that exploits the powerful features and class information learned by a neural network on a large training dataset. Probabilistic models are then used to transfer information in the network weights to new classes. Experiments on miniImageNet using a simple Gaussian model within our framework achieve state-of-the-art for 1-shot and 5-shot learning by a wide margin, and at the same time return well calibrated predictions. This finding is contrary to the current belief that episodic training is necessary to learn good $\mathbf { k }$ -shot features and puts the success of recent complex deep learning approaches to $\mathbf { k }$ -shot learning into context. The new approach is flexible and extensible, being applicable to general discriminative models and $\mathbf { k } .$ - shot learning paradigms. For example, preliminary results on online $\mathbf { k }$ -shot learning indicate that the probabilistic framework mitigates catastrophic forgetting by automatically balancing performance on the new and old classes.
+
+The Gaussian model is closely related to regularised logistic regression, but provides a principled and fully automatic way to regularise. This is particularly important in $\mathbf { k }$ -shot learning, as it is a low-data regime, in which cross-validation performs poorly and where it is important to train on all available data, rather than using validation sets.
+
+# REFERENCES
+
+Bart Bakker and Tom Heskes (2003). Task Clustering and Gating for Bayesian Multitask Learning. Journal of Machine Learning Research 4, pp. 83–99.
+
+Jordan Burgess, James Robert Lloyd, and Zoubin Ghahramani (2016). One-Shot Learning in Discriminative Neural Networks. NIPS Bayesian Deep Learning workshop.
+
+Robert M French (1999). Catastrophic forgetting in connectionist networks. Trends in cognitive sciences 3.4, pp. 128–135.
+
+Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger (2017). On Calibration of Modern Neural Networks. arXiv e-print: 1706.04599.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun (2016). Deep Residual Learning for Image Recognition. In: The IEEE Conference on Computer Vision and Pattern Recognition (CVPR). eprint: 1512.03385.
+
+Gregory Koch, Richard Zemel, and Ruslan Salakhutdinov (2015). Siamese neural networks for oneshot image recognition. Deep Learning workshop, International Conference of Machine Learning.
+
+Brenden Lake, Ruslan Salakhutdinov, and Joshua Tenenbaum (2015). Human-level concept learning through probabilistic program induction. Science 350.6266, pp. 1332–1338.
+
+Siyuan Qiao, Chenxi Liu, Wei Shen, and Alan Yuille (2017). Few-Shot Image Recognition by Predicting Parameters from Activations. arXiv e-print: 1706.03466.
+
+Sachin Ravi and Hugo Larochelle (2017). Optimization as a model for few-shot learning. In: International Conference on Learning Representations. Vol. 1. 2, p. 6.
+
+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei (2015). ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJCV) 115.3, pp. 211–252. DOI: 10.1007/s11263-015-0816-y.
+
+Karen Simonyan and Andrew Zisserman (2014). Very deep convolutional networks for large-scale image recognition. arXiv e-print:1409.1556.
+
+Jake Snell, Kevin Swersky, and Richard Zemel (2017). Prototypical Networks for Few-shot Learning. arXiv e-print: 1703.05175.
+
+Nitish Srivastava and Ruslan R Salakhutdinov (2013). Discriminative transfer learning with treebased priors. In: Advances in Neural Information Processing Systems, pp. 2094–2102.
+
+Laurens Van der Maaten and Geoffrey Hinton (2008). Visualizing Data using t-SNE. Journal of Machine Learning Research 9, pp. 2579–2605.
+
+Oriol Vinyals, Charles Blundell, Tim Lillicrap, Daan Wierstra, et al. (2016). Matching networks for one shot learning. In: Advances in Neural Information Processing Systems, pp. 3630–3638.
+
+# Appendix to “Discriminative $\mathbf { k }$ -shot learning using probabilistic models”
+
+A DETAILS ON THE DERIVATION AND APPROXIMATIONS FROM SEC. 2.1
+
+As stated in the main text, the probabilistic $\mathbf { k }$ -shot learning approach comprises four phases mirroring the dataflow:
+
+Phase 1: Representational learning. The large dataset $\widetilde { \mathcal { D } }$ is used to train the CNN $\Phi _ { \varphi }$ using standard deep learning optimisation approaches. This involves learning both the parameters $\varphi$ of the feature extractor up to the last hidden layer, as well as the softmax weights $\widetilde { \mathrm { W } }$ . The network parameters $\varphi$ are fixed from this point on and shared across phases. This is a standard setup for multitask learning and in the present case it ensures that the features derived from the representational learning can be leveraged for $\mathbf { k }$ -shot learning.
+
+Phase 2: Concept learning. The softmax weights $\widetilde { \mathrm { W } }$ are effectively used as data for concept learning by training a probabilistic model that detects structure in these weights which can be transferred for $\mathbf { k }$ -shot learning. This approach will be justified in the next section. For the moment, we consider a general class of probabilistic models in which the two sets of weights are generated from shared hyperparameters $\theta$ , so that $p ( \widetilde { \mathrm { W } } , \mathrm { W } , \theta ) = p ( \theta ) p ( \widetilde { \mathrm { W } } | \theta ) p ( \mathrm { W } | \theta )$ (see Fig. 1).
+
+Phases 3 and ${ \bf 4 } \colon { \bf k }$ -shot learning and testing. Probabilistic $\mathbf { k }$ -shot learning leverages the learned representation $\Phi _ { \varphi }$ from phase 1 and the probabilistic model $p ( \widetilde { \mathrm { W } } , \mathrm { W } , \theta )$ from phase 2 to build a (posterior) predictive model for unseen new examples using examples from the small dataset $\mathcal { D }$ .
+
+PROBABILISTIC MODEL OF THE WEIGHTS
+
+Given the dataflow and the assumed probabilistic model in Fig. 1 (right), a completely probabilistic approach would involve the following steps.
+
+In the concept learning phase, the initial dataset would be used to form the posterior distribution over the concept hyperparameters $p ( \boldsymbol { \theta } \mid \widetilde { \mathcal { D } } )$ . The $\mathbf { k }$ -shot learning phase combines the information about the new weights provided by $\widetilde { \mathcal { D } }$ with the information in the k-shot dataset $\mathcal { D }$ to form the posterior distribution
+
+$$
+p ( \mathrm { W } \mid \mathcal { D } , \widetilde { \mathcal { D } } ) \propto p ( \mathrm { W } \mid \widetilde { \mathcal { D } } ) \prod _ { n } p ( y _ { n } \mid \mathbf { x } _ { n } , \mathrm { W } ) \mathrm { w h e r e } p ( \mathrm { W } \mid \widetilde { \mathcal { D } } ) = \int p ( \mathrm { W } \mid \theta ) p ( \theta \mid \widetilde { \mathcal { D } } ) \mathrm { d } \theta .
+$$
+
+To see this, notice that
+
+$$
+\begin{array} { r } { p ( \mathrm { W } | \mathcal { D } , \widetilde { \mathcal { D } } ) \propto p ( \mathrm { W } , \mathcal { D } , \widetilde { \mathcal { D } } ) = p ( \widetilde { \mathcal { D } } ) p ( W | \widetilde { \mathcal { D } } ) p ( \mathcal { D } | \widetilde { \mathcal { D } } , \mathrm { W } ) . } \end{array}
+$$
+
+The graphical model in Fig. 1 entails that $\mathcal { D }$ is conditionally independent from $\widetilde { \mathcal { D } }$ given $\mathrm { W }$ , such that
+
+$$
+p ( \mathcal { D } \mid \mathrm { W } , \widetilde { \mathcal { D } } ) = p ( \mathcal { D } \mid \mathrm { W } ) = \prod _ { n } p ( y _ { n } \mid x _ { n } , \mathrm { W } ) .
+$$
+
+We recover Eq. (6) by adding $p ( \widetilde { \mathcal { D } } )$ to the constant of proportionality.
+
+Inference in this model is generally intractable and requires approximations. The main challenge is computing the posterior distribution over the hyper-parameters given the initial dataset. However, progress can be made if we assume that the posterior distribution over the weights can be well approximated by the MAP value $p ( \widetilde { \mathrm { W } } | \widetilde { \mathcal { D } } ) \approx \delta ( \widetilde { \mathrm { W } } - \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } )$ . This is an arguably justifiable assumption as the initial dataset is large and so the posterior will concentrate on narrow modes (with similar predictive performance). In this case $p ( \theta | \mathcal { \widetilde { D } } ) \approx p ( \theta | \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } )$ and, due to the structure of the probabilistic model, all instances of $\widetilde { \mathcal { D } }$ in Eq. (6) and Eq. (1) can be replaced by the analogous expressions involving ${ \widetilde { \mathrm { W } } } ^ { \mathrm { M A P } }$ . This greatly simplifies the learning pipeline as the probabilistic modelling only needs to have access to the weights returned by representational learning. Remaining intractabilities involve only a small number of data points $\mathcal { D }$ and can be handled using standard approximate inference tools. The following summarizes the approximations and computational steps for each phase of training.
+
+Phase 1: Representational learning. Deep learning is used to train a CNN. The representation of input images at the last hidden layer, $\mathbf { x } = \Phi _ { \varphi } ( \mathbf { u } )$ , is used in subsequent phases. The final layer softmax weights are assumed to be MAP estimates $\widetilde { \mathrm { W } } ^ { \mathrm { M A P } }$ .
+
+Phase 2: Concept learning. A probabilistic model is fit directly to the MAP weights $p ( \theta | \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } ) \propto \bar { p ( \theta ) } p ( \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } | \theta ) .$ . For conjugate models a full posterior can be retained, otherwise a MAP estimate $p ( \theta | \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } ) \approx \delta ( \theta - \theta ^ { \mathrm { M A P } } )$ is used.
+
+Phase 3: $\mathbf { k }$ -shot learning. The posterior distribution over the new softmax weights $\begin{array} { r } { p ( \mathrm { W } | \mathcal { D } , \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } ) \propto p ( \mathrm { W } | \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } ) \prod _ { n = 1 } ^ { \tilde { N } } p ( y _ { n } | \mathbf { x } _ { n } , \mathrm { W } ) } \end{array}$ is generally intractable. The posterior can, however, be approximated using the MAP estimate $p ( \mathrm { W } | \mathcal { D } , \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } ) \approx \delta ( \mathrm { W } - \mathrm { W } ^ { \mathrm { M A P } } )$ or through sampling $\mathrm { W } _ { m } \sim p ( \mathrm { W } | \mathcal { D } , \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } )$ . Note that $\begin{array} { r } { p ( \mathrm { W } | \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } ) = \int p ( \mathrm { W } | \boldsymbol { \theta } ) p ( \boldsymbol { \theta } | \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } ) \mathrm { d } \boldsymbol { \theta } } \end{array}$ is analytic for conjugate models and, if instead a MAP estimate for $\theta$ is provided by the concept modelling stage, then $\bar { p } ( \mathrm { W } | \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } ) \approx p ( \mathrm { W } | \theta ^ { \mathrm { M A P } } )$ .
+
+Phase 4: $\mathbf { k }$ -shot testing. Approximate inference is used to compute $p ( y ^ { * } \mid \mathbf { x } ^ { * } , \mathcal { D } , \widetilde { \mathrm { W } } ^ { \mathrm { M A P } } ) \ =$ $\begin{array} { r } { \int p ( y ^ { * } \mid \mathbf { x ^ { * } } , \operatorname { W } ) p ( \operatorname { W } \mid \mathcal { D } , \widetilde { \operatorname { W } } ^ { \mathrm { M A P } } ) \mathrm { d W } } \end{array}$ . If the $\mathbf { k }$ -shot learning phase provides a MAP estimate of W then $p ( y ^ { * } \mid \mathbf { x } ^ { * } , \mathcal { D } , \widetilde { \mathbf { W } } ^ { \mathrm { M A P } } ) \ \approx \ p ( y ^ { * } \mid \mathbf { x } ^ { * } , \mathbf { W } ^ { \mathrm { M A P } } )$ . If samples are returned then $\begin{array} { r } { p ( y ^ { * } \mid \mathbf { x } ^ { * } , \mathcal { D } , \widetilde { \mathbf { W } } ^ { \mathrm { M A P } } ) \approx \frac { 1 } { M } \sum _ { m = 1 } ^ { M } p ( y ^ { * } \mid \mathbf { x } ^ { * } , \mathbf { W } _ { m } ) } \end{array}$ .
+
+# B APPROXIMATE INFERENCE METHODS
+
+In this section we briefly discuss different inference methods for the probabilistic models. In the main text we only considered MAP inference as we found that other more complicated inference schemes do not yield a practical benefit. However, in Appendix E.4 we provide a detailed model comparison, in which we also consider other approximate inference methods.
+
+In all cases the gradients of the densities w.r.t. W can be computed, enabling MAP inference in the k-shot learning phase to be efficiently performed via gradient-based optimisation using L-BFGS (Liu & Nocedal, 1989). Alternatively, Markov Chain Monte Carlo (MCMC) sampling can be performed to approximate the associated integral, see Eq. (1). Due to the high dimensionality of the space and as gradients are available, we employ Hybrid Monte Carlo (HMC) (Neal et al., 2011) sampling in the form of the recently proposed NUTS sampler that automatically tunes the HMC parameters (step size and number of leapfrog steps) (Hoffman & Gelman, 2014). For the GMMs we employed pymc3 (Salvatier et al., 2016) to perform MAP inference.
+
+# C MODELS FOR THE PRIOR ON THE WEIGHTS
+
+As discussed in Sec. 2.1, we specify our model through $p ( \mathrm { W } , \widetilde { \mathrm { W } } , \theta )$ thus defining $p ( \mathrm { W } \vert \widetilde { \mathrm { W } } ^ { M A P } )$ in Eq. (2). This section analyses different priors on the weights: (i) Gaussian models, (ii) Gaussian mixture models, and (iii) Laplace distribution. In the main paper, we only use a Gaussian model with MAP inference, as we saw no significant advantage in using other, more complex models. However, we provide an extensive comparison of the different models in Appendix E.4.
+
+# C.1 GAUSSIAN MODEL
+
+Possibly the simplest approach consists of modelling $p ( \mathrm { W } | \widetilde { \mathrm { W } } )$ as a Gaussian distribution:
+
+$$
+p ( \mathbf { W } \mid \widetilde { \mathbf { W } } ) = \int \mathcal { N } ( \mathbf { W } \mid \boldsymbol { \mu } , \boldsymbol { \Sigma } ) p ( \boldsymbol { \mu } , \boldsymbol { \Sigma } \mid \widetilde { \mathbf { W } } ) \mathrm { d } \boldsymbol { \mu } \mathrm { d } \boldsymbol { \Sigma } .
+$$
+
+Details for this section can be found in Murphy, 2012. The normal-inverse-Wishart distribution for $\mu$ and $\Sigma$ is a conjugate prior for the Gaussian, which allows for the posterior to be written in closed form. More precisely,
+
+$$
+\begin{array} { l } { { \displaystyle p ( \mu , \Sigma ) = \mathcal { N } T \mathcal { W } ( \mu , \Sigma \mid \mu _ { 0 } , \kappa _ { 0 } , \Lambda _ { 0 } , \nu _ { 0 } ) } } \\ { { \displaystyle \quad = \frac { 1 } { Z } | \Sigma | ^ { - ( \nu _ { 0 } + p ) / 2 + 1 } e ^ { - \frac { 1 } { 2 } t r ( \Lambda _ { 0 } \Sigma ^ { - 1 } ) - \frac { \kappa _ { 0 } } { 2 } ( \mu - \mu _ { 0 } ) ^ { t } \Sigma ^ { - 1 } ( \mu - \mu _ { 0 } ) } , } } \end{array}
+$$
+
+where $Z$ is the normalising constant. The posterior $p ( \mu , \Sigma | \widetilde { \mathrm { W } } )$ also follows a normal-inverse-Wishart distribution:
+
+$$
+p ( \mu , \Sigma | \widetilde { \mathbf { W } } ) = { \mathcal { N } } \widetilde { \cal T } \mathcal { W } ( \mu , \Sigma | \mu _ { \widetilde { N } } , \kappa _ { \widetilde { N } } , \Lambda _ { \widetilde { N } } , \nu _ { \widetilde { N } } ) ,
+$$
+
+where
+
+$$
+\begin{array} { r l } & { \displaystyle \mu _ { \widetilde { N } } = \frac { \kappa _ { 0 } } { \kappa _ { 0 } + \widetilde { N } } \mu _ { 0 } + \frac { \widetilde { N } } { \kappa _ { 0 } + \widetilde { N } } \overline { { \widetilde { \mathrm { W } } } } } \\ & { \displaystyle \kappa _ { \widetilde { N } } = \kappa _ { 0 } + \widetilde { N } } \\ & { \displaystyle \Lambda _ { \widetilde { N } } = \Lambda _ { 0 } + S + \frac { \kappa _ { 0 } \widetilde { N } } { \kappa _ { 0 } + \widetilde { N } } ( \overline { { \widetilde { \mathrm { W } } } } - \mu _ { 0 } ) ( \overline { { \widetilde { \mathrm { W } } } } - \mu _ { 0 } ) ^ { t } } \\ & { \displaystyle \nu _ { \widetilde { N } } = \nu _ { 0 } + \widetilde { N } , } \end{array}
+$$
+
+and $S$ is the sample covariance of $\widetilde { \mathrm { W } }$ .
+
+For this model, we can integrate (9) in closed form, which results in the following multivariate Student $t$ -distribution:
+
+$$
+p ( \mathrm { W } | \widetilde { \mathrm { W } } ) = t _ { \nu _ { \widetilde { N } } - p + 1 } \left( \mu _ { \widetilde { N } } , \frac { \Lambda _ { \widetilde { N } } ( \kappa _ { \widetilde { N } } + 1 ) } { \kappa _ { \widetilde { N } } ( \nu _ { \widetilde { N } } - p + 1 ) } \right) .
+$$
+
+As with other approaches, one can also compute the MAP solutions for the mean $\mu _ { \mathrm { M A P } }$ and covariance $\Sigma _ { \mathrm { M A P } }$ , such that $p ( \mathrm { W } \mid \widetilde { \mathrm { W } } ) = \mathcal { N } ( \mathrm { W } \mid \mu _ { \mathrm { M A P } } , \Sigma _ { \mathrm { M A P } } )$ .
+
+For both the analytic posterior and the MAP approximation, $p ( \mathrm { W } | \widetilde { \mathrm { W } } )$ depends on the hyperparameters of the normal-inverse-Wishart distribution: $\mu _ { 0 } , \nu _ { 0 } , \kappa _ { 0 }$ and $\Lambda _ { 0 }$ . There are different ways to choose these hyperparameters. One way would be by optimising the log probability of held out training weights, see Appendix E.4 for a brief discussion. In practise, it is common to choose uninformative or data dependent priors as discussed by Murphy (2012, Chapter 4).
+
+# C.2 MIXTURE OF GAUSSIANS (GMM)
+
+A Gaussian mixture model can potentially leverage cluster structure in the weights (animal classes might have similar weights, for example). This is related to the tree-based prior proposed in Srivastava & Salakhutdinov (2013). MAP inference is performed because exact inference is intractable. Similarly to the Gaussian case, different structures for the covariance of each cluster were tested. In our experiments, we fit the parameters of the GMM via maximum likelihood using the EM algorithm. GMM consists on modelling $p ( \mathrm { W } | \widetilde { \mathrm { W } } )$ as a mixture of Gaussians with $S$ components:
+
+$$
+\begin{array} { r } { \rho ( \mathbf { W } \mid \widetilde { \mathbf { W } } ) = \displaystyle \int \left( \displaystyle \sum _ { s = 1 } ^ { S } \pi _ { s } \mathcal { N } ( \mathbf { W } \mid \mu _ { s } , \Sigma _ { s } ) \right) p ( \mu _ { 1 } , \dots , \mu _ { S } , \Sigma _ { 1 } , \dots , \Sigma _ { S } \mid \widetilde { \mathbf { W } } ) \mathrm { d } \mu _ { 1 } \dots \mathrm { d } \mu _ { S } \mathrm { d } \Sigma _ { 1 } \dots \mathrm { d } \Sigma _ { S } , } \end{array}
+$$
+
+where cluster $\textstyle \sum _ { s = 1 } ^ { S } \pi _ { s } = 1$ . In this work, we only compute the MAP mean and covariance for each of theto averaging over the parameters of the mixture. The resulting posterior is
+
+$$
+p ( \mathbf { W } \mid \widetilde { \mathbf { W } } ) = \sum _ { s = 1 } ^ { S } \pi _ { s } \mathcal { N } ( \mathbf { W } \mid \mu _ { M A P , s } , \Sigma _ { M A P , s } ) .
+$$
+
+The components of the mixture are fit in two ways. For CIFAR-100, the classes are grouped into 20 superclasses, each containing 5 of the 100 classes. One option is therefore to initialize 20 components, each fit with the data points in the corresponding superclass. For each such individual Gaussian, the MAP inference method presented in the previous section can be used. In order to increase the number of weight examples in each superclass, we merge the original superclasses into 9 larger superclasses. The merging of the superclasses is the following:
+
+Aquatic mammals $^ +$ fish
+
+• flowers $^ +$ fruit and vegetables $^ +$ trees   
+• insects $^ +$ non-insect invertebrates $^ +$ reptiles   
+medium-sized mammals $^ +$ small mammals   
+• large carnivores $^ +$ large omnivores and herbivores   
+• people   
+• large man-made outdoor things $^ +$ large natural outdoor things food containers $^ +$ household electrical devices $^ +$ household furniture • Vehicles $1 +$ Vehicles 2.
+
+The parameters of the mixture can also be fit using maximum likelihood with EM. We use the implementation of EM in scikit-learn. Both 3 and 10 clusters are considered in CIFAR-100. Weight log-likelihoods under this model and k-shot performance can be found in Appendix E.4.
+
+Note that, similarly to the Gaussian model, we consider isotropic, diagonal or full covariance models for the covariance matrices.
+
+# C.3 LAPLACE DISTRIBUTION
+
+Sparsity is an attractive feature which could be helpful for modelling the weights. Indeed, it is reasonable to assume that each class uses a set of characteristic features which drive classification accuracy, while others are irrelevant. Sparse models would then provide sensible regularization. As such, we consider a product of independent Laplace distribution. Sec. 2.3 highlights the relation between a Gaussian prior on the weights and $L _ { 2 }$ regularised logistic regression. One can similarly show that the Laplace prior is related to $L _ { 1 }$ regularised logistic regression, which is well known for encouraging sparse weight vectors.
+
+We consider a prior which factors along the feature dimensions:
+
+$$
+p ( \widetilde { \mathbf { W } } | \{ \mu _ { j } \} , \{ \lambda _ { j } \} ) = \prod _ { j } ^ { p } \frac { 1 } { 2 \lambda _ { j } } \exp \left( - \sum _ { i } ^ { \widetilde { C } } \frac { | \widetilde { \mathbf { W } } _ { i j } - \boldsymbol { \mu } _ { j } | } { \lambda _ { j } } \right) .
+$$
+
+where the product over $j$ is along the feature dimensions and the sum over $i$ is across the classes. We fit the parameters $\mu$ and $\lambda$ via maximum likelihood:
+
+$$
+\begin{array} { l } { \displaystyle \mu _ { \mathrm { M L E } , j } = \mathrm { m e d i a n } _ { i } ( \widetilde { \mathrm { W } } _ { i j } ) } \\ { \displaystyle \lambda _ { \mathrm { M L E } , j } = \frac { 1 } { N } \sum _ { i } | \widetilde { \mathrm { W } } _ { i j } - \mu _ { j } | , } \end{array}
+$$
+
+such that
+
+$$
+p ( \mathrm { W } | \widetilde { \mathrm { W } } ) = \prod _ { j } ^ { p } \frac { 1 } { 2 \lambda _ { \mathrm { M L E } , j } } \exp \left( - \sum _ { i } ^ { C } \frac { | \mathrm { W } _ { i j } - \mu _ { \mathrm { M L E } , j } | } { \lambda _ { \mathrm { M L E } , j } } \right) .
+$$
+
+An isotropic Laplace model with mean $\mu$ and scale $\lambda$ is also considered:
+
+$$
+p ( \widetilde { \mathrm { W } } \left| \mu , \lambda \right) = \frac { 1 } { 2 \lambda } \exp \left( - \frac { \sum _ { i j } \left| \widetilde { \mathrm { W } } _ { i j } - \mu \right| } { \lambda } \right) ,
+$$
+
+where
+
+$$
+\begin{array} { l } { \displaystyle \mu _ { \mathrm { M L E } } = \mathrm { m e d i a n } ( \widetilde { \mathrm { W } } ) } \\ { \displaystyle \lambda _ { \mathrm { M L E } } = \frac { 1 } { N p } \sum _ { i j } \big | \widetilde { \mathrm { W } } _ { i j } - \mu \big | , } \end{array}
+$$
+
+# D TRAINING AND EVALUATION PROCEDURE DETAILS
+
+# D.1 miniIMAGENET
+
+To construct miniImageNet we use the same classes as initially proposed by Ravi & Larochelle (2017) and used in (Snell et al., 2017), which is split into 64 training classes (cf. Tab. 2), 16 validation classes (cf. Tab. 3), and 20 test classes (cf. Tab. 4). We will make a full list of image files available.
+
+As we do not require a validation set, we combine the training and validation set to form an extended training set. We extract 600 images per class from the ImageNet 2012 Challange dataset (Krizhevsky et al., 2012), scale the shorter side to 84 pixels and then centrally crop to $8 4 \times 8 4$ pixels, that is, we preserve the original aspect ratio of the image content. We use these coloured $8 4 \times 8 4 \times 3$ images as input for representational and $\mathbf { k }$ -shot learning and testing.
+
+In order to train very deep models, such as a ResNet, we need to perform data augmentation as is the case when training full ImageNet. We use the following standard data augmentation from ImageNet that we adapt to the size of the input images:
+
+• random horizontal flipping • randomly paste image into $1 0 0 \times 1 0 0$ frame and cut out central $8 4 \times 8 4$ pixels • randomly change brightness, contrast, saturation and lighting
+
+We highlight that we do not perform any data augmentation for the k-shot learning and $\mathbf { k }$ -shot testing but use the original $8 4 \times 8 4$ colour images as input to the feature extractor.
+
+# D.2 CIFAR-100
+
+CIFAR-100 consists of 100 classes each with 500 training and 100 test images of size $3 2 \times 3 2$ . The classes are grouped into 20 superclasses with 5 classes each. For example, the superclass ”fish” contains the classes aquarium fish, flatfish, ray, shark, and trout. Unless otherwise stated, we used a random split into 80 base classes and $2 0 \mathrm { k }$ -shot learning classes.
+
+For $\mathbf { k }$ -shot learning and testing, we split the 100 classes into 80 base classes used for network training and $2 0 \mathrm { k \Omega }$ -shot learning classes.
+
+classes_base $\begin{array} { r l } { \mathbf { \Psi } } & { { } = \mathbf { \Psi } \left[ \begin{array} { l } { \mathbf { \Psi } } \end{array} \right. } \end{array}$ 0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 15, 16, 17, 18, 19, 21, 22, 24, 25, 27, 28, 32, 34, 35, 36, 38, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99   
+]   
+classes_heldout $\begin{array} { r l r } { \mathrm { ~  ~ \psi ~ } } & { { } = } & { [ } \end{array}$ 8, 11, 12, 20, 23, 26, 29, 30, 31, 33, 37, 39, 41, 47, 57, 68, 71, 72, 81, 84   
+]
+
+We provide an exhaustive comparison of different probabilistic models for this $\mathbf { k }$ -shot learning task in Appendix E.4.
+
+# D.3 NETWORK ARCHITECTURE AND TRAINING: RESNET INSPIRED
+
+The network architecture is inspired by the ResNet-34 architecture for ImageNet (He et al., 2016) that uses convolution blocks, with two convolutions each, that are bridged by skip connections. As a base, we utilise the example code2 provided by tensorpack (https://github.com/ppwwyyxx/tensorpack), a neural network training library built on top of tensorflow (Mart´ın Abadi et al., 2015). We adapt the number of features as well as the size of the last fully connected layer to account for the smaller number of training samples and training classes. The final architecture is detailed in Tab. 5.
+
+n03400231 frying pan, frypan, skillet   
+n02108551 Tibetan mastiff   
+n02687172 aircraft carrier, carrier, flattop, attack aircraft carrier   
+n04296562 stage   
+n13133613 ear, spike, capitulum   
+n02165456 ladybug, ladybeetle, lady beetle, ladybird, ladybird beetle   
+n03337140 file, file cabinet, filing cabinet   
+n02966193 carousel, carrousel, merry-go-round, roundabout, whirligig   
+n02074367 dugong, Dugong dugon   
+n02105505 komondor   
+n04389033 tank, army tank, armored combat vehicle, armoured combat vehicle   
+n09246464 cliff, drop, drop-off   
+n03924679 photocopier   
+n03527444 holster   
+n04612504 yawl   
+n01749939 green mamba   
+n04251144 snorkel   
+n03347037 fire screen, fireguard   
+n04067472 reel   
+n03998194 prayer rug, prayer mat   
+n13054560 bolete   
+n02747177 ashcan, trash can, garbage can, wastebin, ash bin, ash-bin, ashbin, dustbin, trash barrel, trash bin   
+n04435653 tile roof   
+n02108089 boxer   
+n03908618 pencil box, pencil case   
+n01770081 harvestman, daddy longlegs, Phalangium opilio   
+n03676483 lipstick, lip rouge   
+n03220513 dome   
+n04515003 upright, upright piano   
+n04258138 solar dish, solar collector, solar furnace   
+n04509417 unicycle, monocycle   
+n01704323 triceratops   
+n04443257 tobacco shop, tobacconist shop, tobacconist   
+n02089867 Walker hound, Walker foxhound   
+n01910747 jellyfish   
+n02111277 Newfoundland, Newfoundland dog   
+n04243546 slot, one-armed bandit   
+n01558993 robin, American robin, Turdus migratorius   
+n03047690 clog, geta, patten, sabot   
+n03854065 organ, pipe organ   
+n03476684 hair slide   
+n02113712 miniature poodle   
+n07747607 orange   
+n03838899 oboe, hautboy, hautbois   
+n07584110 consomme   
+n02795169 barrel, cask   
+n03017168 chime, bell, gong   
+n04275548 spider web, spider’s web   
+n04604644 worm fence, snake fence, snake-rail fence, Virginia fence   
+n02606052 rock beauty, Holocanthus tricolor   
+n01843383 toucan   
+n02457408 three-toed sloth, ai, Bradypus tridactylus   
+n03062245 cocktail shaker   
+n03207743 dishrag, dishcloth   
+n02108915 French bulldog   
+n06794110 street sign   
+n02823428 beer bottle   
+n03888605 parallel bars, bars   
+n04596742 wok   
+n02091831 Saluki, gazelle hound   
+n02101006 Gordon setter   
+n02120079 Arctic fox, white fox, Alopex lagopus   
+n01532829 house finch, linnet, Carpodacus mexicanus   
+n07697537 hotdog, hot dog, red hot
+
+Table 2: Training classes for miniImageNet as proposed by Ravi & Larochelle (2017)
+
+n03075370 combination lock   
+n02971356 carton   
+n03980874 poncho   
+n02114548 white wolf, Arctic wolf, Canis lupus tundrarum   
+n03535780 horizontal bar, high bar   
+n03584254 iPod   
+n02981792 catamaran   
+n03417042 garbage truck, dustcart   
+n03770439 miniskirt, mini   
+n02091244 Ibizan hound, Ibizan Podenco   
+n02174001 rhinoceros beetle   
+n09256479 coral reef   
+n02950826 cannon   
+n01855672 goose   
+n02138441 meerkat, mierkat   
+n03773504 missiles   
+n02116738 African hunting dog, hyena dog, Cape hunting dog, Lycaon pictus   
+n02110063 malamute, malemute, Alaskan malamute   
+n02443484 black-footed ferret, ferret, Mustela nigripes   
+n03146219 cuirass   
+n03775546 mixing bowl   
+n03544143 hourglass   
+n04149813 scoreboard   
+n03127925 crate   
+n04418357 theater curtain, theatre curtain   
+n02099601 golden retriever   
+n02219486 ant, emmet, pismire   
+n03272010 electric guitar   
+n04146614 school bus   
+n02129165 lion, king of beasts, Panthera leo   
+n04522168 vase   
+n07613480 trifle   
+n02871525 bookshop, bookstore, bookstall   
+n01981276 king crab, Alaska crab, Alaskan king crab, Alaska king crab, Paralithodes camtschatica   
+n02110341 dalmatian, coach dog, carriage dog   
+n01930112 nematode, nematode worm, roundworm
+
+ResNet-34 inspired for miniImageNet   
+
+<table><tr><td>Output size</td><td colspan="2">Layers</td></tr><tr><td>84×84×3</td><td colspan="2">Input patch</td></tr><tr><td>42 × 42 × 32</td><td colspan="2">5 × 5,32, stride 2</td></tr><tr><td>42 × 42 × 32</td><td>[3 × 3,32] 3 × 3,32</td><td rowspan="2">×3</td></tr><tr><td>21 × 21×64</td><td>[3 × 3,64] ×4 3× 3,64</td></tr><tr><td>11 × 11 × 128</td><td>[3 × 3,128] 3 ×3,128</td><td rowspan="2">×6 ×3</td></tr><tr><td>6×6×256</td><td>[3 × 3,256] 3 ×3,256</td></tr><tr><td>256</td><td colspan="2"> global average pooling</td></tr><tr><td>C</td><td colspan="2">fully connected, softmax</td></tr></table>
+
+Table 5: Network architecture. All unnamed layers are 2D convolutions with stated kernel size and padding SAME; the output of the shaded layer corresponds to $\Phi _ { \varphi } ( \mathbf { u } )$ , the feature space representation of the image u, which is used as input for probabilistic $\mathbf { k }$ -shot learning.
+
+The network is trained using a decaying learning rate schedule and momentum SGD and is implemented in tensorpack using tensorflow.
+
+VGG-style Network for CIFAR-100   
+
+<table><tr><td>Output size</td><td>Layers</td></tr><tr><td>32 × 32×3</td><td>Input patch</td></tr><tr><td>16 ×16×64 8×8×64</td><td>2×(Conv2D,ELU),Pool 2×(Conv2D,ELU),Pool</td></tr><tr><td>4×4×128</td><td>2×(Conv2D,ELU),Pool</td></tr><tr><td>2×2×128</td><td>2×(Conv2D,ELU),Pool</td></tr><tr><td>2×2×128</td><td>Dropout (0.5)</td></tr><tr><td>256</td><td>FullyConnected, ELU</td></tr><tr><td>256</td><td>Dropout (0.5)</td></tr><tr><td>128</td><td>FullyConnected, ELU</td></tr><tr><td>C</td><td>FullyConnected, SoftMax</td></tr></table>
+
+D.4 NETWORK ARCHITECTURE AND TRAINING: VGG INSPIRED   
+VGG-style Network for miniImageNet   
+Table 6: Network architectures. All 2D convolutions have kernel size $3 \times 3$ and padding SAME; max-pooling is performed with stride 2. The output of the shaded layer corresponds to $\Phi _ { \varphi } ( u )$ , the feature space representation of the image $u$ , which is used as input for probabilistic k-shot learning   
+
+<table><tr><td>Output size</td><td>Layers</td></tr><tr><td>84×84×3 42 × 42 × 32 21 × 21×64</td><td>Input patch 2×(Conv2D,ELU),Pool 2×(Conv2D,ELU),Pool 11 × 11 × 128 2×(Conv2D,ELU),Pool</td></tr><tr><td>6×6×128 3×3×128 3×3×128</td><td>2×(Conv2D,ELU),Pool 2×(Conv2D,ELU),Pool Dropout (0.5)</td></tr><tr><td>512</td><td>FullyConnected, ELU</td></tr><tr><td>512 256</td><td>Dropout (0.5)</td></tr><tr><td>C</td><td>FullyConnected, ELU FullyConnected, SoftMax</td></tr></table>
+
+The network architecture was inspired by the VGG networks Simonyan & Zisserman, 2014, but does not employ batch normalisation Ioffe & Szegedy, 2015. To speed up training, we employ exponential linear units (ELUs), which have been reported to lead to faster convergence as compared to ordinary ReLUs Clevert et al., 2015. To regularise the networks, we employ dropout (Srivastava, Hinton, et al., 2014) and regularisation of the weights in the fully connected layers. The networks are trained with the ADAM optimiser Kingma & Ba, 2014 with decaying learning rate.
+
+The network is implemented in tensorpack using tensorflow.
+
+![](images/24aeedb39ec86a94aaff7d6f4c45a90304c6181dbeb70488554d9ac9c1e2d287.jpg)  
+Figure 6: t-SNE embedding of the CIFAR-100 weights $\widetilde { \mathrm { W } }$ trained using a VGG style architecture. The points are coloured according to their respective superclass. The colouring by superclass makes the structure in the weights evident, as t-SNE overall recovers the structure in the dataset. For instance, oak tree, palm tree, willow tree and pine tree form a cluster on the bottom right. This structure motivates our approach, as the training weights contain information which may be useful at $\mathbf { k }$ -shot time, for instance given a few example from chestnut trees.
+
+![](images/9a7bf6dd84c6f01fc75e5535055295f2dde23f3ffc3010ff2244284f61a71424.jpg)  
+Figure 7: t-SNE embedding of the miniImageNet weights trained using a ResNet-34 architecture. Structure is still present and we observe meaningful patterns, even though the classes in miniImageNet are more unique than in CIFAR-100. For instance, goose, house finch, toucan, Arctic fox, green mamba and other animals are clustered on the top, with birds close to each other. Examples of other small clusters include poncho and miniskirt, or organ and oboe. For readability, not all class names are plotted.
+
+# E EXTENDED EXPERIMENTS
+
+# E.1 T-SNE EMBEDDING OF THE WEIGHTS
+
+We provide t-SNE embeddings for the weights of a VGG network trained in CIFAR-100 and a ResNet34 trained on miniImageNet. A structure in the weights is apparent and provides motivation for our framework. The results can be seen in Fig. 6 and Fig. 7.
+
+# E.2 EXTENDED RESULTS ON miniIMAGENET
+
+Fig. 8 provides extended results on $\mathbf { k }$ -shot learning for the miniImageNet dataset for different network architectures. We investigate the influence of different feature extractors of increasing complexity and training data size on performance on: i) a VGG style network trained on 500 images per class, ii) a ResNet-34 trained on 500 examples per class, and iii) a ResNet-34 trained on all 600 examples per class.
+
+# E.3 CHOICE OF REGULARISATION CONSTANT
+
+Fig. 9 reports accuracy and calibration in terms of Expected Calibration Error (ECE) (lower is better) and log likelihoods (higher is better) for different regularisations of logistic regression for all three model architectures considered.
+
+![](images/2fa9dcedc3e2ad9bb81730a425916fb0df31a973f5aecfa9f8ba854d9fa434ad.jpg)  
+Figure 8: Extended results for the miniImageNet dataset utilising different network architectures and representational training. top: a ResNet-34 trained with all 600 examples per class; middle: a ResNet-34 trained with 500 images per class; bottom: a VGG style network trained with 500 images per class. We highlight that for all three architectures the order of the different methods as well as the main messages are the same. However, the general performance in terms of accuracy and calibration differ between the architectures. The more complex architecture trained on most images performs best in terms of accuracy, indicating that it learns better features for $\mathbf { k }$ -shot learning. Both ResNets behave very similarly on calibration whereas the VGG-style network performs better (lower ECE and higher log likelihood as well as more diagonal calibration curve). This is in line with observations by Guo et al. (2017) that calibration of deep architectures gets worse as depth and complexity increase.
+
+![](images/84f491fdb00b6cb7bec2df1c1ea54cc5e57068ca1a00f2d9ce7d626b5c50a6c3.jpg)  
+Figure 9: Choice of regularisation constant for logistic regression on k-shot learning. Note that all three rows use the same raw data that are only visualised differently. Top: Summary of accuracy and calibration in terms of log likelihood and Expected Calibration Error (ECE). Middle: detailed plot of ECE vs. accuracy. Bottom: detailed plot of log likelihood vs. accuracy. Results for $C _ { \mathrm { r e g } } = \overset { * } { 2 } \sigma _ { \widetilde { \mathrm { W } } } ^ { 2 }$ fare drawn as black triangles. Dashed lines correspond to logistic regression with cross-validated (changing) regularisation constant. Colour brightness of the markers ranges from dark $( C = 1 0 ^ { - 5 }$ ) to bright ( $C = 1 0 $ ). In addition to Fig. 4 we also provide results for calibration in terms of ECE (lower is better), which are consistent with log likelihoods (higher is better): The Bayesian inspired choice of the regularisation parameter strikes a good balance between accuracy and calibration and consistently outperforms cross-validated choice of the parameter.
+
+# E.4 MODEL ASSESSMENT IN CIFAR-100
+
+This section reports an extensive model comparison on CIFAR-100, both for the model of the weights $p ( \mathrm { W } | \widetilde { \mathrm { W } } )$ and for the inference procedure at $\mathbf { k }$ -shot time (MAP or Hybrid Monte Carlo (HMC) sampling using NUTS (Hoffman & Gelman, 2014), see the description of approximate inference algorithms in Appendix B). We report log-likelihood of the weights under different models, as well as accuracy, log-likelihood and calibration in a k-shot learning task. Tab. 7 and Tab. 8 show descriptions of the methods analysed for respectively phase 2 (concept learning) and phase 3 ( $\mathbf { k }$ -shot learning) of our $\mathbf { k }$ -shot pipeline described in Sec. 2.1.
+
+<table><tr><td rowspan="2">Method name</td><td colspan="2">Phase 2: Concept learning</td></tr><tr><td>Priordistribution</td><td>Inference</td></tr><tr><td>Gauss (iso)</td><td>Gaussian isotropic covariance</td><td>MAP</td></tr><tr><td>Gauss (MAP prior)</td><td>Gaussian isotropic covariance</td><td>MAP</td></tr><tr><td>Gauss (integr. prior)</td><td>Gaussian full covariance</td><td>Integrated</td></tr><tr><td>GMM (supercl.)</td><td>GMM on superclasses iso.cov.</td><td>MAP</td></tr><tr><td>GMM (3, iso)</td><td>GMM on 3 clusters iso. cov.</td><td>MLE</td></tr><tr><td>GMM (3, diag)</td><td>GMM on 3 clusters diagonal cov.</td><td>MLE</td></tr><tr><td>GMM (10, iso)</td><td>GMM on 10 clusters iso. cov.</td><td>MLE</td></tr><tr><td>Laplace (diag)</td><td>Laplace diagonal covariance</td><td>MLE</td></tr></table>
+
+Table 7: Description of the inference for the parameters of the prior in phase 2 (concept learning) for the models in from Fig. 10. This specifies the inference procedure for $\theta$ in $p ( \mathbf { w } \mid \boldsymbol { \theta } )$ after observing the training weights $\widetilde { \mathrm { W } }$ .
+
+<table><tr><td rowspan="2">Method name</td><td colspan="2">Phase 3: k-shot learning</td></tr><tr><td>Prior distribution</td><td>Inference</td></tr><tr><td>Gauss (iso) MAP</td><td>Gaussian</td><td>MAP</td></tr><tr><td>Gauss (MAP prior) MAP</td><td>Gaussian</td><td>MAP</td></tr><tr><td>Gauss (MAP prior) HMC</td><td>Gaussian</td><td>HMC</td></tr><tr><td>Gauss ( ：(integr. prior)MAP</td><td>Gaussian</td><td>MAP</td></tr><tr><td>Gauss (integr. prior) HMC</td><td>Gaussian</td><td>HMC</td></tr><tr><td>GMM (supercl.) MAP</td><td>GMM on superclasses</td><td>MAP</td></tr><tr><td>GMM (3,iso) MAP</td><td>GMM on 3 isotropic comp.</td><td>MAP</td></tr><tr><td>Laplace (diag) HMC</td><td>Laplace (diagonal)</td><td>HMC</td></tr><tr><td>Laplace (diag) MAP</td><td>Laplace (diagonal)</td><td>MAP</td></tr></table>
+
+Table 8: Methods and inference procedure during phase 3 $\mathbf { k }$ -shot learning) for the models used in Fig. 10. This specifies the inference procedure used when computing $p ( \mathrm { W } | \mathrm { \widehat { \mathcal { D } } } , \widetilde { \mathrm { W } } )$ for the specified prior distribution.
+
+In the main text, we only consider an isotropic Gaussian model with MAP inference since we do not observe benefits from using alternative methods in terms of $\mathbf { k }$ -shot performance and calibration. Moreover, while we report results on a VGG-like architecture, we could also use a ResNet architecture, and preliminary results point to the same conclusion as experiments on miniImageNet when switching from VGG to ResNet: the deeper features consistently lead to higher $\mathbf { k }$ -shot performance on all methods whereas the ordering of the methods stays roughly the same.
+
+Analysis of the models on held-out training weights. First, we analyse how well the different prior models for the new softmax weights are able to fit the $\widetilde { C }$ training weights $\widetilde { \mathrm { W } }$ . We randomly excluded 10 of those weights and evaluated their held-out negative log likelihood given the remaining $C - 1 0$ weights. We emphasise that this approach also constitutes a principled way to set the hyperparameters of the prior and, critically, relies on an explicit probabilistic model.
+
+The negative log probabilities are averaged over 50 random splits and results of best optimised values w.r.t. hyperparameters are shown in Tab. 9 for CIFAR-100 (lower is better). We find that all models
+
+![](images/953c6f23804e6219e6ece708a197cd141af454587feeceb14d0180c671a22efa.jpg)
+
+Table 9: Held-out log probabilities on random 70/10-splits of the training weights for the different models on CIFAR-100. Values are averaged over 50 splits.
+
+![](images/a8e6e0bdc72e26a75fe56a83c9f0f11fdb2663e07ddffdf549944f8e983d295f.jpg)  
+Figure 10: Results on CIFAR-100 for VGG style architecture. We report accuracy, log-likelihood and calibration for the methods and inference procedures presented in Tab. 8. With the exception of GMM (10, iso) and Laplace, all methods are similar terms of accuracy and log-likelihood. Gauss (integr. prior) HMC and Gauss (MAP) HMC are slightly better calibrated than our proposed Gauss (MAP) iso, but require significantly more computation for the sampling procedure.
+
+behave very similar but that multivariate Gaussian models generally outperform other models. We attribute the good performance of the simpler models to the small number of data points $( C - 1 0 = 7 0 $ training weights) and the high dimensionality of the space, which entail that fitting even simple models is difficult. Thus, more complicated models cannot improve over them.
+
+$\mathbf { k }$ -shot performance in CIFAR-100. Accuracies are measured on a 5-way classification task on the $\mathbf { k }$ -shot classes for $k \in \{ 1 , 5 , 1 0 \}$ . Results were averaged two-fold: (i) 20 random splits of the 5 $\mathbf { k }$ -shot classes; (ii) 10 repetitions of each split with different $\mathbf { k }$ -shot training examples. Among our models, no statistically significant difference in accuracy is observed, with the exception of Laplace MAP and GMM (iso), which consistently underperforms. These findings are consistent in terms of log-likelihoods, see the first and second plots in Fig. 10.
+
+Finally, our methods are generally well calibrated, with Gaussian models generally better than Laplace models. Moreover, all methods (with the exception of Laplace and GMM (10, iso) have low ECE and high accuracy, see the third and fourth plots of Fig. 10. While Gauss (integr. prior) HMC and Gauss (MAP) HMC are sightly better calibrated than our proposed method in the main paper, Gauss (MAP) iso, we believe the gain in calibration is not worth the significant increase in computational resources needed for the sampling procedure. Interestingly, both GMM approaches are not able to outperform the other, simpler models. This is in line with the previous observation that the simpler models are better able to explain the weights. Again, we attribute this inability of mixture models to use their larger expressivity/capacity to the small number of data points and the high-dimensionality of weight-space which means learning even simple models is difficult. These observations suggest that the use of mixture models in this type of $\mathbf { k }$ -shot learning framework is not beneficial and is in contrast to the approach of Srivastava & Salakhutdinov (2013), who employ a tree-structured mixture model. The authors show compare a model in which the assignments to the superclasses in the tree are optimized over against a model with a naive initialisation of the superclass assignments, and show that the first outperforms the second. However, they do not compare against a simpler baseline, e.g., a single Gaussian model.
+
+Overall, we observe that there is no significant benefit of more complex methods over the simple isotropic Gaussian, either in terms of accuracy, log-likelihood or calibration. Thus, our recommendation is that practitioners should use simple models and employ simple inference schemes to estimate all free parameters thereby avoiding expending valuable data on validation sets
+
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+Jake Snell, Kevin Swersky, & Richard Zemel (2017). Prototypical Networks for Few-shot Learning. arXiv e-print: 1703.05175.
+
+Nitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, & Ruslan Salakhutdinov (2014). Dropout: a simple way to prevent neural networks from overfitting. Journal of machine learning research 15.1, pp. 1929–1958.
+
+Nitish Srivastava & Ruslan R Salakhutdinov (2013). Discriminative transfer learning with tree-based priors. In: Advances in Neural Information Processing Systems, pp. 2094–2102.

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+# MEMORY-BASED PARAMETER ADAPTATION
+
+Pablo Sprechmann\*, Siddhant M. Jayakumar\*, Jack W. Rae, Alexander Pritzel   
+Adria Puigdom \` enech Badia, Benigno Uria, Oriol Vinyals \`   
+Demis Hassabis, Razvan Pascanu, Charles Blundell   
+DeepMind   
+London, UK   
+{psprechmann, sidmj, jwrae, apritzel,   
+adriap, buria, vinyals,   
+dhcontact, razp, cblundell}@google.com
+
+# ABSTRACT
+
+Deep neural networks have excelled on a wide range of problems, from vision to language and game playing. Neural networks very gradually incorporate information into weights as they process data, requiring very low learning rates. If the training distribution shifts, the network is slow to adapt, and when it does adapt, it typically performs badly on the training distribution before the shift. Our method, Memory-based Parameter Adaptation, stores examples in memory and then uses a context-based lookup to directly modify the weights of a neural network. Much higher learning rates can be used for this local adaptation, reneging the need for many iterations over similar data before good predictions can be made. As our method is memory-based, it alleviates several shortcomings of neural networks, such as catastrophic forgetting, fast, stable acquisition of new knowledge, learning with an imbalanced class labels, and fast learning during evaluation. We demonstrate this on a range of supervised tasks: large-scale image classification and language modelling.
+
+# 1 INTRODUCTION
+
+Neural networks have been proven to be powerful function approximators, as shown in a long list of successful applications: image classification (e.g. Krizhevsky et al., 2012), audio processing (e.g. Oord et al., 2016), game playing (e.g. Mnih et al., 2015; Silver et al., 2017), and machine translation (e.g. Wu et al., 2016). Typically these applications apply batch training to large or near-infinite data sets, requiring many iterations to obtain satisfactory performance.
+
+Humans and animals are able to incorporate new knowledge quickly from single examples, continually throughout much of their lifetime. In contrast, neural network-based models rely on the data distribution being stationary and the training procedure using low learning rates and many passes through the training data to obtain good generalisation. This limits their application to life-long learning or dynamic environments and tasks.
+
+Problems in continual learning with neural networks commonly manifest as the phenomenon of catastrophic forgetting (McCloskey & Cohen, 1989; French, 1999): a neural network performs badly on old tasks having been trained to perform well on a new task. Several recent approaches have proven promising at overcoming this, such as elastic weight consolidation (Kirkpatrick et al., 2017). Recent work in language modelling has demonstrated how popular neural language models may appropriately be adapted to take advantage of rare, recently seen words, as in the neural cache (Grave et al., 2016), pointer sentinel networks (Merity et al., 2016) and learning to remember rare events (Kaiser et al., 2017). Our work generalises these approaches and we present experimental results where we apply our model to both continual or incremental learning tasks, as well as language modelling.
+
+We propose Memory-based Parameter Adaptation (MbPA), a method for augmenting neural networks with an episodic memory to allow for rapid acquisition of new knowledge while preserving the high performance and good generalisation of standard deep models. It combines desirable properties of many existing few-shot, continual learning and language models. We draw inspiration from the theory of complementary learning systems (CLS: McClelland et al., 1995; Leibo et al., 2015; Kumaran et al., 2016), where effective continual, life-long learning necessitates two complementary systems: one that allows for the gradual acquisition of structured knowledge, and another that allows rapid learning of the specifics of individual experiences. As such, MbPA consists of two components: a parametric component (a standard neural network) and a non-parametric component (a neural network augmented with a memory containing previous problem instances). The parametric component learns slowly but generalises well, whereas the non-parametric component rapidly adapts the weights of the parametric component. The non-parametric, instance-based adaptation of the weights is local, in the sense the modification is directly dictated by the inputs to the parametric component. The local adaptation is discarded after the model produces its output, avoiding long term consequences of strong local adaptation (such as overfitting), allowing the weights of the parametric model to learn slowly leading to strong performance and generalisation.
+
+![](images/bf5b55954aebbcf68a9dff8d751a4460e5d644785664fd22a45c2c3cf7131728.jpg)  
+Figure 1: Architecture for the MbPA model. Left: Training usage. The parametric network is used directly and experiences are stored in the memory. Right: Testing setting. The embedding is used to query the episodic memory, the retrieved context is used to adapt the parameters of the output network.
+
+The contributions of our work are: $( i )$ proposing an architecture for enhancing powerful parametric models with a fast adaptation mechanism to efficiently cope with changes in the task at hand; (ii) establish connections between our method and attention mechanisms frequently used for querying memories; (iii) present a Bayesian interpretation of the method allowing a principled form of regularisation; $( i \nu )$ evaluating the method on a range of different tasks: continual learning, incremental learning and data distribution shifts, obtaining promising results.
+
+# 2 MODEL-BASED PARAMETER ADAPTATION
+
+Our models consist of three components: an embedding network, $f _ { \gamma }$ , a memory $M$ and an output network $g _ { \theta }$ . The embedding network, $f _ { \gamma }$ , and the output network, $g _ { \theta }$ , are standard parametric (feed forward or recurrent) neural networks with parameters $\gamma$ and $\theta$ , respectively. The memory $M$ is a dynamically-sized memory module that stores key and value pairs, $M = \{ ( h _ { i } , v _ { i } ) \}$ . Keys $\{ h _ { i } \}$ are given by the embedding network. The values $\dot { \{ { v } _ { i } \} }$ correspond to the desired output $y _ { i }$ . For classification, $y _ { i }$ would simply be the true class label, whereas for regression, $y _ { i }$ would be the true regression target. Hence, upon observing the $j$ -th example, we append the pair $( h _ { j } , v _ { j } )$ to the memory $M$ , where:
+
+$$
+\begin{array} { l } { h _ { j }  f _ { \gamma } ( x _ { j } ) , } \\ { v _ { j }  y _ { j } . } \end{array}
+$$
+
+The memory has a fixed size and acts as a circular buffer: when it is full, the oldest data is overwritten first. Retrieval from the memory $M$ uses $K$ -nearest neighbour search on the keys $\{ h _ { i } \}$ with Euclidean distance to obtain the $K$ most similar keys and associated values.
+
+Our model is used differently in the training and testing phases. During training, for a given input $x$ , we parametrise the conditional likelihood with a deep neural network given by the composition
+
+# Algorithm 1 Model-based Parameter Adaptation
+
+<table><tr><td>Aigoritnm1Model-basedParameterAdaptation procedure MBPA-TRAIN</td></tr><tr><td>Sample mini-batch of training examples B = {(xb, yb)}b from training data.</td></tr><tr><td>Calculate the embedded mini-batch B&#x27;= {(fγ(xb),yb) : xb,yb ∈B}.</td></tr><tr><td>Update 0,γ by maximising the likelihood (1) of θ and γ with respect to mini-batch B</td></tr><tr><td>Add the embedded mini-batch examples B&#x27; to memory M: M ← MU B&#x27;.</td></tr><tr><td>procedure MBPA-TEST(test input: x, output prediction: y)</td></tr><tr><td>Calculate embedding q = fγ(x),and △total ← 0.</td></tr><tr><td>(x） ，(x) )k=1</td></tr><tr><td>for each step of MbPA do</td></tr><tr><td>Calculate △m(x,θ +△total) according to (4)</td></tr><tr><td>△total ←△total +△M(x).</td></tr><tr><td>Output prediction y = g0+△tota (h)</td></tr></table>
+
+of the embedding and output networks. Namely,
+
+$$
+p _ { \mathrm { t r a i n } } ( y | x , \gamma , \theta ) = g _ { \theta } ( f _ { \gamma } ( x ) ) .
+$$
+
+In the case of classification, the last layer of $g _ { \theta }$ is a softmax layer. The parameters $\{ \theta , \gamma \}$ are estimated by maximum likelihood estimation. The memory is updated with new entries, as they are seen, however no local adaptation is performed on the model. Figure 1 (left) shows a diagram of the training setting and Algorithm 1 (MbPA-Train) shows the algorithm for updating MbPA during training.
+
+On the other hand, at test time, it temporarily adapts the parameters of the output network based upon the current input and the contents of the memory $M$ . That is, it uses the exact same parametrisation as (1), but with a different set of parameters in the output network.
+
+Let the context $C$ of an input $x$ be the keys, values and associated weights of the $K$ nearest neighbours to query $q \ = \ f _ { \gamma } ( x )$ in the memory $M$ : $C = \{ ( h _ { k } ^ { ( x ) } , v _ { k } ^ { ( x ) } , \bar { w } _ { k } ^ { ( x ) } ) \} _ { k = 1 } ^ { K }$ The coefficients $w _ { k } ^ { ( x ) } \propto \ker ( h _ { k } ^ { ( x ) } , q )$ are weightings of each of the retrieved neighbours according to their closeness to the query $f _ { \gamma } ( { \boldsymbol { x } } ) . \operatorname { k e r n } ( h , q )$ is a kernel function which, following (Pritzel et al., 2017), we choose as $\begin{array} { r } { \ker ( h , q ) = \frac { 1 } { \epsilon + \lVert h - q \rVert _ { 2 } ^ { 2 } } } \end{array}$ . The parametrisation of the likelihood takes the form,
+
+$$
+\begin{array} { r } { p ( y | x , \theta ^ { x } ) = p ( y | x , \theta ^ { x } , C ) = g _ { \theta ^ { x } } ( f _ { \gamma } ( x ) ) , } \end{array}
+$$
+
+as opposed to the standard parametric approach $g _ { \theta } ( f _ { \gamma } ( x ) )$ , where $\theta ^ { x } \ = \ \theta + \Delta _ { M } ( x , \theta )$ with $\Delta _ { M } \bar { ( \boldsymbol { x } , \boldsymbol { \theta } ) }$ being a contextual (it is based upon the input $x$ ) update of the parameters of the output network. The MbPA adaptation corresponds to decreasing the weighted average negative loglikelihood over the retrieved neighbours in $C$ . Figure 1 (right) shows a diagram of the testing setting and Algorithm 1(MbPA-Test) shows the algorithm for using MbPA during testing.
+
+An interesting property of the model is that the correction $\Delta _ { M } ( x , \theta )$ is such that, as the parametric model becomes better at fitting the training data (and consequently the episodic memories), it selfregulates and diminishes. In the CLS theory, this process is referred to as consolidation, when the parametric model can reliably perform predictions without relying on episodic memories.
+
+# 2.1 MAXIMUM A POSTERIORI INTERPRETATION OF MBPA
+
+We can now derive $\Delta _ { M } ( x , \theta )$ , motivated by considering the posterior distribution on the parameters $\theta ^ { x }$ . Let $x$ correspond to the input with context $C = \{ h _ { k } , v _ { k } , w _ { k } ^ { ( x ) } \} _ { k = 1 } ^ { K }$ w(x)k }Kk=1. The maximum a posteriori over the context $C$ , given the parameters obtained after training $\theta$ , can be written as:
+
+$$
+\operatorname* { m a x } _ { \theta ^ { x } } \log p ( \theta ^ { x } | \theta ) + \sum _ { k = 1 } ^ { K } w _ { k } ^ { ( x ) } \log p ( v _ { k } ^ { ( x ) } | h _ { k } ^ { ( x ) } , \theta ^ { x } , x ) ,
+$$
+
+where the second term is a weighted likelihood of the data in C and log p(θx|θ) ∝ − ||θx−θ||222αM ( i.e. a Gaussian prior on $\theta ^ { x }$ centred at $\theta$ ) can be thought as a regularisation term that prevents overfitting. See Appendix $\mathrm { D }$ for details of this derivation.
+
+![](images/49827038181af68573ad0a8242f56be9014f4ca296954e0177473e0a3499cf4b.jpg)  
+Figure 2: Illustrative diagram of the local fitting on a regression task. Given a query (blue), we retrieve the context from memory showed in red.
+
+Equation (3) does not have a closed form solution, and requires fitting a large number of parameters at inference time. This can be costly and susceptible to overfitting. We can avoid this problem by adapting the reference parameters $\theta$ . Specifically, we perform a fixed number of gradient descent steps to minimise (3). One step of gradient descent to the loss in (3) with respect to $\theta ^ { x }$ yields
+
+$$
+{ \Delta } _ { M } ( x , \theta ) = - \alpha _ { M } \left. \nabla _ { \theta } \sum _ { k = 1 } ^ { K } w _ { k } ^ { ( x ) } \log p ( v _ { k } ^ { ( x ) } | h _ { k } ^ { ( x ) } , \theta ^ { x } , x ) \right| _ { \theta } - \beta ( \theta - \theta ^ { x } ) ,
+$$
+
+where $\beta$ is a scalar hyper-parameter. These adapted parameters are used for output computation but discarded thereafter, as described in Algorithm 1.
+
+# 2.2 FROM ATTENTION TO LOCAL FITTING
+
+A standard formulation of memory augmented networks is in the form of attention (Bahdanau et al., 2014), i.e. query memory to use a weighted average based on some similarity metric.
+
+We can now show that an attention-based procedure is a particular case of local adaptation or MbPA. The details of this are discussed in Appendix E. Effectively, attention can be viewed as fitting a constant function the neighbourhood of memories, whereas MbPA generalises to fit a function parameterised by the output network of our model.
+
+The diagram in Figure 2 illustrates an example in a regression task for simplicity. Given a query (blue), the retrieved memories and their corresponding values are depicted in red. The predictions of an attention based model are shown in orange. We can see that the prediction is biased towards the value of the neighbours with higher functional value. In magenta we represent the predictions made by the model $g _ { \theta }$ . We can see that it is not able to explain all memories equally well. This could be either because the problem is too difficult, poor training, or because the a prediction needs to be made while assimilating new information. The green curve show the prediction obtained after adapting the parameters to better explain the episodic memories.
+
+# 3 RELATED WORK
+
+A key component of MbPA is the non-parametric, episodic memory. Many recent works have looked at augmenting neural network systems with memories to allow for fast adaptation or incorporation of new knowledge. Variants of this architecture have been successfully used in the context of classification (Vinyals et al., 2016; Santoro et al., 2016; Kaiser et al., 2017), language modelling (Merity et al., 2016; Grave et al., 2016), reinforcement learning (Blundell et al., 2016; Pritzel et al., 2017), machine translation (Bahdanau et al., 2014), and question answering (Weston et al., 2014), to name a few. For the MbPA experiments below, we use a memory architecture similar to the Differentiable Neural Dictionary (DND) used in Neural Episodic Control (NEC) (Pritzel et al., 2017). One key difference is that we do not train the embedding network through the gradients from the memories (as they are not used at training time).
+
+While many of these approaches share a contextual memory lookup system, MbPA is distinct in the method by which the memories are used. Matching Networks (Vinyals et al., 2016) use a nonparametric network to map from a few examples to a target class via a kernel weighted average. Prototypical Networks (Snell et al., 2017) extend this and use a linear model instead of a nearest neighbour method.
+
+MbPA is further related to meta-learning approaches for few shot learning. In the context of learning invariant representations for object recognition, Anselmi et al. (2014) proposed a method that can invariantly and discriminatively represent objects using a single sample, even of a new class. In their method, instead of training via gradient descent, image templates are stored in the weights of simple-complex cell networks while objects undergo transformations. Optimisation as a model of few shot learning (Ravi & Larochelle, 2016) proposes using a meta-learner LSTM to control the gradient updates of another network, while Model-Agnostic Meta-Learning (MAML Finn et al. (2017)) proposes a way of doing meta-learning over a distribution of tasks. These methods extend the classic fine-tuning technique used in domain adaptation type of ideas (e.g. fit a given neural network to a small set of new data). The MAML algorithm (particularly related to our work) aims at learning an easily adaptable set of weights, such that given a small amount of training data for a given task following the training distribution, the fine-tuning procedure would effectively adapt the weights to this particular task. Their work does not use any memory or per-example adaptation and is not based on a continual (life-long) learning setting. In contrast, our work, aims at augmenting a powerful neural network with a fine-tuning procedure that is used at inference only. The idea is to enhance the performance of the parametric model while maintaining its full training.
+
+Recent approaches to addressing the continual learning problem have included elastic weight consolidation (Kirkpatrick et al., 2017), where a penalty term is added to the loss for deviations far from previous weights, and learning without forgetting (Li & Hoiem, 2016; Furlanello et al., 2016), where distillation (Hinton et al., 2015) from previously trained models is used to keep old knowledge available. Gradient Episodic Memory for Continual Learning (Lopez-Paz & Ranzato, 2017) attempts to solve the problem by storing data from previous tasks and taking gradient updates when learning new tasks that do not increase the training loss on examples stored in memory.
+
+There has been recent work in applying attention to quickly adapt a subset of fast weights (Ba et al., 2016). A number of recent works in language modelling have augmented prediction with attention over recent examples to account for the distributional shift between training and testing settings. Works in this direction include neural cache (Grave et al., 2016) and pointer sentinel networks (Merity et al., 2016). Learning to remember rare events (Kaiser et al., 2017) augments an LSTM with a key-value memory structure, and meta networks (Munkhdalai & Yu, 2017) combines fast weights with regular weights. Our model shares this flavour of attention and fast weights, while providing a model agnostic memory-based method that applies beyond language modelling.
+
+Works in the context of machine translation relate to MbPA. Gu et al. (2017) explore how to incorporate information from memory into the final model predictions. The authors find that shallow mixing works best. We show in this paper that MbPA is another competitive strategy to shallow mixing, and often working better (PTB for language modelling, ImageNet for image classification). The work by Li et al. (2016) shares the focus on fast-adaptation during inference with our work. Given a test example, the translation model is fine-tuned by fitting similar sentences from the training set. MbPA can be viewed as a generalisation of such approach: it relies on an episodic memory (rather than the training set), contextual lookup and similarity based weighting scheme to fine-tune the original model. Collectively, these allow MbPA to be a powerful domain-agnostic algorithm, which allows it to handle continual and incremental learning.
+
+Finally, we mention that our work is closely related to the local regression and adaptive coefficient models literature, see Loader (2006) and references therein. Locally adaptive methods achieved relatively modest success in high-dimensional classification problems, as fitting many parameters to a few neighbours often leads to over fitting. We attempt to counter this with contextual lookups and a local modification of only a subset of model parameters.
+
+# 4 EXPERIMENTS AND RESULTS
+
+Our scheme unifies elements from traditional approaches to continual, one-shot, and incremental or life-long learning. Models that solve these problems must have certain fundamental attributes in common: the ability to negate the effects of catastrophic forgetting, unbalanced and scarce data, while displaying rapid acquisition of knowledge and good generalisation.
+
+In essence, these problems require the ability to deal with changes and shifts in data distributions. We demonstrate that MbPA provides a way to address this. More concretely, due to the robustness of the local adaptation, the model can deal with shifts in domain distribution (e.g. train vs test set in language), the task label set (e.g. incremental learning) or sequential distributional shifts (e.g. continual learning). Further, MbPA is agnostic to both task domain (e.g. image or language) and choice of underlying parametric model, e.g. convolutional neural networks (LeCun et al., 1998) or LSTM (Hochreiter & Schmidhuber, 1997).
+
+To this end, our experiments focus on displaying the advantages of MbPA on widely used tasks and datasets, comparing with competing deep learning methods and baselines. We start by looking at the continual learning framework, followed by incremental learning, the problems of unbalanced data and test time distributional changes.
+
+# 4.1 CONTINUAL LEARNING: SEQUENTIAL DISTRIBUTIONAL SHIFT
+
+In this set of experiments, we explored the effects of MbPA on continual learning, i.e. when dealing with the problem of sequentially learning multiple tasks without the ability to revisit a task.
+
+We considered the permuted MNIST setup (Goodfellow et al., 2013). In this setting, each task was given by a different random permutation of the pixels of the MNIST dataset. We explored a chaining of 20 different tasks (20 different permutations) trained sequentially. The model was tested on all tasks it had been trained on thus far.
+
+We trained all models using 10,000 examples per task, comparing to elastic weight consolidation (EWC; Kirkpatrick et al., 2017) and regular gradient descent training. In all cases we rely on a two layer MLP and use Adam (Kingma & Ba, 2014) as the optimiser. The EWC penalty cost was chosen using a grid search, as was the local MbPA learning rate (between 0.0 and 1.0) and number of optimisation steps for MbPA (between 1 and 20).
+
+Figure 3 compares our approach with that of the baselines. For this particular task we worked directly on pixels as our embedding, i.e. $f _ { \gamma }$ is the identity function, and explored regimes where the episodic memory is small. A key takeaway of this experiment is that once a task is catastrophically forgotten, we find that only a few gradient steps on carefully selected data from memory are sufficient to recover performance, as MbPA does. Considering the number of updates required to reach the solution from random initialisation, this fact itself might seem surprising. MbPA provides a principled and effective way of performing these updates. The naive approach of performing updates on memories chosen at random from the entire memory is considerably less useful.
+
+We outperformed the MLP, and were superior to EWC for all but one memory size (when storing only a 100 examples per task). Further, the performance of our model grew with the number of examples stored, ceteris paribus. Crucially, our memory requirements are much lower than that of EWC, which requires storing model parameters and Fisher matrices for all tasks seen so far. Unlike EWC we do not store any tasks identifiers, merely appending the memory with a few examples. Further, MbPA does not use knowledge of exact task boundaries or identities of tasks switched to, unlike EWC and other methods. This allows for frequent switches that would otherwise hamper the Fisher calculations needed for models like EWC.
+
+Our method can be combined with any other algorithm such as standard replay from the memory buffer or EWC, providing further improvement. In Figure 3 (right) we combine MbPA and EWC.
+
+4.2 INCREMENTAL LEARNING: SHIFTS IN TASK LABEL DISTRIBUTIONS
+
+The goal of this section was to evaluate the model in the context of incremental learning. We considered a classification scenario where a model pre-trained on a subset of classes, was introduced to novel, previously unseen classes. The aim was to incorporate the new related knowledge, as quickly as possible, while preserving knowledge from the previous set. This was as opposed to the continual learning problem where there are distinct tasks without the ability to revisit old data.
+
+![](images/2cbcd709adb66ca8bb71574fd1fa498a3aa36147913e630b30df692412fe95b6.jpg)  
+Figure 3: (Left) Results on Permuted MNIST comparing baselines with MbPA using different memory sizes. (Right) Results augmenting MbPA with EWC, showing the flexibility and complementarity of MbPA.
+
+Specifically we considered the problem of image classification on the ImageNet dataset (Russakovsky et al., 2015). As a parametric model we used a ResnetV1 model (He et al., 2016). This was pre-trained on a random subset of the ImageNet dataset containing half of the classes. We then presented all 1000 classes and evaluated how quickly the network can acquire this knowledge (i.e. perform well across all 1000 classes).
+
+For MbPA, we used the penultimate layer of the network as the embedding network $f _ { \gamma }$ , forming the key $h$ and query $q$ for our episodic memory $M$ . The last fully connected layer was used to initialise the parametric model $g _ { \theta }$ . MbPA was applied at test time, using RMSprop with a local learning rate $\alpha _ { M }$ and the number of optimisation steps (as in Algorithm 1) tuned as hyper-parameters.
+
+A natural baseline was to simply fine-tune the last layer of the parametric model with the new training set. We also evaluated a mixture model, combining the classifications of the parametric model and the non-parametric model at decision level in the following manner:
+
+$$
+p ( y | q ) = \lambda p _ { \mathrm { p a r a m } } ( y | q ) + ( 1 - \lambda ) p _ { \mathrm { m e m } } ( y | q ) ,
+$$
+
+where the parameter $\lambda \in [ 0 , 1 ]$ controls the contribution of each model (this model was proposed by Grave et al. (2016) in the context of language modelling). We created five random splits in new and old classes. Hyperparameters were tuned for all models using the first split and the validation set, and we report the average performance on the remaining splits evaluated on the test set.
+
+Figure 4 shows the test set performance for all models, split by new and old classes.
+
+While the mixture model provides a large improvement over the plain parametric model, MbPA significantly outperforms both of them both in speed and performance. This is particularly noticeable in the new classes, where MbPA acquires knowledge from very few examples. Table 1 shows a quantitative analysis of these observations. After around 30 epoches the parametric model matches the performance of MbPA. In the appendix we explore sensitivity of MbPA on this task to various hyperparameters (memory size, learning rate).
+
+# 4.2.1 UNBALANCED DATASETS
+
+We further explored the incremental introduction of new classes, specifically in the context of unbalanced datasets. Most real world data are unbalanced, whereas standard datasets (like ImageNet) are artificially balanced to play well with deep learning methods.
+
+We replicated the setting from the ImageNet experiments in the previous section, where new classes were introduced to a pre-trained model. However, we only showed a tenth of the data for half the new classes and all data for the other half. We report performance on the full balanced validation set. Once again, we compared the parametric model with MbPA and a memory based mixture model.
+
+Results are summarised in Figure 5 (left). After 20 epochs of training, MbPA outperformed both baselines, with a wider gap in performance than the previous experiment. Further, the mixture model, though equipped with memory, did significantly worse than MbPA, leading us to conclude that the inductive bias in the local adaptation process was well suited to deal with data scarcity.
+
+Table 1: Quantitative evaluation of the learning dynamics for the Imagenet experiment. We compare a parametric model, non-parametric model (prediction based on memory only (9)), a mixture model and MbPA. We report the top 1 accuracy as well as the area under the curve (AUC) at different points in training.   
+
+<table><tr><td rowspan="2">Subset</td><td rowspan="2">Model</td><td colspan="3">Top 1 (at epochs)</td><td colspan="3">AUC (at epochs)</td></tr><tr><td>0.1</td><td>1</td><td>3</td><td>0.1</td><td>1</td><td>3</td></tr><tr><td rowspan="4">Novel</td><td>MbPA</td><td>46.2 %</td><td>64.5 %</td><td>65.7 %</td><td>27.4 %</td><td>57.7 %</td><td>63.0 %</td></tr><tr><td>Non-Parametric</td><td>40.0 %</td><td>53.3 %</td><td>52.9 %</td><td>28.3 %</td><td>47.9 %</td><td>51.8 %</td></tr><tr><td>Mixture</td><td>31.6 %</td><td>56.0 %</td><td>59.1 %</td><td>18.6 %</td><td>47.4 %</td><td>54.7 %</td></tr><tr><td>Parametric</td><td>16.2 %</td><td>53.6 %</td><td>57.9 %</td><td>5.7 %</td><td>41.7 %</td><td>51.9 %</td></tr><tr><td rowspan="4">Pre Trained</td><td>MbPA</td><td>68.5 %</td><td>70.9 %</td><td>70.9 %</td><td>71.4 %</td><td>70.3 %</td><td>70.3 %</td></tr><tr><td>Non-Parametric</td><td>62.7 %</td><td>69.4 %</td><td>70.0 %</td><td>45.9 %</td><td>65.8 %</td><td>68.7 %</td></tr><tr><td>Mixture</td><td>71.9 %</td><td>70.3 %</td><td>70.2 %</td><td>74.8 %</td><td>70.6 %</td><td>70.1 %</td></tr><tr><td>Parametric</td><td>71.4 %</td><td>68.1 %</td><td>68.8 %</td><td>76.0 %</td><td>68.6 %</td><td>68.3 %</td></tr></table>
+
+![](images/c5cc3b65b2160373872d43edecda7c4405134c93ce04e390e100a6f22e2a9201.jpg)  
+Figure 4: The figure compares the performance of MbPA (blue) against two baselines: the parametric model (green) and the mixture of experts (red). (Left) Aggregated performance (Right) disentangled performance evaluated on new (dashed) and old (solid) classes.
+
+# 4.3 LANGUAGE MODELLING: DOMAIN SHIFTS
+
+Finally we considered how MbPA can be used at test time to further improve the performance of language modelling. Given the general formulation of MbPA, this could be applied to any problem where there is a shift in distribution at test time — we focus on language modelling, where using recent information has proved promising, such as neural cache and dynamic evaluation (Grave et al., 2016; Krause et al., 2017).
+
+We considered two datasets with established performance benchmarks, Penn Treebank (PTB; Marcus et al., 1993) and WikiText-2 (Merity et al., 2016). We pre-trained an LSTM and apply MbPA to the weights and biases of the output softmax layer. The memory stores the past LSTM outputs and associated class labels observed during evaluation. Full model details and hyper-parameters are detailed in Appendix B.
+
+Penn Treebank is a small text corpus containing 887,521 train tokens, 70,390 validation tokens, and 78,669 test tokens; with a vocabulary size of 10,000. The LSTM obtained a test perplexity of 59.6 and this dropped by 4.3 points when interpolated with the neural cache. When we interpolated an LSTM with MbPA we were able to improve on the LSTM baseline by 5.3 points (an additional one from the cache model). We also attempted a dynamic evaluation scheme in a similar style to Krause et al. (2017), where we loaded the Adam optimisation parameters obtained during training and evaluated with training of the LSTM enabled, using a BPTT window of 5 steps. However we did not manage to obtain gains above 1 perplexity from baseline, and so we did not try it for WikiText-2.
+
+WikiText-2 is a larger text corpus than PTB, derived from Wikipedia articles. It contains 2,088,628 train tokens, 217,646 validation tokens, and 245,569 test tokens, with a vocabulary of 33,278. Our LSTM baseline obtained a test perplexity of 65.9, and this is improved by 14.6 points when mixed with a neural cache. Combining the baseline LSTM with an LSTM fit with MbPA we see a drop of 9.9 points, however the combination of all three models (LSTM baseline $+ \mathrm { M b P A } +$ cache) produced
+
+![](images/c41b5c92fb5bee6179520c30df8f051177432b71df5f70a6ac505a33db4aacce.jpg)  
+Figure 5: (Left) MbPA outperformed both parametric and memory-based mixture baselines, in the presence of unbalanced data on previously unseen classes (dashed lines). (Right) Example of MbPA. Query (shown larger in the top-right corner) of class “TV” and neighbourhood (all other images) for a specific case. Mixture and parametric models fail to classify the image while MbPA succeeds. 8 different classes in the closest 20 neighbours (e.g. “desktop computer”, “monitor”, “CRT screen”). Accuracy went from $2 5 \%$ to $7 5 \%$ after local adaptation.
+
+the largest drop of 15.9 points. Comparing the perplexity word-by-word between LSTM $^ +$ cache and LSTM $^ +$ cache $^ +$ MbPA, we see that MbPA improves predictions for rarer words (Figure 8).   
+
+<table><tr><td></td><td>Valid</td><td>PTB Test △Test</td><td></td><td>WikiText-2 Valid Test</td><td></td><td>△Test</td></tr><tr><td>CharCNN (Zhang et al., 2015) Variational LSTM (Aharoni et al., 2017) LSTM + cache (Grave et al., 2016) LSTM (Melis et al., 2017) AWD-LSTM (Merity et al., 2017) AWD-LSTM + cache (Merity et al., 2017) AWD-LSTM (reprod.) (Krause et al., 2017) AWD-LSTM+ dyn eval (Krause et al., 2017)</td><td>74.6 60.9 60.0 53.9 59.8 51.6</td><td>78.9 61.7 72.1 58.3 57.3 52.8 57.7 51.1</td><td>- 4.5 - 6.6</td><td>72.1 69.1 68.6 53.8 68.9 46.4</td><td>68.9 65.9 65.8 52.0 66.1 44.3</td><td>- 13.8 - 21.8</td></tr><tr><td>LSTM (ours) LSTM + cache (ours) LSTM+MbPA LSTM+MbPA+cache</td><td>61.8 55.7 54.8 54.8</td><td>59.6 55.3 54.3 54.4</td><td>-4.3 -5.3 -5.2</td><td>69.3 53.2 58.4 51.8</td><td>65.9 51.3 56.0 49.4</td><td>-14.6 -9.9 -16.5</td></tr></table>
+
+Table 2: Table with PTB and WikiText-2 perplexities. $\Delta$ Test denotes improvement of model on the test set relative to the corresponding baseline.
+
+# 5 CONCLUSION
+
+We have described Memory-based Parameter Adaptation (MbPA), a scheme for using an episodic memory structure to locally adapt the parameters of a neural network based upon the current input context. MbPA works well on a wide range of supervised learning tasks in several incremental, lifelong learning settings: image classification, language modelling. Our experiments show that MbPA improves performance in continual learning experiments, comparable to or in many cases exceeding the performance of EWC. We also demonstrated that MbPA allows neural networks to rapidly adapt to previously unseen classes in large-scale image classification problems using the ImageNet dataset. Furthermore, MbPA can use the local, contextual updates from memory to counter and alleviate the effect of imbalanced classification data, where some new classes are over-represented at train time whilst others are underrepresented. Finally we demonstrated on two language modelling tasks that MbPA is able to adapts to shifts in word distribution common in language modelling tasks, achieving significant improvements in performance compared to LSTMs and building on methods like neural cache (Grave et al., 2016).
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Gabor Melis for providing the LSTM baselines on the language tasks. We would also like to thank Dharshan Kumaran, Jonathan Hunt, Olivier Tieleman, Koray Kavukcuoglu, Daan Wierstra, Sam Ritter, Jane Wang, Alistair Muldal, Nando de Frietas, Tim Harley, Jacob Menick and Steven Hansen for many helpful comments and invigorating discussions.
+
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+
+# A MBPA HYPERPARAMETERS FOR INCREMENTAL LEARNING IMAGENET TASK
+
+MbPA was robust and the inductive bias of the MbPA correction adapts the performance of the model on novel classes. This is shown in Figure 6 (right) where MbPA manages to achieve high performance almost at the same rate, regardless of the learning rate of the underlying parametric component.
+
+In Figure 6 (left) we explore the influence in performance when changing the size of the episodic memory. We can see that the performance on the new classes is more sensitive to this parameter but it quickly saturates after about 400,000 entries. We repeat the above experiment by changing now the number of neighbours retrieved. The results are shown in Figure 7. We can observe that using more neighbours is better, but again, performance saturates quickly after 50 neighbours.
+
+![](images/ab01ba261a99d787dd0def5a6ca5ee84d38dade4234cd054cb9fc310f6986fc4.jpg)  
+Figure 6: Left: Performance of MbPA when varying the dictionary size. Right: Performance of the parametric, mixture and MbPA models varying the learning rate of the parametric model. The colour code is the same as in Figure 4 and the thickness of the lines indicate the learning rate used.
+
+![](images/be4e608fd488811d04262abc34f0deb32ad3937ac3b4f8fbba0d1e59efe44179.jpg)  
+Figure 7: Performance of MbPA when varying the number of nearest neighours used for performing the local adaptation.
+
+# B MODEL DETAILS LANGUAGE MODELLING TASKS
+
+For both datasets we used a single-layer LSTM baseline trained with Adam (Kingma & Ba, 2014) using the regularisation techniques described in Melis et al. (2017).
+
+In this application of MbPA the test set is small (e.g. $< 8 0 { , } 0 0 0$ words for PTB), and so it was easy to overfit to the retrieved points. To remedy this, we tuned an L2 penalty $\beta | | \theta ^ { x } - \theta | | _ { 2 }$ term in our MbPA loss (7), where $\theta$ were the parameters derived from the training set and $\beta$ was a scalar hyper-parameter.
+
+We swept over the following hyper-parameters:
+
+• Memory size: $N \in \{ 5 0 0 , 1 0 0 0 , 5 0 0 0 \}$ • Nearest neighbours: $K \in \{ 2 5 6 , 5 1 2 \}$ • Cache interpolation: $\lambda _ { c a c h e } \in \{ 0 , 0 . 0 5 , 0 . 1 , 0 . 1 5 \}$ • MbPA interpolation: $\lambda _ { m b p a } \in \{ 0 , 0 . 0 5 , 0 . 1 , 0 . 1 5 \}$ • Number of MbPA optimisation steps: $T \in \{ 1 , 5 , 1 0 \}$ • MbPA optimization learning rate: $\alpha \in \{ 0 . 0 1 , 0 . 1 , 0 . 1 5 , 0 . 2 , 0 . 5 , 1 \}$
+
+Where memory size refers to both the MbPA memory size, and the size of the neural cache for comparison, and the $\lambda$ interpolation parameters refer to the mixing of model outputs, alike to Eq. 5. The optimal parameters were: $N = 5 0 0 0$ , $K = 2 5 6$ , $\lambda _ { c a c h e } = 0 . 1 5$ , $\lambda _ { m b p a } = 0 . 1$ , $T = 1$ , $\alpha =$ 0.15.
+
+For Penn Treebank, we used a pre-trained LSTM baseline containing roughly $1 0 M$ parameters with a hidden size of 1194 and a word embedding size of 268. For WikiText-2, we used a pretrained LSTM baseline containing roughly 24M parameters with a hidden size of 1,853 and a word embedding size of 241.
+
+# C COMPARISON OF CACHE VS MBPA FOR WIKITEXT-2
+
+The comparative benefit of MbPA is investigated, when combined with the $\mathrm { L S T M + }$ cache model. By computing the perplexity on a per-word basis and comparing whether the inclusion of MbPA improves (lowers) the perplexity, we can understand what types of words are better predicted. Anecdotal samples were not sufficient to understand the trend, however when the words were bucketed by their training frequency, we see a tend of improved performance for less frequent words.
+
+This improved performance for rare words may be because the cache model has a prior to boost all recent words. Specifically, the cache probabilities are obtained from summing the attention for each instance of a word in memory, and so frequently occurring recent words that are not very contextually relevant will still be boosted. As MbPA does not do this, it appears to be more sensitive to infrequently occurring words.
+
+![](images/838d5f5049ec97c6771c3354bf946be0c2a2c765025d883590e0205c1abe7b0e.jpg)  
+Figure 8: Percent improvement when MbPA is included with the LSTM baseline and neural cache, split by training word frequency into five equally sized buckets. The bucket 1 contains the most frequent words, and bucket 5 contains the least frequent words. The average improvement $\pm 1$ standard deviation are shown. MbPA provides a directional improvement for less frequent words.
+
+# D MAP INTERPRETATION OF MBPA AND DERIVATION OF CONTEXTUAL UPDATE
+
+Let $x _ { c }$ correspond to the input of the $h _ { c } , v _ { c }$ key-value pair in the context $C$ of a given input $x$ . In other words, $h _ { c }$ was computed by feeding $x _ { c }$ to the embedding network. Then the posterior given
+
+this pair and the parameter obtained after training $\theta$ , can be written as:
+
+$$
+p ( \theta ^ { x } | \theta , x _ { c } , v _ { c } , x ) = \frac { p ( v _ { c } | x _ { c } , \theta ^ { x } , x ) p ( \theta ^ { x } | \theta ) } { p ( v _ { c } | \theta , x _ { c } , x ) } .
+$$
+
+If we maximise the posterior over the context $C$ with respect to $\theta ^ { x }$ .
+
+$$
+\begin{array} { r l } { \arg \underset { \theta ^ { \alpha } } { \operatorname* { m a x } } \mathbb { E } _ { C } \left\{ \log p ( \theta ^ { x } | \theta , x _ { c } , v _ { c } , x ) \right\} = \arg \underset { \theta ^ { x } } { \operatorname* { m a x } } } & { \log p ( \theta ^ { x } | \theta ) + \mathbb { E } _ { C } \left\{ \log p ( v _ { c } | x _ { c } , \theta ^ { x } , x ) \right\} } \\ & { \qquad = \arg \underset { \theta ^ { \alpha } } { \operatorname* { m a x } } \log p ( \theta ^ { x } | \theta ) + \displaystyle \sum _ { k = 1 } ^ { K } w _ { k } ^ { ( x ) } \log p ( v _ { k } ^ { ( x ) } | h _ { k } ^ { ( x ) } , \theta ^ { x } , x ) . } \end{array}
+$$
+
+Let $\begin{array} { r } { \log p ( \theta ^ { x } | \theta ) \propto - \frac { | | \theta ^ { x } - \theta | | _ { 2 } ^ { 2 } } { 2 \alpha _ { M } } } \end{array}$ (i.e. a Gaussian prior on $\theta ^ { x }$ centred at $\theta$ ) be thought as a regularisation term that prevents the local adaptation to move $\theta ^ { x }$ too far from $\theta$ , preventing overfitting.
+
+Another interpretation of (7) is that when the prior is taken to be a Gaussian, it is a form of elastic weight regularisation (similar to Kirkpatrick et al. (2017)) and the second term corresponds to the log likelihood of $\theta ^ { x }$ on the data in the context $C$ . This can also be seen as posterior sharpening (Fortunato et al., 2017), where we can think of the second term as an approximation of log $p ( \boldsymbol { y } _ { t } | \boldsymbol { x } _ { t } , \boldsymbol { \theta } )$ . Thus a view of MbPA is it is a form of local elastic weight consolidation on a context dataset $C$ .
+
+Equation (7) does not have a closed form solution, and requires fitting a large number of parameters at inference time. This can be costly and susceptible to overfitting. We can avoid this problem by simply adapting the reference parameters $\theta$ . Specifically, we perform a fixed number of gradient descent steps (or any of its popular variants) to minimise (7). One step of gradient descent to the loss in (7) with respect to $\theta ^ { x }$ yields
+
+$$
+{ \Delta } _ { M } ( x , \theta ) = - \alpha _ { M } \left. \nabla _ { \theta } \sum _ { k = 1 } ^ { K } w _ { k } ^ { ( x ) } \log p ( v _ { k } ^ { ( x ) } | h _ { k } ^ { ( x ) } , \theta ^ { x } , x ) \right| _ { \theta } - \beta ( \theta - \theta ^ { x } ) ,
+$$
+
+where $\beta$ is a scalar hyper-parameter. These adapted parameters are used for output computation but discarded thereafter.
+
+# E ATTENTION AS A SPECIAL CASE OF MBPA
+
+Let $C = \{ ( w _ { i } , h _ { i } , v _ { i } ) \} _ { i = 1 } ^ { k }$ be the neighbourhood retrieved from memory given a query $q$ . The likelihood prediction based on attention is given by
+
+$$
+p _ { \mathrm { m e m } } ( y = j | q ) = \frac { \sum _ { i = 1 } ^ { k } w _ { i } \delta ( v _ { i } = j ) } { \sum _ { i = 1 } ^ { k } w _ { i } } ,
+$$
+
+where the Kronecker $\delta$ is one when the equality holds and zero otherwise. We now show how the attention-based prediction given in (9) can be seen as particular case of local adaptation.
+
+For classification with $c$ classes, we parameterise $p _ { \mathrm { m e m } }$ via its logits, $z \in \mathbb { R } ^ { c }$ , with $p _ { \mathrm { m e m } } ( v | q ) =$ softmax $( z )$ . One good candidate $z$ is the one that is the most consistent with context $C$ . Specifically, the logit vector that minimises the weighted average negative log-likelihood (NLL) of the memories in context $C$ :
+
+$$
+z _ { q } = \underset { z } { \operatorname { a r g m i n } } \sum _ { i = 1 } ^ { N } w _ { i } \left( z _ { v _ { i } } - \log ( \sum _ { k = 1 } ^ { c } e ^ { z _ { k } } ) \right) .
+$$
+
+The attention weights scale the importance of each memory in the neighbour given its similarity to the query. This matches the loss (7) (ignoring the prior term). If we differentiate the above equation with respect to a $z _ { j }$ and set to zero, we obtain exactly the same expression as in (9).
+
+Effectively, in (10) we are fitting a constant function to the context retrieved from the episodic memory. This is a particular case of a local likelihood model (Loader, 2006). The update also is the same as applying a k-nn, see Figure 2.
+
+Note that this interpretation is not limited to classification tasks, the exact same reasoning (and result) could be done for a regression task, simply by changing the loss function to be Mean Squared Error (MSE).
+
+In this context, we can think of MbPA as a generalisation of the attention mechanism, in which the function used for the local fitting is given by the output network. Moreover, the parameters of the model are used as a prior for solving the local fitting problem and only change slightly to prevent overfitting.

parse/train/rkfOvGbCW/rkfOvGbCW_content_list.json

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+        "text": "MEMORY-BASED PARAMETER ADAPTATION ",
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+    {
+        "type": "text",
+        "text": "Pablo Sprechmann\\*, Siddhant M. Jayakumar\\*, Jack W. Rae, Alexander Pritzel   \nAdria Puigdom \\` enech Badia, Benigno Uria, Oriol Vinyals \\`   \nDemis Hassabis, Razvan Pascanu, Charles Blundell   \nDeepMind   \nLondon, UK   \n{psprechmann, sidmj, jwrae, apritzel,   \nadriap, buria, vinyals,   \ndhcontact, razp, cblundell}@google.com ",
+        "bbox": [
+            183,
+            143,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+            454,
+            292,
+            544,
+            308
+        ],
+        "page_idx": 0
+    },
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+        "text": "Deep neural networks have excelled on a wide range of problems, from vision to language and game playing. Neural networks very gradually incorporate information into weights as they process data, requiring very low learning rates. If the training distribution shifts, the network is slow to adapt, and when it does adapt, it typically performs badly on the training distribution before the shift. Our method, Memory-based Parameter Adaptation, stores examples in memory and then uses a context-based lookup to directly modify the weights of a neural network. Much higher learning rates can be used for this local adaptation, reneging the need for many iterations over similar data before good predictions can be made. As our method is memory-based, it alleviates several shortcomings of neural networks, such as catastrophic forgetting, fast, stable acquisition of new knowledge, learning with an imbalanced class labels, and fast learning during evaluation. We demonstrate this on a range of supervised tasks: large-scale image classification and language modelling. ",
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+            324,
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    },
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+        "text": "Neural networks have been proven to be powerful function approximators, as shown in a long list of successful applications: image classification (e.g. Krizhevsky et al., 2012), audio processing (e.g. Oord et al., 2016), game playing (e.g. Mnih et al., 2015; Silver et al., 2017), and machine translation (e.g. Wu et al., 2016). Typically these applications apply batch training to large or near-infinite data sets, requiring many iterations to obtain satisfactory performance. ",
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+        "type": "text",
+        "text": "Humans and animals are able to incorporate new knowledge quickly from single examples, continually throughout much of their lifetime. In contrast, neural network-based models rely on the data distribution being stationary and the training procedure using low learning rates and many passes through the training data to obtain good generalisation. This limits their application to life-long learning or dynamic environments and tasks. ",
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+        "text": "Problems in continual learning with neural networks commonly manifest as the phenomenon of catastrophic forgetting (McCloskey & Cohen, 1989; French, 1999): a neural network performs badly on old tasks having been trained to perform well on a new task. Several recent approaches have proven promising at overcoming this, such as elastic weight consolidation (Kirkpatrick et al., 2017). Recent work in language modelling has demonstrated how popular neural language models may appropriately be adapted to take advantage of rare, recently seen words, as in the neural cache (Grave et al., 2016), pointer sentinel networks (Merity et al., 2016) and learning to remember rare events (Kaiser et al., 2017). Our work generalises these approaches and we present experimental results where we apply our model to both continual or incremental learning tasks, as well as language modelling. ",
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+        "text": "We propose Memory-based Parameter Adaptation (MbPA), a method for augmenting neural networks with an episodic memory to allow for rapid acquisition of new knowledge while preserving the high performance and good generalisation of standard deep models. It combines desirable properties of many existing few-shot, continual learning and language models. We draw inspiration from the theory of complementary learning systems (CLS: McClelland et al., 1995; Leibo et al., 2015; Kumaran et al., 2016), where effective continual, life-long learning necessitates two complementary systems: one that allows for the gradual acquisition of structured knowledge, and another that allows rapid learning of the specifics of individual experiences. As such, MbPA consists of two components: a parametric component (a standard neural network) and a non-parametric component (a neural network augmented with a memory containing previous problem instances). The parametric component learns slowly but generalises well, whereas the non-parametric component rapidly adapts the weights of the parametric component. The non-parametric, instance-based adaptation of the weights is local, in the sense the modification is directly dictated by the inputs to the parametric component. The local adaptation is discarded after the model produces its output, avoiding long term consequences of strong local adaptation (such as overfitting), allowing the weights of the parametric model to learn slowly leading to strong performance and generalisation. ",
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+            "Figure 1: Architecture for the MbPA model. Left: Training usage. The parametric network is used directly and experiences are stored in the memory. Right: Testing setting. The embedding is used to query the episodic memory, the retrieved context is used to adapt the parameters of the output network. "
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+        "text": "The contributions of our work are: $( i )$ proposing an architecture for enhancing powerful parametric models with a fast adaptation mechanism to efficiently cope with changes in the task at hand; (ii) establish connections between our method and attention mechanisms frequently used for querying memories; (iii) present a Bayesian interpretation of the method allowing a principled form of regularisation; $( i \\nu )$ evaluating the method on a range of different tasks: continual learning, incremental learning and data distribution shifts, obtaining promising results. ",
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+        "text": "2 MODEL-BASED PARAMETER ADAPTATION ",
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+        "text": "Our models consist of three components: an embedding network, $f _ { \\gamma }$ , a memory $M$ and an output network $g _ { \\theta }$ . The embedding network, $f _ { \\gamma }$ , and the output network, $g _ { \\theta }$ , are standard parametric (feed forward or recurrent) neural networks with parameters $\\gamma$ and $\\theta$ , respectively. The memory $M$ is a dynamically-sized memory module that stores key and value pairs, $M = \\{ ( h _ { i } , v _ { i } ) \\}$ . Keys $\\{ h _ { i } \\}$ are given by the embedding network. The values $\\dot { \\{ { v } _ { i } \\} }$ correspond to the desired output $y _ { i }$ . For classification, $y _ { i }$ would simply be the true class label, whereas for regression, $y _ { i }$ would be the true regression target. Hence, upon observing the $j$ -th example, we append the pair $( h _ { j } , v _ { j } )$ to the memory $M$ , where: ",
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+        "img_path": "images/2a26b5cf89949dc2dff4e59a2844b14092469742968dca1c7aab38a6df16ca5b.jpg",
+        "text": "$$\n\\begin{array} { l } { h _ { j }  f _ { \\gamma } ( x _ { j } ) , } \\\\ { v _ { j }  y _ { j } . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The memory has a fixed size and acts as a circular buffer: when it is full, the oldest data is overwritten first. Retrieval from the memory $M$ uses $K$ -nearest neighbour search on the keys $\\{ h _ { i } \\}$ with Euclidean distance to obtain the $K$ most similar keys and associated values. ",
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+        "type": "text",
+        "text": "Our model is used differently in the training and testing phases. During training, for a given input $x$ , we parametrise the conditional likelihood with a deep neural network given by the composition ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Model-based Parameter Adaptation ",
+        "text_level": 1,
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+        "type": "table",
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+        "table_caption": [],
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+        "table_body": "<table><tr><td>Aigoritnm1Model-basedParameterAdaptation procedure MBPA-TRAIN</td></tr><tr><td>Sample mini-batch of training examples B = {(xb, yb)}b from training data.</td></tr><tr><td>Calculate the embedded mini-batch B&#x27;= {(fγ(xb),yb) : xb,yb ∈B}.</td></tr><tr><td>Update 0,γ by maximising the likelihood (1) of θ and γ with respect to mini-batch B</td></tr><tr><td>Add the embedded mini-batch examples B&#x27; to memory M: M ← MU B&#x27;.</td></tr><tr><td>procedure MBPA-TEST(test input: x, output prediction: y)</td></tr><tr><td>Calculate embedding q = fγ(x),and △total ← 0.</td></tr><tr><td>(x） ，(x) )k=1</td></tr><tr><td>for each step of MbPA do</td></tr><tr><td>Calculate △m(x,θ +△total) according to (4)</td></tr><tr><td>△total ←△total +△M(x).</td></tr><tr><td>Output prediction y = g0+△tota (h)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "of the embedding and output networks. Namely, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/88c902b34ff122d5497b466ad260f5bdf0d17e88cccd4c4ba14d555778d593d2.jpg",
+        "text": "$$\np _ { \\mathrm { t r a i n } } ( y | x , \\gamma , \\theta ) = g _ { \\theta } ( f _ { \\gamma } ( x ) ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "In the case of classification, the last layer of $g _ { \\theta }$ is a softmax layer. The parameters $\\{ \\theta , \\gamma \\}$ are estimated by maximum likelihood estimation. The memory is updated with new entries, as they are seen, however no local adaptation is performed on the model. Figure 1 (left) shows a diagram of the training setting and Algorithm 1 (MbPA-Train) shows the algorithm for updating MbPA during training. ",
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+        "type": "text",
+        "text": "On the other hand, at test time, it temporarily adapts the parameters of the output network based upon the current input and the contents of the memory $M$ . That is, it uses the exact same parametrisation as (1), but with a different set of parameters in the output network. ",
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+        "type": "text",
+        "text": "Let the context $C$ of an input $x$ be the keys, values and associated weights of the $K$ nearest neighbours to query $q \\ = \\ f _ { \\gamma } ( x )$ in the memory $M$ : $C = \\{ ( h _ { k } ^ { ( x ) } , v _ { k } ^ { ( x ) } , \\bar { w } _ { k } ^ { ( x ) } ) \\} _ { k = 1 } ^ { K }$ The coefficients $w _ { k } ^ { ( x ) } \\propto \\ker ( h _ { k } ^ { ( x ) } , q )$ are weightings of each of the retrieved neighbours according to their closeness to the query $f _ { \\gamma } ( { \\boldsymbol { x } } ) . \\operatorname { k e r n } ( h , q )$ is a kernel function which, following (Pritzel et al., 2017), we choose as $\\begin{array} { r } { \\ker ( h , q ) = \\frac { 1 } { \\epsilon + \\lVert h - q \\rVert _ { 2 } ^ { 2 } } } \\end{array}$ . The parametrisation of the likelihood takes the form, ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/42813a0dd295386a636ca7795db365b04cc3f41f7d878219932618a0eb974015.jpg",
+        "text": "$$\n\\begin{array} { r } { p ( y | x , \\theta ^ { x } ) = p ( y | x , \\theta ^ { x } , C ) = g _ { \\theta ^ { x } } ( f _ { \\gamma } ( x ) ) , } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "as opposed to the standard parametric approach $g _ { \\theta } ( f _ { \\gamma } ( x ) )$ , where $\\theta ^ { x } \\ = \\ \\theta + \\Delta _ { M } ( x , \\theta )$ with $\\Delta _ { M } \\bar { ( \\boldsymbol { x } , \\boldsymbol { \\theta } ) }$ being a contextual (it is based upon the input $x$ ) update of the parameters of the output network. The MbPA adaptation corresponds to decreasing the weighted average negative loglikelihood over the retrieved neighbours in $C$ . Figure 1 (right) shows a diagram of the testing setting and Algorithm 1(MbPA-Test) shows the algorithm for using MbPA during testing. ",
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+        "type": "text",
+        "text": "An interesting property of the model is that the correction $\\Delta _ { M } ( x , \\theta )$ is such that, as the parametric model becomes better at fitting the training data (and consequently the episodic memories), it selfregulates and diminishes. In the CLS theory, this process is referred to as consolidation, when the parametric model can reliably perform predictions without relying on episodic memories. ",
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+    {
+        "type": "text",
+        "text": "2.1 MAXIMUM A POSTERIORI INTERPRETATION OF MBPA ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We can now derive $\\Delta _ { M } ( x , \\theta )$ , motivated by considering the posterior distribution on the parameters $\\theta ^ { x }$ . Let $x$ correspond to the input with context $C = \\{ h _ { k } , v _ { k } , w _ { k } ^ { ( x ) } \\} _ { k = 1 } ^ { K }$ w(x)k }Kk=1. The maximum a posteriori over the context $C$ , given the parameters obtained after training $\\theta$ , can be written as: ",
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+        "type": "equation",
+        "img_path": "images/100d0dcca61d39bddf5081f857152111cc0a7e0d41794611e07179b8baa2a817.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\theta ^ { x } } \\log p ( \\theta ^ { x } | \\theta ) + \\sum _ { k = 1 } ^ { K } w _ { k } ^ { ( x ) } \\log p ( v _ { k } ^ { ( x ) } | h _ { k } ^ { ( x ) } , \\theta ^ { x } , x ) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where the second term is a weighted likelihood of the data in C and log p(θx|θ) ∝ − ||θx−θ||222αM ( i.e. a Gaussian prior on $\\theta ^ { x }$ centred at $\\theta$ ) can be thought as a regularisation term that prevents overfitting. See Appendix $\\mathrm { D }$ for details of this derivation. ",
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+        "type": "image",
+        "img_path": "images/49827038181af68573ad0a8242f56be9014f4ca296954e0177473e0a3499cf4b.jpg",
+        "image_caption": [
+            "Figure 2: Illustrative diagram of the local fitting on a regression task. Given a query (blue), we retrieve the context from memory showed in red. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Equation (3) does not have a closed form solution, and requires fitting a large number of parameters at inference time. This can be costly and susceptible to overfitting. We can avoid this problem by adapting the reference parameters $\\theta$ . Specifically, we perform a fixed number of gradient descent steps to minimise (3). One step of gradient descent to the loss in (3) with respect to $\\theta ^ { x }$ yields ",
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+        "img_path": "images/6af173c29bae8db4056e431d772e408d1ab4de3ac7de6ce6a06ebe7661cf592e.jpg",
+        "text": "$$\n{ \\Delta } _ { M } ( x , \\theta ) = - \\alpha _ { M } \\left. \\nabla _ { \\theta } \\sum _ { k = 1 } ^ { K } w _ { k } ^ { ( x ) } \\log p ( v _ { k } ^ { ( x ) } | h _ { k } ^ { ( x ) } , \\theta ^ { x } , x ) \\right| _ { \\theta } - \\beta ( \\theta - \\theta ^ { x } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\beta$ is a scalar hyper-parameter. These adapted parameters are used for output computation but discarded thereafter, as described in Algorithm 1. ",
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+    {
+        "type": "text",
+        "text": "2.2 FROM ATTENTION TO LOCAL FITTING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A standard formulation of memory augmented networks is in the form of attention (Bahdanau et al., 2014), i.e. query memory to use a weighted average based on some similarity metric. ",
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+        "type": "text",
+        "text": "We can now show that an attention-based procedure is a particular case of local adaptation or MbPA. The details of this are discussed in Appendix E. Effectively, attention can be viewed as fitting a constant function the neighbourhood of memories, whereas MbPA generalises to fit a function parameterised by the output network of our model. ",
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+    {
+        "type": "text",
+        "text": "The diagram in Figure 2 illustrates an example in a regression task for simplicity. Given a query (blue), the retrieved memories and their corresponding values are depicted in red. The predictions of an attention based model are shown in orange. We can see that the prediction is biased towards the value of the neighbours with higher functional value. In magenta we represent the predictions made by the model $g _ { \\theta }$ . We can see that it is not able to explain all memories equally well. This could be either because the problem is too difficult, poor training, or because the a prediction needs to be made while assimilating new information. The green curve show the prediction obtained after adapting the parameters to better explain the episodic memories. ",
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+    {
+        "type": "text",
+        "text": "3 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A key component of MbPA is the non-parametric, episodic memory. Many recent works have looked at augmenting neural network systems with memories to allow for fast adaptation or incorporation of new knowledge. Variants of this architecture have been successfully used in the context of classification (Vinyals et al., 2016; Santoro et al., 2016; Kaiser et al., 2017), language modelling (Merity et al., 2016; Grave et al., 2016), reinforcement learning (Blundell et al., 2016; Pritzel et al., 2017), machine translation (Bahdanau et al., 2014), and question answering (Weston et al., 2014), to name a few. For the MbPA experiments below, we use a memory architecture similar to the Differentiable Neural Dictionary (DND) used in Neural Episodic Control (NEC) (Pritzel et al., 2017). One key difference is that we do not train the embedding network through the gradients from the memories (as they are not used at training time). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While many of these approaches share a contextual memory lookup system, MbPA is distinct in the method by which the memories are used. Matching Networks (Vinyals et al., 2016) use a nonparametric network to map from a few examples to a target class via a kernel weighted average. Prototypical Networks (Snell et al., 2017) extend this and use a linear model instead of a nearest neighbour method. ",
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+        "type": "text",
+        "text": "MbPA is further related to meta-learning approaches for few shot learning. In the context of learning invariant representations for object recognition, Anselmi et al. (2014) proposed a method that can invariantly and discriminatively represent objects using a single sample, even of a new class. In their method, instead of training via gradient descent, image templates are stored in the weights of simple-complex cell networks while objects undergo transformations. Optimisation as a model of few shot learning (Ravi & Larochelle, 2016) proposes using a meta-learner LSTM to control the gradient updates of another network, while Model-Agnostic Meta-Learning (MAML Finn et al. (2017)) proposes a way of doing meta-learning over a distribution of tasks. These methods extend the classic fine-tuning technique used in domain adaptation type of ideas (e.g. fit a given neural network to a small set of new data). The MAML algorithm (particularly related to our work) aims at learning an easily adaptable set of weights, such that given a small amount of training data for a given task following the training distribution, the fine-tuning procedure would effectively adapt the weights to this particular task. Their work does not use any memory or per-example adaptation and is not based on a continual (life-long) learning setting. In contrast, our work, aims at augmenting a powerful neural network with a fine-tuning procedure that is used at inference only. The idea is to enhance the performance of the parametric model while maintaining its full training. ",
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+    {
+        "type": "text",
+        "text": "Recent approaches to addressing the continual learning problem have included elastic weight consolidation (Kirkpatrick et al., 2017), where a penalty term is added to the loss for deviations far from previous weights, and learning without forgetting (Li & Hoiem, 2016; Furlanello et al., 2016), where distillation (Hinton et al., 2015) from previously trained models is used to keep old knowledge available. Gradient Episodic Memory for Continual Learning (Lopez-Paz & Ranzato, 2017) attempts to solve the problem by storing data from previous tasks and taking gradient updates when learning new tasks that do not increase the training loss on examples stored in memory. ",
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+    {
+        "type": "text",
+        "text": "There has been recent work in applying attention to quickly adapt a subset of fast weights (Ba et al., 2016). A number of recent works in language modelling have augmented prediction with attention over recent examples to account for the distributional shift between training and testing settings. Works in this direction include neural cache (Grave et al., 2016) and pointer sentinel networks (Merity et al., 2016). Learning to remember rare events (Kaiser et al., 2017) augments an LSTM with a key-value memory structure, and meta networks (Munkhdalai & Yu, 2017) combines fast weights with regular weights. Our model shares this flavour of attention and fast weights, while providing a model agnostic memory-based method that applies beyond language modelling. ",
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+    {
+        "type": "text",
+        "text": "Works in the context of machine translation relate to MbPA. Gu et al. (2017) explore how to incorporate information from memory into the final model predictions. The authors find that shallow mixing works best. We show in this paper that MbPA is another competitive strategy to shallow mixing, and often working better (PTB for language modelling, ImageNet for image classification). The work by Li et al. (2016) shares the focus on fast-adaptation during inference with our work. Given a test example, the translation model is fine-tuned by fitting similar sentences from the training set. MbPA can be viewed as a generalisation of such approach: it relies on an episodic memory (rather than the training set), contextual lookup and similarity based weighting scheme to fine-tune the original model. Collectively, these allow MbPA to be a powerful domain-agnostic algorithm, which allows it to handle continual and incremental learning. ",
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+        "text": "Finally, we mention that our work is closely related to the local regression and adaptive coefficient models literature, see Loader (2006) and references therein. Locally adaptive methods achieved relatively modest success in high-dimensional classification problems, as fitting many parameters to a few neighbours often leads to over fitting. We attempt to counter this with contextual lookups and a local modification of only a subset of model parameters. ",
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+        "type": "text",
+        "text": "4 EXPERIMENTS AND RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our scheme unifies elements from traditional approaches to continual, one-shot, and incremental or life-long learning. Models that solve these problems must have certain fundamental attributes in common: the ability to negate the effects of catastrophic forgetting, unbalanced and scarce data, while displaying rapid acquisition of knowledge and good generalisation. ",
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+        "type": "text",
+        "text": "In essence, these problems require the ability to deal with changes and shifts in data distributions. We demonstrate that MbPA provides a way to address this. More concretely, due to the robustness of the local adaptation, the model can deal with shifts in domain distribution (e.g. train vs test set in language), the task label set (e.g. incremental learning) or sequential distributional shifts (e.g. continual learning). Further, MbPA is agnostic to both task domain (e.g. image or language) and choice of underlying parametric model, e.g. convolutional neural networks (LeCun et al., 1998) or LSTM (Hochreiter & Schmidhuber, 1997). ",
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+        "text": "To this end, our experiments focus on displaying the advantages of MbPA on widely used tasks and datasets, comparing with competing deep learning methods and baselines. We start by looking at the continual learning framework, followed by incremental learning, the problems of unbalanced data and test time distributional changes. ",
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+        "type": "text",
+        "text": "4.1 CONTINUAL LEARNING: SEQUENTIAL DISTRIBUTIONAL SHIFT ",
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+        "text": "In this set of experiments, we explored the effects of MbPA on continual learning, i.e. when dealing with the problem of sequentially learning multiple tasks without the ability to revisit a task. ",
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+        "text": "We considered the permuted MNIST setup (Goodfellow et al., 2013). In this setting, each task was given by a different random permutation of the pixels of the MNIST dataset. We explored a chaining of 20 different tasks (20 different permutations) trained sequentially. The model was tested on all tasks it had been trained on thus far. ",
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+        "type": "text",
+        "text": "We trained all models using 10,000 examples per task, comparing to elastic weight consolidation (EWC; Kirkpatrick et al., 2017) and regular gradient descent training. In all cases we rely on a two layer MLP and use Adam (Kingma & Ba, 2014) as the optimiser. The EWC penalty cost was chosen using a grid search, as was the local MbPA learning rate (between 0.0 and 1.0) and number of optimisation steps for MbPA (between 1 and 20). ",
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+        "type": "text",
+        "text": "Figure 3 compares our approach with that of the baselines. For this particular task we worked directly on pixels as our embedding, i.e. $f _ { \\gamma }$ is the identity function, and explored regimes where the episodic memory is small. A key takeaway of this experiment is that once a task is catastrophically forgotten, we find that only a few gradient steps on carefully selected data from memory are sufficient to recover performance, as MbPA does. Considering the number of updates required to reach the solution from random initialisation, this fact itself might seem surprising. MbPA provides a principled and effective way of performing these updates. The naive approach of performing updates on memories chosen at random from the entire memory is considerably less useful. ",
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+        "text": "We outperformed the MLP, and were superior to EWC for all but one memory size (when storing only a 100 examples per task). Further, the performance of our model grew with the number of examples stored, ceteris paribus. Crucially, our memory requirements are much lower than that of EWC, which requires storing model parameters and Fisher matrices for all tasks seen so far. Unlike EWC we do not store any tasks identifiers, merely appending the memory with a few examples. Further, MbPA does not use knowledge of exact task boundaries or identities of tasks switched to, unlike EWC and other methods. This allows for frequent switches that would otherwise hamper the Fisher calculations needed for models like EWC. ",
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+        "type": "text",
+        "text": "Our method can be combined with any other algorithm such as standard replay from the memory buffer or EWC, providing further improvement. In Figure 3 (right) we combine MbPA and EWC. ",
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+        "text": "4.2 INCREMENTAL LEARNING: SHIFTS IN TASK LABEL DISTRIBUTIONS ",
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+        "text": "The goal of this section was to evaluate the model in the context of incremental learning. We considered a classification scenario where a model pre-trained on a subset of classes, was introduced to novel, previously unseen classes. The aim was to incorporate the new related knowledge, as quickly as possible, while preserving knowledge from the previous set. This was as opposed to the continual learning problem where there are distinct tasks without the ability to revisit old data. ",
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+        "image_caption": [
+            "Figure 3: (Left) Results on Permuted MNIST comparing baselines with MbPA using different memory sizes. (Right) Results augmenting MbPA with EWC, showing the flexibility and complementarity of MbPA. "
+        ],
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+        "type": "text",
+        "text": "Specifically we considered the problem of image classification on the ImageNet dataset (Russakovsky et al., 2015). As a parametric model we used a ResnetV1 model (He et al., 2016). This was pre-trained on a random subset of the ImageNet dataset containing half of the classes. We then presented all 1000 classes and evaluated how quickly the network can acquire this knowledge (i.e. perform well across all 1000 classes). ",
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+        "type": "text",
+        "text": "For MbPA, we used the penultimate layer of the network as the embedding network $f _ { \\gamma }$ , forming the key $h$ and query $q$ for our episodic memory $M$ . The last fully connected layer was used to initialise the parametric model $g _ { \\theta }$ . MbPA was applied at test time, using RMSprop with a local learning rate $\\alpha _ { M }$ and the number of optimisation steps (as in Algorithm 1) tuned as hyper-parameters. ",
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+        "type": "text",
+        "text": "A natural baseline was to simply fine-tune the last layer of the parametric model with the new training set. We also evaluated a mixture model, combining the classifications of the parametric model and the non-parametric model at decision level in the following manner: ",
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+        "type": "equation",
+        "img_path": "images/c51e28aa4f694e7b13af291a101f5a2d2be0dddbe3dfe4f753c83a115bf9c0ce.jpg",
+        "text": "$$\np ( y | q ) = \\lambda p _ { \\mathrm { p a r a m } } ( y | q ) + ( 1 - \\lambda ) p _ { \\mathrm { m e m } } ( y | q ) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where the parameter $\\lambda \\in [ 0 , 1 ]$ controls the contribution of each model (this model was proposed by Grave et al. (2016) in the context of language modelling). We created five random splits in new and old classes. Hyperparameters were tuned for all models using the first split and the validation set, and we report the average performance on the remaining splits evaluated on the test set. ",
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+        "type": "text",
+        "text": "Figure 4 shows the test set performance for all models, split by new and old classes. ",
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+        "text": "While the mixture model provides a large improvement over the plain parametric model, MbPA significantly outperforms both of them both in speed and performance. This is particularly noticeable in the new classes, where MbPA acquires knowledge from very few examples. Table 1 shows a quantitative analysis of these observations. After around 30 epoches the parametric model matches the performance of MbPA. In the appendix we explore sensitivity of MbPA on this task to various hyperparameters (memory size, learning rate). ",
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+    {
+        "type": "text",
+        "text": "4.2.1 UNBALANCED DATASETS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We further explored the incremental introduction of new classes, specifically in the context of unbalanced datasets. Most real world data are unbalanced, whereas standard datasets (like ImageNet) are artificially balanced to play well with deep learning methods. ",
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+        "type": "text",
+        "text": "We replicated the setting from the ImageNet experiments in the previous section, where new classes were introduced to a pre-trained model. However, we only showed a tenth of the data for half the new classes and all data for the other half. We report performance on the full balanced validation set. Once again, we compared the parametric model with MbPA and a memory based mixture model. ",
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+    {
+        "type": "text",
+        "text": "Results are summarised in Figure 5 (left). After 20 epochs of training, MbPA outperformed both baselines, with a wider gap in performance than the previous experiment. Further, the mixture model, though equipped with memory, did significantly worse than MbPA, leading us to conclude that the inductive bias in the local adaptation process was well suited to deal with data scarcity. ",
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+        "img_path": "images/99497f1b12644cc7e86512b2bc4122a11791cdf6e6e98e4b373a67551a68c716.jpg",
+        "table_caption": [
+            "Table 1: Quantitative evaluation of the learning dynamics for the Imagenet experiment. We compare a parametric model, non-parametric model (prediction based on memory only (9)), a mixture model and MbPA. We report the top 1 accuracy as well as the area under the curve (AUC) at different points in training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Subset</td><td rowspan=\"2\">Model</td><td colspan=\"3\">Top 1 (at epochs)</td><td colspan=\"3\">AUC (at epochs)</td></tr><tr><td>0.1</td><td>1</td><td>3</td><td>0.1</td><td>1</td><td>3</td></tr><tr><td rowspan=\"4\">Novel</td><td>MbPA</td><td>46.2 %</td><td>64.5 %</td><td>65.7 %</td><td>27.4 %</td><td>57.7 %</td><td>63.0 %</td></tr><tr><td>Non-Parametric</td><td>40.0 %</td><td>53.3 %</td><td>52.9 %</td><td>28.3 %</td><td>47.9 %</td><td>51.8 %</td></tr><tr><td>Mixture</td><td>31.6 %</td><td>56.0 %</td><td>59.1 %</td><td>18.6 %</td><td>47.4 %</td><td>54.7 %</td></tr><tr><td>Parametric</td><td>16.2 %</td><td>53.6 %</td><td>57.9 %</td><td>5.7 %</td><td>41.7 %</td><td>51.9 %</td></tr><tr><td rowspan=\"4\">Pre Trained</td><td>MbPA</td><td>68.5 %</td><td>70.9 %</td><td>70.9 %</td><td>71.4 %</td><td>70.3 %</td><td>70.3 %</td></tr><tr><td>Non-Parametric</td><td>62.7 %</td><td>69.4 %</td><td>70.0 %</td><td>45.9 %</td><td>65.8 %</td><td>68.7 %</td></tr><tr><td>Mixture</td><td>71.9 %</td><td>70.3 %</td><td>70.2 %</td><td>74.8 %</td><td>70.6 %</td><td>70.1 %</td></tr><tr><td>Parametric</td><td>71.4 %</td><td>68.1 %</td><td>68.8 %</td><td>76.0 %</td><td>68.6 %</td><td>68.3 %</td></tr></table>",
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+    {
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+        "img_path": "images/c5cc3b65b2160373872d43edecda7c4405134c93ce04e390e100a6f22e2a9201.jpg",
+        "image_caption": [
+            "Figure 4: The figure compares the performance of MbPA (blue) against two baselines: the parametric model (green) and the mixture of experts (red). (Left) Aggregated performance (Right) disentangled performance evaluated on new (dashed) and old (solid) classes. "
+        ],
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+        "text": "4.3 LANGUAGE MODELLING: DOMAIN SHIFTS ",
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+        "type": "text",
+        "text": "Finally we considered how MbPA can be used at test time to further improve the performance of language modelling. Given the general formulation of MbPA, this could be applied to any problem where there is a shift in distribution at test time — we focus on language modelling, where using recent information has proved promising, such as neural cache and dynamic evaluation (Grave et al., 2016; Krause et al., 2017). ",
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+        "type": "text",
+        "text": "We considered two datasets with established performance benchmarks, Penn Treebank (PTB; Marcus et al., 1993) and WikiText-2 (Merity et al., 2016). We pre-trained an LSTM and apply MbPA to the weights and biases of the output softmax layer. The memory stores the past LSTM outputs and associated class labels observed during evaluation. Full model details and hyper-parameters are detailed in Appendix B. ",
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+        "text": "Penn Treebank is a small text corpus containing 887,521 train tokens, 70,390 validation tokens, and 78,669 test tokens; with a vocabulary size of 10,000. The LSTM obtained a test perplexity of 59.6 and this dropped by 4.3 points when interpolated with the neural cache. When we interpolated an LSTM with MbPA we were able to improve on the LSTM baseline by 5.3 points (an additional one from the cache model). We also attempted a dynamic evaluation scheme in a similar style to Krause et al. (2017), where we loaded the Adam optimisation parameters obtained during training and evaluated with training of the LSTM enabled, using a BPTT window of 5 steps. However we did not manage to obtain gains above 1 perplexity from baseline, and so we did not try it for WikiText-2. ",
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+        "type": "text",
+        "text": "WikiText-2 is a larger text corpus than PTB, derived from Wikipedia articles. It contains 2,088,628 train tokens, 217,646 validation tokens, and 245,569 test tokens, with a vocabulary of 33,278. Our LSTM baseline obtained a test perplexity of 65.9, and this is improved by 14.6 points when mixed with a neural cache. Combining the baseline LSTM with an LSTM fit with MbPA we see a drop of 9.9 points, however the combination of all three models (LSTM baseline $+ \\mathrm { M b P A } +$ cache) produced ",
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+    {
+        "type": "image",
+        "img_path": "images/c41b5c92fb5bee6179520c30df8f051177432b71df5f70a6ac505a33db4aacce.jpg",
+        "image_caption": [
+            "Figure 5: (Left) MbPA outperformed both parametric and memory-based mixture baselines, in the presence of unbalanced data on previously unseen classes (dashed lines). (Right) Example of MbPA. Query (shown larger in the top-right corner) of class “TV” and neighbourhood (all other images) for a specific case. Mixture and parametric models fail to classify the image while MbPA succeeds. 8 different classes in the closest 20 neighbours (e.g. “desktop computer”, “monitor”, “CRT screen”). Accuracy went from $2 5 \\%$ to $7 5 \\%$ after local adaptation. "
+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c3f7e93b44563e06c85ee44cb1c0ce7ad0017050a10703e7a9161807d8ebe7e8.jpg",
+        "table_caption": [
+            "the largest drop of 15.9 points. Comparing the perplexity word-by-word between LSTM $^ +$ cache and LSTM $^ +$ cache $^ +$ MbPA, we see that MbPA improves predictions for rarer words (Figure 8). "
+        ],
+        "table_footnote": [
+            "Table 2: Table with PTB and WikiText-2 perplexities. $\\Delta$ Test denotes improvement of model on the test set relative to the corresponding baseline. "
+        ],
+        "table_body": "<table><tr><td></td><td>Valid</td><td>PTB Test △Test</td><td></td><td>WikiText-2 Valid Test</td><td></td><td>△Test</td></tr><tr><td>CharCNN (Zhang et al., 2015) Variational LSTM (Aharoni et al., 2017) LSTM + cache (Grave et al., 2016) LSTM (Melis et al., 2017) AWD-LSTM (Merity et al., 2017) AWD-LSTM + cache (Merity et al., 2017) AWD-LSTM (reprod.) (Krause et al., 2017) AWD-LSTM+ dyn eval (Krause et al., 2017)</td><td>74.6 60.9 60.0 53.9 59.8 51.6</td><td>78.9 61.7 72.1 58.3 57.3 52.8 57.7 51.1</td><td>- 4.5 - 6.6</td><td>72.1 69.1 68.6 53.8 68.9 46.4</td><td>68.9 65.9 65.8 52.0 66.1 44.3</td><td>- 13.8 - 21.8</td></tr><tr><td>LSTM (ours) LSTM + cache (ours) LSTM+MbPA LSTM+MbPA+cache</td><td>61.8 55.7 54.8 54.8</td><td>59.6 55.3 54.3 54.4</td><td>-4.3 -5.3 -5.2</td><td>69.3 53.2 58.4 51.8</td><td>65.9 51.3 56.0 49.4</td><td>-14.6 -9.9 -16.5</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We have described Memory-based Parameter Adaptation (MbPA), a scheme for using an episodic memory structure to locally adapt the parameters of a neural network based upon the current input context. MbPA works well on a wide range of supervised learning tasks in several incremental, lifelong learning settings: image classification, language modelling. Our experiments show that MbPA improves performance in continual learning experiments, comparable to or in many cases exceeding the performance of EWC. We also demonstrated that MbPA allows neural networks to rapidly adapt to previously unseen classes in large-scale image classification problems using the ImageNet dataset. Furthermore, MbPA can use the local, contextual updates from memory to counter and alleviate the effect of imbalanced classification data, where some new classes are over-represented at train time whilst others are underrepresented. Finally we demonstrated on two language modelling tasks that MbPA is able to adapts to shifts in word distribution common in language modelling tasks, achieving significant improvements in performance compared to LSTMs and building on methods like neural cache (Grave et al., 2016). ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Gabor Melis for providing the LSTM baselines on the language tasks. We would also like to thank Dharshan Kumaran, Jonathan Hunt, Olivier Tieleman, Koray Kavukcuoglu, Daan Wierstra, Sam Ritter, Jane Wang, Alistair Muldal, Nando de Frietas, Tim Harley, Jacob Menick and Steven Hansen for many helpful comments and invigorating discussions. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            205,
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+    {
+        "type": "text",
+        "text": "A MBPA HYPERPARAMETERS FOR INCREMENTAL LEARNING IMAGENET TASK ",
+        "text_level": 1,
+        "bbox": [
+            171,
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+    {
+        "type": "text",
+        "text": "MbPA was robust and the inductive bias of the MbPA correction adapts the performance of the model on novel classes. This is shown in Figure 6 (right) where MbPA manages to achieve high performance almost at the same rate, regardless of the learning rate of the underlying parametric component. ",
+        "bbox": [
+            173,
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+    {
+        "type": "text",
+        "text": "In Figure 6 (left) we explore the influence in performance when changing the size of the episodic memory. We can see that the performance on the new classes is more sensitive to this parameter but it quickly saturates after about 400,000 entries. We repeat the above experiment by changing now the number of neighbours retrieved. The results are shown in Figure 7. We can observe that using more neighbours is better, but again, performance saturates quickly after 50 neighbours. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/ab01ba261a99d787dd0def5a6ca5ee84d38dade4234cd054cb9fc310f6986fc4.jpg",
+        "image_caption": [
+            "Figure 6: Left: Performance of MbPA when varying the dictionary size. Right: Performance of the parametric, mixture and MbPA models varying the learning rate of the parametric model. The colour code is the same as in Figure 4 and the thickness of the lines indicate the learning rate used. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/be4e608fd488811d04262abc34f0deb32ad3937ac3b4f8fbba0d1e59efe44179.jpg",
+        "image_caption": [
+            "Figure 7: Performance of MbPA when varying the number of nearest neighours used for performing the local adaptation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            535,
+            630,
+            671
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B MODEL DETAILS LANGUAGE MODELLING TASKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            752,
+            622,
+            768
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For both datasets we used a single-layer LSTM baseline trained with Adam (Kingma & Ba, 2014) using the regularisation techniques described in Melis et al. (2017). ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In this application of MbPA the test set is small (e.g. $< 8 0 { , } 0 0 0$ words for PTB), and so it was easy to overfit to the retrieved points. To remedy this, we tuned an L2 penalty $\\beta | | \\theta ^ { x } - \\theta | | _ { 2 }$ term in our MbPA loss (7), where $\\theta$ were the parameters derived from the training set and $\\beta$ was a scalar hyper-parameter. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We swept over the following hyper-parameters: ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• Memory size: $N \\in \\{ 5 0 0 , 1 0 0 0 , 5 0 0 0 \\}$ • Nearest neighbours: $K \\in \\{ 2 5 6 , 5 1 2 \\}$ • Cache interpolation: $\\lambda _ { c a c h e } \\in \\{ 0 , 0 . 0 5 , 0 . 1 , 0 . 1 5 \\}$ • MbPA interpolation: $\\lambda _ { m b p a } \\in \\{ 0 , 0 . 0 5 , 0 . 1 , 0 . 1 5 \\}$ • Number of MbPA optimisation steps: $T \\in \\{ 1 , 5 , 1 0 \\}$ • MbPA optimization learning rate: $\\alpha \\in \\{ 0 . 0 1 , 0 . 1 , 0 . 1 5 , 0 . 2 , 0 . 5 , 1 \\}$ ",
+        "bbox": [
+            217,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            215,
+            102,
+            669,
+            199
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Where memory size refers to both the MbPA memory size, and the size of the neural cache for comparison, and the $\\lambda$ interpolation parameters refer to the mixing of model outputs, alike to Eq. 5. The optimal parameters were: $N = 5 0 0 0$ , $K = 2 5 6$ , $\\lambda _ { c a c h e } = 0 . 1 5$ , $\\lambda _ { m b p a } = 0 . 1$ , $T = 1$ , $\\alpha =$ 0.15. ",
+        "bbox": [
+            176,
+            210,
+            821,
+            266
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For Penn Treebank, we used a pre-trained LSTM baseline containing roughly $1 0 M$ parameters with a hidden size of 1194 and a word embedding size of 268. For WikiText-2, we used a pretrained LSTM baseline containing roughly 24M parameters with a hidden size of 1,853 and a word embedding size of 241. ",
+        "bbox": [
+            173,
+            272,
+            825,
+            329
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C COMPARISON OF CACHE VS MBPA FOR WIKITEXT-2 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            349,
+            651,
+            366
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The comparative benefit of MbPA is investigated, when combined with the $\\mathrm { L S T M + }$ cache model. By computing the perplexity on a per-word basis and comparing whether the inclusion of MbPA improves (lowers) the perplexity, we can understand what types of words are better predicted. Anecdotal samples were not sufficient to understand the trend, however when the words were bucketed by their training frequency, we see a tend of improved performance for less frequent words. ",
+        "bbox": [
+            174,
+            381,
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+            452
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "This improved performance for rare words may be because the cache model has a prior to boost all recent words. Specifically, the cache probabilities are obtained from summing the attention for each instance of a word in memory, and so frequently occurring recent words that are not very contextually relevant will still be boosted. As MbPA does not do this, it appears to be more sensitive to infrequently occurring words. ",
+        "bbox": [
+            173,
+            459,
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+            529
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/838d5f5049ec97c6771c3354bf946be0c2a2c765025d883590e0205c1abe7b0e.jpg",
+        "image_caption": [
+            "Figure 8: Percent improvement when MbPA is included with the LSTM baseline and neural cache, split by training word frequency into five equally sized buckets. The bucket 1 contains the most frequent words, and bucket 5 contains the least frequent words. The average improvement $\\pm 1$ standard deviation are shown. MbPA provides a directional improvement for less frequent words. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            310,
+            550,
+            653,
+            746
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D MAP INTERPRETATION OF MBPA AND DERIVATION OF CONTEXTUAL UPDATE ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            844,
+            787,
+            878
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Let $x _ { c }$ correspond to the input of the $h _ { c } , v _ { c }$ key-value pair in the context $C$ of a given input $x$ . In other words, $h _ { c }$ was computed by feeding $x _ { c }$ to the embedding network. Then the posterior given ",
+        "bbox": [
+            173,
+            895,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "this pair and the parameter obtained after training $\\theta$ , can be written as: ",
+        "bbox": [
+            173,
+            103,
+            633,
+            119
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/471b6a16a99d208af3a333aedabd52a3def0822ddf52f78530eaf115d9726864.jpg",
+        "text": "$$\np ( \\theta ^ { x } | \\theta , x _ { c } , v _ { c } , x ) = \\frac { p ( v _ { c } | x _ { c } , \\theta ^ { x } , x ) p ( \\theta ^ { x } | \\theta ) } { p ( v _ { c } | \\theta , x _ { c } , x ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            126,
+            645,
+            161
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "If we maximise the posterior over the context $C$ with respect to $\\theta ^ { x }$ . ",
+        "bbox": [
+            173,
+            167,
+            614,
+            183
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/db10435e2ad640445039da641296f12f4039d5a053d9d73298177c4254c4d2a3.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\arg \\underset { \\theta ^ { \\alpha } } { \\operatorname* { m a x } } \\mathbb { E } _ { C } \\left\\{ \\log p ( \\theta ^ { x } | \\theta , x _ { c } , v _ { c } , x ) \\right\\} = \\arg \\underset { \\theta ^ { x } } { \\operatorname* { m a x } } } & { \\log p ( \\theta ^ { x } | \\theta ) + \\mathbb { E } _ { C } \\left\\{ \\log p ( v _ { c } | x _ { c } , \\theta ^ { x } , x ) \\right\\} } \\\\ & { \\qquad = \\arg \\underset { \\theta ^ { \\alpha } } { \\operatorname* { m a x } } \\log p ( \\theta ^ { x } | \\theta ) + \\displaystyle \\sum _ { k = 1 } ^ { K } w _ { k } ^ { ( x ) } \\log p ( v _ { k } ^ { ( x ) } | h _ { k } ^ { ( x ) } , \\theta ^ { x } , x ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            184,
+            189,
+            812,
+            258
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Let $\\begin{array} { r } { \\log p ( \\theta ^ { x } | \\theta ) \\propto - \\frac { | | \\theta ^ { x } - \\theta | | _ { 2 } ^ { 2 } } { 2 \\alpha _ { M } } } \\end{array}$ (i.e. a Gaussian prior on $\\theta ^ { x }$ centred at $\\theta$ ) be thought as a regularisation term that prevents the local adaptation to move $\\theta ^ { x }$ too far from $\\theta$ , preventing overfitting. ",
+        "bbox": [
+            176,
+            281,
+            825,
+            316
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Another interpretation of (7) is that when the prior is taken to be a Gaussian, it is a form of elastic weight regularisation (similar to Kirkpatrick et al. (2017)) and the second term corresponds to the log likelihood of $\\theta ^ { x }$ on the data in the context $C$ . This can also be seen as posterior sharpening (Fortunato et al., 2017), where we can think of the second term as an approximation of log $p ( \\boldsymbol { y } _ { t } | \\boldsymbol { x } _ { t } , \\boldsymbol { \\theta } )$ . Thus a view of MbPA is it is a form of local elastic weight consolidation on a context dataset $C$ . ",
+        "bbox": [
+            174,
+            321,
+            825,
+            393
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Equation (7) does not have a closed form solution, and requires fitting a large number of parameters at inference time. This can be costly and susceptible to overfitting. We can avoid this problem by simply adapting the reference parameters $\\theta$ . Specifically, we perform a fixed number of gradient descent steps (or any of its popular variants) to minimise (7). One step of gradient descent to the loss in (7) with respect to $\\theta ^ { x }$ yields ",
+        "bbox": [
+            173,
+            398,
+            825,
+            470
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a301b79939a87fccff6028effe1f529f9d7de35c16e178c5631f8f3c6b0121dd.jpg",
+        "text": "$$\n{ \\Delta } _ { M } ( x , \\theta ) = - \\alpha _ { M } \\left. \\nabla _ { \\theta } \\sum _ { k = 1 } ^ { K } w _ { k } ^ { ( x ) } \\log p ( v _ { k } ^ { ( x ) } | h _ { k } ^ { ( x ) } , \\theta ^ { x } , x ) \\right| _ { \\theta } - \\beta ( \\theta - \\theta ^ { x } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            478,
+            730,
+            522
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where $\\beta$ is a scalar hyper-parameter. These adapted parameters are used for output computation but discarded thereafter. ",
+        "bbox": [
+            173,
+            529,
+            825,
+            558
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E ATTENTION AS A SPECIAL CASE OF MBPA ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            579,
+            560,
+            597
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Let $C = \\{ ( w _ { i } , h _ { i } , v _ { i } ) \\} _ { i = 1 } ^ { k }$ be the neighbourhood retrieved from memory given a query $q$ . The likelihood prediction based on attention is given by ",
+        "bbox": [
+            174,
+            611,
+            821,
+            641
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4125b3abe2d229a4776ade99e7badf301e7af07f0f9db97642ea250154c21431.jpg",
+        "text": "$$\np _ { \\mathrm { m e m } } ( y = j | q ) = \\frac { \\sum _ { i = 1 } ^ { k } w _ { i } \\delta ( v _ { i } = j ) } { \\sum _ { i = 1 } ^ { k } w _ { i } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            647,
+            622,
+            689
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where the Kronecker $\\delta$ is one when the equality holds and zero otherwise. We now show how the attention-based prediction given in (9) can be seen as particular case of local adaptation. ",
+        "bbox": [
+            173,
+            695,
+            821,
+            724
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For classification with $c$ classes, we parameterise $p _ { \\mathrm { m e m } }$ via its logits, $z \\in \\mathbb { R } ^ { c }$ , with $p _ { \\mathrm { m e m } } ( v | q ) =$ softmax $( z )$ . One good candidate $z$ is the one that is the most consistent with context $C$ . Specifically, the logit vector that minimises the weighted average negative log-likelihood (NLL) of the memories in context $C$ : ",
+        "bbox": [
+            173,
+            731,
+            825,
+            786
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9b53940a47c00e8d1f2848625b43856ba914fb5de12cc56c26f20ed3e495413b.jpg",
+        "text": "$$\nz _ { q } = \\underset { z } { \\operatorname { a r g m i n } } \\sum _ { i = 1 } ^ { N } w _ { i } \\left( z _ { v _ { i } } - \\log ( \\sum _ { k = 1 } ^ { c } e ^ { z _ { k } } ) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            785,
+            648,
+            829
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The attention weights scale the importance of each memory in the neighbour given its similarity to the query. This matches the loss (7) (ignoring the prior term). If we differentiate the above equation with respect to a $z _ { j }$ and set to zero, we obtain exactly the same expression as in (9). ",
+        "bbox": [
+            174,
+            832,
+            825,
+            876
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Effectively, in (10) we are fitting a constant function to the context retrieved from the episodic memory. This is a particular case of a local likelihood model (Loader, 2006). The update also is the same as applying a k-nn, see Figure 2. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Note that this interpretation is not limited to classification tasks, the exact same reasoning (and result) could be done for a regression task, simply by changing the loss function to be Mean Squared Error (MSE). ",
+        "bbox": [
+            174,
+            103,
+            823,
+            145
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In this context, we can think of MbPA as a generalisation of the attention mechanism, in which the function used for the local fitting is given by the output network. Moreover, the parameters of the model are used as a prior for solving the local fitting problem and only change slightly to prevent overfitting. ",
+        "bbox": [
+            174,
+            152,
+            825,
+            208
+        ],
+        "page_idx": 15
+    }
+]

parse/train/yhjpeuWepoj/yhjpeuWepoj.md

@@ -0,0 +1,303 @@
+# Exploiting the Intrinsic Neighborhood Structure for Source-free Domain Adaptation
+
+Shiqi Yang1, Yaxing Wang1,2∗, Joost van de Weijer1, Luis Herranz1, Shangling Jui3 1 Computer Vision Center, Universitat Autonoma de Barcelona, Barcelona, Spain 2 PCALab, Nanjing University of Science and Technology, China 3 Huawei Kirin Solution, Shanghai, China   
+{syang,yaxing,joost,lherranz}@cvc.uab.es, jui.shangling@huawei.com
+
+# Abstract
+
+Domain adaptation (DA) aims to alleviate the domain shift between source domain and target domain. Most DA methods require access to the source data, but often that is not possible (e.g. due to data privacy or intellectual property). In this paper, we address the challenging source-free domain adaptation (SFDA) problem, where the source pretrained model is adapted to the target domain in the absence of source data. Our method is based on the observation that target data, which might no longer align with the source domain classifier, still forms clear clusters. We capture this intrinsic structure by defining local affinity of the target data, and encourage label consistency among data with high local affinity. We observe that higher affinity should be assigned to reciprocal neighbors, and propose a self regularization loss to decrease the negative impact of noisy neighbors. Furthermore, to aggregate information with more context, we consider expanded neighborhoods with small affinity values. In the experimental results we verify that the inherent structure of the target features is an important source of information for domain adaptation. We demonstrate that this local structure can be efficiently captured by considering the local neighbors, the reciprocal neighbors, and the expanded neighborhood. Finally, we achieve state-of-the-art performance on several 2D image and 3D point cloud recognition datasets. Code is available in https://github.com/Albert0147/SFDA_neighbors.
+
+# 1 Introduction
+
+Most deep learning methods rely on training on large amount of labeled data, while they cannot generalize well to a related yet different domain. One research direction to address this issue is Domain Adaptation (DA), which aims to transfer learned knowledge from a source to a target domain. Most existing DA methods demand labeled source data during the adaptation period, however, it is often not practical that source data are always accessible, such as when applied on data with privacy or property restrictions. Therefore, recently, there have emerged a few works [16, 17, 20, 21] tackling a new challenging DA scenario where instead of source data only the source pretrained model is available for adapting, i.e., source-free domain adaptation (SFDA). Among these methods, USFDA [16] addresses universal DA [57] and SF [17] addresses open-set DA [36]. In both universal and open-set DA the label set is different for source and target domains. SHOT [21] and 3C-GAN [20] are for closed-set DA where source and target domains have the same categories. 3C-GAN [20] is based on target-style image generation with a conditional GAN, and SHOT [21] is based on mutual information maximization and pseudo labeling. Finally, BAIT [56] extends MCD [35] to the SFDA setting. However, these methods ignore the intrinsic neighborhood structure of the target data in feature space which can be very valuable to tackle SFDA.
+
+![](images/343b4b429f7a34d25d762b61bf6977f13ba8eaac4be276860b5d2170fa2f18be.jpg)  
+Figure 1: (a) t-SNE visualization of target features by source model. (b) Ratio of different type of nearest neighbor features of which: the predicted label is the same as the feature, K is the number of nearest neighbors. The features in (a) and (b) are on task $\mathrm { A r } {  } \mathrm { R w }$ of Office-Home. (c) Illustration of our method. In the left shows we distinguish reciprocal and non-reciprocal neighbors. The adaptation is achieved by pushed the features towards reciprocal neighbors heavily.
+
+In this paper, we focus on closed-set source-free domain adaptation. Our main observation is that current DA methods do not exploit the intrinsic neighborhood structure of the target data. We use this term to refer to the fact that, even though the target data might have shifted in the feature space (due to the covariance shift), target data of the same class is still expected to form a cluster in the embedding space. This can be implied to some degree from the t-SNE visualization of target features on the source model which suggests that significant cluster structure is preserved (see Fig. 1 (a)). This assumption is implicitly adopted by most DA methods, as instantiated by a recent DA work [42]. A well-established way to assess the structure of points in high-dimensional spaces is by considering the nearest neighbors of points, which are expected to belong to the same class. However, this assumption is not true for all points; the blue curve in Figure 1(b) shows that around $7 5 \%$ of the nearest neighbors has the correct label. In this paper, we observe that this problem can be mitigated by considering reciprocal nearest neighbors (RNN); the reciprocal neighbors of a point have the point as their neighbor. Reciprocal neighbors have been studied before in different contexts [14, 31, 60]. The reason why reciprocal neighbors are more trustworthy is illustrated in Fig. 1(c). Fig. 1(b) shows the ratio of neighbors which have the correct prediction for different kinds of nearest neighbors. The curves show that reciprocal neighbors indeed have more chances to predict the true label than non-reciprocal nearest neighbors (nRNN).
+
+The above observation and analysis motivate us to assign different weights to the supervision from nearest neighbors. Our method, called Neighborhood Reciprocity Clustering (NRC), achieves sourcefree domain adaptation by encouraging reciprocal neighbors to concord in their label prediction. In addition, we will also consider a weaker connection to the non-reciprocal neighbors. We define affinity values to describe the degree of connectivity between each data point and its neighbors, which is also utilized to encourage class-consistency between neighbors, and we propose to use a self-regularization to decrease the negative impact of potential noisy neighbors. Furthermore, inspired by recent graph based methods [1, 3, 61] which show that the higher order neighbors can provide relevant context, and also considering neighbors of neighbors is more likely to provide datapoints that are close on the data manifold [43]. Thus, to aggregate wider local information, we further retrieve the expanded neighbors, i.e, neighbor of the nearest neighbors, for auxiliary supervision.
+
+Our contributions can be summarized as follows, to achieve source-free domain adaptation: (i) we explicitly exploit the fact that same-class data forms cluster in the target embedding space, we do this by considering the predictions of neighbors and reciprocal neighbors, (ii) we further show that considering an extended neighborhood of data points further improves results (iii) the experiments results on three 2D image datasets and one 3D point cloud dataset show that our method achieves state-of-the-art performance compared with related methods.
+
+# 2 Related Work
+
+Domain Adaptation. Most DA methods tackle domain shift by aligning the feature distributions. Early DA methods such as [23, 41, 45] adopt moment matching to align feature distributions. And in recent years, plenty of works have emerged that achieve alignment by adversarial training. DANN [7] formulates domain adaptation as an adversarial two-player game. The adversarial training of CDAN [24] is conditioned on several sources of information. DIRT-T [40] performs domain adversarial training with an added term that penalizes violations of the cluster assumption. Additionally, [18, 26, 35] adopts prediction diversity between multiple learnable classifiers to achieve local or category-level feature alignment between source and target domains. AFN [52] shows that the erratic discrimination of target features stems from much smaller norms than those found in the source features. SRDC [42] proposes to directly uncover the intrinsic target discrimination via discriminative clustering to achieve adaptation. More related, [27] resorts to K-means clustering for open-set adaptation while considering global structure. Our method instead only focuses on nearest neighbors (local structure) for source-free adaptation.
+
+Source-free Domain Adaptation. Source-present methods need supervision from the source domain during adaptation. Recently, there are several methods investigating source-free domain adaptation. USFDA [16] and FS [17] explore source-free universal DA [57] and open-set DA [36], and they propose to synthesize extra training samples to make the decision boundary compact, thereby allowing to recognise the open classes. For closed-set DA setting. SHOT [21] proposes to fix the source classifier and match the target features to the fixed classifier by maximizing mutual information and a proposed pseudo label strategy which considers global structure. 3C-GAN [20] synthesizes labeled target-style training images based on the conditional GAN to provide supervision for adaptation. Finally, SFDA [22] is for segmentation based on synthesizing fake source samples.
+
+Graph Clustering. Our method shares some similarities with graph clustering work such as [38, 48, 54, 55] by utilizing neighborhood information. However, our methods are fundamentally different. Unlike those works which require labeled data to train the graph network for estimating the affinity, we instead adopt reciprocity to assign affinity.
+
+# 3 Method
+
+Notation. We denote the labeled source domain data with $n _ { s }$ samples as $\mathcal { D } _ { s } = \{ ( x _ { i } ^ { s } , y _ { i } ^ { s } ) \} _ { i = 1 } ^ { n _ { s } }$ , where $y _ { i } ^ { s }$ orresponding label of . Both domains have t $x _ { i } ^ { s }$ , andsame e unlabeled target domain data with classes (closed-set setting). Under the $n _ { t }$ samples asFDA setting $\mathcal { D } _ { t } \overset { \vartriangle } { = } \{ x _ { j } ^ { t } \} _ { j = 1 } ^ { n _ { t } }$ $C$ $\mathcal { D } _ { s }$ is only available for model pretraining. Our method is based on a neural network, which we split into two parts: a feature extractor $f$ , and a classifier $g$ . The feature output by the feature extractor is denoted as $z ( x ) = f \left( x \right)$ , the output of network is denoted as $p ( x ) = \bar { \delta } ( g ( \dot { z } ) ) \in \mathcal { R } ^ { C }$ where $\delta$ is the softmax function, for readability we will abandon the input and use $z , p$ in the following sections.
+
+Overview. We assume that the source pretrained model has already been trained. As discusses in the introduction, the target features output by the source model form clusters. We exploit this intrinsic structure of the target data for SFDA by considering the neighborhood information, and the adaptation is achieved with the following objective:
+
+$$
+\mathcal { L } = - \frac { 1 } { n _ { t } } \sum _ { x _ { i } \in \mathcal { D } _ { t } } \sum _ { x _ { j } \in \mathrm { N e i g h } ( x _ { i } ) } \frac { D _ { s i m } ( p _ { i } , p _ { j } ) } { D _ { d i s } ( x _ { i } , x _ { j } ) }
+$$
+
+where the $\mathrm { { N e i g h } } ( x _ { i } )$ means the nearest neighbors of $x _ { i }$ , $D _ { s i m }$ computes the similarity between predictions, and $D _ { d i s }$ is a constant measuring the semantic distance (dissimilarity) between data. The principle behind the objective is to push the data towards their semantically close neighbors by encouraging similar predictions. In the next sections, we will define $D _ { s i m }$ and $D _ { d i s }$ .
+
+# 3.1 Encouraging Class-Consistency with Neighborhood Affinity
+
+To achieve adaptation without source data, we use the prediction of the nearest neighbor to encourage prediction consistency. While the target features from the source model are not necessarily totally intrinsic discriminative, meaning some neighbors belong to different class and will provide the wrong supervision. To decrease the potentially negative impact of those neighbors, we propose to weigh the supervision from neighbors according to the connectivity (semantic similarity). We define affinity values to signify the connectivity between the neighbor and the feature, which corresponds to the $\frac { 1 } { D _ { d i s } }$ in Eq. 1 indicating the semantic similarity.
+
+To retrieve the nearest neighbors for batch training, similar to [33, 50, 62], we build two memory banks: $\mathcal { F }$ stores all target features, and $s$ stores corresponding prediction scores:
+
+$$
+\mathcal { F } = [ z _ { 1 } , z _ { 2 } , \dotsc , z _ { n _ { t } } ] \mathrm { a n d } \ S = [ p _ { 1 } , p _ { 2 } , \dotsc , p _ { n _ { t } } ]
+$$
+
+We use the cosine similarity for nearest neighbors retrieving. The difference between ours and [33, 50] lies in the fact that we utilize the memory bank to retrieve nearest neighbors while [33, 50] adopts the memory bank to compute the instance discrimination loss. Before every mini-batch training, we simply update the old items in the memory banks corresponding to current mini-batch. Note that updating the memory bank is only done to replace the old low-dimension vectors with new ones computed by the model, and does not require any additional computation.
+
+We then use the prediction of the neighbors to supervise the training weighted by the affinity values, with the following objective adapted from Eq. 1:
+
+$$
+\mathcal { L } _ { \mathcal { N } } = - \frac { 1 } { n _ { t } } \sum _ { i } \sum _ { k \in \mathcal { N } _ { K } ^ { i } } A _ { i k } \boldsymbol { S } _ { k } ^ { \top } \boldsymbol { p } _ { i }
+$$
+
+where we use the dot product to compute the similarity between predictions, corresponding to $D _ { s i m }$ in Eq.1, the $k$ is the index of the $k$ -th nearest neighbors of $z _ { i }$ , $\scriptstyle { S _ { k } }$ is the $k$ -th item in memory bank $s$ , $A _ { i k }$ is the affinity value of $k$ -th nearest neighbors of feature $z _ { i }$ . Here the $\mathcal { N } _ { K } ^ { i }$ is the index $\mathrm { { \dot { s e t } } } ^ { 2 }$ of the $K$ -nearest neighbors of feature $z _ { i }$ . Note that all neighbors are retrieved from the feature bank $\mathcal { F }$ . With the affinity value as weight, this objective pushes the features to their neighbors with strong connectivity and to a lesser degree to those with weak connectivity.
+
+To assign larger affinity values to semantic similar neighbors, we divide the nearest neighbors retrieved into two groups: reciprocal nearest neighbors (RNN) and non-reciprocal nearest neighbors (nRNN). The feature $z _ { j }$ is regarded as the RNN of the feature $z _ { i }$ if it meets the following condition:
+
+$$
+j \in \mathcal { N } _ { K } ^ { i } \wedge i \in \mathcal { N } _ { M } ^ { j }
+$$
+
+Other neighbors which do not meet the above condition are nRNN. Note that the normal definition of reciprocal nearest neighbors [31] applies $K = M$ , while in this paper $K$ and $M$ can be different. We find that reciprocal neighbors have a higher potential to belong to the same cluster as the feature (Fig. 1(b)). Thus, we assign a high affinity value to the RNN features. Specifically for feature $z _ { i }$ , the affinity value of its $j$ -th $\mathrm { K }$ -nearest neighbor is defined as:
+
+$$
+A _ { i , j } = { \left\{ \begin{array} { l l } { 1 } & { { \mathrm { i f ~ } } j \in { \mathcal { N } } _ { K } ^ { i } \land i \in { \mathcal { N } } _ { M } ^ { j } } \\ { r } & { { \mathrm { o t h e r w i s e . } } } \end{array} \right. }
+$$
+
+where $r$ is a hyperparameter. If not specified $r$ is set to 0.1.
+
+To further reduce the potential impact of noisy neighbors in $\mathcal { N } _ { K }$ , which belong to the different class but still are RNN, we propose a simply yet effective way dubbed self-regularization, that is, to not ignore the current prediction of ego feature:
+
+$$
+\mathcal { L } _ { s e l f } = - \frac { 1 } { n _ { t } } \sum _ { i } ^ { n _ { t } } S _ { i } ^ { \top } p _ { i }
+$$
+
+where $s _ { i }$ means the stored prediction in the memory bank, note this term is a constant vector and is identical to the $p _ { i }$ since we update the memory banks before the training, here the loss is only back-propagated for variable $p _ { i }$ .
+
+Require: $\mathcal { D } _ { s }$ (only for source model training), $\mathcal { D } _ { t }$
+
+1: Pre-train model on $\mathcal { D } _ { s }$   
+2: Build feature bank $\mathcal { F }$ and score bank $s$ for $\mathcal { D } _ { t }$   
+3: while Adaptation do   
+4: Sample batch $\tau$ from $\mathcal { D } _ { t }$   
+5: Update $\mathcal { F }$ and $s$ corresponding to current batch $\tau$   
+6: Retrieve nearest neighbors $\mathcal { N }$ for each of $\tau$   
+7: Compute affinity value $A$   
+8: Retrieve expanded neighborhoods $E$ for each of $\mathcal { N }$   
+9: Compute loss and update the model   
+10: end while
+
+. Eq.5 . Eq. 9
+
+To avoid the degenerated solution [8, 39] where the model predicts all data as some specific classes (and does not predict other classes for any of the target data), we encourage the prediction to be balanced. We adopt the prediction diversity loss which is widely used in clustering [8, 9, 13] and also in several domain adaptation works [21, 39, 42]:
+
+$$
+\mathcal { L } _ { d i v } = \sum _ { c = 1 } ^ { C } \mathrm { K L } ( \bar { p } _ { c } | | q _ { c } ) , \mathrm { w i t h } \bar { p } _ { c } = \frac { 1 } { n _ { t } } \sum _ { i } p _ { i } ^ { ( c ) } , \mathrm { a n d } q _ { \{ c = 1 , . . , C \} } = \frac { 1 } { C }
+$$
+
+where the $p _ { i } ^ { ( c ) }$ is the score of the $c$ -th class and $\bar { p } _ { c }$ is the empirical label distribution, it represents the predicted possibility of class $c$ and q is a uniform distribution.
+
+# 3.2 Expanded Neighborhood Affinity
+
+As mentioned in Sec. 1, a simple way to achieve the aggregation of more information is by considering more nearest neighbors. However, a drawback is that larger neighborhoods are expected to contain more datapoint from multiple classes, defying the purpose of class consistency. A better way to include more target features is by considering the $M$ -nearest neighbor of each neighbor in $\mathcal { N } _ { K }$ of $z _ { i }$ in Eq. 4, i.e., the expanded neighbors. These target features are expected to be closer on the target data manifold than the features that are included by considering a larger number of nearest neighbors [43]. The expanded neighbors of feature $z _ { i }$ are defined as $\bar { E _ { M } } ( z _ { i } ) \bar { = } \mathcal { N } _ { M } ( z _ { j } ) \forall j \in \mathcal { N } _ { K } ( z _ { i } \bar { ) }$ , note that $E _ { M } ( z _ { i } )$ is still an index set and $i$ (ego feature) $\not \in E _ { M } ( z _ { i } )$ . We directly assign a small affinity value $r$ to those expanded neighbors, since they are further than nearest neighbors and may contain noise. We utilize the prediction of those expanded neighborhoods for training:
+
+$$
+\mathcal { L } _ { E } = - \frac { 1 } { n _ { t } } \sum _ { i } \sum _ { k \in \mathcal { N } _ { K } ^ { i } } \sum _ { m \in E _ { M } ^ { k } } r \mathcal { S } _ { m } ^ { \top } p _ { i }
+$$
+
+where $E _ { M } ^ { k }$ contain the $M$ -nearest neighbors of neighbor $k$ in $\mathcal { N } _ { K }$
+
+Although the affinity values of all expanded neighbors are the same, it does not necessarily mean that they have equal importance. Taking a closer look at the expanded neighbors $E _ { M } ( z _ { i } )$ , some neighbors will show up more than once, for example $z _ { m }$ can be the nearest neighbor of both $z _ { h }$ and $z _ { j }$ where $h , j \in \mathcal N _ { K } ( \bar { z } _ { i } )$ , and the nearest neighbors can also serve as expanded neighbor. It implies that those neighbors form compact cluster, and we posit that those duplicated expanded neighbors have potential to be semantically closer to the ego-feature $z _ { i }$ . Thus, we do not remove duplicated features in $E _ { M } ( z _ { i } )$ , as those can lead to actually larger affinity value for those expanded neighbors. This is one advantage of utilizing expanded neighbors instead of more nearest neighbors, we will verify the importance of maintaining the duplicated features in the experimental section.
+
+Final objective. Our method, called Neighborhood Reciprocity Clustering (NRC), is illustrated in Algorithm. 1. The final objective for adaptation is:
+
+$$
+\mathcal { L } = \mathcal { L } _ { d i v } + \mathcal { L } _ { \mathcal { N } } + \mathcal { L } _ { E } + \mathcal { L } _ { s e l f } .
+$$
+
+# 4 Experiments
+
+Datasets. We use three 2D image benchmark datasets and a 3D point cloud recognition dataset.   
+Office-31 [32] contains 3 domains (Amazon, Webcam, DSLR) with 31 classes and 4,652 images.
+
+Table 1: Accuracies $( \% )$ on Office-31 for ResNet50-based methods.   
+
+<table><tr><td>Method</td><td>SF</td><td>A→D</td><td>A→W</td><td>D→W</td><td>W→D</td><td>D→A</td><td>W→A</td><td>Avg</td></tr><tr><td>MCD [35]</td><td>X</td><td>92.2</td><td>88.6</td><td>98.5</td><td>100.0</td><td>69.5</td><td>69.7</td><td>86.5</td></tr><tr><td>CDAN [24]</td><td>X</td><td>92.9</td><td>94.1</td><td>98.6</td><td>100.0</td><td>71.0</td><td>69.3</td><td>87.7</td></tr><tr><td>MDD [59]</td><td>X</td><td>90.4</td><td>90.4</td><td>98.7</td><td>99.9</td><td>75.0</td><td>73.7</td><td>88.0</td></tr><tr><td>BNM[4]</td><td>X</td><td>90.3</td><td>91.5</td><td>98.5</td><td>100.0</td><td>70.9</td><td>71.6</td><td>87.1</td></tr><tr><td>DMRL [49]</td><td>X</td><td>93.4</td><td>90.8</td><td>99.0</td><td>100.0</td><td>73.0</td><td>71.2</td><td>87.9</td></tr><tr><td>BDG[53]</td><td>X</td><td>93.6</td><td>93.6</td><td>99.0</td><td>100.0</td><td>73.2</td><td>72.0</td><td>88.5</td></tr><tr><td>MCC[15]</td><td>X</td><td>95.6</td><td>95.4</td><td>98.6</td><td>100.0</td><td>72.6</td><td>73.9</td><td>89.4</td></tr><tr><td>SRDC[42]</td><td>X</td><td>95.8</td><td>95.7</td><td>99.2</td><td>100.0</td><td>76.7</td><td>77.1</td><td>90.8</td></tr><tr><td>RWOT[51]</td><td>X</td><td>94.5</td><td>95.1</td><td>99.5</td><td>100.0</td><td>77.5</td><td>77.9</td><td>90.8</td></tr><tr><td>RSDA-MSTN[10]</td><td>X</td><td>95.8</td><td>96.1</td><td>99.3</td><td>100.0</td><td>77.4</td><td>78.9</td><td>91.1</td></tr><tr><td>SHOT [21]</td><td>√</td><td>94.0</td><td>90.1</td><td>98.4</td><td>99.9</td><td>74.7</td><td>74.3</td><td>88.6</td></tr><tr><td>3C-GAN[20]</td><td>【</td><td>92.7</td><td>93.7</td><td>98.5</td><td>99.8</td><td>75.3</td><td>77.8</td><td>89.6</td></tr><tr><td>NRC</td><td></td><td>96.0</td><td>90.8</td><td>99.0</td><td>100.0</td><td>75.3</td><td>75.0</td><td>89.4</td></tr></table>
+
+Table 2: Accuracies $( \% )$ on Office-Home for ResNet50-based methods.   
+
+<table><tr><td>Method</td><td></td><td>SFAr-&gt;CIAr-→PrAr-→RwC1-→ArCI-→PrCI-→&gt;RwPr-→ArPr-→&gt;CIPr-→RwRw-→ArRw-→CIRw-→PrAvg</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MCD [35]</td><td>xxxxxxxxxx</td><td>48.9 68.3</td><td>74.6</td><td>61.3</td><td>67.6</td><td>68.8</td><td>57.0</td><td>47.1</td><td>75.1</td><td>69.1</td><td>52.2</td><td>79.6</td><td>64.1</td></tr><tr><td>CDAN [24]</td><td></td><td>50.7</td><td>70.6 76.0</td><td>57.6</td><td>70.0</td><td>70.0</td><td>57.4</td><td>50.9</td><td>77.3</td><td>70.9</td><td>56.7</td><td>81.6</td><td>65.8</td></tr><tr><td>SAFN [52]</td><td></td><td>52.0</td><td>71.7 76.3</td><td>64.2</td><td>69.9</td><td>71.9</td><td>63.7</td><td>51.4</td><td>77.1</td><td>70.9</td><td>57.1</td><td>81.5</td><td>67.3</td></tr><tr><td>Symnets [58]</td><td></td><td>47.7 72.9</td><td>78.5</td><td>64.2</td><td>71.3</td><td>74.2</td><td>64.2</td><td>48.8</td><td>79.5</td><td>74.5</td><td>52.6</td><td>82.7</td><td>67.6</td></tr><tr><td>MDD [59]</td><td></td><td>54.9 73.7</td><td>77.8</td><td>60.0</td><td>71.4</td><td>71.8</td><td>61.2</td><td>53.6</td><td>78.1</td><td>72.5</td><td>60.2</td><td>82.3</td><td>68.1</td></tr><tr><td>TADA [47]</td><td></td><td>53.1</td><td>72.3 77.2</td><td>59.1</td><td>71.2</td><td>72.1</td><td>59.7</td><td>53.1</td><td>78.4</td><td>72.4</td><td>60.0</td><td>82.9</td><td>67.6</td></tr><tr><td>BNM[4]</td><td></td><td>52.3</td><td>73.9 80.0</td><td>63.3</td><td>72.9</td><td>74.9</td><td>61.7</td><td>49.5</td><td>79.7</td><td>70.5</td><td>53.6</td><td>82.2</td><td>67.9</td></tr><tr><td>BDG [53]</td><td></td><td>51.5 73.4</td><td>78.7</td><td>65.3</td><td>71.5</td><td>73.7</td><td>65.1</td><td>49.7</td><td>81.1</td><td>74.6</td><td>55.1</td><td>84.8</td><td>68.7</td></tr><tr><td>SRDC [42]</td><td></td><td>52.3 76.3</td><td>81.0</td><td>69.5</td><td>76.2</td><td>78.0</td><td>68.7</td><td>53.8</td><td>81.7</td><td>76.3</td><td>57.1</td><td>85.0</td><td>71.3</td></tr><tr><td>RSDA-MSTN[10]</td><td></td><td>53.2</td><td>77.7 81.3</td><td>66.4</td><td>74.0</td><td>76.5</td><td>67.9</td><td>53.0</td><td>82.0</td><td>75.8</td><td>57.8</td><td>85.4</td><td>70.9</td></tr><tr><td>SHOT [21]</td><td></td><td>57.1</td><td>78.1 81.5</td><td>68.0</td><td>78.2</td><td>78.1</td><td>67.4</td><td>54.9</td><td>82.2</td><td>73.3</td><td>58.8</td><td>84.3</td><td>71.8</td></tr><tr><td>NRC</td><td>区</td><td>57.7</td><td>80.3 82.0</td><td>68.1</td><td>79.8</td><td>78.6</td><td>65.3</td><td>56.4</td><td>83.0</td><td>71.0</td><td>58.6</td><td>85.6</td><td>72.2</td></tr></table>
+
+Office-Home [46] contains 4 domains (Real, Clipart, Art, Product) with 65 classes and a total of 15,500 images. VisDA [28] is a more challenging dataset, with 12-class synthetic-to-real object recognition tasks, its source domain contains of $1 5 2 \mathrm { k }$ synthetic images while the target domain has 55k real object images. PointDA-10 [30] is the first 3D point cloud benchmark specifically designed for domain adaptation, it has 3 domains with 10 classes, denoted as ModelNet-10, ShapeNet-10 and ScanNet-10, containing approximately $2 7 . 7 \mathrm { k }$ training and 5.1k testing images together.
+
+Evaluation. We compare with existing source-present and source-free DA methods. All results are the average on three random runs. SF in the tables denotes source-free.
+
+Model details. For fair comparison with related methods, we also adopt the backbone of ResNet-50 [11] for Office-Home and ResNet-101 for VisDA, and PointNet [29] for PointDA10. Specifically, for 2D image datasets, we use the same network architecture as SHOT [21], i.e., the final part of the network is: fully connected layer − Batch Normalization [12] − fully connected layer with weight normalization [37]. And for PointDA-10 [29], we use the code released by the authors for fair comparison with PointDAN [29], and only use the backbone without any of their proposed modules. To train the source model, we also adopt label smoothing as SHOT does. We adopt SGD with momentum 0.9 and batch size of 64 for all 2D datasets, and Adam for PointDA-10. The learning rate for Office-31 and Office-Home is set to 1e-3 for all layers, except for the last two newly added fc layers, where we apply 1e-2. Learning rates are set 10 times smaller for VisDA. Learning rate for PointDA-10 is set to 1e-6. We train 30 epochs for Office-31 and OfficeHome while 15 epochs for VisDA, and 100 for PointDA-10. For the number of nearest neighbors (K) and expanded neighborhoods (M), we use 3,2 for Office-31, Office-Home and PointDA-10, since VisDA is much larger we set K, M to 5. Experiments are conducted on a TITAN Xp.
+
+# 4.1 Results
+
+2D image datasets. We first evaluate the target performance of our method compared with existing DA and SFDA methods on three 2D image datasets. As shown in Table 1-3, the top part shows results for the source-present methods with access to source data during adaptation. The bottom shows results for the source-free DA methods. On Office-31, our method gets similar results compared with source-free method 3C-GAN and lower than source-present method RSDA-MSTN. And our method achieves state-of-the-art performance on Office-Home and VisDA, especially on VisDA our method surpasses the source-free method SHOT and source-present method RWOT by a wide margin $3 \%$ and $1 . 9 \%$ respectively). The reported results clearly demonstrate the efficiency of the proposed method for source-free domain adaptation. Interestingly, like already observed in the SHOT paper, source-free methods outperform methods that have access to source data during adaptation.
+
+Table 3: Accuracies $( \% )$ on VisDA-C (Synthesis Real) for ResNet101-based methods.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>SF</td><td rowspan=1 colspan=1>[SF|plane bcycl bus car horse knife mcycl person plant sktbrd train truck Per-class</td></tr><tr><td rowspan=5 colspan=1>ADR [34]CDAN [24]CDAN+BSP[2]SAFN [52]SWD[19]MDD [59]DMRL [49]MCC[15]STAR [26]RWOT[51]</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>94.248.584.0 72.990.174.292.6 72.580.861.882.2 28.8  73.5</td></tr><tr><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>85.266.983.0 50.884.274.988.1 74.583.476.081.9 38.0  73.9</td></tr><tr><td rowspan=3 colspan=1>×</td><td rowspan=2 colspan=1>61.081.0 57.5 89.080.690.1 77.084.277.982.1 38.4  75.9</td></tr><tr><td rowspan=1 colspan=1>92.493.690.8</td></tr><tr><td rowspan=1 colspan=1>93.661.384.1 70.6 94.179.091.8 79.689.955.689.0 24.4  76.190.882.5 81.7 70.5 91.769.586.3 77.587.463.685.6 29.2  76.4-    1    1   1   1   1    1     1    1    -    1   1    74.6-                =     =     =     =     =     =     =          =    75.588.780.3 80.5 71.5 90.1 93.285.0 71.689.473.8 85.0 36.9  78.895.084.084.6 73.0 91.691.885.9 78.494.484.787.0 42.2  82.795.180.383.7 90.092.468.092.5 82.287.978.490.4 68.2  84.0</td></tr><tr><td rowspan=3 colspan=1>3C-GAN [20]SHOT[21]NRC</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>94.873.468.8 74.893.195.488.6 84.7 89.184.783.5 48.1  81.6</td></tr><tr><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>94.388.580.1 57.3 93.194.980.7 80.391.589.186.3 58.2  82.9</td></tr><tr><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>96.891.382.4 62.4 96.295.986.1 80.694.894.190.4 59.7  85.9</td></tr></table>
+
+Table 4: Accuracies $( \% )$ on PointDA-10. The results except ours are from PointDAN [30].   
+
+<table><tr><td colspan="2"></td><td colspan="4">|SF|Model-→Shape Model-→Scan Shape-→Model Shape-&gt;Scan Scan→Model Scan-→Shape Avg</td></tr><tr><td>MMD [25]</td><td></td><td>57.5 27.9</td><td>40.7</td><td>26.7</td><td>47.3</td><td>54.8</td><td>42.5</td></tr><tr><td>DANN [6]</td><td>xxxxx</td><td>58.7 29.4</td><td>42.3</td><td>30.5</td><td>48.1</td><td>56.7</td><td>44.2</td></tr><tr><td>ADDA [44]</td><td></td><td>61.0 30.5</td><td>40.4</td><td>29.3</td><td>48.9</td><td>51.1</td><td>43.5</td></tr><tr><td>MCD [35]</td><td></td><td>62.0 31.0</td><td>41.4</td><td>31.3</td><td>46.8</td><td>59.3</td><td>45.3</td></tr><tr><td>PointDAN [30]</td><td></td><td>64.2 33.0</td><td>47.6</td><td>33.9</td><td>49.1</td><td>64.1</td><td>48.7</td></tr><tr><td>Source-only</td><td></td><td>43.1</td><td>17.3 40.0</td><td>15.0</td><td>33.9</td><td>47.1</td><td>32.7</td></tr><tr><td>NRC</td><td>&lt;</td><td>64.8</td><td>25.8 59.8</td><td>26.9</td><td>70.1</td><td>68.1</td><td>52.6</td></tr></table>
+
+3D point cloud dataset. We also report the result for the PointDA-10. As shown in Table 4, our method outperforms PointDA [30], which demands source data for adaptation and is specifically tailored for point cloud data with extra attention modules, by a large margin $(4 \% )$ .
+
+# 4.2 Analysis
+
+Ablation study on neighbors $\mathcal { N }$ , $E$ and affinity $A$ . In the first two tables of Table 5, we conduct the ablation study on Office-Home and VisDA. The 1-st row contains results from the source model and the 2-nd row from only training with the diversity loss $\mathcal { L } _ { d i v }$ . From the remaining rows, several conclusions can be drawn.
+
+First, the original supervision, which considers all neighbors equally can lead to a decent performance (67.1 on Office-Home). Second, considering higher affinity values for reciprocal neighbors leads to a large performance gain (69.1 on Office-Home). Last but not the least, the expanded neighborhoods can also be helpful, but only when combined with the affinity values $A$ (72.2 on Office-Home). Using expanded neighborhoods without affinity obtains bad performance (65,2 on Office-Home). We conjecture that those expanded neighborhoods, especially those neighbors of nRNN, may be noisy as discussed in Sec. 3.2. Removing the affinity $A$ means we treat all those neighbors equally, which is not reasonable.
+
+Table 5: Ablation study of different modules on Office-Home (left) and VisDA (middle), comparison between using expanded neighbors and larger nearest neighbors (right).   
+
+<table><tr><td>Ldiv</td><td>LN</td><td>LE LEA</td><td>Avg</td><td>Ldiv</td><td>LN</td><td></td><td>LE LEA</td><td></td><td>Acc</td></tr><tr><td></td><td></td><td></td><td>59.5</td><td></td><td></td><td></td><td></td><td></td><td>44.6</td></tr><tr><td></td><td></td><td></td><td>62.1</td><td></td><td></td><td></td><td></td><td></td><td>47.8</td></tr><tr><td></td><td></td><td></td><td>67.1</td><td></td><td></td><td></td><td></td><td></td><td>74.6</td></tr><tr><td></td><td></td><td></td><td>√ 69.1</td><td></td><td></td><td></td><td></td><td>√</td><td>81.5</td></tr><tr><td></td><td></td><td></td><td>65.2</td><td></td><td></td><td></td><td></td><td></td><td>61.2</td></tr><tr><td></td><td></td><td></td><td>72.2</td><td></td><td></td><td></td><td></td><td>！</td><td>85.9</td></tr><tr><td></td><td></td><td></td><td>69.1 [</td><td></td><td></td><td></td><td></td><td></td><td>82.0</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>Method&amp;Dataset</td><td rowspan=1 colspan=1>Acc</td></tr><tr><td rowspan=1 colspan=1>VisDA (K=M=5)VisDA w/o E (K=30)</td><td rowspan=1 colspan=1>85.984.0</td></tr><tr><td rowspan=1 colspan=1>OH(K=3,M=2)OH w/o E (K=9)</td><td rowspan=1 colspan=1>72.269.5</td></tr></table>
+
+Table 6: Runtime analysis on SHOT and our method. For SHOT, pseudo labels are computed at each epoch. $20 \%$ , $10 \%$ and $5 \%$ denote the percentage of target features which are stored in the memory bank.   
+
+<table><tr><td>VisDA</td><td colspan="2">Runtime (s/epoch)Per-class (%)</td></tr><tr><td>SHOT</td><td>618.82</td><td>82.9</td></tr><tr><td>NRC</td><td>540.89</td><td>85.9</td></tr><tr><td>NRC(20%) 6formemorybank)</td><td>507.15</td><td>85.3</td></tr><tr><td>NRC(10% for memory bank)</td><td>499.49</td><td>85.2</td></tr><tr><td>NRC(5% for memory bank)</td><td>499.28</td><td>85.1</td></tr></table>
+
+![](images/e1748829bba37f8f3b6d43d90dfdbea6be91a7b5cbb5e87bd18f0bf9bd522aa6.jpg)  
+Figure 2: (Left and middle) Ablation study of $\mathcal { L } _ { s e l f }$ on Office-Home and VisDA respectively. (Right) Performance with different $r$ on VisDA.
+
+We also show that duplication in the expanded neighbors is important in the last row of Table 5, where the $\mathcal { L } _ { \hat { E } }$ means we remove duplication in Eq. 8. The results show that the performance will degrade significantly when removing them, implying that the duplicated expanded neighbors are indeed more important than others.
+
+Next we ablate the importance of the expanded neighborhood in the right of Table5. We show that if we increase the number of datapoints considered for class-consistency by simply considering a larger K, we obtain significantly lower scores. We have chosen $K$ so that the total number of points considered is equal to our method (i.e. $5 { + } 5 ^ { * } 5 { = } 3 0$ and $3 + 3 ^ { * } 2 { = } 9 ,$ ). Considering neighbors of neighbors is more likely to provide datapoints that are close on the data manifold [43], and are therefore more likely to share the class label with the ego feature.
+
+Runtime analysis. Instead of storing all feature vectors in the memory bank, we follow the same memory bank setting as in [5] which is for nearest neighbor retrieval. The method only stores a fixed number of target features, we update the memory bank at the end of each iteration by taking the $n$ (batch size) embeddings from the current training iteration and concatenating them at the end of the memory bank, and discard the oldest $n$ elements from the memory bank. We report the results with this type of memory bank of different buffer size in the Table 6. The results show that indeed this could be an efficient way to reduce computation on very large datasets.
+
+Ablation study on self-regularization. In the left and middle of Fig 2, we show the results with and without self-regularization $\mathcal { L } _ { s e l f }$ . The $\mathcal { L } _ { s e l f }$ can improve the performance when adopting only nearest neighbors $\mathcal { N }$ or all neighbors $\mathcal { N } + E$ . The results imply that self-regularization can effectively reduce the negative impact of the potential noisy neighbors, especially on the Office-Home dataset.
+
+Sensitivity to hyperparameter. There are three hyperparameters in our method: K and M which are the number of nearest neighbors and expanded neighbors, $r$ which is the affinity value assigned to nRNN. We show the results with different $r$ in the right of Fig. 2. Note we keep the affinity of expanded neighbors as 0.1. $r = 1$ means no affinity. $r = - 1$ means treating supervision of nRNN feature as totally wrong, which is not always the case and will lead to quite lower result. $r = 0$ can also achieve good performance, signifying RNN can already work well. Results with $r = 0 . 1 / 0 . 1 5 / 0 . 2$ show that our method is not sensitive to the choice of a reasonable $r$ . Note in DA, there is no validation set for hyperparameter tuning, we show the results varying the number of neighbors in the right of Tab. 3, demonstrating the robustness to the choice of $K$ and $M$ .
+
+![](images/52c51cde4275a75094e8c62c8d4fd77f70977cdffb6dcc8e5fcf1796cfa99e42.jpg)  
+Figure 3: (Left) The three curves are (on VisDA): target accuracy (Blue), ratio of features which have 5-nearest neighbors all sharing the same predicted label (dashed Red), and ratio of features which have 5-nearest neighbors all sharing the same and correct predicted label (dashed Black). (Right) Ablation study on choice of K and M on VisDA.
+
+![](images/fd8c9cc81f0a4968ed7f3124861b6deeb77a38f4952bd4cbfedaa30cfe215f98.jpg)  
+Figure 4: (Left) Ratio of different type of nearest neighbor features which have the correct predicted label, before and after adaptation. (Right) Visualization of target features after adaptation.
+
+Training curve. We show the evolution of several statistics during adaptation on VisDA in the left of Tab. 3. The blue curve is the target accuracy. The dashed red and black curves are the ratio of features which have 5-nearest neighbors all sharing the same (dashed Red), or the same and also correct (dashed Black) predicted label. The curves show that the target features are clustering during the training. Another interesting finding is that the curve ’Per Shared’ correlates with the accuracy curve, which might therefore be used to determine training convergence.
+
+Accuracy of supervision from neighbors. We also show the accuracy of supervision from neighbors on task $\mathrm { A r } {  } \mathrm { R w }$ of Office-Home in Fig. 4(left). It shows that after adaptation, the ratio of all types of neighbors having more correct predicted label, proving the effectiveness of the method.
+
+t-SNE visualization. We show the t-SNE feature visualization on task $\mathrm { A r } {  } \mathrm { R w }$ of target features before (Fig. 1(a)) and after (Fig. 4(right)) adaptation. After adaptation, the features are more compactly clustered.
+
+# 5 Conclusions
+
+We introduce a source-free domain adaptation (SFDA) method by uncovering the intrinsic target data structure. We propose to achieve the adaptation by encouraging label consistency among local target features. We differentiate between nearest neighbors, reciprocal neighbors and expanded neighborhood. Experimental results verify the importance of considering the local structure of the target features. Finally, our experimental results on both 2D image and 3D point cloud datasets testify the efficacy of our method.
+
+Acknowledgement We acknowledge the support from Huawei Kirin Solution, and the project PID2019-104174GB-I00 (MINECO, Spain) and RTI2018-102285-A-I00 (MICINN, Spain), Ramón y Cajal fellowship RYC2019-027020-I, and the CERCA Programme of Generalitat de Catalunya.
+
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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]   
+(b) Did you describe the limitations of your work? [No]   
+(c) Did you discuss any potential negative societal impacts of your work? [No]   
+(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? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+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] We attach the code in the supplemental material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] As in the model details in Sec.4   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] All main results are average over three running with random seeds.   
+(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]
+
+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? [No]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No]
+
+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/yhjpeuWepoj/yhjpeuWepoj_content_list.json

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+        "text": "Exploiting the Intrinsic Neighborhood Structure for Source-free Domain Adaptation ",
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+        "text": "Shiqi Yang1, Yaxing Wang1,2∗, Joost van de Weijer1, Luis Herranz1, Shangling Jui3 1 Computer Vision Center, Universitat Autonoma de Barcelona, Barcelona, Spain 2 PCALab, Nanjing University of Science and Technology, China 3 Huawei Kirin Solution, Shanghai, China   \n{syang,yaxing,joost,lherranz}@cvc.uab.es, jui.shangling@huawei.com ",
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+        "text": "Abstract ",
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+        "text": "Domain adaptation (DA) aims to alleviate the domain shift between source domain and target domain. Most DA methods require access to the source data, but often that is not possible (e.g. due to data privacy or intellectual property). In this paper, we address the challenging source-free domain adaptation (SFDA) problem, where the source pretrained model is adapted to the target domain in the absence of source data. Our method is based on the observation that target data, which might no longer align with the source domain classifier, still forms clear clusters. We capture this intrinsic structure by defining local affinity of the target data, and encourage label consistency among data with high local affinity. We observe that higher affinity should be assigned to reciprocal neighbors, and propose a self regularization loss to decrease the negative impact of noisy neighbors. Furthermore, to aggregate information with more context, we consider expanded neighborhoods with small affinity values. In the experimental results we verify that the inherent structure of the target features is an important source of information for domain adaptation. We demonstrate that this local structure can be efficiently captured by considering the local neighbors, the reciprocal neighbors, and the expanded neighborhood. Finally, we achieve state-of-the-art performance on several 2D image and 3D point cloud recognition datasets. Code is available in https://github.com/Albert0147/SFDA_neighbors. ",
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+        "text": "1 Introduction ",
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+        "text": "Most deep learning methods rely on training on large amount of labeled data, while they cannot generalize well to a related yet different domain. One research direction to address this issue is Domain Adaptation (DA), which aims to transfer learned knowledge from a source to a target domain. Most existing DA methods demand labeled source data during the adaptation period, however, it is often not practical that source data are always accessible, such as when applied on data with privacy or property restrictions. Therefore, recently, there have emerged a few works [16, 17, 20, 21] tackling a new challenging DA scenario where instead of source data only the source pretrained model is available for adapting, i.e., source-free domain adaptation (SFDA). Among these methods, USFDA [16] addresses universal DA [57] and SF [17] addresses open-set DA [36]. In both universal and open-set DA the label set is different for source and target domains. SHOT [21] and 3C-GAN [20] are for closed-set DA where source and target domains have the same categories. 3C-GAN [20] is based on target-style image generation with a conditional GAN, and SHOT [21] is based on mutual information maximization and pseudo labeling. Finally, BAIT [56] extends MCD [35] to the SFDA setting. However, these methods ignore the intrinsic neighborhood structure of the target data in feature space which can be very valuable to tackle SFDA. ",
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+        "image_caption": [
+            "Figure 1: (a) t-SNE visualization of target features by source model. (b) Ratio of different type of nearest neighbor features of which: the predicted label is the same as the feature, K is the number of nearest neighbors. The features in (a) and (b) are on task $\\mathrm { A r } {  } \\mathrm { R w }$ of Office-Home. (c) Illustration of our method. In the left shows we distinguish reciprocal and non-reciprocal neighbors. The adaptation is achieved by pushed the features towards reciprocal neighbors heavily. "
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+        "text": "In this paper, we focus on closed-set source-free domain adaptation. Our main observation is that current DA methods do not exploit the intrinsic neighborhood structure of the target data. We use this term to refer to the fact that, even though the target data might have shifted in the feature space (due to the covariance shift), target data of the same class is still expected to form a cluster in the embedding space. This can be implied to some degree from the t-SNE visualization of target features on the source model which suggests that significant cluster structure is preserved (see Fig. 1 (a)). This assumption is implicitly adopted by most DA methods, as instantiated by a recent DA work [42]. A well-established way to assess the structure of points in high-dimensional spaces is by considering the nearest neighbors of points, which are expected to belong to the same class. However, this assumption is not true for all points; the blue curve in Figure 1(b) shows that around $7 5 \\%$ of the nearest neighbors has the correct label. In this paper, we observe that this problem can be mitigated by considering reciprocal nearest neighbors (RNN); the reciprocal neighbors of a point have the point as their neighbor. Reciprocal neighbors have been studied before in different contexts [14, 31, 60]. The reason why reciprocal neighbors are more trustworthy is illustrated in Fig. 1(c). Fig. 1(b) shows the ratio of neighbors which have the correct prediction for different kinds of nearest neighbors. The curves show that reciprocal neighbors indeed have more chances to predict the true label than non-reciprocal nearest neighbors (nRNN). ",
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+        "text": "The above observation and analysis motivate us to assign different weights to the supervision from nearest neighbors. Our method, called Neighborhood Reciprocity Clustering (NRC), achieves sourcefree domain adaptation by encouraging reciprocal neighbors to concord in their label prediction. In addition, we will also consider a weaker connection to the non-reciprocal neighbors. We define affinity values to describe the degree of connectivity between each data point and its neighbors, which is also utilized to encourage class-consistency between neighbors, and we propose to use a self-regularization to decrease the negative impact of potential noisy neighbors. Furthermore, inspired by recent graph based methods [1, 3, 61] which show that the higher order neighbors can provide relevant context, and also considering neighbors of neighbors is more likely to provide datapoints that are close on the data manifold [43]. Thus, to aggregate wider local information, we further retrieve the expanded neighbors, i.e, neighbor of the nearest neighbors, for auxiliary supervision. ",
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+        "text": "Our contributions can be summarized as follows, to achieve source-free domain adaptation: (i) we explicitly exploit the fact that same-class data forms cluster in the target embedding space, we do this by considering the predictions of neighbors and reciprocal neighbors, (ii) we further show that considering an extended neighborhood of data points further improves results (iii) the experiments results on three 2D image datasets and one 3D point cloud dataset show that our method achieves state-of-the-art performance compared with related methods. ",
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+        "text": "2 Related Work ",
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+        "text": "Domain Adaptation. Most DA methods tackle domain shift by aligning the feature distributions. Early DA methods such as [23, 41, 45] adopt moment matching to align feature distributions. And in recent years, plenty of works have emerged that achieve alignment by adversarial training. DANN [7] formulates domain adaptation as an adversarial two-player game. The adversarial training of CDAN [24] is conditioned on several sources of information. DIRT-T [40] performs domain adversarial training with an added term that penalizes violations of the cluster assumption. Additionally, [18, 26, 35] adopts prediction diversity between multiple learnable classifiers to achieve local or category-level feature alignment between source and target domains. AFN [52] shows that the erratic discrimination of target features stems from much smaller norms than those found in the source features. SRDC [42] proposes to directly uncover the intrinsic target discrimination via discriminative clustering to achieve adaptation. More related, [27] resorts to K-means clustering for open-set adaptation while considering global structure. Our method instead only focuses on nearest neighbors (local structure) for source-free adaptation. ",
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+        "type": "text",
+        "text": "Source-free Domain Adaptation. Source-present methods need supervision from the source domain during adaptation. Recently, there are several methods investigating source-free domain adaptation. USFDA [16] and FS [17] explore source-free universal DA [57] and open-set DA [36], and they propose to synthesize extra training samples to make the decision boundary compact, thereby allowing to recognise the open classes. For closed-set DA setting. SHOT [21] proposes to fix the source classifier and match the target features to the fixed classifier by maximizing mutual information and a proposed pseudo label strategy which considers global structure. 3C-GAN [20] synthesizes labeled target-style training images based on the conditional GAN to provide supervision for adaptation. Finally, SFDA [22] is for segmentation based on synthesizing fake source samples. ",
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+        "text": "Graph Clustering. Our method shares some similarities with graph clustering work such as [38, 48, 54, 55] by utilizing neighborhood information. However, our methods are fundamentally different. Unlike those works which require labeled data to train the graph network for estimating the affinity, we instead adopt reciprocity to assign affinity. ",
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+    {
+        "type": "text",
+        "text": "3 Method ",
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+    {
+        "type": "text",
+        "text": "Notation. We denote the labeled source domain data with $n _ { s }$ samples as $\\mathcal { D } _ { s } = \\{ ( x _ { i } ^ { s } , y _ { i } ^ { s } ) \\} _ { i = 1 } ^ { n _ { s } }$ , where $y _ { i } ^ { s }$ orresponding label of . Both domains have t $x _ { i } ^ { s }$ , andsame e unlabeled target domain data with classes (closed-set setting). Under the $n _ { t }$ samples asFDA setting $\\mathcal { D } _ { t } \\overset { \\vartriangle } { = } \\{ x _ { j } ^ { t } \\} _ { j = 1 } ^ { n _ { t } }$ $C$ $\\mathcal { D } _ { s }$ is only available for model pretraining. Our method is based on a neural network, which we split into two parts: a feature extractor $f$ , and a classifier $g$ . The feature output by the feature extractor is denoted as $z ( x ) = f \\left( x \\right)$ , the output of network is denoted as $p ( x ) = \\bar { \\delta } ( g ( \\dot { z } ) ) \\in \\mathcal { R } ^ { C }$ where $\\delta$ is the softmax function, for readability we will abandon the input and use $z , p$ in the following sections. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Overview. We assume that the source pretrained model has already been trained. As discusses in the introduction, the target features output by the source model form clusters. We exploit this intrinsic structure of the target data for SFDA by considering the neighborhood information, and the adaptation is achieved with the following objective: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f5201ec763543d11770c0d5cb15e97a623ee3202a252898dd2e123a078de00d7.jpg",
+        "text": "$$\n\\mathcal { L } = - \\frac { 1 } { n _ { t } } \\sum _ { x _ { i } \\in \\mathcal { D } _ { t } } \\sum _ { x _ { j } \\in \\mathrm { N e i g h } ( x _ { i } ) } \\frac { D _ { s i m } ( p _ { i } , p _ { j } ) } { D _ { d i s } ( x _ { i } , x _ { j } ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the $\\mathrm { { N e i g h } } ( x _ { i } )$ means the nearest neighbors of $x _ { i }$ , $D _ { s i m }$ computes the similarity between predictions, and $D _ { d i s }$ is a constant measuring the semantic distance (dissimilarity) between data. The principle behind the objective is to push the data towards their semantically close neighbors by encouraging similar predictions. In the next sections, we will define $D _ { s i m }$ and $D _ { d i s }$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "3.1 Encouraging Class-Consistency with Neighborhood Affinity ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To achieve adaptation without source data, we use the prediction of the nearest neighbor to encourage prediction consistency. While the target features from the source model are not necessarily totally intrinsic discriminative, meaning some neighbors belong to different class and will provide the wrong supervision. To decrease the potentially negative impact of those neighbors, we propose to weigh the supervision from neighbors according to the connectivity (semantic similarity). We define affinity values to signify the connectivity between the neighbor and the feature, which corresponds to the $\\frac { 1 } { D _ { d i s } }$ in Eq. 1 indicating the semantic similarity. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To retrieve the nearest neighbors for batch training, similar to [33, 50, 62], we build two memory banks: $\\mathcal { F }$ stores all target features, and $s$ stores corresponding prediction scores: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/77d28a00bc4b339a6c125a9e814ebeed0f4f2af38733b940ab6aa2d9c7ee636a.jpg",
+        "text": "$$\n\\mathcal { F } = [ z _ { 1 } , z _ { 2 } , \\dotsc , z _ { n _ { t } } ] \\mathrm { a n d } \\ S = [ p _ { 1 } , p _ { 2 } , \\dotsc , p _ { n _ { t } } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use the cosine similarity for nearest neighbors retrieving. The difference between ours and [33, 50] lies in the fact that we utilize the memory bank to retrieve nearest neighbors while [33, 50] adopts the memory bank to compute the instance discrimination loss. Before every mini-batch training, we simply update the old items in the memory banks corresponding to current mini-batch. Note that updating the memory bank is only done to replace the old low-dimension vectors with new ones computed by the model, and does not require any additional computation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We then use the prediction of the neighbors to supervise the training weighted by the affinity values, with the following objective adapted from Eq. 1: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ba2c563d21377f4062e4a0eb8b3c9ea39a500ab7b58d34337af1aa5687d187a1.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathcal { N } } = - \\frac { 1 } { n _ { t } } \\sum _ { i } \\sum _ { k \\in \\mathcal { N } _ { K } ^ { i } } A _ { i k } \\boldsymbol { S } _ { k } ^ { \\top } \\boldsymbol { p } _ { i }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            402,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where we use the dot product to compute the similarity between predictions, corresponding to $D _ { s i m }$ in Eq.1, the $k$ is the index of the $k$ -th nearest neighbors of $z _ { i }$ , $\\scriptstyle { S _ { k } }$ is the $k$ -th item in memory bank $s$ , $A _ { i k }$ is the affinity value of $k$ -th nearest neighbors of feature $z _ { i }$ . Here the $\\mathcal { N } _ { K } ^ { i }$ is the index $\\mathrm { { \\dot { s e t } } } ^ { 2 }$ of the $K$ -nearest neighbors of feature $z _ { i }$ . Note that all neighbors are retrieved from the feature bank $\\mathcal { F }$ . With the affinity value as weight, this objective pushes the features to their neighbors with strong connectivity and to a lesser degree to those with weak connectivity. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To assign larger affinity values to semantic similar neighbors, we divide the nearest neighbors retrieved into two groups: reciprocal nearest neighbors (RNN) and non-reciprocal nearest neighbors (nRNN). The feature $z _ { j }$ is regarded as the RNN of the feature $z _ { i }$ if it meets the following condition: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a9d9b4a442e9d237d920ac30ca38baaa15d164bc7139d9d474453e1248ee9ec9.jpg",
+        "text": "$$\nj \\in \\mathcal { N } _ { K } ^ { i } \\wedge i \\in \\mathcal { N } _ { M } ^ { j }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Other neighbors which do not meet the above condition are nRNN. Note that the normal definition of reciprocal nearest neighbors [31] applies $K = M$ , while in this paper $K$ and $M$ can be different. We find that reciprocal neighbors have a higher potential to belong to the same cluster as the feature (Fig. 1(b)). Thus, we assign a high affinity value to the RNN features. Specifically for feature $z _ { i }$ , the affinity value of its $j$ -th $\\mathrm { K }$ -nearest neighbor is defined as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7c68533c93cb96b6505a8381ce377b48d20e35c00d1231df58593a8ccd6a27f2.jpg",
+        "text": "$$\nA _ { i , j } = { \\left\\{ \\begin{array} { l l } { 1 } & { { \\mathrm { i f ~ } } j \\in { \\mathcal { N } } _ { K } ^ { i } \\land i \\in { \\mathcal { N } } _ { M } ^ { j } } \\\\ { r } & { { \\mathrm { o t h e r w i s e . } } } \\end{array} \\right. }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $r$ is a hyperparameter. If not specified $r$ is set to 0.1. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To further reduce the potential impact of noisy neighbors in $\\mathcal { N } _ { K }$ , which belong to the different class but still are RNN, we propose a simply yet effective way dubbed self-regularization, that is, to not ignore the current prediction of ego feature: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/06558638152092a6f3c523e045df68d43ae228dd6b46d434df06a6d7d21bf3ce.jpg",
+        "text": "$$\n\\mathcal { L } _ { s e l f } = - \\frac { 1 } { n _ { t } } \\sum _ { i } ^ { n _ { t } } S _ { i } ^ { \\top } p _ { i }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            799,
+            562,
+            839
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $s _ { i }$ means the stored prediction in the memory bank, note this term is a constant vector and is identical to the $p _ { i }$ since we update the memory banks before the training, here the loss is only back-propagated for variable $p _ { i }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Require: $\\mathcal { D } _ { s }$ (only for source model training), $\\mathcal { D } _ { t }$ ",
+        "bbox": [
+            176,
+            111,
+            472,
+            123
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1: Pre-train model on $\\mathcal { D } _ { s }$   \n2: Build feature bank $\\mathcal { F }$ and score bank $s$ for $\\mathcal { D } _ { t }$   \n3: while Adaptation do   \n4: Sample batch $\\tau$ from $\\mathcal { D } _ { t }$   \n5: Update $\\mathcal { F }$ and $s$ corresponding to current batch $\\tau$   \n6: Retrieve nearest neighbors $\\mathcal { N }$ for each of $\\tau$   \n7: Compute affinity value $A$   \n8: Retrieve expanded neighborhoods $E$ for each of $\\mathcal { N }$   \n9: Compute loss and update the model   \n10: end while ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": ". Eq.5 . Eq. 9 ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To avoid the degenerated solution [8, 39] where the model predicts all data as some specific classes (and does not predict other classes for any of the target data), we encourage the prediction to be balanced. We adopt the prediction diversity loss which is widely used in clustering [8, 9, 13] and also in several domain adaptation works [21, 39, 42]: ",
+        "bbox": [
+            173,
+            273,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/573d90daac2f84185a8ae91be8ed3864534da43c9506209ed0e2158432214e91.jpg",
+        "text": "$$\n\\mathcal { L } _ { d i v } = \\sum _ { c = 1 } ^ { C } \\mathrm { K L } ( \\bar { p } _ { c } | | q _ { c } ) , \\mathrm { w i t h } \\bar { p } _ { c } = \\frac { 1 } { n _ { t } } \\sum _ { i } p _ { i } ^ { ( c ) } , \\mathrm { a n d } q _ { \\{ c = 1 , . . , C \\} } = \\frac { 1 } { C }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            256,
+            333,
+            709,
+            376
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the $p _ { i } ^ { ( c ) }$ is the score of the $c$ -th class and $\\bar { p } _ { c }$ is the empirical label distribution, it represents the predicted possibility of class $c$ and q is a uniform distribution. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 Expanded Neighborhood Affinity ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As mentioned in Sec. 1, a simple way to achieve the aggregation of more information is by considering more nearest neighbors. However, a drawback is that larger neighborhoods are expected to contain more datapoint from multiple classes, defying the purpose of class consistency. A better way to include more target features is by considering the $M$ -nearest neighbor of each neighbor in $\\mathcal { N } _ { K }$ of $z _ { i }$ in Eq. 4, i.e., the expanded neighbors. These target features are expected to be closer on the target data manifold than the features that are included by considering a larger number of nearest neighbors [43]. The expanded neighbors of feature $z _ { i }$ are defined as $\\bar { E _ { M } } ( z _ { i } ) \\bar { = } \\mathcal { N } _ { M } ( z _ { j } ) \\forall j \\in \\mathcal { N } _ { K } ( z _ { i } \\bar { ) }$ , note that $E _ { M } ( z _ { i } )$ is still an index set and $i$ (ego feature) $\\not \\in E _ { M } ( z _ { i } )$ . We directly assign a small affinity value $r$ to those expanded neighbors, since they are further than nearest neighbors and may contain noise. We utilize the prediction of those expanded neighborhoods for training: ",
+        "bbox": [
+            173,
+            444,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/191bdd0d5b01784262feba9ccaf69858692f4d6b2b63d4147a304c98893c147e.jpg",
+        "text": "$$\n\\mathcal { L } _ { E } = - \\frac { 1 } { n _ { t } } \\sum _ { i } \\sum _ { k \\in \\mathcal { N } _ { K } ^ { i } } \\sum _ { m \\in E _ { M } ^ { k } } r \\mathcal { S } _ { m } ^ { \\top } p _ { i }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            585,
+            602,
+            627
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $E _ { M } ^ { k }$ contain the $M$ -nearest neighbors of neighbor $k$ in $\\mathcal { N } _ { K }$ ",
+        "bbox": [
+            173,
+            631,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Although the affinity values of all expanded neighbors are the same, it does not necessarily mean that they have equal importance. Taking a closer look at the expanded neighbors $E _ { M } ( z _ { i } )$ , some neighbors will show up more than once, for example $z _ { m }$ can be the nearest neighbor of both $z _ { h }$ and $z _ { j }$ where $h , j \\in \\mathcal N _ { K } ( \\bar { z } _ { i } )$ , and the nearest neighbors can also serve as expanded neighbor. It implies that those neighbors form compact cluster, and we posit that those duplicated expanded neighbors have potential to be semantically closer to the ego-feature $z _ { i }$ . Thus, we do not remove duplicated features in $E _ { M } ( z _ { i } )$ , as those can lead to actually larger affinity value for those expanded neighbors. This is one advantage of utilizing expanded neighbors instead of more nearest neighbors, we will verify the importance of maintaining the duplicated features in the experimental section. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Final objective. Our method, called Neighborhood Reciprocity Clustering (NRC), is illustrated in Algorithm. 1. The final objective for adaptation is: ",
+        "bbox": [
+            171,
+            782,
+            821,
+            810
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ae8aa11a8013cac6ffb5c6ceca6f5aefe7cec77b25048817ebdc48852e90b837.jpg",
+        "text": "$$\n\\mathcal { L } = \\mathcal { L } _ { d i v } + \\mathcal { L } _ { \\mathcal { N } } + \\mathcal { L } _ { E } + \\mathcal { L } _ { s e l f } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            852,
+            312,
+            869
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Datasets. We use three 2D image benchmark datasets and a 3D point cloud recognition dataset.   \nOffice-31 [32] contains 3 domains (Amazon, Webcam, DSLR) with 31 classes and 4,652 images. ",
+        "bbox": [
+            173,
+            882,
+            825,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/d30542c42975f40b60b9b182d2926ef49a24c6a1af57e99813de1d1fb8002c91.jpg",
+        "table_caption": [
+            "Table 1: Accuracies $( \\% )$ on Office-31 for ResNet50-based methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>SF</td><td>A→D</td><td>A→W</td><td>D→W</td><td>W→D</td><td>D→A</td><td>W→A</td><td>Avg</td></tr><tr><td>MCD [35]</td><td>X</td><td>92.2</td><td>88.6</td><td>98.5</td><td>100.0</td><td>69.5</td><td>69.7</td><td>86.5</td></tr><tr><td>CDAN [24]</td><td>X</td><td>92.9</td><td>94.1</td><td>98.6</td><td>100.0</td><td>71.0</td><td>69.3</td><td>87.7</td></tr><tr><td>MDD [59]</td><td>X</td><td>90.4</td><td>90.4</td><td>98.7</td><td>99.9</td><td>75.0</td><td>73.7</td><td>88.0</td></tr><tr><td>BNM[4]</td><td>X</td><td>90.3</td><td>91.5</td><td>98.5</td><td>100.0</td><td>70.9</td><td>71.6</td><td>87.1</td></tr><tr><td>DMRL [49]</td><td>X</td><td>93.4</td><td>90.8</td><td>99.0</td><td>100.0</td><td>73.0</td><td>71.2</td><td>87.9</td></tr><tr><td>BDG[53]</td><td>X</td><td>93.6</td><td>93.6</td><td>99.0</td><td>100.0</td><td>73.2</td><td>72.0</td><td>88.5</td></tr><tr><td>MCC[15]</td><td>X</td><td>95.6</td><td>95.4</td><td>98.6</td><td>100.0</td><td>72.6</td><td>73.9</td><td>89.4</td></tr><tr><td>SRDC[42]</td><td>X</td><td>95.8</td><td>95.7</td><td>99.2</td><td>100.0</td><td>76.7</td><td>77.1</td><td>90.8</td></tr><tr><td>RWOT[51]</td><td>X</td><td>94.5</td><td>95.1</td><td>99.5</td><td>100.0</td><td>77.5</td><td>77.9</td><td>90.8</td></tr><tr><td>RSDA-MSTN[10]</td><td>X</td><td>95.8</td><td>96.1</td><td>99.3</td><td>100.0</td><td>77.4</td><td>78.9</td><td>91.1</td></tr><tr><td>SHOT [21]</td><td>√</td><td>94.0</td><td>90.1</td><td>98.4</td><td>99.9</td><td>74.7</td><td>74.3</td><td>88.6</td></tr><tr><td>3C-GAN[20]</td><td>【</td><td>92.7</td><td>93.7</td><td>98.5</td><td>99.8</td><td>75.3</td><td>77.8</td><td>89.6</td></tr><tr><td>NRC</td><td></td><td>96.0</td><td>90.8</td><td>99.0</td><td>100.0</td><td>75.3</td><td>75.0</td><td>89.4</td></tr></table>",
+        "bbox": [
+            209,
+            116,
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+            292
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/0cdd1c9003321317d28d8b4559d03bcd25da016f57b307b23642e9f94c25ffba.jpg",
+        "table_caption": [
+            "Table 2: Accuracies $( \\% )$ on Office-Home for ResNet50-based methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td></td><td>SFAr-&gt;CIAr-→PrAr-→RwC1-→ArCI-→PrCI-→&gt;RwPr-→ArPr-→&gt;CIPr-→RwRw-→ArRw-→CIRw-→PrAvg</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MCD [35]</td><td>xxxxxxxxxx</td><td>48.9 68.3</td><td>74.6</td><td>61.3</td><td>67.6</td><td>68.8</td><td>57.0</td><td>47.1</td><td>75.1</td><td>69.1</td><td>52.2</td><td>79.6</td><td>64.1</td></tr><tr><td>CDAN [24]</td><td></td><td>50.7</td><td>70.6 76.0</td><td>57.6</td><td>70.0</td><td>70.0</td><td>57.4</td><td>50.9</td><td>77.3</td><td>70.9</td><td>56.7</td><td>81.6</td><td>65.8</td></tr><tr><td>SAFN [52]</td><td></td><td>52.0</td><td>71.7 76.3</td><td>64.2</td><td>69.9</td><td>71.9</td><td>63.7</td><td>51.4</td><td>77.1</td><td>70.9</td><td>57.1</td><td>81.5</td><td>67.3</td></tr><tr><td>Symnets [58]</td><td></td><td>47.7 72.9</td><td>78.5</td><td>64.2</td><td>71.3</td><td>74.2</td><td>64.2</td><td>48.8</td><td>79.5</td><td>74.5</td><td>52.6</td><td>82.7</td><td>67.6</td></tr><tr><td>MDD [59]</td><td></td><td>54.9 73.7</td><td>77.8</td><td>60.0</td><td>71.4</td><td>71.8</td><td>61.2</td><td>53.6</td><td>78.1</td><td>72.5</td><td>60.2</td><td>82.3</td><td>68.1</td></tr><tr><td>TADA [47]</td><td></td><td>53.1</td><td>72.3 77.2</td><td>59.1</td><td>71.2</td><td>72.1</td><td>59.7</td><td>53.1</td><td>78.4</td><td>72.4</td><td>60.0</td><td>82.9</td><td>67.6</td></tr><tr><td>BNM[4]</td><td></td><td>52.3</td><td>73.9 80.0</td><td>63.3</td><td>72.9</td><td>74.9</td><td>61.7</td><td>49.5</td><td>79.7</td><td>70.5</td><td>53.6</td><td>82.2</td><td>67.9</td></tr><tr><td>BDG [53]</td><td></td><td>51.5 73.4</td><td>78.7</td><td>65.3</td><td>71.5</td><td>73.7</td><td>65.1</td><td>49.7</td><td>81.1</td><td>74.6</td><td>55.1</td><td>84.8</td><td>68.7</td></tr><tr><td>SRDC [42]</td><td></td><td>52.3 76.3</td><td>81.0</td><td>69.5</td><td>76.2</td><td>78.0</td><td>68.7</td><td>53.8</td><td>81.7</td><td>76.3</td><td>57.1</td><td>85.0</td><td>71.3</td></tr><tr><td>RSDA-MSTN[10]</td><td></td><td>53.2</td><td>77.7 81.3</td><td>66.4</td><td>74.0</td><td>76.5</td><td>67.9</td><td>53.0</td><td>82.0</td><td>75.8</td><td>57.8</td><td>85.4</td><td>70.9</td></tr><tr><td>SHOT [21]</td><td></td><td>57.1</td><td>78.1 81.5</td><td>68.0</td><td>78.2</td><td>78.1</td><td>67.4</td><td>54.9</td><td>82.2</td><td>73.3</td><td>58.8</td><td>84.3</td><td>71.8</td></tr><tr><td>NRC</td><td>区</td><td>57.7</td><td>80.3 82.0</td><td>68.1</td><td>79.8</td><td>78.6</td><td>65.3</td><td>56.4</td><td>83.0</td><td>71.0</td><td>58.6</td><td>85.6</td><td>72.2</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Office-Home [46] contains 4 domains (Real, Clipart, Art, Product) with 65 classes and a total of 15,500 images. VisDA [28] is a more challenging dataset, with 12-class synthetic-to-real object recognition tasks, its source domain contains of $1 5 2 \\mathrm { k }$ synthetic images while the target domain has 55k real object images. PointDA-10 [30] is the first 3D point cloud benchmark specifically designed for domain adaptation, it has 3 domains with 10 classes, denoted as ModelNet-10, ShapeNet-10 and ScanNet-10, containing approximately $2 7 . 7 \\mathrm { k }$ training and 5.1k testing images together. ",
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+    {
+        "type": "text",
+        "text": "Evaluation. We compare with existing source-present and source-free DA methods. All results are the average on three random runs. SF in the tables denotes source-free. ",
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+    {
+        "type": "text",
+        "text": "Model details. For fair comparison with related methods, we also adopt the backbone of ResNet-50 [11] for Office-Home and ResNet-101 for VisDA, and PointNet [29] for PointDA10. Specifically, for 2D image datasets, we use the same network architecture as SHOT [21], i.e., the final part of the network is: fully connected layer − Batch Normalization [12] − fully connected layer with weight normalization [37]. And for PointDA-10 [29], we use the code released by the authors for fair comparison with PointDAN [29], and only use the backbone without any of their proposed modules. To train the source model, we also adopt label smoothing as SHOT does. We adopt SGD with momentum 0.9 and batch size of 64 for all 2D datasets, and Adam for PointDA-10. The learning rate for Office-31 and Office-Home is set to 1e-3 for all layers, except for the last two newly added fc layers, where we apply 1e-2. Learning rates are set 10 times smaller for VisDA. Learning rate for PointDA-10 is set to 1e-6. We train 30 epochs for Office-31 and OfficeHome while 15 epochs for VisDA, and 100 for PointDA-10. For the number of nearest neighbors (K) and expanded neighborhoods (M), we use 3,2 for Office-31, Office-Home and PointDA-10, since VisDA is much larger we set K, M to 5. Experiments are conducted on a TITAN Xp. ",
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+    {
+        "type": "text",
+        "text": "4.1 Results ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2D image datasets. We first evaluate the target performance of our method compared with existing DA and SFDA methods on three 2D image datasets. As shown in Table 1-3, the top part shows results for the source-present methods with access to source data during adaptation. The bottom shows results for the source-free DA methods. On Office-31, our method gets similar results compared with source-free method 3C-GAN and lower than source-present method RSDA-MSTN. And our method achieves state-of-the-art performance on Office-Home and VisDA, especially on VisDA our method surpasses the source-free method SHOT and source-present method RWOT by a wide margin $3 \\%$ and $1 . 9 \\%$ respectively). The reported results clearly demonstrate the efficiency of the proposed method for source-free domain adaptation. Interestingly, like already observed in the SHOT paper, source-free methods outperform methods that have access to source data during adaptation. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6a311b2328d40c53ef20883419217b4dd44bd0f693fb7b8a722536b379b6e88c.jpg",
+        "table_caption": [
+            "Table 3: Accuracies $( \\% )$ on VisDA-C (Synthesis Real) for ResNet101-based methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>SF</td><td rowspan=1 colspan=1>[SF|plane bcycl bus car horse knife mcycl person plant sktbrd train truck Per-class</td></tr><tr><td rowspan=5 colspan=1>ADR [34]CDAN [24]CDAN+BSP[2]SAFN [52]SWD[19]MDD [59]DMRL [49]MCC[15]STAR [26]RWOT[51]</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>94.248.584.0 72.990.174.292.6 72.580.861.882.2 28.8  73.5</td></tr><tr><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>85.266.983.0 50.884.274.988.1 74.583.476.081.9 38.0  73.9</td></tr><tr><td rowspan=3 colspan=1>×</td><td rowspan=2 colspan=1>61.081.0 57.5 89.080.690.1 77.084.277.982.1 38.4  75.9</td></tr><tr><td rowspan=1 colspan=1>92.493.690.8</td></tr><tr><td rowspan=1 colspan=1>93.661.384.1 70.6 94.179.091.8 79.689.955.689.0 24.4  76.190.882.5 81.7 70.5 91.769.586.3 77.587.463.685.6 29.2  76.4-    1    1   1   1   1    1     1    1    -    1   1    74.6-                =     =     =     =     =     =     =          =    75.588.780.3 80.5 71.5 90.1 93.285.0 71.689.473.8 85.0 36.9  78.895.084.084.6 73.0 91.691.885.9 78.494.484.787.0 42.2  82.795.180.383.7 90.092.468.092.5 82.287.978.490.4 68.2  84.0</td></tr><tr><td rowspan=3 colspan=1>3C-GAN [20]SHOT[21]NRC</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>94.873.468.8 74.893.195.488.6 84.7 89.184.783.5 48.1  81.6</td></tr><tr><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>94.388.580.1 57.3 93.194.980.7 80.391.589.186.3 58.2  82.9</td></tr><tr><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>96.891.382.4 62.4 96.295.986.1 80.694.894.190.4 59.7  85.9</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/332a783b5902ad7cfa38d86058bd58c58d53c04b7c617a4ab838c6053a0fe857.jpg",
+        "table_caption": [
+            "Table 4: Accuracies $( \\% )$ on PointDA-10. The results except ours are from PointDAN [30]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td colspan=\"4\">|SF|Model-→Shape Model-→Scan Shape-→Model Shape-&gt;Scan Scan→Model Scan-→Shape Avg</td></tr><tr><td>MMD [25]</td><td></td><td>57.5 27.9</td><td>40.7</td><td>26.7</td><td>47.3</td><td>54.8</td><td>42.5</td></tr><tr><td>DANN [6]</td><td>xxxxx</td><td>58.7 29.4</td><td>42.3</td><td>30.5</td><td>48.1</td><td>56.7</td><td>44.2</td></tr><tr><td>ADDA [44]</td><td></td><td>61.0 30.5</td><td>40.4</td><td>29.3</td><td>48.9</td><td>51.1</td><td>43.5</td></tr><tr><td>MCD [35]</td><td></td><td>62.0 31.0</td><td>41.4</td><td>31.3</td><td>46.8</td><td>59.3</td><td>45.3</td></tr><tr><td>PointDAN [30]</td><td></td><td>64.2 33.0</td><td>47.6</td><td>33.9</td><td>49.1</td><td>64.1</td><td>48.7</td></tr><tr><td>Source-only</td><td></td><td>43.1</td><td>17.3 40.0</td><td>15.0</td><td>33.9</td><td>47.1</td><td>32.7</td></tr><tr><td>NRC</td><td>&lt;</td><td>64.8</td><td>25.8 59.8</td><td>26.9</td><td>70.1</td><td>68.1</td><td>52.6</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "3D point cloud dataset. We also report the result for the PointDA-10. As shown in Table 4, our method outperforms PointDA [30], which demands source data for adaptation and is specifically tailored for point cloud data with extra attention modules, by a large margin $(4 \\% )$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 Analysis ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Ablation study on neighbors $\\mathcal { N }$ , $E$ and affinity $A$ . In the first two tables of Table 5, we conduct the ablation study on Office-Home and VisDA. The 1-st row contains results from the source model and the 2-nd row from only training with the diversity loss $\\mathcal { L } _ { d i v }$ . From the remaining rows, several conclusions can be drawn. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "First, the original supervision, which considers all neighbors equally can lead to a decent performance (67.1 on Office-Home). Second, considering higher affinity values for reciprocal neighbors leads to a large performance gain (69.1 on Office-Home). Last but not the least, the expanded neighborhoods can also be helpful, but only when combined with the affinity values $A$ (72.2 on Office-Home). Using expanded neighborhoods without affinity obtains bad performance (65,2 on Office-Home). We conjecture that those expanded neighborhoods, especially those neighbors of nRNN, may be noisy as discussed in Sec. 3.2. Removing the affinity $A$ means we treat all those neighbors equally, which is not reasonable. ",
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+    {
+        "type": "table",
+        "img_path": "images/83cbcee8914a7e800eb1b8208228d67d3d85ccbbfa65a9ddc5cc8f8ec7844b0b.jpg",
+        "table_caption": [
+            "Table 5: Ablation study of different modules on Office-Home (left) and VisDA (middle), comparison between using expanded neighbors and larger nearest neighbors (right). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Ldiv</td><td>LN</td><td>LE LEA</td><td>Avg</td><td>Ldiv</td><td>LN</td><td></td><td>LE LEA</td><td></td><td>Acc</td></tr><tr><td></td><td></td><td></td><td>59.5</td><td></td><td></td><td></td><td></td><td></td><td>44.6</td></tr><tr><td></td><td></td><td></td><td>62.1</td><td></td><td></td><td></td><td></td><td></td><td>47.8</td></tr><tr><td></td><td></td><td></td><td>67.1</td><td></td><td></td><td></td><td></td><td></td><td>74.6</td></tr><tr><td></td><td></td><td></td><td>√ 69.1</td><td></td><td></td><td></td><td></td><td>√</td><td>81.5</td></tr><tr><td></td><td></td><td></td><td>65.2</td><td></td><td></td><td></td><td></td><td></td><td>61.2</td></tr><tr><td></td><td></td><td></td><td>72.2</td><td></td><td></td><td></td><td></td><td>！</td><td>85.9</td></tr><tr><td></td><td></td><td></td><td>69.1 [</td><td></td><td></td><td></td><td></td><td></td><td>82.0</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method&amp;Dataset</td><td rowspan=1 colspan=1>Acc</td></tr><tr><td rowspan=1 colspan=1>VisDA (K=M=5)VisDA w/o E (K=30)</td><td rowspan=1 colspan=1>85.984.0</td></tr><tr><td rowspan=1 colspan=1>OH(K=3,M=2)OH w/o E (K=9)</td><td rowspan=1 colspan=1>72.269.5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/84a0df3fa2021c0d9528204175fa9c78feacdd9407706d9d82014cc0299e96ee.jpg",
+        "table_caption": [
+            "Table 6: Runtime analysis on SHOT and our method. For SHOT, pseudo labels are computed at each epoch. $20 \\%$ , $10 \\%$ and $5 \\%$ denote the percentage of target features which are stored in the memory bank. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>VisDA</td><td colspan=\"2\">Runtime (s/epoch)Per-class (%)</td></tr><tr><td>SHOT</td><td>618.82</td><td>82.9</td></tr><tr><td>NRC</td><td>540.89</td><td>85.9</td></tr><tr><td>NRC(20%) 6formemorybank)</td><td>507.15</td><td>85.3</td></tr><tr><td>NRC(10% for memory bank)</td><td>499.49</td><td>85.2</td></tr><tr><td>NRC(5% for memory bank)</td><td>499.28</td><td>85.1</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/e1748829bba37f8f3b6d43d90dfdbea6be91a7b5cbb5e87bd18f0bf9bd522aa6.jpg",
+        "image_caption": [
+            "Figure 2: (Left and middle) Ablation study of $\\mathcal { L } _ { s e l f }$ on Office-Home and VisDA respectively. (Right) Performance with different $r$ on VisDA. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We also show that duplication in the expanded neighbors is important in the last row of Table 5, where the $\\mathcal { L } _ { \\hat { E } }$ means we remove duplication in Eq. 8. The results show that the performance will degrade significantly when removing them, implying that the duplicated expanded neighbors are indeed more important than others. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Next we ablate the importance of the expanded neighborhood in the right of Table5. We show that if we increase the number of datapoints considered for class-consistency by simply considering a larger K, we obtain significantly lower scores. We have chosen $K$ so that the total number of points considered is equal to our method (i.e. $5 { + } 5 ^ { * } 5 { = } 3 0$ and $3 + 3 ^ { * } 2 { = } 9 ,$ ). Considering neighbors of neighbors is more likely to provide datapoints that are close on the data manifold [43], and are therefore more likely to share the class label with the ego feature. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Runtime analysis. Instead of storing all feature vectors in the memory bank, we follow the same memory bank setting as in [5] which is for nearest neighbor retrieval. The method only stores a fixed number of target features, we update the memory bank at the end of each iteration by taking the $n$ (batch size) embeddings from the current training iteration and concatenating them at the end of the memory bank, and discard the oldest $n$ elements from the memory bank. We report the results with this type of memory bank of different buffer size in the Table 6. The results show that indeed this could be an efficient way to reduce computation on very large datasets. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Ablation study on self-regularization. In the left and middle of Fig 2, we show the results with and without self-regularization $\\mathcal { L } _ { s e l f }$ . The $\\mathcal { L } _ { s e l f }$ can improve the performance when adopting only nearest neighbors $\\mathcal { N }$ or all neighbors $\\mathcal { N } + E$ . The results imply that self-regularization can effectively reduce the negative impact of the potential noisy neighbors, especially on the Office-Home dataset. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Sensitivity to hyperparameter. There are three hyperparameters in our method: K and M which are the number of nearest neighbors and expanded neighbors, $r$ which is the affinity value assigned to nRNN. We show the results with different $r$ in the right of Fig. 2. Note we keep the affinity of expanded neighbors as 0.1. $r = 1$ means no affinity. $r = - 1$ means treating supervision of nRNN feature as totally wrong, which is not always the case and will lead to quite lower result. $r = 0$ can also achieve good performance, signifying RNN can already work well. Results with $r = 0 . 1 / 0 . 1 5 / 0 . 2$ show that our method is not sensitive to the choice of a reasonable $r$ . Note in DA, there is no validation set for hyperparameter tuning, we show the results varying the number of neighbors in the right of Tab. 3, demonstrating the robustness to the choice of $K$ and $M$ . ",
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+    {
+        "type": "image",
+        "img_path": "images/52c51cde4275a75094e8c62c8d4fd77f70977cdffb6dcc8e5fcf1796cfa99e42.jpg",
+        "image_caption": [
+            "Figure 3: (Left) The three curves are (on VisDA): target accuracy (Blue), ratio of features which have 5-nearest neighbors all sharing the same predicted label (dashed Red), and ratio of features which have 5-nearest neighbors all sharing the same and correct predicted label (dashed Black). (Right) Ablation study on choice of K and M on VisDA. "
+        ],
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+            "Figure 4: (Left) Ratio of different type of nearest neighbor features which have the correct predicted label, before and after adaptation. (Right) Visualization of target features after adaptation. "
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+        "text": "Training curve. We show the evolution of several statistics during adaptation on VisDA in the left of Tab. 3. The blue curve is the target accuracy. The dashed red and black curves are the ratio of features which have 5-nearest neighbors all sharing the same (dashed Red), or the same and also correct (dashed Black) predicted label. The curves show that the target features are clustering during the training. Another interesting finding is that the curve ’Per Shared’ correlates with the accuracy curve, which might therefore be used to determine training convergence. ",
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+        "text": "Accuracy of supervision from neighbors. We also show the accuracy of supervision from neighbors on task $\\mathrm { A r } {  } \\mathrm { R w }$ of Office-Home in Fig. 4(left). It shows that after adaptation, the ratio of all types of neighbors having more correct predicted label, proving the effectiveness of the method. ",
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+        "text": "t-SNE visualization. We show the t-SNE feature visualization on task $\\mathrm { A r } {  } \\mathrm { R w }$ of target features before (Fig. 1(a)) and after (Fig. 4(right)) adaptation. After adaptation, the features are more compactly clustered. ",
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+        "text": "5 Conclusions ",
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+        "text": "We introduce a source-free domain adaptation (SFDA) method by uncovering the intrinsic target data structure. We propose to achieve the adaptation by encouraging label consistency among local target features. We differentiate between nearest neighbors, reciprocal neighbors and expanded neighborhood. Experimental results verify the importance of considering the local structure of the target features. Finally, our experimental results on both 2D image and 3D point cloud datasets testify the efficacy of our method. ",
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+        "text": "Acknowledgement We acknowledge the support from Huawei Kirin Solution, and the project PID2019-104174GB-I00 (MINECO, Spain) and RTI2018-102285-A-I00 (MICINN, Spain), Ramón y Cajal fellowship RYC2019-027020-I, and the CERCA Programme of Generalitat de Catalunya. ",
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+        "text": "References ",
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