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+# LEARNING HIERARCHICAL DISCRETE LINGUISTIC UNITS FROM VISUALLY-GROUNDED SPEECH
+
+David Harwath∗, Wei-Ning $\mathbf { H s u } ^ { * }$ , and James Glass
+
+Computer Science and Artificial Intelligence Lab Massachusetts Institute of Technology Cambridge, MA 02139, USA {dharwath,wnhsu,glass}@csail.mit.edu
+
+# ABSTRACT
+
+In this paper, we present a method for learning discrete linguistic units by incorporating vector quantization layers into neural models of visually grounded speech. We show that our method is capable of capturing both word-level and sub-word units, depending on how it is configured. What differentiates this paper from prior work on speech unit learning is the choice of training objective. Rather than using a reconstruction-based loss, we use a discriminative, multimodal grounding objective which forces the learned units to be useful for semantic image retrieval. We evaluate the sub-word units on the ZeroSpeech 2019 challenge, achieving a $2 7 . 3 \%$ reduction in ABX error rate over the top-performing submission, while keeping the bitrate approximately the same. We also present experiments demonstrating the noise robustness of these units. Finally, we show that a model with multiple quantizers can simultaneously learn phone-like detectors at a lower layer and word-like detectors at a higher layer. We show that these detectors are highly accurate, discovering 279 words with an F1 score of greater than 0.5.
+
+# 1 INTRODUCTION
+
+By 8 months of age, human infants learn to recognize not only the names of their caregivers and common objects, but also the contrast between the different vowels and consonants which comprise these words (Dupoux, 2018). Nearly all toddlers learn to carry a conversation long before they can read and write. Humans learn to model the discrete, hierarchical, and compositional nature of their native language not from written text, but from speech audio - a continuous, time-varying waveform which is the product not only of the underlying words which were spoken, but also the physical properties of the speaker’s vocal tract, the speaker’s health and emotional state, and the noise and reverberation present in the environment. The question of how such a complex symbolic system is inferred from continuous and noisy sensory input data is of interest not only to the cognitive science community, but also to machine learning researchers who aim to reproduce this ability with computers. A more comprehensive understanding of human language acquisition has practical significance in real-world applications, such as automatic speech recognition (ASR) and natural language understanding (NLU) systems. In the past several decades, enormous progress has been made in speech recognition research, and nowadays ASR systems are able to achieve human-level accuracy in many domains (Chiu et al., 2018). Unfortunately, the techniques that have been developed to achieve these levels of performance are extremely data-hungry, requiring many thousands of hours of speech audio recordings for training. Since supervised machine learning algorithms form the basis of ASR training, the data also needs to be annotated by expert humans. Due to the immense cost of collecting and annotating speech data, ASR technology currently exists for approximately 120 (Google, 2019) out of the nearly 7,000 (Lewis et al., 2016) human languages spoken worldwide. It is highly unlikely that purely supervised machine learning techniques will be able to scale to include all human languages, necessitating the development of alternative methods by researchers which are able to function with far fewer annotations, or even no annotations at all. Because human beings provide an existence proof of language acquisition from speech completely without language supervision, it is plausible that this ability could be replicated by a machine learning algorithm.
+
+In this paper, we present a method for discovering discrete and hierarchical representations of speech units both at the sub-word level and the word level. Previously proposed linguistic unit discovery methods have only leveraged the speech audio modality in isolation, relying on objective functions that attempt to capture statistical regularities within the speech signal. The key innovation in our work is that we discover units by training models with explicit discretization layers to associate speech waveforms with visual images using a cross-modal grounding objective. This forces our models to learn representations which capture semantic information at the highest layers of the network. Because semantics are predominantly carried by words, and words are composed of subword units (such as phones and syllables), the visual grounding objective indirectly forces the model to learn speaker- and noise-invariant representations of speech units. By incorporating trainable quantization layers into our networks, we are able to capture these units in discrete inventories. Whether these units correspond to word-like or sub-word units depends on where the quantization layers are inserted, and how they are trained.
+
+# 2 RELATED WORK
+
+Prior work on unsupervised modeling of the speech signal has generally focused on learning representations which either disentangle or isolate the latent factors that are of interest for downstream tasks. In most cases the primary latent factor of interest is the phonetic or lexical identity of a given segment of speech, but other factors, such as the identity of the speaker, are sometimes of interest as well. Because the factors of interest are often inherently discrete (e.g. words and phones), many of the proposed approaches attempt to perform segmentation and clustering of the surface features in one way or another. One family of techniques is based upon Segmental Dynamic Time Warping (S-DTW) (Park & Glass, 2005; 2008; Jansen et al., 2010; Jansen & Van Durme, 2011), which uses a self-comparison algorithm to identify relatively long duration (on the order of a second) patterns which frequently reoccur in a speech corpus; these patterns tend to capture words or short phrases. A different line of work employs probabilistic graphical models to jointly segment and cluster the speech signal (Varadarajan et al., 2008; Zhang & Glass, 2009; Gish et al., 2009; Lee & Glass, 2012; Siu et al., 2014; Lee et al., 2015; Ondel et al., 2016; Kamper et al., 2016; 2017a). With an appropriately designed model, it is possible to learn multiple, hierarchical categories of speech units. However, in order to enable efficient inference, the conditional distributions of these models tend to be simple and therefore have limited modeling power.
+
+Deep neural network models have been successfully used to learn powerful speech representations using weakly or unsupervised objectives (Thiolliere et al., 2015; Kamper et al., 2015; Hsu et al., 2017a;b; Hsu & Glass, 2018; Holzenberger et al., 2018; Milde & Biemann, 2018; van den Oord et al., 2018; Chung et al., 2019; Pascual et al., 2019). These representations have predominantly been continuous in nature, as discrete latent variables are not trivially compatible with backpropagation. To obtain discrete representations, a post-hoc clustering step can be applied to the continuous representations (Kamper et al., 2017b; Feng et al., 2019). More recently, several papers have proposed ways of directly incorporating discrete variables into neural network models, including using Gumbel-Softmax (Eloff et al., 2019b) or straight-through estimators (van den Oord et al., 2017; Chorowski et al., 2019; Razavi et al., 2019).
+
+A different method for learning meaningful representations of speech is via a multimodal grounding objective, which encourages the learning of speech representations that are predictive of the contextual information contained in a separate but accompanying modality, such as vision. Visual grounding of speech is a form of self-supervised learning (Virginia de Sa, 1994), which is powerful in part because it offers a way of training models with a discriminative objective that does not depend on traditional transcriptions or annotations. The first work in this direction relied on phone strings to represent the speech (Roy & Pentland, 2002; Roy, 2003), but more recently this learning has been shown to be possible directly on the speech signal (Synnaeve et al., 2014; Harwath & Glass, 2015; Harwath et al., 2016). Subsequent work on visually-grounded models of speech has investigated improvements and alternatives to the modeling or training algorithms (Leidal et al., 2017; Kamper et al., 2017c; Havard et al., 2019a; Merkx et al., 2019; Chrupała et al., 2017; Scharenborg et al., 2018; Kamper et al., 2019b;a; Sur´ıs et al., 2019; Ilharco et al., 2019; Eloff et al., 2019a), application to multilingual settings (Harwath et al., 2018a; Kamper & Roth, 2017; Azuh et al., 2019; Havard et al., 2019a), analysis of the linguistic abstractions, such as words and phones, which are learned by the models (Harwath & Glass, 2017; Harwath et al., 2018b; Drexler & Glass, 2017; Alishahi et al.,
+
+2017; Harwath et al., 2019; Harwath & Glass, 2019; Havard et al., 2019b), and the impact of jointly training with textual input (Holzenberger et al., 2019; Chrupała, 2019; Pasad et al., 2019). Representations learned by models of visually grounded speech are also well-suited for transfer learning to supervised tasks, being highly robust to noise and domain shift (Hsu et al., 2019).
+
+# 3 DATA AND MODELS
+
+# 3.1 DATASET
+
+For training our models, we utilize the MIT Places 205 dataset (Zhou et al., 2014) and their accompanying spoken audio captions (Harwath et al., 2016; 2018b). The caption dataset contains approximately 400,000 spoken audio captions, each of which describes a different Places image. These captions are free-form spontaneous speech, collected from over 2,500 different speakers and covering a 40,000 word vocabulary. The average caption duration is approximately 10 seconds, and each caption contains on average 20 words. For vetting our models during training, we use a held-out validation set of 1,000 image-caption pairs.
+
+# 3.2 NEURAL MODELS OF VISUALLY-GROUNDED SPEECH
+
+We base our model upon the Residual Deep Audio-Visual Embedding network (ResDAVEnet) architecture (Harwath et al., 2019), which contains two branches of fully convolutional networks, one for images and the other for audio. Each branch encodes samples of the corresponding modality into a $d$ -dimensional space, regardless of the original dimensionality of the samples. This is achieved by applying global spatial mean pooling and global temporal mean pooling to the image branch output and the audio branch output, respectively. The image branch is adapted from ResNet50 (He et al., 2016), where the final softmax layer and the preceding fully-connected layers are removed, replaced with a 1x1 linear convolutional layer in order to project the feature map to the desired dimension. To model the audio inputs, a 17-layer fully convolutional network with residual connections is used. The input is a log Mel-frequency spectrogram with 40 frequency bins and $2 5 ~ \mathrm { m s }$ -wide, Hammingwindowed frames with a shift of $1 0 ~ \mathrm { m s }$ . The first layer of this network is a 1-D convolution that spans the entire frequency axis of the spectrogram, while the remaining 16 convolutional layers are 1-D across the time axis. These 16 layers are divided into four residual blocks of 4 layers each, and downsampling between these blocks is accomplished by applying the first convolution of each block with a stride of 2. For full details of the model, refer to Harwath et al. (2019).
+
+# 3.3 LEARNING HIERARCHICAL DISCRETE UNITS WITH VECTOR QUANTIZING LAYERS
+
+Previous analyses reveal that ResDAVEnet-like models learn linguistic abstractions at different levels, including words (Harwath & Glass, 2017) and robust phonetic features (Harwath & Glass, 2019; Hsu et al., 2019). To explicitly learn hierarchical discrete linguistic units within this framework, we propose to incorporate multiple vector quantization (VQ) layers (van den Oord et al., 2017) into the ResDAVEnet audio branch; we refer to this new architecture as ResDAVEnet-VQ.
+
+VQ layers can be understood as a type of bottleneck, which constrain the amount of information that can flow through. While these layers have been used to learn discrete sub-word units (van den Oord et al., 2017; Chorowski et al., 2019; Razavi et al., 2019), previous work injects VQ layers into autoencoders that are trained with a reconstruction loss. As a result, the embedding dimension of each code and the number of codes need to be carefully tuned (Liu et al., 2019). When the embedding dimension is too low or the codebook size too small, the model does not have enough expressive power to capture linguistic variability. When it is too large, the model starts to encode non-linguistic information in order to improve reconstruction. In contrast, the learning signal of ResDAVEnet-VQ is provided by the visual-semantic grounding objective. Rather than encoding as much information about input as possible, the learned codes in ResDAVEnet-VQ only need to capture semantic information. Since semantics in speech are predominantly transmitted by words, and words are composed of sub-word units like phones, the grounding objective places pressure on the model to robustly infer both from speech. Since words and phones are inherently discrete symbols, representing them with learned discrete units may not even hurt the grounding performance.
+
+Figure 1 illustrates the proposed ResDAVEnet-VQ model. We add a quantization layer after each of the first two residual blocks of the ResDAVEnet-VQ model, denoted as VQ2 and VQ3, respectively, with the intention that they should capture discrete sub-word-like and word-like units. A VQ layer is defined as $\pmb { { \cal E } } \in \mathbb { R } ^ { K \times \check { D } }$ , where $K$ represents the codebook size, and $D$ represents the output dimensionality of the input features to the codebook. Denoting the $t ^ { t h }$ temporal frame of the input to the quantization layer as $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ , quantization is performed according to $\begin{array} { r } { \mathbf q _ { t } ~ = ~ \mathbf E _ { k , : } } \end{array}$ , where $k \mathbf { \Psi } =$ a $\begin{array} { r } { \operatorname { r g m i n } _ { j } | | \pmb { x } _ { t } - \pmb { E } _ { j , : } | | _ { 2 } } \end{array}$ The quantized output is then fed as input to the subsequent residual block. As in van den Oord et al. (2017), we use the straight-through estimator (Bengio et al., 2013) to compute the gradient passed from $\pmb q _ { t }$ to $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ . We use the exponential moving average (EMA) codebook updates proposed by van den Oord et al. (2017).
+
+![](images/19ce5553db00c8d36eb035d8c0e99b07adf264a1d88df2fd584d4391d903bb99.jpg)  
+Figure 1: Diagram of the ResDAVEnet-VQ model. On the left, we show the placement of the vector quantization blocks in the audio branch. Note that each “Res” block is comprised of a stack of multiple sub-layers (see Harwath et al. (2019) for details). The right half of the figure depicts the quantization mechanism of each VQ block, as well as the bypass path when the block is disabled.
+
+# 3.4 CODEBOOK LEARNING SCHEDULES
+
+We include multiple VQ layers in the ResDAVEnet-VQ model, each of which can be independently enabled or bypassed without changing the rest of the architecture configuration. When all model weights, including the VQ codebooks, are trained jointly in a single training run we call this a “coldstart” model. Alternatively, a model can be “warm-started” by copying the weights from another trained model that has fewer (or no) VQ layers enabled, and randomly initializing the codebook of the newly activated VQ layer(s). This gives rise to the questions of how many quantizers should be used and in what order they should be enabled. It is unclear whether models with the same VQ layers activated would learn the same representation at each layer regardless of the training curriculum. Let $A _ { m }$ denote a subset of all VQ layers, and $A _ { m - 1 } \subset A _ { m }$ . We use $\mathrm { } ^ { \ast } A _ { 1 }  \dots  A _ { M } ^ { \prime \prime }$ to denote a model that is obtained by sequentially training models $^ { * * } A _ { 1 } \to \dots \to A _ { m } ^ { \phantom { * } , }$ initialized from $^ {  } A _ { 1 } \to \dots \to A _ { m - 1 } { } ^ { , , }$ , where the model $A _ { 1 }$ is initialized from scratch, and the final model would have VQ layers in $A _ { M }$ activated. For instance, a model initialized from scratch with no VQ layers enabled is denoted as $" \boldsymbol { Q } ^ { \flat }$ , and a model initialized with that and with both layers enabled is denoted as $\cdot \cal { O }  \{ 2 , 3 \} ^ { \cdots }$ .
+
+# 3.5 TRAINING WITH THE TRIPLET LOSS
+
+We train our models using the same loss function as Harwath et al. (2019). This loss function blends two triplet loss terms (Weinberger & Saul, 2009), one based on random sampling of negative examples, and the other based on semi-hard negative mining (Jansen et al., 2018), in order to find more challenging negative samples. Specifically, let the sets of output embedding vectors for a minibatch of $B$ audio/image training pairs respectively be $\mathbb { A } = \{ \pmb { a } _ { 1 } , \dotsc , \pmb { a } _ { B } \}$ and $\mathbb { I } \stackrel { - } { = } \{ i _ { 1 } , \dotsc , i _ { B } \}$ . To compute the randomly-sampled triplet loss term, we select impostor examples for the $j ^ { t h }$ input according to $\bar { \mathbf { } } _ { j } \sim \mathbf { \delta }$ UniformCategorical $( \{ \pmb { a } _ { 1 } , \dots , \pmb { a } _ { B } \} \backslash \pmb { a } _ { j } )$ and $\bar { i } _ { j } \sim$ UniformCategorical $\big ( \{ i _ { 1 } , \ldots , i _ { B } \} \backslash i _ { j } \big )$ . The randomly-sampled triplet loss is then computed as:
+
+$$
+\mathcal { L } _ { s } = \sum _ { j = 1 } ^ { B } \left( \operatorname* { m a x } ( 0 , i _ { j } ^ { T } \bar { a } _ { j } - i _ { j } ^ { T } a _ { j } + 1 ) + \operatorname* { m a x } ( 0 , \bar { i } _ { j } ^ { T } a _ { j } - i _ { j } ^ { T } a _ { j } + 1 ) \right)
+$$
+
+For the semi-hard negative triplet loss, we first define the sets of impostor candidates for the $j ^ { t h }$ example as $\hat { \mathbb { A } } _ { j } = \{ \pmb { a } \in \mathbb { A } | \pmb { i } _ { j } ^ { T } \pmb { a } < \pmb { i } _ { j } ^ { T } \pmb { a } _ { j } \}$ and $\hat { \mathbb { I } } _ { j } = \lbrace i \in \mathbb { I } \vert i ^ { T } { \pmb { a } } _ { j } < i _ { j } ^ { T } { \pmb { a } } _ { j } \rbrace$ . The semi-hard negative
+
+loss is then computed as:
+
+$$
+\mathcal { L } _ { h } = \sum _ { j = 1 } ^ { B } \Big ( \operatorname* { m a x } ( 0 , \operatorname* { m a x } ( i _ { j } ^ { T } \hat { a } ) - i _ { j } ^ { T } a _ { j } + 1 ) + \operatorname* { m a x } ( 0 , \operatorname* { m a x } ( \hat { i } ^ { T } a _ { j } ) - i _ { j } ^ { T } a _ { j } + 1 ) \Big )
+$$
+
+Finally, the overall loss function is computed by combining the two above losses, $\mathcal { L } = \mathcal { L } _ { s } + \mathcal { L } _ { h }$ , which was found by (Harwath et al., 2019) to outperform either loss on its own.
+
+# 3.6 IMPLEMENTATION DETAILS
+
+All of our models were trained for 180 epochs using the Adam optimizer (Kingma & Ba, 2014) with a batch size of 80. We used an exponentially decaying learning rate schedule, with an initial value of 2e-4 that decayed by a factor of 0.95 every 3 epochs. Following van den Oord et al. (2017), we use an EMA decay factor of $\gamma = . 9 9$ for training each VQ codebook. Our core experimental results all use a codebook size of 1024 vectors for all quantizers, but in the supplementary material we include experiments with smaller and larger codebooks. Following Chorowski et al. (2019), the jitter probability hyperparameter for each quantization layer was fixed at 0.12. While we do not apply data augmentation to the input spectrograms, during training we perform standard data augmentation techniques to the images. We resize each raw image so that its smallest dimension is 256 pixels, and then we apply an Inception-style random crop which is resized to 224 pixels square. During training, we also flip each image horizontally with a probability of 0.5. During evaluation, the center 224 pixel square crop is always taken from the image. Finally, the RGB pixel values are mean and variance normalized. We trained each model on the Places audio caption train split, and computed the image and caption recall at 10 $( \mathbb { R } ^ { \ @ 1 0 ) }$ scores on the validation split of the Places audio captions after each training epoch. The model snapshot that achieved the highest average $\mathbb { R } \ @ 1 0$ score on the validation set from each training is used for all evaluation. To extract embeddings and units from our models, we simply perform a forward pass through the speech branch of the ResDAVEnet-VQ network and retain the outputs from the target layer at a uniform frame-rate. The frame-rate is determined by the downsampling factor at the target layer relative to the input. For non-quantized layers, these outputs will be continuous embeddings. For quantized layers, these will be quantized embedding retrieved from the assigned entry in the codebook.
+
+# 4 EXPERIMENTS
+
+# 4.1 SUB-WORD UNIT LEARNING ON THE ZEROSPEECH 2019 ABX TASK
+
+Evaluation metrics Learning unsupervised speech representations that are indicative of phonetic content is of high interest to the speech community, and recently has been the focus of the ZeroSpeech Challenge (Versteegh et al., 2015; Dunbar et al., 2017; 2019). One of the core evaluations is the minimal-pair ABX task (Schatz et al., 2013), which aims to benchmark representations in terms of their discriminability between different sub-word speech units. In this task, a model is tasked with extracting representations for a triplet of speech waveform segments denoted by $A , B ,$ , and $X$ . $A$ and $B$ are constrained to be a triphone minimal pair; that is, both segments capture three phones, but differ only in the identity of their center phone. The third segment, $X$ is chosen to contain the same underlying triphone sequence as $A$ . Supposing $f ( \cdot )$ denotes the model’s mapping function from a waveform segment to a sequence of embedding vectors, the ABX error rate under a given similarity metric $ { \left. { \cal { S } } ( \cdot , \cdot ) \right. }$ is defined as the fraction of ABX triples in which $S ( f ( A ) , f ( X ) ) \bar { > } S ( f ( B ) , f ( \bar { X } ) )$ . An ABX error rate of $50 \%$ indicates random assignment, while an ABX of $0 \%$ reflects perfect phone discriminability. In the ZeroSpeech challenge, $\bar { S ( \cdot , \cdot ) }$ is implemented using Dynamic Time Warping (DTW) with various distance measures (cosine, KL, etc.). In our evaluation, we use the cosine distance.
+
+The ZeroSpeech 2019 challenge in particular emphasizes on discovering an inventory of discrete sub-word units, rather than continuous representations. Therefore, in addition to an ABX error rate, a bitrate is also computed for each model which reflects the amount of information carried by the learned units. A lower bitrate can be achieved by having a more compact inventory of learned units or having a smaller number of codes per second. The full details of the evaluation can be found in Dunbar et al. (2019). To be clear, all of our ResDAVEnet-VQ models were not trained on the
+
+Table 1: Comparison of $\mathrm { R @ 1 0 }$ , ABX scores, and bit-rates between different configurations and baseline models trained on ZeroSpeech 2019 data or Places Audio Caption. All quantizers reflected in this table used a codebook size of 1,024 vectors. We do not compute RLE or segment scores for the FHVAE-DPGMM model, since we did not re-implement that model.   
+
+<table><tr><td>Model ID</td><td>Layer</td><td>R@10</td><td>ABX</td><td>Frame-Based Bitrate</td><td>RLE Bitrate</td><td>Segment-Based ABX</td><td>Bitrate</td></tr><tr><td>FHVAE-DPGMM (ZS)</td><td>N/A</td><td>N/A</td><td>21.67</td><td>413.23</td><td></td><td></td><td></td></tr><tr><td>WaveNet-VQ (ZS)</td><td>N/A</td><td>N/A</td><td>19.98</td><td>151.55</td><td>136.74</td><td>20.48</td><td>126.17</td></tr><tr><td>WaveNet-VQ (PA)</td><td>N/A</td><td>N/A</td><td>24.87</td><td>149.00</td><td>136.27</td><td>25.23</td><td>126.22</td></tr><tr><td>“g”</td><td>Res2 Res3</td><td>.735</td><td>11.35 10.86</td><td>N/A N/A</td><td>N/A N/A</td><td>N/A N/A</td><td>N/A N/A</td></tr><tr><td></td><td>VQ2</td><td>.753</td><td>12.33</td><td>433.30</td><td>361.09</td><td>12.78</td><td>332.86</td></tr><tr><td>“ →{2}”</td><td>VQ2</td><td>.760</td><td>11.79</td><td>390.61</td><td>317.66</td><td>12.66</td><td>289.11</td></tr><tr><td>3</td><td>VQ3</td><td>.734</td><td>38.21</td><td>213.92</td><td>129.65</td><td>38.68</td><td>108.84</td></tr><tr><td></td><td>VQ3</td><td>.794</td><td>15.04</td><td>182.93</td><td>140.04</td><td>16.53</td><td>121.26</td></tr><tr><td>&quot;{2,3}&quot;</td><td>VQ2</td><td>.667</td><td>25.62</td><td>408.75</td><td>258.37</td><td>26.32</td><td>217.58</td></tr><tr><td></td><td>VQ3</td><td></td><td>32.23</td><td>218.76</td><td>156.69</td><td>32.49</td><td>136.90</td></tr><tr><td>“→{2,3}”</td><td>VQ2</td><td>.787</td><td>13.15</td><td>405.43</td><td>334.39</td><td>13.30</td><td>303.03</td></tr><tr><td></td><td>VQ3</td><td></td><td>14.95</td><td>199.91</td><td>172.05</td><td>15.60</td><td>159.07</td></tr><tr><td>“{2}→{2,3}”</td><td>VQ2</td><td>.764</td><td>12.51</td><td>415.13</td><td>341.85</td><td>13.06</td><td>311.82</td></tr><tr><td></td><td>VQ3</td><td></td><td>14.52</td><td>167.84</td><td>136.11</td><td>15.68</td><td>121.17</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>“{3}→{2,3}”</td><td>VQ2</td><td>.760</td><td>13.55</td><td>421.23</td><td>271.91</td><td>14.38</td><td>232.87</td></tr><tr><td></td><td>VQ3</td><td></td><td>33.70</td><td>208.63</td><td>117.37</td><td>33.58</td><td>98.29</td></tr></table>
+
+ZeroSpeech training data, but instead on the Places audio captions, thus there is a domain mismatch between training and testing these models.
+
+In addition to the frame-based bitrate and ABX scores computed by the ZeroSpeech 2019 evaluation toolkit, we implement our own extensions to these metrics. Because it is common for successive frames to be assigned to the same codebook entry and phonetic information is not encoded at a fixed frame rate, lossless run length encoding (RLE) can be a more reasonable measure of the bitrate of a frame-based model. RLE does not change the ABX score since it can be trivially inverted, but it does change the bitrate. For computing the RLE bitrate, we modify the bitrate calculation specified in Dunbar et al. (2019) so that a unique symbol is defined as the tuple (unit, length) where length is the number of frames assigned to a given unit with in a segment. We also consider segment-based ABX and bitrate, which is similar to the RLE metrics except in this case we outright discard the frame length information. This typically results in an even greater reduction in bitrate, but also an accompanying deterioration in ABX score.
+
+Baseline models In Table 1, we compare our results to those derived from two of the topperforming submissions to the ZeroSpeech 2019 challenge: a re-implementation of WaveNetVQ (Chorowski et al., 2019) provided by Cho et al. (2019) and FHVAE-DPGMM (Feng et al., 2019). Using the code accompanied with the WaveNet-VQ submission, we were able to train their model on the set of 400,000 Places audio captions to make a fairer comparison with our ResDAVEnet-VQ models in terms of the amount of speech data used. In addition, when trying to reproduce the reported WaveNet-VQ results, we obtain better performance than previously reported by training for more steps. Table 1 shows that WaveNet-VQ achieves similar bitrates regardless of the training data. However, ABX deteriorates from 19.98 to 24.87, implying the model cannot utilize data of a larger scale but out-of-domain relative to the test set. A similar degradation when testing on out-of-domain data with FHVAE models was observed in Hsu et al. (2019). We did not re-train the model submitted by Feng et al. (2019), and instead compare against the scores reported in Dunbar et al. (2019).
+
+ABX discrimination without using quantization Our first experiment investigates exactly which layer in the ResDAVEnet-VQ model is most suited for ABX phone discrimination, and would thus make a good candidate for learning of quantized sub-word units. The leftmost plot in Figure 2 shows that layers 2 and 3 of a ResDAVEnet-VQ model without any quantization enabled perform the best in terms of ABX error rate on the ZeroSpeech 2019 English test set; the exact numbers for this model are displayed in the caption of Figure 2. Because layers 2 and 3 achieve the lowest ABX error rates without quantization, we focus our attention on the impact of quantization there.
+
+![](images/aa6cb902d02b2c9832557b19f7fc98058e7abfec7e4f1fd1f89b65f92f3a6ea4.jpg)  
+Figure 2: $\mathrm { R @ 1 0 }$ and ABX tracked at various training epochs. The “ $\mathcal { D }$ ” model achieves a final $\mathbb { R } \ @ 1 0$ of .735, with ABX scores of 19.77, 11.35, 10.86, and 14.05 for the conv1, res2, res3, and res4 layers.
+
+Quantizing one layer When quantizing only one layer, we examine quantization of layer 2 vs. layer 3, and using cold-start training vs. warm-start initialization from model “ $\chi ^ { \prime }$ . The ABX and bitrate results for these models, as well as the $\mathbb { R } \ @ 1 0$ scores on the Places validation set, are shown in Table 1. In all cases, quantization applied at the output of layer 2 achieves a better ABX score than quantization at layer 3, but VQ3 achieves a better bitrate. Quantization barely impacts the performance of layer 2, whose ABX score very slightly rises from 11.35 to 11.79. Warm-start initialization is beneficial to $\mathrm { R @ 1 0 }$ and ABX score in both cases, but we notice an intriguing anomaly when applying cold-start quantization to layer 3: the ABX score deteriorates significantly, rising from 10.86 in the case of the non-quantized model to 38.21. This indicates that while VQ2 is capable of learning a finite inventory of units that are highly predictive of phonetic identity from either a warm-start or cold-start initialization, cold-start training of VQ3 results in very little phonetic information captured by the quantizer. Interestingly, this model is still learning to infer visual semantics from the speech signal, as evidenced by a high $\mathbb { R } \ @ 1 0$ score; we later show in Section 4.2 that the reason for this anomaly is because cold-start training of VQ3 results in the learning of word detectors. In all cases except for model $\{ 3 \} ^ { \flat }$ , we note that the ABX scores achieved by our models are significantly better than the baselines. Our best model in terms of ABX $\ " \infty  \{ 2 \} \ " )$ achieves a $4 1 . 0 \%$ reduction in ABX over the WaveNet-VQ baseline, at a cost of a $1 3 2 . 3 \%$ increase in RLE bitrate; however, model “ $\mathrm { \Phi ^ { \prime } } \mathcal { O } \to \{ 3 \} ^ { \flat }$ achieves a $2 4 . 7 \%$ reduction in ABX error rate with only a $2 . 4 \%$ increase in RLE bitrate. These results do not constitute a fair comparison, however, because the WaveNet-VQ and ResDAVEnet-VQ models were trained on different datasets; when training the WaveNet-VQ model on the same set of audio captions used to train ResDAVEnet-VQ (but without the accompanying images, since WaveNet-VQ is not a multimodal model), the ABX error rate increases to $2 4 . 8 7 \%$ , tipping the results even more in favor of the ResDAVEnet-VQ models.
+
+Quantizing two layers Quantizing multiple layers at once offers the possibility of learning a hierarchy of units. Thus, we aim to capture phonetic information in a lower layer quantizer and word-level information at a higher layer quantizer. Cold-start training of two quantizers $( ^ { 6 6 } \{ 2 , 3 \} ^ { 5 } )$ results in a significant drop in ABX performance for both VQ2 and VQ3, but also a drop in $\mathrm { R @ 1 0 }$ on the Places validation set. We see much better results in terms of $\textrm { R @ 1 0 }$ and ABX for the remaining 3 models which were initialized from the “ $\varnothing$ ” model or a model with only one quantizer enabled; for example, model “ $\{ 2 \}  \{ 2 , 3 \} ^ { \prime }$ achieves an ABX of 14.52 with an RLE bitrate of 136.11, representing a $2 7 . 3 \%$ ABX improvement over the best baseline while keeping the bitrate approximately the same. We see in model “ $\{ 3 \}  \{ 2 , 3 \} ^ { \prime }$ that the same phenomenon observed with model $\{ 3 \} ^ { \flat }$ persists: VQ3 achieves relatively poor ABX, despite a high overall $\textrm { R @ 1 0 }$ and strong ABX with VQ2 at $1 3 . 5 5 \%$ . We confirm in Section 4.2 that the VQ3 layer of model ${ } ^ { * * } \{ 3 \}  \{ 2 , 3 \bar  \} ^ { * }$ does indeed capture word-level information, indicating that this model has successfully localized phonetic unit identity in the second layer and lexical unit identity in the third layer. Overall, our results suggest that when learning hierarchical quantized representations with a ResDAVEnet-VQ model, the nature of the representations learned is highly dependent on the training curriculum.
+
+Table 2: ABX scores and RLE bitrates for various SNRs on the noisy ZeroSpeech19 English test set. “R-B” stands for “RLE-Bitrate,” and (n) denotes a model trained on the noisy Places Audio dataset. For the WaveNet-VQ models, (ZS) and (PA) respectively denote training on the ZeroSpeech 19 English training set, and the clean Places Audio dataset.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Layer</td><td colspan="2">Clean</td><td colspan="2">20-30 dB</td><td colspan="2">10-20 dB</td><td colspan="2">0-10 dB</td></tr><tr><td>ABX</td><td>R-B</td><td>ABX</td><td>R-B</td><td>ABX</td><td>R-B</td><td>ABX</td><td>R-B</td></tr><tr><td colspan="2">WaveNet-VQ (ZS) WaveNet-VQ (PA)</td><td>N/A N/A</td><td>19.98 24.87</td><td>136.74 136.27</td><td>21.22 27.18</td><td>141.07 137.70</td><td>27.51</td><td>144.28 132.34</td><td>42.55</td><td>126.96 110.50</td></tr><tr><td colspan="2">“0”</td><td>Res2</td><td></td><td>N/A</td><td>11.63</td><td></td><td>33.29</td><td></td><td>42.67</td><td></td></tr><tr><td colspan="2">“g</td><td>Res3</td><td>11.35</td><td>N/A</td><td>11.16</td><td>N/A N/A</td><td>13.17 12.96</td><td>N/A N/A</td><td>19.44 19.43</td><td>N/A N/A</td></tr><tr><td colspan="2">“→{2}”</td><td></td><td>10.86 11.79</td><td>317.66</td><td>12.15</td><td>325.40</td><td>14.62</td><td>332.21</td><td>23.96</td><td>327.15</td></tr><tr><td colspan="2">→ {2,3}”</td><td>VQ2</td><td>12.51</td><td>341.85</td><td>12.56</td><td>350.28</td><td>14.82</td><td>362.73</td><td>25.02</td><td>330.54</td></tr><tr><td colspan="2">“ 2 →</td><td>VQ2 VQ3</td><td>14.52</td><td>136.11</td><td>14.73</td><td>137.68</td><td>17.44</td><td>143.14</td><td>27.68</td><td>133.13</td></tr><tr><td colspan="2">{2,3” 3 → {2,3”</td><td>VQ2</td><td>13.55</td><td>271.91</td><td>13.65</td><td>272.46</td><td>15.69</td><td>267.70</td><td>24.06</td><td>244.52</td></tr><tr><td colspan="2">“3 → {2,3}”</td><td>VQ3</td><td>33.70</td><td>117.37</td><td>32.56</td><td>118.22</td><td>34.65</td><td>115.40</td><td>39.82</td><td>102.48</td></tr><tr><td colspan="2">“g”(n)</td><td>Res2</td><td>13.32</td><td>N/A</td><td>12.30</td><td>N/A</td><td>12.97</td><td>N/A</td><td>16.91</td><td>N/A</td></tr><tr><td colspan="2">“0”(n)</td><td>Res3</td><td>11.85</td><td>N/A</td><td>11.90</td><td>N/A</td><td>12.44</td><td>N/A</td><td>16.09</td><td>N/A</td></tr><tr><td colspan="2">“→{2}”(n）</td><td>VQ2</td><td>12.64</td><td>342.53</td><td>12.20</td><td>348.57</td><td>13.34</td><td>359.43</td><td></td><td></td></tr><tr><td colspan="2">“{2} → {2,3}”(n）</td><td></td><td>13.42</td><td>365.89</td><td>13.71</td><td>359.14</td><td>14.57</td><td></td><td>18.82</td><td>373.60</td></tr><tr><td colspan="2">“2</td><td>VQ2</td><td>14.39</td><td>179.19</td><td>14.92</td><td>180.36</td><td>15.38</td><td>370.67</td><td>18.78</td><td>392.10</td></tr><tr><td colspan="2">{2,3}&quot;(n) “3 →</td><td>VQ3</td><td>16.52</td><td>223.28</td><td>16.47</td><td>223.61</td><td></td><td>182.27</td><td>19.58</td><td>188.32</td></tr><tr><td colspan="2">{2,3}&quot;(n） “3}</td><td>VQ2</td><td></td><td></td><td></td><td></td><td>17.75</td><td>225.72</td><td>22.68</td><td>230.01</td></tr><tr><td colspan="2">→ {2,3}&quot;(n)</td><td>VQ3</td><td>26.21</td><td>187.31</td><td>25.88</td><td>187.92</td><td>26.34</td><td>188.49</td><td>31.26</td><td>191.28</td></tr></table>
+
+Training and testing on noisy data In Hsu et al. (2019), it was shown that representations learned by a ResDAVEnet model were far more robust to train/test domain mismatch in terms of background noise, channel characteristics, and speaker identity than standard spectral features when training a supervised speech recognizer. Here, we examine whether this robustness is also exemplified by the quantized versions of this model. We construct three additional test sets using the ZeroSpeech 2019 English testing data by adding noise sampled from the AudioSet (Jansen et al., 2018) dataset. For each ZeroSpeech testing waveform, we randomly sampled an AudioSet waveform of the same duration and performed linear mixing with a signal-to-noise ratio (SNR) selected randomly within a specified range. We construct low, medium, and high noise testing sets, corresponding to SNRs of 20-30 dB, 10-20 dB, and 0-10 dB. We then perform the ABX discrimination task on these noisy waveforms, displaying the results in Table 2. We find that for all models, a worsening SNR results in a deterioration in ABX performance. However, the ResDAVEnet-VQ models prove to be far more noise robust than the Wavenet-VQ model; even in the high noise testing set, the best ResDAVEnetVQ model achieves an ABX of $2 3 . 9 6 \%$ , while the WaveNet-VQ models degrade to nearly-random ABX scores of $4 2 . 5 5 \%$ and $4 2 . 6 7 \%$ .
+
+Given that a ResDAVEnet-VQ model trained on the “clean” Places Audio captions is highly robust to additive noise on the ABX discrimination task, we investigated whether adding noise to the Places Audio captions themselves would result in an even higher degree of noise robustness. To that end, we followed a similar data augmentation approach to create a noisy version of the Places Audio captions, where the SNR of each caption was randomly chosen to sit within the range of 0-30 dB. The bottom half of Table 2 shows the results of training several ResDAVEnet-VQ models on the noisy Places Audio captions and testing on the clean and noisy ZeroSpeech ABX tasks. In general, we observe a degradation ABX score in the clean conditions, but with a significantly higher degree of noise robustness in the noisier conditions.
+
+Visualization of learned units To better measure the correspondence between the VQ units and English phones, we compute corpus-level co-occurrence statistics (at the frame-level) across the TIMIT training set, excluding the sa dialect sentences. To facilitate visualization, we use the $^ { \cdot } \sigma  \{ 2 \} ^  \cdot \}$ model with a codebook size of 128. We display the conditional probability matrix P (phone|unit) in Figure 3, with the rows and columns ordered via spectral co-clustering with 10 clusters in order to group together phones that share similar sets of VQ codes. Visually, there is a strong mapping between TIMIT phone labels and ResDAVEnet-VQ codes. In some cases, redundant codes are used for the same phone label (this is especially the case for the silence label), and in other cases we see that phones belonging to the same manner class often tend to share codebook units. We can numerically quantify the mapping between the phone and unit labels with the normalized mutual information measure (NMI), which we found to be .378 in this case. We also include several caption spectrograms with their time-aligned unit sequences in Figures 5, 6, and 7 in the supplementary material.
+
+![](images/97a1905d8097fd9ab06289ecabc5bc6d96d1bed7168174f0e95d55783b875fdd.jpg)  
+Figure 3: Conditional probability matrix displaying $P ( p h o n e | u n i t )$ using the “ $\cdot \sigma  \{ 2 \} ^ { \ast }$ model with a VQ2 codebook size of 128. For visualization, we saturate the color scaling at probability 0.5.
+
+Table 3: Performance of the VQ3 layer from the $\mathrm { \cdot \{ 3 \}  \{ 2 , 3 \} ^ { \bullet } }$ ” model when codes are treated as word detectors. Codes are ranked by the highest F1 score among the retrieved words for a given code. Word hypotheses for a given code are ranked by the F1 score. P denotes precision, R recall, and occ the number of co-occurrences of the code and word in the data.   
+
+<table><tr><td>code</td><td></td><td colspan="4">Top Hypotheses</td><td colspan="6">Second Hypotheses</td></tr><tr><td>rank</td><td></td><td>word</td><td>F1</td><td>P</td><td>R</td><td>occ</td><td>word</td><td>F1</td><td>P</td><td>R</td><td>occ</td></tr><tr><td>1</td><td>918</td><td>pantry</td><td>90.67</td><td>88.29</td><td>93.18</td><td>41</td><td>spice</td><td>3.96</td><td>2.20</td><td>20.00</td><td>1</td></tr><tr><td>2</td><td>596</td><td>kitchen</td><td>90.08</td><td>91.59</td><td>88.63</td><td>304</td><td>countertop</td><td>1.64</td><td>0.84</td><td>29.63</td><td>8</td></tr><tr><td>3</td><td>88</td><td>classroom</td><td>88.97</td><td>89.05</td><td>88.89</td><td>72</td><td>classrooms</td><td>5.01</td><td>2.57</td><td>100.00</td><td>2</td></tr><tr><td>4</td><td>58</td><td>baseball</td><td>88.71</td><td>88.63</td><td>88.78</td><td>182</td><td>player</td><td>3.01</td><td>1.65</td><td>17.11</td><td>13</td></tr><tr><td>5</td><td>706</td><td>background</td><td>87.86</td><td>91.93</td><td>84.14</td><td>838</td><td>ground</td><td>0.58</td><td>0.39</td><td>1.18</td><td>4</td></tr><tr><td>198</td><td>237</td><td>lobby</td><td>68.43</td><td>56.77</td><td>86.11</td><td>31</td><td> waiting</td><td>9.93</td><td>7.86</td><td>13.46</td><td>14</td></tr><tr><td>199</td><td>829</td><td>shirt</td><td>68.41</td><td>71.49</td><td>65.58</td><td>322</td><td>shirts</td><td>18.28</td><td>10.37</td><td>76.79</td><td>43</td></tr><tr><td>200</td><td>59</td><td>grass</td><td>68.31</td><td>56.53</td><td>86.28</td><td>503</td><td>grassy</td><td>15.30</td><td>8.67</td><td>65.35</td><td>83</td></tr></table>
+
+# 4.2 FROM PHONES TO WORDS: LEARNING A HIERARCHY OF UNITS
+
+As shown in Table 1, all of the ResDAVEnet-VQ models which underwent cold-start training of VQ3 exhibited a similar phenomenon in which the ABX error rate of that layer was particularly high, despite the model performing well at the image-caption retrieval task. We hypothesized that this could be due to VQ3 learning to recognize higher level linguistic units, such as words. To examine this empirically, we inferred the VQ3 unit sequence for every audio caption in the Places Audio training set according to several different models. Using the estimated word-level transcriptions of the utterances (provided by the Google SpeechRecognition API), we computed precision, recall, and F1 scores for every unique (word, VQ3 code) pair for a given model and quantization layer. We then ranked the VQ codes in descending order according to their maximum F1 score for any word in the vocabulary. Table 3 shows a sampling of these statistics for model ${ \bf \dot { \theta } } \{ 3 \}  \{ 2 , 3 \} ^ { \bf \theta }$ . In the supplementary material, we include many more examples for this model in Table 7, as well as examples for the $^ { \bullet } \{ 2 \}  \{ 2 , 3 \} ^ { \bullet }$ model (which did not learn VQ3 word detectors) in Table 8. It should be emphasized that these models are exactly the same in all respects, except for the order in which their quantizers were trained.
+
+![](images/67b478058c146c396767c83603f1d2f574f0a6db7715c9dde45a531cef1ad2f1.jpg)  
+Figure 4: Visualization of the precision, recall, and F1 scores of individual VQ3 codes when treated as word detectors on the Places Audio captions.
+
+We examine the overall performance of VQ3 as a word detector for these models in Figure 4. The right hand side of Figure 4 displays the number of VQ3 codes whose maximum F1 score is above a given threshold, while the left hand side shows the distribution of precision and recall scores for the top 250 words ranked by F1. This gives an approximate indication of how many VQ3 codes have learned to specialize as detectors for a specific word. We see that the VQ3 layer of model $\cdot \{ 3 \}  \{ 2 , 3 \} ^ { \prime }$ learns 279 codebook entries with an F1 score above 0.5. In contrast, the VQ3 layer of model $^ { \cdot } \{ 2 \bar  \}  \{ 2 , 3 \} ^ { \prime }$ learns only a handful of word-detecting codebook entries with an F1 of greater than 0.5. This experiment supports the notion that the reason for the poor ABX performances of the VQ3 layer in models $\ " \{ 3 \} \ "$ and ${ \bf \cdot } \{ 3 \}  \{ 2 , 3 \} ^ { \bf \cdot }$ is in fact due to its specialization for detecting specific words, and that this specialization only emerges when the VQ3 layer is learned before the VQ2 layer. Section A.2 in the supplementary material examines this phenomenon in greater experimental detail.
+
+# 5 CONCLUSIONS
+
+In this paper, we demonstrated that the neural vector quantization layers proposed by van den Oord et al. (2017) can be integrated into the visually-grounded speech models proposed by Harwath et al. (2019). This resulted in the ability of the speech model to directly represent speech units, such as phones and words, as discrete latent variables. We presented extensive experiments and analysis of these learned representations, demonstrating significant improvements in phone discrimination ability over the current state-of-the-art models for sub-word speech unit discovery. We demonstrated that these units are also far more robust to noise and domain shift than units derived from previously proposed models. These results supported the notion that semantic supervision via a discriminative, multimodal grounding objective has the potential to be more powerful than reconstruction-based objectives typically used in unsupervised speech models.
+
+We also showed how multiple vector quantizers could be employed simultaneously within a single ResDAVEnet-VQ model, and that these quantizers could be made to specialize in learning a hierarchy of speech units: specifically, phones in the lower quantizer and words in the upper quantizer. Our analysis showed that hundreds of codebooks in the upper quantizer learned to perform as word detectors, and that these detectors were highly accurate. Our experiments also revealed that this behavior only emerged when VQ3 was trained before VQ2. These results suggest the importance of the learning curriculum, which should be more deeply investigated in future work. Future work should attempt to make explicit what kind of compositional rules are implicitly encoded by these models when mapping sequences of codes from the lower quantizer to word-level units in the upper quantizer; the automatic derivation of a sub-word unit inventory, vocabulary, and pronunciation lexicon could serve as the starting point for a fully unsupervised speech recognition system. Future work should also investigate whether layers above VQ3 could be made to learn even higher-level linguistic abstractions, such as grammar, syntax, and compositional reasoning.
+
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+
+# A APPENDIX
+
+# A.1 VARYING THE CODEBOOK SIZE.
+
+In Table 4, we examine the impact of varying the codebook size of model $\^ { \bullet } \cal { O }  \{ 2 \} ^ { \bullet }$ from 128 through 2048. We find that the ABX score is best for 1024 codebook vectors, although the performance is quite good for all models. Unsurprisingly, models with smaller codebooks also achieve lower bitrates.
+
+Table 4: ABX scores and bitrates for various codebook sizes on the clean ZeroSpeech19 English test set, using the $^ { \cdot \cdot } \mathcal { O }  \{ 2 \} ^ { \cdot \cdot }$ model.   
+
+<table><tr><td>Codebook size</td><td>R@10</td><td>ABX</td><td>Bitrate</td><td>RLE-Bitrate</td><td>Segment-ABX</td><td>Segment-Bitrate</td></tr><tr><td>128</td><td>.772</td><td>14.25</td><td>295.65</td><td>212.27</td><td>15.42</td><td>179.38</td></tr><tr><td>256</td><td>.756</td><td>12.95</td><td>341.18</td><td>260.10</td><td>14.21</td><td>228.07</td></tr><tr><td>512</td><td>.761</td><td>12.59</td><td>363.95</td><td>288.64</td><td>13.10</td><td>259.94</td></tr><tr><td>1024</td><td>.760</td><td>11.79</td><td>390.61</td><td>317.66</td><td>12.66</td><td>289.11</td></tr><tr><td>2048</td><td>.768</td><td>12.41</td><td>360.04</td><td>283.68</td><td>13.15</td><td>254.23</td></tr></table>
+
+A.2 THE IMPACT OF THE VQ TRAINING CURRICULUM ON THE LOCALIZATION OF WORD DETECTORS
+
+In Section 4.2, we showed that cold-start training of the VQ3 layer caused its codebook vectors to specialize as word detectors, whereas warm-start training did not. Our subsequent experiments (Table 6) revealed that when adding a third quantization layer at the Res4 position to a model that did not learn word detectors at VQ3, the VQ4 layer did in fact learn many word detectors. This suggests implicit word recognition ability can be localized at different layers in the ResDAVEnet audio model, and exactly where it emerges depends upon the VQ training curriculum. We hypothesize that this is due in part to two factors:
+
+Table 5: ABX scores on the ZeroSpeech 19 English test set using features derived from the output of the Res3 block of the ResDAVEnet audio branch (pre-quantization).   
+
+<table><tr><td>Model ID</td><td>Res3 ABX</td></tr><tr><td>“Q”</td><td>10.86</td></tr><tr><td>“Q→{2}”</td><td>11.61</td></tr><tr><td>“→{3}”</td><td>10.91</td></tr><tr><td>“→{2,3}”</td><td>12.68</td></tr><tr><td>“{2}”</td><td>11.45</td></tr><tr><td>“{2}→{2,3}”</td><td>11.37</td></tr><tr><td>“{3}”</td><td>32.24</td></tr><tr><td>“{3}→{2,3}”</td><td>28.33</td></tr></table>
+
+1. In a warm-start model, whatever type of information (subword-like, word-like) the continuous model learned to encode at a particular convolutional layer (or residual block) does not change after a quantizer is appended to that layer.   
+2. In a cold-start model, each active quantization layer forms a potential bottleneck, restricting the amount of information that is able to pass through to subsequent layers.
+
+According to this hypothesis, if word-level recognition tends to emerge at a particular layer in an unconstrained network with no quantization bottleneck, it will stay there when quantization is introduced for fine-tuning. However, when a quantization bottleneck is introduced from the very beginning of training, the gradient flowing down into the lower network layers is more constrained during the initial training epochs (when the gradient tends to be the largest). This may have the effect of steering the optimizer into a different part of the parameter space, in which word recognition occurs at a different layer than it otherwise would.
+
+We present results from two experiments that support this view. In Table 5, we show the ABX scores of the Res3 layer prior to quantization (if present) during the course of three different training curricula resulting in a $^ { 6 6 } \{ 2 , 3 \}$ ” final model. We observe that the ABX error rate changes very little within each individual curriculum. This indicates that the initial model sets the stage for which layer learns to capture phonetic information with the highest salience, and that subsequent training steps do not tend to move this information elsewhere.
+
+Table 6: The number of codebook vectors at a particular VQ layer that learned to be a detector for any word with an F1 score greater than 0.5.
+
+$$
+\begin{array} { r l } &  \underbrace { \frac { \mathcal { H } \mathrm { ~ o u n i t r e s t ~ L y e r s } } { 1 } } _ { \begin{array} { c } { \times } \\ \\ { \times } \\ \\ { \frac { \mathcal { H } \cdot \sqrt { 2 } \cdot 3 ^ { \nu } } { \begin{array} { c } { \times \cos \left( 3 \right) ^ { \nu } } \\ { \frac { 3 } { \sqrt { 3 } } } \\ { \frac { \mathcal { H } \cdot \sqrt { 2 } \cdot 3 ^ { \nu } } { \sqrt { 3 } + \left( 2 \cdot 3 \right) ^ { \nu } } } \end{array} } } & { \begin{array} { c } { \mathrm { ~ V o r ~ L y e r e ~ \mathcal { } ~ \mathcal { H } ~ w o r d r o s ~ ( 7 1 \cdot 5 \cdot 0 . 5 ) } } \\ { \frac { 3 } { 2 0 } } \\ { \frac { 3 } { 2 0 } } \end{array} } } \\ { \geq } & { \begin{array} { c } { \frac { \mathcal { H } \cdot \sqrt { 2 } \cdot 3 ^ { \nu } } { \begin{array} { c } { \times \left( 2 \cdot 3 \right) ^ { \nu } } \\ { \frac { \nu } { \sqrt { 3 } } + \left( 2 \cdot 3 \right) ^ { \nu } } \end{array} } } & { \begin{array} { c } { \frac { 3 } { 3 } } \\ { \frac { 3 } { 2 } } \\ { \frac { \pi } { \sqrt { 3 } } } \end{array} } & { \begin{array} { c } { 1 0 } \\ { 1 0 } \end{array} } \\ { \left( \begin{array} { c } { \frac { \pi } { \sqrt { 3 } + \left( 2 \cdot 3 \right) ^ { \nu } } - \frac { 2 } { 3 } \frac { \pi } { \sqrt { 3 } } } \\ { \frac { \pi } { \sqrt { 3 } } - \frac { 2 } { 3 } \cdot 3 ^ { \nu } } \end{array} \right) } \\ { \geq } & { \begin{array} { c } { \frac { 3 } { \sqrt { 3 } + 3 - \left( 2 \cdot 3 \right) ^ { \nu } } } \\ { 3 } \end{array} } \\ { \left( \begin{array} { c } { \frac { \pi } { \sqrt { 3 } + \left( 2 \cdot 3 \right) ^ { \nu } } - \frac { 2 } { 3 } \frac { \pi } { \sqrt { 3 } } } \\ { \frac { \pi } { \sqrt { 3 } } } \end{array} \right) } \end{array} } \end{array} \end{array}
+$$
+
+Table 6 displays the number of word detectors learned by various VQ layers across different training curricula. Here, we claim that a codebook vector belonging to a particular VQ layer has learned to be a word detector if its F1 score for any word appearing in the test set exceeds 0.5 (as measured on the test set). There are several interesting things to note here. First, we only observe a significant number of word detectors at the VQ3 layer when that layer is trained from a cold-start. Even when adding a second VQ layer as in the ${ } ^ { * * } \{ 3 \} \stackrel { \cdot } { }  \{ 2 , 3 \} ^ { \prime }$ model, these detectors remain at VQ3.
+
+Jointly training VQ2 and VQ3 together from a cold-start results in the word detectors being divided between those layers. While this experiment demonstrates that it is possible to jointly train two quantizers at once, the $\{ 2 , 3 \}$ model learned the smallest total number of word detectors of any model. Additionally, we were unable to successfully train a cold-start $\{ 2 , 3 , 4 \}$ model; these experiments suggest that training quantizers one by one may be easier in general.
+
+For all models beginning from a “ $\overrightarrow { Q }$ ” or “ $\{ 2 \} ^ { \ast }$ initial model, we do not observe any word detectors at either VQ2 or VQ3. However, a third quantizer at the VQ4 position in the $^ { \infty } \{ \stackrel { . . } { 2 } \}  \{ 2 , 3 \} $ $\{ 2 , 3 , 4 \} ^ { \prime }$ model was able to capture words. We hypothesize that in all models not trained from a “ $\{ 3 \} ^ { \ast }$ initialization, word recognition is implicitly learned by the Res4 layer, and adding a quantizer to the output of this layer serves to make the categorical nature of those representations explicit.
+
+# A.3 WORD DETECTOR TABLES FOR VARIOUS MODELS
+
+In Table 7, we show a sampling of 50 word-detecting codebook entries from the VQ layer of the $^ { \bullet } \{ 3 \}  \{ 2 , 3 \} ^ { \ast }$ model (many word detectors learned). Analagous results for the $\^ { * } \{ 2 \} \stackrel { \cdot } {  } \{ 2 , 3 \} ^ { * }$ model (few word detectors learned) are shown in Table 8.
+
+# A.4 UNIT VISUALIZATION FOR INDIVIDUAL CAPTION SPECTROGRAMS
+
+To provide a better intuitive understanding of what the units learned by our models look like, in Figures 5, 6, and 7, we display speech spectrograms for several Places caption fragments. Along with each spectrogram we display the time-aligned, ground-truth, word-level text (top transcription), the inferred unit sequence for the VQ4 layer (middle transcription), and the unit sequence for the VQ3 (bottom transcription) layer. All VQ unit alignments in these figures are derived from the “ $\{ 2 \}  \{ 2 , 3 \}  \{ 2 , \bar { 3 } , 4 \} ^ { , , }$ model.
+
+Table 7: Performance of the VQ3 layer from the $\{ 3 \}  \{ 2 , 3 \} ^ { \prime }$ ” model when codes are treated as word detectors. Codes are ranked by the highest F1 score among the retrieved words for a given code. Word hypotheses for a given code are ranked by the F1 score.   
+
+<table><tr><td rowspan=1 colspan=12>Top Hypotheses                            Second Hypothesesrank codeword       F1     P     R   occ    word       F1     P      R  oCC</td></tr><tr><td rowspan=1 colspan=12>1   918    pantry    90.67 88.29 93.18   41     spice      3.96  2.20  20.00    1</td></tr><tr><td rowspan=1 colspan=2>2   596    kitchen</td><td rowspan=1 colspan=1>90.08</td><td rowspan=1 colspan=1>91.59</td><td rowspan=1 colspan=8>88.63   304  countertop    1.64   0.84  29.63    8</td></tr><tr><td rowspan=1 colspan=2>3    88   classroom</td><td rowspan=1 colspan=1>88.97</td><td rowspan=1 colspan=1>89.05</td><td rowspan=1 colspan=8>88.89   72  classrooms   5.01  2.57 100.00    2</td></tr><tr><td rowspan=1 colspan=2>4    58    baseball</td><td rowspan=1 colspan=1>88.71</td><td rowspan=1 colspan=1>88.63</td><td rowspan=1 colspan=8>88.78  182    player     3.01   1.65  17.11   13</td></tr><tr><td rowspan=1 colspan=2>5    706  background</td><td rowspan=1 colspan=1>87.86</td><td rowspan=1 colspan=1>91.93</td><td rowspan=1 colspan=8>84.14  838   ground     0.58  0.39   1.18    4</td></tr><tr><td rowspan=1 colspan=2>6   736    museum</td><td rowspan=1 colspan=1>87.35</td><td rowspan=1 colspan=1>93.44</td><td rowspan=1 colspan=8>82.00   41   museums    5.47  2.81 100.00    1</td></tr><tr><td rowspan=1 colspan=2>7   274    subway</td><td rowspan=1 colspan=1>87.26</td><td rowspan=1 colspan=1>88.34</td><td rowspan=1 colspan=8>86.21    75   assembly    5.32  2.85  40.00    4</td></tr><tr><td rowspan=1 colspan=2>8    116  construction</td><td rowspan=1 colspan=1>87.07</td><td rowspan=1 colspan=1>89.78</td><td rowspan=1 colspan=8>84.52  131  constructed   2.43  1.25  38.46    5</td></tr><tr><td rowspan=1 colspan=2>9   892    walking</td><td rowspan=1 colspan=1>87.06</td><td rowspan=1 colspan=1>87.57</td><td rowspan=1 colspan=8>86.55  412     walk      7.07  3.97  31.94  23</td></tr><tr><td rowspan=1 colspan=2>10   557    concrete</td><td rowspan=1 colspan=1>86.53</td><td rowspan=1 colspan=1>90.98</td><td rowspan=1 colspan=2>82.50</td><td rowspan=1 colspan=1>99</td><td rowspan=1 colspan=3>concur     1.21  0.61</td><td rowspan=1 colspan=2>100.00    1</td></tr><tr><td rowspan=1 colspan=2>11   48     desert</td><td rowspan=1 colspan=1>86.50</td><td rowspan=1 colspan=1>90.30</td><td rowspan=1 colspan=2>83.01</td><td rowspan=1 colspan=1>171</td><td rowspan=1 colspan=3>dozen     2.76  1.49</td><td rowspan=1 colspan=2>18.18    2</td></tr><tr><td rowspan=1 colspan=2>12   534  background</td><td rowspan=1 colspan=1>86.18</td><td rowspan=1 colspan=1>81.95</td><td rowspan=1 colspan=2>90.86</td><td rowspan=1 colspan=1>905</td><td rowspan=1 colspan=1>back</td><td rowspan=1 colspan=1>8.95</td><td rowspan=1 colspan=2>5.83  19.34</td><td rowspan=1 colspan=1>64</td></tr><tr><td rowspan=1 colspan=2>13   44     patio</td><td rowspan=1 colspan=1>85.82</td><td rowspan=1 colspan=1>90.87</td><td rowspan=1 colspan=2>81.29</td><td rowspan=1 colspan=1>113</td><td rowspan=1 colspan=1>patios</td><td rowspan=1 colspan=1>1.56</td><td rowspan=1 colspan=1>0.79</td><td rowspan=1 colspan=1>100.00</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=2>14   625  background</td><td rowspan=1 colspan=1>85.17</td><td rowspan=1 colspan=1>92.92</td><td rowspan=1 colspan=3>78.61   783</td><td rowspan=1 colspan=2>back      1.63</td><td rowspan=1 colspan=1>1.01</td><td rowspan=1 colspan=2>4.23   14</td></tr><tr><td rowspan=1 colspan=2>15   732     closet</td><td rowspan=1 colspan=1>84.92</td><td rowspan=1 colspan=1>94.64</td><td rowspan=1 colspan=3>77.01    67</td><td rowspan=1 colspan=2>closets     4.96</td><td rowspan=1 colspan=1>2.68</td><td rowspan=1 colspan=2>33.33</td></tr><tr><td rowspan=1 colspan=2>16   30    waterfall</td><td rowspan=1 colspan=1>84.90</td><td rowspan=1 colspan=1>75.73</td><td rowspan=1 colspan=3>96.61    57</td><td rowspan=1 colspan=2>waterfalls   14.26</td><td rowspan=1 colspan=1>7.68</td><td rowspan=1 colspan=2>100.00    7</td></tr><tr><td rowspan=1 colspan=2>17   388   courtyard</td><td rowspan=1 colspan=1>84.89</td><td rowspan=1 colspan=1>92.16</td><td rowspan=1 colspan=3>78.69   48</td><td rowspan=1 colspan=2> graveyard   5.50</td><td rowspan=1 colspan=1>3.24</td><td rowspan=1 colspan=2>18.18    4</td></tr><tr><td rowspan=1 colspan=2>18   560   hospital</td><td rowspan=1 colspan=1>84.70</td><td rowspan=1 colspan=1>91.48</td><td rowspan=1 colspan=3>78.85   41</td><td rowspan=1 colspan=2>horses     1.55</td><td rowspan=1 colspan=1>1.92</td><td rowspan=1 colspan=2>1.30    1</td></tr><tr><td rowspan=1 colspan=2>19   18    driveway</td><td rowspan=1 colspan=1>84.56</td><td rowspan=1 colspan=1>90.10</td><td rowspan=1 colspan=3>79.66   47</td><td rowspan=1 colspan=1>driveways</td><td rowspan=1 colspan=1>3.52</td><td rowspan=1 colspan=1>1.82</td><td rowspan=1 colspan=1>50.00</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=2>20   598    palm</td><td rowspan=1 colspan=1>84.39</td><td rowspan=1 colspan=1>82.08</td><td rowspan=1 colspan=3>86.84   99</td><td rowspan=1 colspan=1>plum</td><td rowspan=1 colspan=1>1.57</td><td rowspan=1 colspan=1>0.79</td><td rowspan=1 colspan=1>100.00</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=2>21    85     yellow</td><td rowspan=1 colspan=1>84.30</td><td rowspan=1 colspan=1>83.93</td><td rowspan=1 colspan=3>84.66  574</td><td rowspan=1 colspan=1>yellowish</td><td rowspan=1 colspan=1>1.98</td><td rowspan=1 colspan=1>1.00</td><td rowspan=1 colspan=1>100.00</td><td rowspan=1 colspan=1>7</td></tr><tr><td rowspan=1 colspan=2>22   584  playground</td><td rowspan=1 colspan=1>84.18</td><td rowspan=1 colspan=1>77.37</td><td rowspan=1 colspan=1>92.31</td><td rowspan=1 colspan=2>36</td><td rowspan=1 colspan=1>play</td><td rowspan=1 colspan=1>6.32</td><td rowspan=1 colspan=1>4.75</td><td rowspan=1 colspan=1>9.43</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=2>23   162    stadium</td><td rowspan=1 colspan=1>83.82</td><td rowspan=1 colspan=1>84.50</td><td rowspan=1 colspan=1>83.15</td><td rowspan=1 colspan=2>74</td><td rowspan=1 colspan=1>boardwalk</td><td rowspan=1 colspan=1>9.12</td><td rowspan=1 colspan=1>4.91</td><td rowspan=1 colspan=1>63.16</td><td rowspan=1 colspan=1>12</td></tr><tr><td rowspan=1 colspan=1>24</td><td rowspan=1 colspan=1>769    bamboo</td><td rowspan=1 colspan=1>83.79</td><td rowspan=1 colspan=1>93.68</td><td rowspan=1 colspan=1>75.79</td><td rowspan=1 colspan=2>72</td><td rowspan=1 colspan=1>baboons</td><td rowspan=1 colspan=1>2.03</td><td rowspan=1 colspan=1>1.03</td><td rowspan=1 colspan=1>100.00</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>25</td><td rowspan=1 colspan=1>193     small</td><td rowspan=1 colspan=1>83.55</td><td rowspan=1 colspan=1>90.46</td><td rowspan=1 colspan=1>77.63</td><td rowspan=1 colspan=2>791</td><td rowspan=1 colspan=1>smaller</td><td rowspan=1 colspan=1>2.15</td><td rowspan=1 colspan=1>1.10</td><td rowspan=1 colspan=1>50.00</td><td rowspan=1 colspan=1>15</td></tr><tr><td rowspan=1 colspan=1>26</td><td rowspan=1 colspan=1>412    podium</td><td rowspan=1 colspan=1>83.53</td><td rowspan=1 colspan=1>76.73</td><td rowspan=1 colspan=1>91.67</td><td rowspan=1 colspan=2>22</td><td rowspan=1 colspan=1>auditorium</td><td rowspan=1 colspan=1>8.69</td><td rowspan=1 colspan=1>5.82</td><td rowspan=1 colspan=1>17.14</td><td rowspan=1 colspan=1>6</td></tr><tr><td rowspan=1 colspan=1>27</td><td rowspan=1 colspan=1>108   highway</td><td rowspan=1 colspan=1>83.52</td><td rowspan=1 colspan=1>79.58</td><td rowspan=1 colspan=1>87.88</td><td rowspan=1 colspan=2>58</td><td rowspan=1 colspan=1>highlights</td><td rowspan=1 colspan=1>5.44</td><td rowspan=1 colspan=1>2.87</td><td rowspan=1 colspan=1>50.00</td><td rowspan=1 colspan=1>3</td></tr><tr><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>394    church</td><td rowspan=1 colspan=1>83.34</td><td rowspan=1 colspan=1>75.98</td><td rowspan=1 colspan=1>92.28</td><td rowspan=1 colspan=2>227</td><td rowspan=1 colspan=1>religious</td><td rowspan=1 colspan=1>6.34</td><td rowspan=1 colspan=1>3.45</td><td rowspan=1 colspan=1>39.39</td><td rowspan=1 colspan=1>13</td></tr><tr><td rowspan=1 colspan=1>29</td><td rowspan=1 colspan=1>661    distance</td><td rowspan=1 colspan=1>83.32</td><td rowspan=1 colspan=1>78.96</td><td rowspan=1 colspan=1>88.19</td><td rowspan=1 colspan=2>351</td><td rowspan=1 colspan=1>lounge</td><td rowspan=1 colspan=1>1.63</td><td rowspan=1 colspan=1>0.86</td><td rowspan=1 colspan=1>14.71</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1>30</td><td rowspan=1 colspan=1>708    distance</td><td rowspan=1 colspan=1>82.97</td><td rowspan=1 colspan=1>96.32</td><td rowspan=1 colspan=1>72.86</td><td rowspan=1 colspan=2>290</td><td rowspan=1 colspan=1>farmland</td><td rowspan=1 colspan=1>1.35</td><td rowspan=1 colspan=1>0.68</td><td rowspan=1 colspan=1>55.56</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1>31</td><td rowspan=1 colspan=1>14      gallery</td><td rowspan=1 colspan=1>82.97</td><td rowspan=1 colspan=1>85.08</td><td rowspan=1 colspan=1>80.95</td><td rowspan=1 colspan=2>17</td><td rowspan=1 colspan=1>art</td><td rowspan=1 colspan=1>11.39</td><td rowspan=1 colspan=1>12.15</td><td rowspan=1 colspan=1>10.71</td><td rowspan=1 colspan=1>9</td></tr><tr><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>996     large</td><td rowspan=1 colspan=1>82.95</td><td rowspan=1 colspan=1>87.05</td><td rowspan=1 colspan=1>79.21</td><td></td><td rowspan=1 colspan=1>1753</td><td rowspan=1 colspan=1>very</td><td rowspan=1 colspan=1>2.83</td><td rowspan=1 colspan=1>1.72</td><td rowspan=1 colspan=1>8.01</td><td rowspan=1 colspan=1>94</td></tr><tr><td rowspan=1 colspan=1>33</td><td rowspan=1 colspan=1>944   cathedral</td><td rowspan=1 colspan=1>82.78</td><td rowspan=1 colspan=1>79.22</td><td rowspan=1 colspan=1>86.67</td><td></td><td rowspan=1 colspan=1>52</td><td rowspan=1 colspan=1>feed</td><td rowspan=1 colspan=1>2.74</td><td rowspan=1 colspan=1>1.41</td><td rowspan=1 colspan=1>50.00</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=2>34   122    purple</td><td rowspan=1 colspan=1>82.63</td><td rowspan=1 colspan=1>91.98</td><td rowspan=1 colspan=1>75.00</td><td rowspan=1 colspan=2>138</td><td rowspan=1 colspan=1>proportion</td><td rowspan=1 colspan=1>1.66</td><td rowspan=1 colspan=1>0.84</td><td rowspan=1 colspan=1>50.00</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=2>35   630     trees</td><td rowspan=1 colspan=1>82.52</td><td rowspan=1 colspan=1>80.39</td><td rowspan=1 colspan=1>84.77</td><td rowspan=1 colspan=2>1258</td><td rowspan=1 colspan=1>tree</td><td rowspan=1 colspan=1>15.48</td><td rowspan=1 colspan=1>9.47</td><td rowspan=1 colspan=1>42.33</td><td rowspan=1 colspan=1>171</td></tr><tr><td rowspan=1 colspan=2>186  375      boy</td><td rowspan=1 colspan=1>69.90</td><td rowspan=1 colspan=1>65.45</td><td rowspan=1 colspan=1>75.00</td><td rowspan=1 colspan=2>93</td><td rowspan=1 colspan=1>boys</td><td rowspan=1 colspan=1>20.87</td><td rowspan=1 colspan=1>13.09</td><td rowspan=1 colspan=1>51.43</td><td rowspan=1 colspan=1>18</td></tr><tr><td rowspan=1 colspan=2>187  634    ground</td><td rowspan=1 colspan=1>69.78</td><td rowspan=1 colspan=1>73.92</td><td rowspan=1 colspan=1>66.08</td><td rowspan=1 colspan=2>224</td><td rowspan=1 colspan=1>playground</td><td rowspan=1 colspan=1>7.45</td><td rowspan=1 colspan=1>3.94</td><td rowspan=1 colspan=1>69.23</td><td rowspan=1 colspan=1>27</td></tr><tr><td rowspan=1 colspan=2>188   69   courtyard</td><td rowspan=1 colspan=1>69.55</td><td rowspan=1 colspan=1>57.98</td><td rowspan=1 colspan=1>86.89</td><td rowspan=1 colspan=2>53</td><td rowspan=1 colspan=1>plaza</td><td rowspan=1 colspan=1>28.89</td><td rowspan=1 colspan=1>19.87</td><td rowspan=1 colspan=1>52.94</td><td rowspan=1 colspan=1>9</td></tr><tr><td rowspan=1 colspan=2>189  281    wooden</td><td rowspan=1 colspan=1>69.50</td><td rowspan=1 colspan=1>57.89</td><td rowspan=1 colspan=1>86.92</td><td rowspan=1 colspan=2>525</td><td rowspan=1 colspan=1>wood</td><td rowspan=1 colspan=1>23.55</td><td rowspan=1 colspan=1>14.52</td><td rowspan=1 colspan=1>62.20</td><td rowspan=1 colspan=1>153</td></tr><tr><td rowspan=1 colspan=2>190  812   lighthouse</td><td rowspan=1 colspan=1>69.41</td><td rowspan=1 colspan=1>59.74</td><td rowspan=1 colspan=1>82.81</td><td rowspan=1 colspan=2>53</td><td rowspan=1 colspan=1>lighthouses</td><td rowspan=1 colspan=1>11.69</td><td rowspan=1 colspan=1>6.21</td><td rowspan=1 colspan=1>100.00</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=2>191  225     house</td><td rowspan=1 colspan=1>69.13</td><td rowspan=1 colspan=1>61.59</td><td rowspan=1 colspan=1>78.78</td><td rowspan=1 colspan=2>516</td><td rowspan=1 colspan=1>houses</td><td rowspan=1 colspan=1>18.29</td><td rowspan=1 colspan=1>10.31</td><td rowspan=1 colspan=1>80.73</td><td rowspan=1 colspan=1>88</td></tr><tr><td rowspan=1 colspan=2>192  705      dark</td><td rowspan=1 colspan=1>69.11</td><td rowspan=1 colspan=1>68.57</td><td rowspan=1 colspan=1>69.66</td><td rowspan=1 colspan=2>186</td><td rowspan=1 colspan=1>darker</td><td rowspan=1 colspan=1>3.05</td><td rowspan=1 colspan=1>1.56</td><td rowspan=1 colspan=1>75.00</td><td rowspan=1 colspan=1>6</td></tr><tr><td rowspan=1 colspan=2>193  980    building</td><td rowspan=1 colspan=1>69.10</td><td rowspan=1 colspan=1>77.48</td><td rowspan=1 colspan=1>62.35</td><td rowspan=1 colspan=2>1161</td><td rowspan=1 colspan=1>buildings</td><td rowspan=1 colspan=1>25.72</td><td rowspan=1 colspan=1>15.71</td><td rowspan=1 colspan=1>70.78</td><td rowspan=1 colspan=1>281</td></tr><tr><td rowspan=1 colspan=2>194  844     grass</td><td rowspan=1 colspan=1>69.01</td><td rowspan=1 colspan=1>61.85</td><td rowspan=1 colspan=1>78.04</td><td rowspan=1 colspan=2>455</td><td rowspan=1 colspan=1>grassy</td><td rowspan=1 colspan=1>21.70</td><td rowspan=1 colspan=1>12.42</td><td rowspan=1 colspan=1>85.83</td><td rowspan=1 colspan=1>109</td></tr><tr><td rowspan=1 colspan=2>195  446      lake</td><td rowspan=1 colspan=1>68.69</td><td rowspan=1 colspan=1>78.63</td><td rowspan=1 colspan=1>60.98</td><td rowspan=1 colspan=2>125</td><td rowspan=1 colspan=1>late</td><td rowspan=1 colspan=1>5.02</td><td rowspan=1 colspan=2>2.90  18.52</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=2>196  182     trash</td><td rowspan=1 colspan=1>68.64</td><td rowspan=1 colspan=1>66.79</td><td rowspan=1 colspan=1>70.59</td><td rowspan=1 colspan=2>48</td><td rowspan=1 colspan=1>boulders</td><td rowspan=1 colspan=1>10.16</td><td rowspan=1 colspan=1>6.27</td><td rowspan=1 colspan=1>26.67</td><td rowspan=1 colspan=1>4</td></tr><tr><td rowspan=1 colspan=2>197  437  photograph</td><td rowspan=1 colspan=1>68.63</td><td rowspan=1 colspan=1>59.06</td><td rowspan=1 colspan=1>81.89</td><td rowspan=1 colspan=2>588</td><td rowspan=1 colspan=1>photographs</td><td rowspan=1 colspan=1>30.56</td><td rowspan=1 colspan=1>18.46</td><td rowspan=1 colspan=1>88.73</td><td rowspan=1 colspan=1>181</td></tr><tr><td rowspan=1 colspan=2>198  237     lobby</td><td rowspan=1 colspan=1>68.43</td><td rowspan=1 colspan=1>56.77</td><td rowspan=1 colspan=1>86.11</td><td rowspan=1 colspan=2>31</td><td rowspan=1 colspan=1>waiting</td><td rowspan=1 colspan=1>9.93</td><td rowspan=1 colspan=1>7.86</td><td rowspan=1 colspan=1>13.46</td><td rowspan=1 colspan=1>14</td></tr><tr><td rowspan=1 colspan=2>199  829     shirt</td><td rowspan=1 colspan=1>68.41</td><td rowspan=1 colspan=1>71.49</td><td rowspan=1 colspan=1>65.58</td><td rowspan=1 colspan=2>322</td><td rowspan=1 colspan=1>shirts</td><td rowspan=1 colspan=1>18.28</td><td rowspan=1 colspan=3>10.37  76.79  43</td></tr><tr><td rowspan=1 colspan=3>200   59      grass    68.31</td><td rowspan=1 colspan=1>56.53</td><td rowspan=1 colspan=8>86.28   503    grassy    15.30  8.67  65.35  83</td></tr></table>
+
+Table 8: Performance of the VQ3 layer from the $\{ 2 \}  \{ 2 , 3 \} ^ { \prime }$ model when codes are treated as word detectors. Codes are ranked by the highest F1 score among the retrieved words for a given code. Word hypotheses for a given code are ranked by the F1 score.   
+
+<table><tr><td>rank</td><td></td><td colspan="5">Top Hypotheses</td><td colspan="5">Second Hypotheses</td></tr><tr><td></td><td>code</td><td>word</td><td>F1</td><td>P</td><td>R</td><td>occ</td><td>word</td><td>F1</td><td>P</td><td>R</td><td>occ</td></tr><tr><td>1</td><td>924</td><td>people</td><td>76.71</td><td>67.49</td><td>88.85</td><td>1665</td><td>computer</td><td>2.17</td><td>1.12</td><td>40.40</td><td>40</td></tr><tr><td>2</td><td>749</td><td>white</td><td>76.47</td><td>66.92</td><td>89.21</td><td>2265</td><td>one</td><td>4.15</td><td>2.50</td><td>12.14</td><td>134</td></tr><tr><td>3</td><td>530</td><td>building</td><td>75.47</td><td>64.84</td><td>90.28</td><td>1681</td><td>buildings</td><td>23.93</td><td>13.81</td><td>89.67</td><td>356</td></tr><tr><td>4</td><td>505</td><td>blue</td><td>59.12</td><td>46.90</td><td>79.96</td><td>1093</td><td>pool</td><td>10.74</td><td>5.89</td><td>60.80</td><td>152</td></tr><tr><td>5</td><td>581</td><td>snow</td><td>57.61</td><td>41.77</td><td>92.83</td><td>466</td><td>snowy</td><td>16.63</td><td>9.12</td><td>94.50</td><td>103</td></tr><tr><td>6</td><td>778</td><td>building</td><td>52.10</td><td>36.71</td><td>89.69</td><td>1670</td><td>buildings</td><td>14.30</td><td>7.78</td><td>88.16</td><td>350</td></tr><tr><td>7</td><td>144</td><td>with</td><td>49.12</td><td>41.58</td><td>59.99</td><td>3386</td><td>wooden</td><td>6.32</td><td>3.34</td><td>60.60</td><td>366</td></tr><tr><td>8</td><td>299</td><td>small</td><td>47.83</td><td>32.78</td><td>88.42</td><td>901</td><td>snow</td><td>30.55</td><td>18.35</td><td>91.24</td><td>458</td></tr><tr><td>9</td><td>550</td><td>large</td><td>45.13</td><td>30.50</td><td>86.76</td><td>1920</td><td>car</td><td>8.57</td><td>4.52</td><td>82.13</td><td>216</td></tr><tr><td>10</td><td>76</td><td>trees</td><td>44.82</td><td>29.76</td><td>90.77</td><td>1347</td><td>tree</td><td>15.21</td><td>8.31</td><td>89.36</td><td>361</td></tr><tr><td>11</td><td>831</td><td>water</td><td>41.84</td><td>27.50</td><td>87.43</td><td>1210</td><td>wall</td><td>17.57</td><td>9.87</td><td>79.97</td><td>491</td></tr><tr><td>12</td><td>1015</td><td>large</td><td>39.59</td><td>26.06</td><td>82.29</td><td>1821</td><td>cars</td><td>6.58</td><td>3.42</td><td>84.21</td><td>224</td></tr><tr><td>13</td><td>80</td><td>red</td><td>39.15</td><td>26.13</td><td>78.05</td><td>992</td><td>bed</td><td>9.14</td><td>4.91</td><td>66.67</td><td>168</td></tr><tr><td>14</td><td>719</td><td>woman</td><td>38.71</td><td>24.95</td><td>86.31</td><td>687</td><td>women</td><td>7.75</td><td>4.09</td><td>72.41</td><td>126</td></tr><tr><td>15</td><td>614</td><td>people</td><td>37.71</td><td>25.10</td><td>75.83</td><td>1421</td><td>table</td><td>22.10</td><td>12.75</td><td>82.93</td><td>656</td></tr><tr><td>16</td><td>816</td><td>water</td><td>37.71</td><td>24.25</td><td>84.75</td><td>1173</td><td>river</td><td>5.72</td><td>2.97</td><td>80.22</td><td>215</td></tr><tr><td>17</td><td>457</td><td>sky</td><td>35.71</td><td>23.20</td><td>77.47</td><td>540</td><td>skies</td><td>11.81</td><td>6.33</td><td>88.12</td><td>141</td></tr><tr><td>18</td><td>480</td><td>has</td><td>34.88</td><td>25.01</td><td>57.64</td><td>1204</td><td>house</td><td>9.88</td><td>5.48</td><td>50.38</td><td>330</td></tr><tr><td>19</td><td>245</td><td>yellow</td><td>34.14</td><td>21.17</td><td>88.05</td><td>597</td><td>flowers</td><td>13.44</td><td>7.27</td><td>89.43</td><td>237</td></tr><tr><td>20</td><td>968</td><td>picture</td><td>34.11</td><td>22.32</td><td>72.33</td><td>1686</td><td>pictures</td><td>16.79</td><td>9.38</td><td>79.77</td><td>698</td></tr><tr><td>21</td><td>985</td><td>trees</td><td>33.88</td><td>22.04</td><td>73.25</td><td>1087</td><td>tree</td><td>10.96</td><td>5.93</td><td>72.52</td><td>293</td></tr><tr><td>22</td><td>536</td><td>man</td><td>33.53</td><td>20.71</td><td>88.06</td><td>1128</td><td> standing</td><td>9.06</td><td>4.86</td><td>66.17</td><td>532</td></tr><tr><td>23</td><td>0</td><td>black</td><td>33.49</td><td>20.81</td><td>85.77</td><td>1163</td><td>background</td><td>21.15</td><td>12.28</td><td>76.20</td><td>759</td></tr><tr><td>24</td><td>815</td><td>with</td><td>33.21</td><td>22.76</td><td>61.36</td><td>3463</td><td>white</td><td>8.60</td><td>4.85</td><td>38.05</td><td>966</td></tr><tr><td>25</td><td>293</td><td>large</td><td>33.13</td><td>23.80</td><td>54.50</td><td>1206</td><td>bridge</td><td>19.09</td><td>10.78</td><td>83.18</td><td>371</td></tr><tr><td>26</td><td>870</td><td>trees</td><td>32.87</td><td>20.79</td><td>78.44</td><td>1164</td><td>train</td><td>12.97</td><td>7.00</td><td>88.03</td><td></td></tr><tr><td>27</td><td>153</td><td>yellow</td><td>32.42</td><td>19.94</td><td>86.73</td><td>588</td><td></td><td>15.39</td><td>9.47</td><td>40.92</td><td>353</td></tr><tr><td>28</td><td>243</td><td>front</td><td></td><td>20.97</td><td>68.69</td><td>895</td><td>area from</td><td>14.34</td><td></td><td></td><td>354</td></tr><tr><td>29</td><td></td><td>black</td><td>32.13 31.40</td><td></td><td></td><td></td><td></td><td></td><td>8.51</td><td>45.49</td><td>358</td></tr><tr><td></td><td>538</td><td>small</td><td></td><td>19.64</td><td>78.24</td><td>1061</td><td>glass</td><td>8.28</td><td>4.36</td><td>82.90</td><td>223</td></tr><tr><td>30</td><td>526</td><td></td><td>31.34</td><td>19.08</td><td>87.83</td><td>895</td><td>iarge</td><td>5.91</td><td>4.04</td><td>11.03</td><td>244</td></tr><tr><td>31</td><td>395</td><td>picture</td><td>31.32</td><td>20.90</td><td>62.46</td><td>1456</td><td>pictures</td><td>13.98</td><td>7.80</td><td>67.54</td><td>591</td></tr><tr><td>32</td><td>133</td><td>white</td><td>29.82</td><td>18.98</td><td>69.55</td><td>1766</td><td>black</td><td>16.34</td><td>9.32 5.89</td><td>66.37 31.29</td><td>900 388</td></tr><tr><td>33 34</td><td>715 39</td><td>white picture</td><td>29.45 29.37</td><td>19.73 22.51</td><td>58.05 42.26</td><td>1474 985</td><td>like like</td><td>9.92 9.82</td></table>
+
+![](images/716be046a55236471d10301f4095bd9c28ab952281f4dcb78c3cb9967fdf410c.jpg)  
+Figure 5: Two different captions containing the phrase “many people.” In both cases, the VQ4 layer infers the same unit sequence (872, 360, 712, middle transcription) beneath the phrase. The VQ3 units are somewhat noisier, but contain the common subsequence (956, 265, 80, 401, 262, 762, 246, 774, 828, 386, bottom transcription).
+
+![](images/147e3c17519983c04eec887526d5ec0fdf8cf72d29c330238c37f37206989848.jpg)  
+Figure 6: Two different captions containing word “train”. In both cases, the VQ4 layer infers the same unit sequence (680, 248, top transcription) surrounding the word “train”. The VQ3 alignments contain the same common subsequence (358, 306, 908, 564, 950, 770, bottom transcription). Notice that the same (358, 306) VQ3 unit sequence is aligned to the $/ \mathrm { { t r } / \Omega }$ phone cluster at the beginning of both instances of the word “train,” as well as the $/ \mathrm { { t r } / \Omega }$ at the beginning of both instances of “tree” in Figure 7. Unit 358 is also found covering the $/ \mathrm { { t r } / \Omega }$ at the beginning of the word “tracks” in the topmost spectrogram (although unit 306 is absent in this case).
+
+![](images/b574aace3d1e58c8e5a3690d435cb445985f356d9ce51c5f744f21ca9ad1ded0.jpg)  
+Figure 7: Two different captions containing word “trees”. In both cases, the VQ4 layer infers the same unit sequence (8, 412, 50, top transcription) surrounding the word “trees”. The VQ3 alignments contain the same common subsequence (358, 306, 648, 677, 730, bottom transcription). Notice that the (358, 306) VQ3 unit sequence is aligned to the $/ \mathrm { { t r } / \Omega }$ phone cluster at the beginning of both instances of “trees,” and this same unit sequence is inferred for the $/ \mathrm { { t r } / \Omega }$ phone sequence in both instances of “train” in Figure 6.

parse/train/BJeOioA9Y7/BJeOioA9Y7.md

@@ -0,0 +1,347 @@
+# KNOWLEDGE FLOW: IMPROVE UPON YOUR TEACH-ERS
+
+Iou-Jen Liu, Jian Peng, Alexander G. Schwing University of Illinois at Urbana-Champaign {iliu3, jpeng, aschwing}@illinois.edu
+
+# ABSTRACT
+
+A zoo of deep nets is available these days for almost any given task, and it is increasingly unclear which net to start with when addressing a new task, or which net to use as an initialization for fine-tuning a new model. To address this issue, in this paper, we develop knowledge flow which moves ‘knowledge’ from multiple deep nets, referred to as teachers, to a new deep net model, called the student. The structure of the teachers and the student can differ arbitrarily and they can be trained on entirely different tasks with different output spaces too. Upon training with knowledge flow the student is independent of the teachers. We demonstrate our approach on a variety of supervised and reinforcement learning tasks, outperforming fine-tuning and other ‘knowledge exchange’ methods.
+
+# 1 INTRODUCTION
+
+Research communities have amassed a sizable number of deep net architectures for different tasks, and new ones are added almost daily. Some of those architectures are trained from scratch while others are fine-tuned, i.e., before training, their weights are initialized using a structurally similar deep net which was trained on different data.
+
+Beyond fine-tuning, particularly in reinforcement learning, teachers have also been considered in one way or another by Rusu et al. (2016b); Fernando et al. (2017); Wang et al. (2017); Li & Hoiem (2016); Bengio et al. (2009); Patel et al. (2015); Chen & Liu (2016); Teh et al. (2017); Parisotto et al. (2016). For instance, progressive neural net (Rusu et al., 2016b) keeps multiple teachers during both training and inference, and learns to extract useful features from the teachers for a new target task. PathNet (Fernando et al., 2017) uses genetic algorithms to choose pathways from a giant network for learning new tasks. ‘Growing a Brain’ (Wang et al., 2017) fine-tunes a neural network while growing the network’s capacity (wider or deeper layers). Actor-mimic (Parisotto et al., 2016) pre-trains a big model on multiple source tasks, then the big model is used as a weight initialization for a new model which will be trained on a new target task. Knowledge distillation (Hinton et al., 2015) distills knowledge from a large ensemble of models to a smaller student model.
+
+However, all the aforementioned techniques have limitations. For example, progressive neural net models (Rusu et al., 2016b) grow with the number of teachers. This large number of parameters limits the number of teachers a progressive neural net can handle, and largely increases the training and testing time. In PathNet (Fernando et al., 2017), searching over a big network for pathways is computationally intensive. For fine-tuning based methods such as ‘Growing a Brain’ (Wang et al., 2017) and actor-mimic (Parisotto et al., 2016), only one pretrained model can be used at a time. Hence, their performance heavily relies on the chosen pretrained model.
+
+To address these shortcomings, we develop knowledge flow which moves ‘knowledge’ of multiple teachers when training a student. Irrespective of how many teachers we use, the student is guaranteed to become independent at the final stage of training and the size of the resulting student net remains constant. In addition, our framework makes no restrictions on the deep net size of the teacher and student, which provides flexibility in choosing teacher models. Importantly, our approach is applicable to a variety of tasks from reinforcement learning to fully-supervised training.
+
+We evaluate knowledge flow on a variety of tasks from reinforcement learning to fully-supervised learning. In particular, we follow Rusu et al. (2016b); Fernando et al. (2017) and compare on the same
+
+Atari games. In addition, we also observed significant top-1 error rate improvements on supervised learning datasets, i.e., CIFAR-10, and CIFAR-100.
+
+# 2 BACKGROUND
+
+Knowledge flow is applicable to a variety of settings from supervised learning to reinforcement learning, which we briefly review to introduce notation.
+
+Supervised Learning recovers the parameters $\theta$ of a mapping $f _ { \theta } : \mathcal { X }  \mathcal { Y }$ from data space $\mathcal { X }$ to output space $\mathcal { V }$ . To this end, a dataset $D = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ containing $n$ pairs $( x _ { i } , y _ { i } )$ (assumed to be sampled i.i.d.) is used, where $x _ { i } \in { \mathcal { X } }$ and $y _ { i } \in \mathcal { V }$ . Given this dataset, the parameters $\theta$ of the mapping $f _ { \theta }$ are learned by minimizing a loss function $\ell _ { ( x , y ) } ( \theta )$ composed of a regularization term $R ( \theta )$ and an empirical risk $\ell ( y , f _ { \boldsymbol { \theta } } ( x ) )$ which compares groundtruth label $y$ and prediction $f _ { \boldsymbol { \theta } } ( \boldsymbol { x } )$ The parameters $\theta$ are obtained by optimizing the following program:
+
+$$
+\operatorname* { m i n } _ { \theta } \mathbb { E } _ { ( x , y ) \sim D } [ \ell _ { ( x , y ) } ( \theta ) ] : = \mathbb { E } _ { ( x , y ) \sim D } [ \ell ( y , f _ { \theta } ( x ) ) ] + R ( \theta ) .
+$$
+
+Hereby, the mapping $f _ { \theta }$ is obtained by maximizing the logits or a corresponding probability distribution $\hat { f } _ { \boldsymbol { \theta } } ( y | \boldsymbol { x } )$ , i.e., $f _ { \theta } = \arg \operatorname* { m a x } _ { y \in \mathcal { V } } \hat { f } _ { \theta } ( y | x )$ . Here and below let the hat $( ^ { 6 } \hat { \cdot } \vec { \cdot } )$ indicate probability distributions over appropriate domains.
+
+Reinforcement Learning considers an agent interacting with an environment according to a policy   
+$\pi _ { \theta _ { \pi } } : \mathcal { X }  \mathcal { A }$ which maps a state $x _ { t } \in \mathcal X$ to an action $a _ { t } \in \mathcal A$ at time $t$ . The policy depends on   
+the parameters a scalar rewardthe discount fa $\theta _ { \pi }$ . After performing action . The discounted return ar. The expected future rew $a _ { t }$ , theime d w agent observeis defined as n observing s $x _ { t + 1 }$ and rece, where wing p esiscy $r _ { t }$ $t$ $\begin{array} { r } { R _ { t } = \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } r _ { t + k } } \end{array}$ $\gamma$ $x$   
+$\pi _ { \theta _ { \pi } }$ is defined as $V ^ { \pi _ { \theta _ { \pi } } } ( x _ { t } ) = \mathbb { E } _ { \tau \sim \pi _ { \theta _ { \pi } } } [ R _ { t } | x _ { t } ]$ , where $\tau = \{ ( x _ { t } , a _ { t } , r _ { t } ) , ( x _ { t + 1 } , a _ { t + 1 } , r _ { t + 1 } ) , \ldots \}$ is a   
+trajectory generated by following $\pi _ { \theta _ { \pi } }$ from state $x _ { t }$ .
+
+The goal of reinforcement learning is to find a policy that maximizes the expected future reward from each state $x _ { t }$ . Without loss of generality, in this paper, we follow the asynchronous advantage actor-critic (A3C) formulation (Mnih et al., 2016). In A3C, the policy mapping $\pi _ { \boldsymbol { \theta } _ { \pi } } ( x ) = \arg \operatorname* { m a x } _ { a \in \mathcal { A } } \hat { \pi } _ { \boldsymbol { \theta } _ { \pi } } ( a \vert x )$ is obtained from a probability distribution over states, where ${ \hat { \pi } } _ { \boldsymbol { \theta } _ { \pi } } ( a | \boldsymbol { x } )$ is modeled by a deep net with parameters $\theta _ { \pi }$ . The value function is also approximated by a deep net $V _ { \theta _ { v } } ( x )$ , having parameters $\theta _ { v }$ .
+
+To optimize the policy parameters $\theta _ { \pi }$ given a state $x _ { t }$ , a loss function based on a scaled negative log-likelihood and a negative entropy regularizer is common:
+
+$$
+\ell _ { \pi } ^ { \tau } ( \theta _ { \pi } ) = \frac { 1 } { | \tau | } \sum _ { t \in \tau } [ - \log \hat { \pi } _ { \theta _ { \pi } } ( a _ { t } | x _ { t } ) ( R _ { t } - V _ { \theta _ { v } } ( x _ { t } ) ) - \beta H ( \hat { \pi } _ { \theta _ { \pi } } ( \cdot | x _ { t } ) ) ] .
+$$
+
+Herestate , $\begin{array} { r } { R _ { t } = \sum _ { i = 0 } ^ { k - 1 } \gamma ^ { i } r _ { t + i } + \gamma ^ { k } V _ { \theta _ { v } } ( x _ { t + k } ) } \end{array}$ isry e empirical generated $k$ -step return following ained when s. The scalar $x _ { t }$ $| \tau |$ $\tau$ $\pi _ { \theta _ { \pi } }$ $\beta \geq 0$ a user-specified constant, and $H ( \hat { \pi } _ { \boldsymbol { \theta } _ { \pi } } ( \cdot | \boldsymbol { x } _ { t } ) )$ is the entropy function, which encourages exploration by favoring a uniform probability distribution ${ \hat { \pi } } _ { \theta _ { \pi } } ( a | x )$ . To optimize the value function $V _ { \theta _ { v } }$ , it is common to use the squared loss $\begin{array} { r } { \ell _ { v } ^ { \tau } ( \theta _ { v } ) = \frac { 1 } { 2 | \tau | } \sum _ { t \in \tau } ( R _ { t } - V _ { \theta _ { v } } ( x _ { t } ) ) ^ { 2 } } \end{array}$ .
+
+By minimizing the empirical expectation of $\ell _ { \pi } ^ { \tau } ( \theta _ { \pi } )$ and $\ell _ { v } ^ { \tau } ( \theta _ { v } )$ , i.e., by addressing
+
+$$
+\operatorname* { m i n } _ { \theta _ { \pi } } \mathbb { E } _ { \tau \sim \pi _ { \theta _ { \pi } } } [ \ell _ { \pi } ^ { \tau } ( \theta _ { \pi } ) ] , \quad \mathrm { ~ a n d ~ } \quad \operatorname* { m i n } _ { \theta _ { v } } \mathbb { E } _ { \tau \sim \pi _ { \theta _ { \pi } } } [ \ell _ { v } ^ { \tau } ( \theta _ { v } ) ] ,
+$$
+
+alternatingly, we learn a policy and a value function that maximize expected return.
+
+# 3 KNOWLEDGE FLOW
+
+Instead of optimizing the programs given in Eq. (1) and Eq. (2) from scratch, the aforementioned warm-start techniques (see Sec. 5 for more) are applicable. To address their mentioned shortcomings, we propose knowledge flow, a framework that moves ‘knowledge’ from an arbitrary number of deep nets, henceforth referred to as ‘teachers’ to a deep net under training, called the ‘student.’
+
+![](images/23ea2fe0c61b350b0c7fa17bfe9b615831f0fafec1d03d55c7e243319a4ca5bc.jpg)  
+Figure 1: (a) Example of a two-teacher knowledge flow. (b) Deep net transformation of knowledge flow. (c) Average normalized weights for teachers’ and the student’s layers. At the beginning of training, the student heavily relies on teacher one. As training progresses, teacher one’s weight decreases, and the student’s weight increases until the student is eventually independent.
+
+# 3.1 OVERVIEW
+
+Knowledge flow is outlined on example deep nets in Fig. 1 (a,b). We train the parameters of the student net which are randomly initialized. To this end we take advantage of teachers, whose parameters are fixed and obtained from pre-trained models on different source tasks by different algorithms. For example, for reinforcement learning, we may consider teachers trained by A3C (Mnih et al., 2016), A2C (Dhariwal et al., 2017) or DQN (Mnih et al., 2015).
+
+‘Knowledge’ of multiple teachers is transferred to a student by adding transformed and scaled intermediate representations from the teacher deep nets to the student net. To achieve this, we modify the student net, i.e., $f _ { \theta }$ in the supervised setting and $\pi _ { \theta _ { \pi } } ( a | x ) , V _ { \theta _ { v } } ( x )$ in the reinforcement learning case. We add teacher representations which are transformed by multiplication with a trainable matrix Q and scaled via a weight $p _ { w }$ that is normalized to sum to one for each student layer and parameterized via trainable parameters $w$ . The normalized weights encode which of the teachers’ or the student’s representation to trust at every layer of the student net. Note that a teacher can help the student at different levels of abstraction with input from different levels of its net.
+
+Importantly, after training, the student model should perform well on the target task without relying on teachers. To achieve this, as training progresses, we increasingly encourage a high normalized weight on the student representation, which forces the student to eventually capture all the ‘knowledge.’ Due to the trainable scaling, at an early stage of training, we observe the student to rely heavily on the ‘knowledge’ of the teacher to quickly obtain better performance. However, as training proceeds, the student is encouraged to become more and more independent. During final stages of training, the student will no longer be able to rely on teachers, which ensures that the student has learned to master the desired task on its own. This is observed in Fig. 1 (c).
+
+To formally encourage this successive transfer we introduce two additional loss functions. The first, referred to as the dependency loss $\ell _ { \mathrm { d e p } } ( w )$ , captures how much a student relies on teachers. It depends on the weight vector $w$ which encodes the strength of the coupling. The second one ensures that a student’s behavior doesn’t change rapidly when the teachers’ influence decreases. We use loss $\ell _ { \mathrm { K L } } ( \cdot , \cdot )$ to capture the change.
+
+By combining student net modifications and additional loss terms, for the supervised task we obtain
+
+$$
+\operatorname* { m i n } _ { \theta , w , Q } \mathbb { E } _ { ( x , y ) } [ \widetilde { \ell } _ { ( x , y ) } ( \theta , w , Q ) + \lambda _ { 1 } \ell _ { \mathrm { d e p } } ( w ) + \lambda _ { 2 } \ell _ { \mathrm { K L } } ( \widetilde { \hat { f } } _ { \theta } , \widetilde { \hat { f } } _ { \theta _ { \mathrm { o l d } } } ) ] ,
+$$
+
+and for reinforcement learning the transformed program reads as follows:
+
+$$
+\left\{ \begin{array} { l l } { \operatorname* { m i n } _ { \theta _ { \pi } , w , Q } \mathbb { E } _ { \tau \sim \tilde { \pi } _ { \theta _ { \pi } } } [ \tilde { \ell } _ { \pi } ^ { \tau } ( \theta _ { \pi } , w , Q ) + \lambda _ { 1 } \ell _ { \mathrm { d e p } } ( w ) + \lambda _ { 2 } \ell _ { \mathrm { K L } } ^ { \tau } ( \tilde { \hat { \pi } } _ { \theta _ { \pi } } , \tilde { \hat { \pi } } _ { \theta _ { \pi _ { \mathrm { o l d } } } } ) ] } \\ { \operatorname* { m i n } _ { \theta _ { v } , w , Q } \mathbb { E } _ { \tau \sim \tilde { \pi } _ { \theta _ { \pi } } } [ \tilde { \ell } _ { v } ^ { \tau } ( \theta _ { v } , w , Q ) ] } \end{array} \right. .
+$$
+
+Loss $\tilde { \ell } \colon ( \theta , w , Q )$ originates from the original loss $\ell \colon ( \theta )$ (Eqs. (1)-(2)) by transforming the deep net to include cross-connections, hence its dependence on $w , Q$ . The tilde $( ^ { 6  } )$ denotes this dependence, also for probability distribution $\tilde { \hat { f } }$ and policy distribution $\tilde { \hat { \pi } }$ . Parameters from the current and a previous iteration are referred to via $\theta$ and $\theta _ { \mathrm { o l d } }$ respectively.
+
+For both supervised and reinforcement learning, $\lambda _ { 1 }$ and $\lambda _ { 2 }$ control the strength which is used to decrease the influence of the teacher. A low $\lambda _ { 1 }$ allows the student to rely on teachers. Close to the end of training, the student should be independent. Therefore, we set $\lambda _ { 1 }$ to a small value at the beginning, and gradually increase its value as training progresses.
+
+Note that we don’t make any assumptions about teachers and student’s objective. If a teacher’s and student’s objective differ, negative transfer may occur initially. However, the proposed method quickly decreases the weight for teacher layers to reduce this effect. Despite differences, students could potentially still benefit from the low level representation of the teachers. We do observe this low level knowledge transfer in our experiments.
+
+In the following we first describe how to modify the deep nets, before we detail the loss functions $\ell _ { \mathrm { d e p } }$ and $\ell _ { \mathrm { K L } }$ , which are used to successively decrease the influence of the teachers.
+
+# 3.2 DEEP NET TRANSFORMATION AND LOSS TERMS
+
+Deep Net Transformation: Knowledge flow enhances the student by adding transformed and scaled intermediate representations from teacher models. To perform the transformation, intermediate representations from teachers are first multiplied by transformation matrices $Q$ . Then the transformed representations from teachers and representations from the student are linearly combined. The weights for this linear combination are determined by a weight $p _ { w }$ which is normalized to sum to one for each student layer.
+
+Let index $m = 0$ denote the student model and let $\theta ^ { ( 0 ) }$ refer to its parameters. Further, let $\theta ^ { ( m ) }$ , $m \in \{ 1 , \ldots , M \}$ denote teacher models. We use $l _ { m } ^ { i }$ to refer to deep net layer $i$ of teacher $m$ , with $i \in \{ 1 , \ldots , L _ { m } \}$ and $L _ { m }$ the number of layers in teacher $m$ . We define layer $j$ of the student model to be $l _ { 0 } ^ { j }$ , where $j \in \{ 1 , \dots , L _ { 0 } \}$ and $L _ { 0 }$ the number of deep net layers in the student model. The output of layer $l _ { m } ^ { k }$ right before and after an activation unit is denoted $z ( l _ { m } ^ { k } )$ and $h ( l _ { m } ^ { k } )$ respectively.
+
+To align a teacher’s layer $l _ { m } ^ { i }$ with a student’s layer $l _ { 0 } ^ { j }$ , we introduce a learnable transformation matrix $Q ^ { j } ( l _ { m } ^ { i } ) \in \mathbb { R } ^ { \dim ( l _ { 0 } ^ { j } ) \times \dim ( l _ { m } ^ { i } ) }$ , where $\dim ( \cdot )$ gives the number of elements in the corresponding layer. The matrix multiplication $Q ^ { j } ( l _ { m } ^ { i } ) z ( l _ { m } ^ { i } )$ aligns the representation from layer $i$ of teacher $m$ with the representation of layer $j$ of the student.
+
+For each layer $j$ in the student model, we define a candidate set $\mathbb { L } ^ { j }$ , which contains $l _ { 0 } ^ { j }$ and all the teachers’ layers to be considered. For example, in Fig. 1 (a), layer one of the student model is combined with layer one of teacher one and layer two of teacher two. Therefore, the candidate set of layer one of the student model is given by $\mathbb { L } ^ { 1 } \dot { = } \{ l _ { 0 } ^ { 1 } , l _ { 1 } ^ { 1 } , l _ { 2 } ^ { 2 } \}$ .
+
+To decide which teachers’ or the student’s representation to trust at every layer of the student net, we introduce a normalized weight $p _ { w } ^ { j } ( l )$ for all $j \in \{ 1 , \ldots , L _ { 0 } \}$ , where $l \in \mathbb { L } ^ { j }$ , summing to one for each layer $j$ in the student deep net, $i . e .$ .,
+
+$$
+\sum _ { l \in \mathbb { L } ^ { j } } p _ { w } ^ { j } ( l ) = 1 , \forall j \in \{ 1 , \dots , L _ { 0 } \} .
+$$
+
+To obtain the combined intermediate representation of layer $j$ for the student model, we use
+
+$$
+h ( l _ { 0 } ^ { j } ) = \sigma \left( \sum _ { l \in \mathbb { L } ^ { j } \backslash l _ { 0 } ^ { j } } p _ { w } ^ { j } ( l ) Q ^ { j } ( l ) z ( l ) + p _ { w } ^ { j } ( l _ { 0 } ^ { j } ) z ( l _ { 0 } ^ { j } ) \right) ,
+$$
+
+where $p _ { w } ^ { j } ( l _ { m } ^ { i } )$ determines how much the student layer $j$ relies on transformed representations of layer $i$ from the $m$ -th teacher. Intuitively, if the transformed representation of the $m$ -th teacher layer $i$ is helpful, $p _ { w } ^ { j } ( l _ { m } ^ { i } )$ will be close to one. We visualize the deep net transformation in Fig. 1 (b).
+
+Note that the intermediate representations of teachers are not changed in our framework. To obtain the output of layer $l _ { m } ^ { k }$ we apply the original activation unit to the original representation $z ( l _ { m } ^ { i } )$ , i.e., $\begin{array} { r } { h ( l _ { m } ^ { i } ) = \sigma ( z ( l _ { m } ^ { i } ) ) , ~ \overset {  } { \forall } m \in \{ 1 , \dots , M \} , j \in \{ 1 , \dots L _ { m } \} } \end{array}$ .
+
+The maximal number of introduced matrices $Q$ in our framework is $\textstyle \sum _ { i = 1 } ^ { M } L _ { i } L _ { 0 }$ . In practice, we don’t link a student’s layer to every layer of a teacher network. Intuitively, a teachers’ bottom layer features are very likely irrelevant to a student’s top layer features. Indeed, we observed that linking a teachers’ bottom layer to a student’s top layer generally doesn’t yield improvements. Therefore, in practice, we recommend to link one teacher layer to one or two student layers, in which case we introduce on the order of $M L _ { 0 }$ matrices Q. Also note that while additional trainable parameters $Q$ and $w$ are introduced in our framework, $Q$ and $w$ are not part of the resulting student network since we ensure $p _ { w } ^ { j } ( l ) \equiv 0 \forall l \in \mathbb { L } ^ { j } \backslash l _ { 0 } ^ { j }$ at the end of training as discussed next. Hence, the additional parameters function as auxiliary knobs that help the student learn faster. In the final stage of training, the student will be independent (see Fig. 1 (c)) and does no longer rely on $Q , w ,$ , or any transformed representations from teachers.
+
+Table 1: Comparison with PathNet (Fernando et al., 2017) and progressive neural network (PNN) (Rusu et al., 2016b). Since PathNet and PNN don’t report exact scores we obtain their numbers from their plots and indicate that with a $\sim$ symbol. The results of the state-of-the-art methods: A3C (Mnih et al., 2016), PPO (Schulman et al., 2017), and ACKTR (Wu et al., 2017) on Atari games are also listed for reference.   
+
+<table><tr><td></td><td colspan="2">w/ Seaquest teacher</td><td colspan="2">w/ Riverraid teacher</td><td colspan="2">w/ Sea. and River. teachers</td><td colspan="3">No teachers</td></tr><tr><td></td><td>Ours</td><td>PathNet</td><td>Ours</td><td>PathNet</td><td>Ours</td><td>PNN</td><td>A3C</td><td>PPO</td><td>ACKTR</td></tr><tr><td>Alien</td><td>1254</td><td>~1700</td><td>1259</td><td>~1800</td><td>1911</td><td>~2000</td><td>182</td><td>1850</td><td>3197</td></tr><tr><td>Asterix</td><td>3982</td><td>~2000</td><td>3823</td><td>~2000</td><td>6012</td><td>~9000</td><td>6723</td><td>4533</td><td>31583</td></tr><tr><td>Boxing</td><td>96</td><td>~70</td><td>96</td><td>~80</td><td>99</td><td>~99</td><td>34</td><td>95</td><td>1</td></tr><tr><td>Gopher</td><td>4152</td><td>~3900</td><td>3820</td><td>~2100</td><td>5233</td><td>~4500</td><td>8443</td><td>2933</td><td>47730</td></tr><tr><td>Hero</td><td>21250</td><td>~12500</td><td>29343</td><td>~12500</td><td>30928</td><td>~30000</td><td>28766</td><td>n/a</td><td>n/a</td></tr><tr><td>James.</td><td>857</td><td>~600</td><td>832</td><td>~600</td><td>1245</td><td>~850</td><td>352</td><td>561</td><td>512</td></tr><tr><td>Krull</td><td>8193</td><td>~7800</td><td>6890</td><td>~7500</td><td>10000</td><td>~9954</td><td>8067</td><td>7942</td><td>9689</td></tr></table>
+
+Decreasing Teachers’ Influence: We successively decrease the influence of the teachers during training by gradually encouraging the normalized weight $p _ { w } ^ { j } ( l _ { 0 } ^ { j } )$ to increase to a value of $1 \forall j \in$ $\{ 1 , 2 , \ldots , L _ { 0 } \}$ . To capture how much the student relies on teachers, we introduce the dependence cost as the negative log probability:
+
+$$
+\ell _ { \mathtt { d e p } } ( w ) = - \frac { 1 } { L _ { 0 } } \sum _ { j \in \{ 1 , 2 , . . . , L _ { 0 } \} } \log p _ { w } ^ { j } ( l _ { 0 } ^ { j } ) .
+$$
+
+By minimizing $\ell _ { \mathrm { d e p } } ( w )$ , we encourage weights for the layers of the student to increase. Hence we encourage the student to become more and more independent. During the final stage of training, $p _ { w } ^ { j } ( l _ { 0 } ^ { j } )$ approaches one for all $j \in \{ 1 , \ldots , L _ { 0 } \}$ , making the student independent of the transformed representation obtained from teachers.
+
+Empirically, we found that a fast decrease of the influence of the teacher can degrade the performance. This is intuitive as it requires some time to find good transformations $Q$ . Moreover, decreasing the influence of a teacher too fast may change the output distribution over labels or actions of the student model too much, and thus lead to performance loss. To prevent changing a student’s output distribution too fast, we found a Kullback-Leibler (KL) regularizer to yield good results. More specifically, in the case of supervised learning we use
+
+$$
+\ell _ { \mathrm { K L } } \big ( \tilde { \hat { f } } _ { \boldsymbol { \theta } } , \tilde { \hat { f } } _ { \boldsymbol { \theta } _ { \mathrm { o l d } } } \big ) = D _ { \mathrm { K L } } \big [ \tilde { \hat { f } } _ { \boldsymbol { \theta } } \big ( \cdot | \boldsymbol { x } \big ) | | \tilde { \hat { f } } _ { \boldsymbol { \theta } _ { \mathrm { o l d } } } \big ( \cdot | \boldsymbol { x } \big ) \big ] .
+$$
+
+Hereby, $\theta$ is the set of current parameters, and $\theta _ { \mathrm { o l d } }$ are the previous ones. In the reinforcement learning case we use $D _ { \mathrm { K L } } [ \tilde { \hat { \pi } } _ { \boldsymbol { \theta } } ( \cdot \vert x _ { t } ) \vert \vert \tilde { \hat { \pi } } _ { \boldsymbol { \theta } _ { \mathrm { o l d } } } ^ { - } ( \cdot \vert x _ { t } ) ]$ .
+
+# 4 EXPERIMENTAL RESULTS
+
+In the following we evaluate knowledge flow on reinforcement and supervised learning tasks. Results are reported by using only the student model to avoid even the smallest influence from any teacher nets.
+
+# 4.1 REINFORCEMENT LEARNING
+
+We evaluate knowledge flow on reinforcement learning using Atari games that were used by Rusu et al. (2016b); Fernando et al. (2017). Following existing work, the input to our agent are raw images from the environment. The agent learns to predict actions only based on the rewards and the input images from the environment. The agent chooses an action every four frames, and the last action is repeated on the skipped four frames. For all teacher models and the student model, we use the fully forward architecture of A3C (Mnih et al., 2016). The model has three hidden layers. The first layer is a convolutional layer with 16 filters of size $8 \mathrm { x } 8$ and stride 4. The second layer is a convolutional layer with 32 filters of size $4 \mathbf { x } 4$ and stride 2. The third layer is a fully connected layer with 256 hidden units. Following the third hidden layer are two sets of output. One is a softmax output that provides a probability distribution over all valid actions. The other one is a scalar output that provides the estimated value function. We use the same hyper-parameter settings as Mnih et al. (2016) except for the learning rate. Mnih et al. (2016) use RMSProp with shared statistics while we use Adam with shared statistics, which we found to give better results when training the baselines. The learning rate is set to $1 0 ^ { - 4 }$ and gradually decreased to zero for all experiments. To select $\lambda _ { 1 }$ and $\lambda _ { 2 }$ in our framework, we follow progressive neural net (Rusu et al., 2016b): randomly sample $\lambda _ { 1 } \in \{ 0 . 0 5 , 0 . 1 , 0 . 5 \}$ and $\lambda _ { 2 } \in \{ 0 . 0 \bar { 0 } 1 , \bar { 0 } . 0 1 , 0 . 0 5 \}$ . Note that $\lambda _ { 1 }$ is set to zero at the beginning of training, and linearly increased to the sampled value at the end of training. Following Rusu et al. (2016b), we repeat each experiment 25 times with different random seeds and randomly sampled $\lambda _ { 1 }$ and $\lambda _ { 2 }$ . The results of the top three out of 25 runs are reported. As A3C, we run 16 agents on 16 CPU cores in parallel.
+
+![](images/e6a6f99c691a883622a6e5d5f9ab027a25423a2f5ee0a355983477af4929eff9.jpg)  
+Figure 2: Comparison with progressive neural network and PathNet.
+
+Evaluation Metrics: We follow the evaluation procedure of Mnih et al. (2015). The trained student models are evaluated by playing each game for 30 episodes. We also follow the ‘no-op’ procedure: at the beginning of each testing episode, the agents perform up to 30 ‘no-op’ actions.
+
+Results: We first compare our framework with PathNet (Fernando et al., 2017) and progressive neural net (PNN) (Rusu et al., 2016b), which are state-of-the-art transfer reinforcement learning frameworks, using their experimental settings. The comparison is summarized in Table 1. The state-of-the-art results (Mnih et al., 2016; Schulman et al., 2017; Wu et al., 2017) on Atari games are also included in Table 1 for reference. Compared to PathNet, a student model trained using our transfer framework with one teacher achieves higher scores in 11 out of 14 experiments. Compared with PNN, for a two-teacher framework, our trained student model has only 0.7M parameters and PNN has 16M parameters. Nonetheless we observe higher scores in five out of the seven experiments. The results demonstrate that knowledge flow effectively transfers knowledge from teachers to the student. Table 1 also indicates that, in our framework, when the number of teachers increases from one to two, the student’s performance improves significantly across all experiments. The training curves for the experiments are shown in Fig. 2. The curve is the average of the top three out of 25 runs. We observe our approach to generally perform very well.
+
+![](images/a5b8d0b83325960e3b46c40527e67f1a90b12eedc7a0c133b8bd1c25c6c5aded.jpg)  
+Figure 3: Comparison with fine-tuning and baseline A3C on different combinations of environment/teacher settings.
+
+To further evaluate knowledge flow, we experiment with different combinations of environment/teacher settings. These settings are not used by PathNet and progressive neural network. The results are summarized in Table 2, where “ours w/ expert” represents that one teacher is expert for the target game; “ours w/ non-expert” represents that both teachers are not experts for the target game; “Fine-tune” represents fine-tuning from a non-expert on a new target game; “A3C baseline” represents our implementation of the A3C baseline; “A3C” represents the scores reported originally (Mnih et al., 2016). Note that our A3C implementation achieves better scores than those reported by Mnih et al. (2016) for most of the games. As shown in Table 2, knowledge flow with expert teacher performs better than the baseline across all experiments, which we interpret as evidence that knowledge flow successfully transfers ‘knowledge’ from an expert teacher to the student. In addition, knowledge flow with non-expert teachers also outperforms fine-tuning on a non-expert teacher. The reasons are twofold: First, a student model in knowledge flow can learn from multiple teachers while the fine-tuning method can only start from one setting. Second, in knowledge flow, the student can avoid the negative impact from insufficiently pretrained teachers, while fine-tuning from an insufficiently pretrained model slows down the training process and may degrade the overall performance. The training curves for the experiments are shown in Fig. 3. More training curves are in the Appendix (Fig. 6). Note that in knowledge flow, the student can benefit from the intermediate representations of the teacher, even if input space, output space and objectives differ. For example, in Fig. 3 (a), the two teachers are Chopper Command and Space Invaders, which are quite different from the target game Seaquest. The student model still benefits from learning from the teachers and achieves scores ten times larger than learning without teacher and fine-tuning from a teacher.
+
+# 4.2 SUPERVISED LEARNING
+
+For supervised learning, we use a variety of image classification benchmarks, including CIFAR10 (Krizhevsky, 2009), CIFAR-100 (Krizhevsky, 2009), STL-10 (Coates et al., 2011), and EMNIST (Cohen et al., 2017). The parameters $\lambda _ { 1 }$ for the dependent cost and $\lambda _ { 2 }$ for the KL cost are determined using the validation set of each dataset.
+
+Evaluation Metrics: To evaluate the trained student model we report top-1 error rate on the test set of each dataset. All plots and reported numbers are the average of three runs obtained using different random seeds.
+
+Table 3: Test Error $( \% )$ on CIFAR-10/100. The parentheses following “Ours” indicates the teachers we use. I.e., ‘Ours (SVHN, C100)’ indicates that we use an SVHN expert and a C100 expert as teachers.   
+
+<table><tr><td></td><td>Baseline Densenet f</td><td>Fine-tune</td><td>Fine-tune t from C100 from SVHN(C100,SVHN)</td><td>Ours</td><td colspan="3">Baseline Fine-tune Fine-tune Ours Densenet from C10 from SVHN(C10,SVHN)</td></tr><tr><td>C10</td><td>4.44</td><td>4.27</td><td>4.58</td><td>3.88</td><td>C100 21.64 20.83</td><td>21.02</td><td>20.78</td></tr><tr><td colspan="6">(a) (b)</td></tr></table>
+
+CIFAR-10/CIFAR-100: CIFAR-10 and CIFAR-100 datasets consist of colored images of size $3 2 \times 3 2$ . CIFAR-10 (C10) has 10 classes and CIFAR-100 (C100) has 100 classes. For both dataset, the training and test sets contain 50,000 and 10,000 images respectively. We perform all experiments on CIFAR-10 and CIFAR-100 with standard data augmentation (Huang et al., 2017).
+
+We use Densenet (Huang et al., 2017) (depth 100, growth rate 24) as a baseline and follow their hyper-parameter settings to train our baseline, teacher and student models. For our approach, we first train teachers on CIFAR-10, CIFAR-100, and SVHN (Netzer et al., 2011). We then train the student model using a different combination of teachers. We compare our results to fine-tuning and the baseline model. As shown in Table 3 (a), for the CIFAR-10 target task, fine-tuning from the CIFAR-100 expert improves $4 \%$ over the baseline. Fine-tuning from the SVHN expert performs worse than the baseline model. Intuitively, for the CIFAR-10 target task, the CIFAR-100 deep net is a good teacher while a deep net trained with SVHN isn’t. Presented with both good and inadequate teachers, knowledge flow improves by $1 3 \%$ over the baseline. This demonstrates that knowledge flow can not only leverage a good teacher’s ‘knowledge,’ but it can also avoid misleading influence. As detailed in Table 3 (b), the results are similar on the CIFAR-100 dataset.
+
+To further demonstrate the properties of knowledge flow, additional results are in the appendix.
+
+# 5 RELATED WORK
+
+As mentioned before, ‘knowledge’ transfer has been considered using a variety of techniques. We briefly discuss related work in contrast to our approach in the following and defer details to Sec. 8.
+
+PathNet (Fernando et al., 2017) enables multiple agents to train the same deep net while reusing parameters and avoiding catastrophic forgetting. In contrast to this formulation we consider availability of multiple pre-trained teacher nets.
+
+Progressive Net (Rusu et al., 2016b) leverages transfer and avoids catastrophic forgetting by introducing lateral connections to previously learned features. Our discussed method uses similar lateral connections. However, in contrast to Rusu et al. (2016b), our method ensures independence of the student upon training, addressing a limitation in (Rusu et al., 2016b) where only a fraction of the capacity of the student is eventually utilized.
+
+Distral a neologism combining ‘distill & transfer learning’ (Teh et al., 2017) considers joint training of multiple tasks. Multiple tasks share a ‘distilled’ policy which encodes common behavior between different tasks. While each worker addresses its own task, a shared policy encourages consistency between the policies. Different from Distral, which is a multi-task learning framework, knowledge flow addresses a single task, while in multi-task learning, multiple tasks are addressed at the same time. Hence, common for multi-task learning and knowledge flow is a transfer of information. However, in multi-task learning, information extracted from different tasks are shared to boost performance, while, in knowledge flow, the information of multiple teachers is leveraged to help a student learn better a single, new, previously unseen task.
+
+Other related work includes actor-mimic (Parisotto et al., 2016), learning without forgetting (Li & Hoiem, 2016), growing a brain (Wang et al., 2017), policy distillation (Rusu et al., 2016a), domain adaptation (Pan & Yang, 2010; Long et al., 2015; Tzeng et al., 2015), knowledge distillation (Hinton et al., 2015) or lifelong learning (Chen & Liu, 2016). A more detailed discussion on related work is provided in Sec. 8 of the supplementary material.
+
+# 6 CONCLUSION
+
+We developed a general knowledge flow approach that permits to train a deep net from any number of teachers. We showed results for reinforcement learning and supervised learning, demonstrating improvements compared to training from scratch and to fine-tuning. In the future we plan to learn when to use which teacher and how to actively swap teachers during training of a student.
+
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+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A. Rusu, Joel Veness, Marc G. Bellemare, Alex Graves, Martin Riedmiller, Andreas K. Fidjeland, Georg Ostrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen King, Dharshan Kumaran, Daan Wierstra, Shane Legg, and Demis Hassabis. Human-level control through deep reinforcement learning. In Nature, 2015.   
+Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy P. Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In Proc. ICML, 2016.   
+Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y Ng. Reading digits in natural images with unsupervised feature learning. 2011.
+
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+Emilio Parisotto, Jimmy Lei Ba, and Ruslan Salakhutdinov. Actor-mimic: Deep multitask and transfer reinforcement learning. In Proc. ICLR, 2016.
+
+Vishal M. Patel, Raghuraman Gopalan, Ruonan Li, and Rama Chellappa. Visual domain adaptation: A survey of recent advances. IEEE Signal Process. Mag., 2015.
+
+Andrei A. Rusu, Sergio Gomez Colmenarejo, Çaglar Gülçehre, Guillaume Desjardins, James Kirkpatrick, Razvan Pascanu, Volodymyr Mnih, Koray Kavukcuoglu, and Raia Hadsell. Policy distillation. In Proc. ICLR, 2016a.
+
+Andrei A. Rusu, Neil C. Rabinowitz, Guillaume Desjardins, Hubert Soyer, James Kirkpatrick, Koray Kavukcuoglu, Razvan Pascanu, and Raia Hadsell. Progressive neural networks. In arXiv preprint arXiv:1606.04671, 2016b.
+
+Paul Ruvolo and Eric Eaton. Ella: An efficient lifelong learning algorithm. In Proc. ICML, 2013.
+
+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.
+
+Yee Teh, Victor Bapst, Wojciech M. Czarnecki, John Quan, James Kirkpatrick, Raia Hadsell, Nicolas Heess, and Razvan Pascanu. Distral: Robust multitask reinforcement learning. In Proc. NIPS, 2017.
+
+Martin Thoma. Analysis and optimization of convolutional neural network architectures. arXiv preprint arXiv:1707.09725, 2017.
+
+Sebastian Thrun. Lifelong learning algorithms. In Learning to Learn. Springer US, 1998.
+
+Eric Tzeng, Judy Hoffman, Trevor Darrell, and Kate Saenko. Simultaneous deep transfer across domains and tasks. In Proc. ICCV, 2015.
+
+Yu-Xiong Wang, Deva Ramanan, and Martial Hebert. Growing a brain: Fine-tuning by increasing model capacity. In Proc. CVPR, 2017.
+
+Yuhuai Wu, Elman Mansimov, Shun Liao, Roger B. Grosse, and Jimmy Ba. Scalable trust-region method for deep reinforcement learning using kronecker-factored approximation. In Proc. NIPS, 2017.
+
+Junbo Jake Zhao, Michaël Mathieu, Ross Goroshin, and Yann LeCun. Stacked what-where autoencoders. arXiv preprint arXiv:1506.02351, 2015.
+
+Table 4: Test error $( \% )$ of distilled student net.   
+
+<table><tr><td></td><td>MNIST</td><td>MNIST w/o digit ‘3&#x27;</td><td>C100</td><td>Imagenet</td></tr><tr><td>Student alone</td><td>1.46</td><td>11.06</td><td>31.87</td><td>30.24</td></tr><tr><td>KD Hinton et al. (2015)</td><td>0.74</td><td>2.06</td><td>30.28</td><td>30.04</td></tr><tr><td>Ours</td><td>0.73</td><td>1.05</td><td>30.07</td><td>29.05</td></tr></table>
+
+Table 5: Our approach on the EMNIST Letters dataset.
+
+<table><tr><td>Model (Teacher)</td><td>Test error(%)</td></tr><tr><td>Cohen et al. (2017)</td><td>14.85</td></tr><tr><td>Fine-tune from EMNIST digits</td><td>9.04</td></tr><tr><td>Baseline</td><td>9.20</td></tr><tr><td>Ours (EMNIST letters)</td><td>7.13</td></tr><tr><td>Ours (EMNIST half letters)</td><td>8.13</td></tr><tr><td>Ours (EMNIST digit)</td><td>8.11</td></tr></table>
+
+# 7 APPENDIX
+
+# 7.1 SUPERVISED LEARNING
+
+Comparison with Knowledge Distillation: We follow knowledge Distillation (KD) (Hinton et al., 2015) to distill knowledge from a larger model (teacher) to a smaller model (student). The student models have $5 0 \% - 5 \%$ parameters of the teacher models. Following their setup, we conduct experiments on MNIST, MNIST with digit $\cdot _ { 3 } \cdot$ missing in the training set, CIFAR-100, and ImageNet. For MNIST and MNIST with digit $\cdot _ { 3 } ,$ missing, following KD, the teacher model is an MLP with two hidden layers of 1200 hidden units, and the student model is an MLP with two hidden layers of 800 hidden units. For CIFAR-100, we use the model from Chen (2017) as teacher model. The student model follows the structure of the teacher, but the number of output channels of each convolutional layer is halved. For ImageNet, the teacher model is a 50-layer ResNet (He et al., 2016), and the student model is a 18-layer ResNet. The test error of the distilled student model are summarize in Table 4. Our framework has consistently better performance than KD, because the student model in our framework benefits not only from the output layer behavior of the teacher but also from intermediate layer representations of the teacher.
+
+# EMNIST:
+
+The ‘EMNIST Letters’ dataset consists of images of size $2 8 \times 2 8$ pixels showing handwritten letters. It has 26 balanced classes. Each class contains lower and upper case letters. The training and test sets contain 124,800 and 20,800 images respectively. The ‘EMNIST Digits’ dataset consists of images of size $2 8 \times 2 8$ pixels showing handwritten digits. It has 10 balanced classes. The training and test sets contain 240,000 and 40,000 images respectively.
+
+In this case we use the MNIST model from Chen (2017) as a baseline, teacher and student model. We trained teachers on EMNIST Digits, EMNIST Letters, and EMNIST Letters with only 13 classes. Our target task is EMNIST Letters. The student model is trained with different teachers and the results are compared to fine-tuning, the baseline model, and the state-of-the-art results on EMNIST. The results are summarized in Table 5. Compared to the baseline and fine-tuning, student learning in our framework with expert teacher (EMNIST Letters), semi-expert teacher (Half EMNIST Letters), and non-expert teacher (EMNIST Digits) all have better performance. In Fig. 4 we illustrate the accuracy over epochs for training of different models.
+
+# STL-10:
+
+The STL-10 dataset consist of colored images of size $9 6 \times 9 6$ pixels. It has 10 balanced classes. The training set contains 5,000 labeled images and 100,000 unlabeled images. The test set contains 8,000 images. In our experiment, we only use the 5,000 labeled images for training.
+
+We use the STL-10 model from Chen (2017) as our baseline, teacher and student model. We trained teachers on CIFAR-10 and CIFAR-100. We compare our results to fine-tuning and the baseline in Table 6. Note that STL-10 is very similar to CIFAR-10 and CIFAR-100. Therefore, both CIFAR-10 and CIFAR-100 are very good teachers. As shown in Table 6, compared to the baseline, fine-tuning a model using weights pretrained on CIFAR-10 and CIFAR-100 reduce test errors by more than $1 0 \%$ . Compared with fine-tuning, student model training in our framework further reduces the test error by $3 \%$ . Note that we only train on the labeled data while other approaches use this data for testing of semi-supervised approaches. Hence our results are obtained using fewer data and may not be directly comparable. We still list their results in Table 6 for reference. In Fig. 5 we illustrate the accuracy over the epochs of training.
+
+![](images/fba38bd486366a25d5fa05fad89dd10a53d2fd19ce8f3f5288b35b8a48b60bde.jpg)  
+Figure 4: Comparison of top-1 accuracy of our approach, fine-tuning and baseline on the EMNIST Letters test dataset.
+
+Table 6: Our approach on the STL-10 dataset (fully supervised).   
+
+<table><tr><td>Test error (%)</td></tr><tr><td>Zhao et al. (2015) Thoma (2017)</td><td>25.20 21.34</td></tr><tr><td>Baseline Fine-tune from C10 Fine-tune from C100</td><td>25.50 14.32</td></tr></table>
+
+# 7.2 REINFORCEMENT LEARNING
+
+We also compare to Distral (Teh et al., 2017), which is the state-of-the-art multi-task reinforcement learning framework. We used $\mathrm { \mathrm { ' K L } } + \mathrm { e n t } \ 1 \ \mathrm { c o l } ^ { \mathrm { ? } }$ , which has a central model $( m _ { 0 } )$ , and a task model $( m _ { i } )$ for each task. We perform the experiments on Atari games. In the experiments, we have three tasks (task 1, task 2, task 3). The teachers of task 2 $\left( m _ { 2 } \right)$ and task 3 $( m _ { 3 } )$ are provided for our framework. Distral is trained for 120M steps (40M steps/task), and our model is trained for $4 0 \mathbf { M }$ steps. For fair comparison, we report results of Distral’s task 1 model $( m _ { 1 } )$ , which is better than its center model $( m _ { 0 } )$ . The results are summarized in Table 7. Distral is suboptimal, because it aims to learn a multi-task agent. In addition, identical action and state space is assumed. When the target task is very different from the source tasks, Distral cannot decrease the teacher influence. In contrast, our framework can decrease a teacher’s influence, and thus reduce negative transfer.
+
+# 7.3 VISUALIZATION OF NORMALIZED WEIGHTS OF TEACHERS AND STUDENT
+
+Following the reviewer’s suggestion, we plot the averaged normalized weight $( p _ { w } )$ for teachers and the student in the C10 experiment, where C100 and SVHN experts are teachers. Intuitively, the C100 teacher should have a higher $p _ { w }$ value than the SVHN teacher, because C100 is more relevant to C10. The plot verifies this intuition. As shown in Fig. 7, $p _ { w }$ of the C100 teacher is higher than that of the SVHN teacher over the entire training. Note, both teachers’ normalized weights approach zero at the end of training.
+
+![](images/20d400a30725b1ed0c1cb4dbb19eb137877a094e43446bf2d42dbd30b584dccd.jpg)  
+Figure 5: Comparison of top-1 accuracy of our approach, fine-tuning and baseline on the STL-10 test dataset.
+
+![](images/a8440e339b32c16866c8a47fe22c32189da764eeb11a61f370f6eabe02666491.jpg)  
+Figure 6: Comparison with fine-tuning and baseline A3C on different combinations of environment/teacher settings.
+
+# 7.4 ABLATION STUDIES
+
+# 7.4.1 UNTRAINED TEACHER MODELS
+
+To verify that the student really benefits from the knowledge of teachers, we conduct an ablation study suggested by a reviewer. We use teacher models that haven’t been trained at all. Intuitively, learning with untrained teachers should have worse performance than learning with knowledgeable teachers. Our experiments verify this intuition. In Fig. 8 (a), where the target task is hero, learning with untrained teachers (‘w/ untrained teachers’) achieves an average reward of 15934. Learning with knowledgeable teachers (‘Ours with seaquest and riverraid teacher’) achieves an average reward of 30928. More results are presented in Figs. 8 (b, c). The results show that knowledge flow achieves higher rewards than training with untrained teachers in different environments and teacher-student settings.
+
+Table 7: Comparison with Distral on Task 1 score.   
+
+<table><tr><td>Task1,Task2,Task3</td><td>Distral Teh et al. (2017)</td><td>Ours</td></tr><tr><td>KungFuMaster, Hero, Seaquest</td><td>27433</td><td>35103</td></tr><tr><td>Hero, Seaquest, Riverraid</td><td>15096</td><td>30928</td></tr><tr><td>James, Seaquest,Riverraid</td><td>550</td><td>1245</td></tr></table>
+
+![](images/cb0990d9b96c9b5636afc93e3de14ccf00f76a022af12d47c435bfd47e2f3fa3.jpg)  
+Figure 7: Normalized weights for the teachers and the student in C10 experiments.
+
+# 7.4.2 TRAINING WITHOUT KL TERM
+
+The KL term prevents the student’s output distribution over actions or labels from drastic changes when the teachers’ influence is decreasing. To investigate the importance of the KL term, we conduct an ablation study where the KL coefficient $\left( \lambda _ { 2 } \right)$ is set to zero. The result is summarized in Fig. 9. Considering Fig. 9 (a), where the target task is MsPacman and the teachers are Riverraid and Seaquest experts. Without the KL term, when a teacher’s influence decreases, the rewards drop drastically. In contrast, with a KL term, we don’t observe performance drops. At the end of training, learning with the KL term achieves an average reward of 2907 and learning without the KL term achieves an average reward of 1215. More results are presented in Fig. 9 (b, c), which shows that training with the KL term achieves higher reward than training without the KL term.
+
+# 7.5 TEACHERS WITH DIFFERENT ARCHITECTURE THAN STUDENT
+
+In additional experiments, following the suggestion of a reviewer, we use architectures for the teacher which differ from the student model. More specifically, we use the model of Mnih et al. (2015) as a teacher model. The teacher model consists of 3 convolutional layers, which have 32, 64, and 64 filters, followed by a hidden fully connected layer which has 512 ReLUs. We use the model of Mnih et al. (2016) as the student model. The student model consists of 2 convolutional layers, which have 16 and 32 filters respectively, followed by a hidden fully connected layer which has 256 ReLUs. Both models’ fully connected layers are followed by two output layers for actions and values. In the experiments, we link each teacher’s first convolutional layer to the student’s first convolutional layer. Moreover, we link each teacher’s third convolutional layer to the student’s second convolutional layer, and each teacher’s fully connected layer to the student’s fully connected layer. In the experiment, the target task is KungFu Master, and the teachers are experts for Seaquest and Riverraid. The results are summarized in Fig. 10. We observed that learning with teachers, whose architecture differs from the student, to have similar performance as learning with teachers which have the same architecture. Consider as an example Fig. 10 (a), where the target task is KungFu Master, and the teachers are experts for Seaquest and Riverraid. At the end of training, learning with teachers of different architectures achieves an average reward of 37520, and learning with teachers of the same architecture achieves an average reward of 35012. More results are shown in Fig. 10 (b, c). The results show that knowledge flow can enable higher rewards, even if the teachers and the student architectures differ.
+
+![](images/c26da5f17450e4711fa09727be150c2d6aec6e942c2bc016144babf6a14e3253.jpg)  
+Figure 8: Ablation study: using untrained teachers.
+
+![](images/679b3e0c12470fc53fef19f217bccea704af2688fdd85988f755fce3e6617e96.jpg)  
+Figure 9: Ablation study regarding KL term. Seaquest and Riverraid experts are used as teachers for all experiments.
+
+# 7.6 AVERAGE NETWORK AS $\theta _ { o l d }$
+
+For the parameters $\theta _ { \mathrm { o l d } }$ an average network can be used. To investigate how usage of an average network to obtain the parameters $\theta _ { \mathrm { o l d } }$ affects the performance, we conduct an experiment where $\theta _ { \mathrm { o l d } }$ is computed using the exponential running average of the model weight. More specifically, $\theta _ { \mathrm { o l d } }$ is updated as follows: $\theta _ { \mathrm { o l d } }  \alpha \cdot \theta _ { \mathrm { o l d } } + ( 1 - \alpha ) \cdot \theta$ , where $\alpha = 0 . 9$ . The results are summarized in Fig. 11. We observe that using an exponential average to compute $\theta _ { \mathrm { o l d } }$ results in very similar performance as using a single model. Consider Fig. 11 (a), where the target task is Boxing and the teacher is a Riverraid expert. At the end of training, using an average network to obtain $\theta _ { \mathrm { o l d } }$ achieves an average reward of 96.2 and using a single network to obtain $\theta _ { \mathrm { o l d } }$ achieves an average reward of 96.0. More results on using an average network are shown in Fig. 11 (b, c).
+
+# 8 RELATED WORK
+
+As mentioned before, variants of ‘knowledge’ transfer have been considered using a variety of techniques, for instance, fine-tuning, progressive neural nets (Rusu et al., 2016b), PathNet (Fernando et al., 2017), ‘Growing a Brain’ (Wang et al., 2017), actor-mimic (Parisotto et al., 2016), learning without forgetting (Li & Hoiem, 2016). Also related are techniques on transfer learning and lifelong learning. We discuss those methods and contrast them to our approach in the following.
+
+PathNet (Fernando et al., 2017) enables multiple agents to train the same giant deep net while reusing parameters and avoiding catastrophic forgetting. To this end, agents embedded in the neural net discover which weights can be reused for new tasks and restrict application of gradients to those parameters. In contrast to this formulation we consider availability of multiple teacher nets, which are trained.
+
+Progressive Net (Rusu et al., 2016b) leverages transfer and avoids catastrophic forgetting by introducing lateral connections to previously learned features. Our discussed method uses similar lateral connections. However, in contrast to Rusu et al. (2016b), we introduce scaling with normalized weights. This ensures independence of the student upon training, addressing a limitation in (Rusu et al., 2016b) where only a fraction of the capacity of the student is eventually utilized.
+
+![](images/52be82be150f4657c771b06c9198d4aef49934be076ba5b84bd90c6094887f41.jpg)  
+Figure 10: Teachers’ architecture differs from the student’s architecture. Seaquest and Riverraid experts are used as teachers for all experiments.
+
+![](images/b1a149c7edaf6b1bdd230bb053fae6d38bdd114f2b767966162b08d69f459959.jpg)  
+Figure 11: Average network to compute $\theta _ { o l d }$ . Riverraid expert is used as teacher for all experiments.
+
+Distral a neologism combining ‘distill & transfer learning’ (Teh et al., 2017) considers joint training of multiple tasks. Multiple tasks share a ‘distilled’ policy which encodes common behavior between different tasks. While each worker addresses its own task, a shared policy encourages consistency between the policies. Different from Distral, which is a multi-task learning framework, knowledge flow addresses a single task, while in multi-task learning, multiple tasks are addressed at the same time. Hence, common for multi-task learning and knowledge flow is a transfer of information. However, in multi-task learning, information extracted from different tasks are shared to boost performance, while, in knowledge flow, the information of multiple teachers is leveraged to help a student learn better a single, new, previously unseen task.
+
+Knowledge distillation (Hinton et al., 2015) distills information form a larger deep net into a smaller one. It assumes both nets are trained on the same dataset. In contrast, our technique allows knowledge transfer between different source and target domains.
+
+Actor-mimic (Parisotto et al., 2016) enables an agent to learn how to address multiple tasks simultaneously and generalize the extracted knowledge to new domains. A single policy net learns how to act in a set of tasks following the guidance of several expert teachers. A combination of feature regression and cross entropy loss is used to encourage the student to produce similar actions and representations. Our proposed technique differs in that we take advantage of a teachers representation at the beginning of training,
+
+Learning without forgetting (Li & Hoiem, 2016) permits to add a new task to a deep net without forgetting the original capabilities. Importantly, only data from the new task is used and the old capabilities are retained by first recording the old networks output on the new data. Similar techniques have been developed by Furlanello et al. (2016); Jung et al. (2016). In contrast, we transfer ‘knowledge’ from teacher networks more explicitly.
+
+Growing a Brain (Wang et al., 2017) analyzes the parameters which change during fine-tuning and points out that more natural model adaptation is obtained when increasing the model capacity, by either extending width or depth. Appropriate normalization is essential to significantly outperform classical fine-tuning. Since this technique is based on fine-tuning, it differs from our student-teacher based approach.
+
+Other related work includes policy distillation (Rusu et al., 2016a), domain adaptation (Pan & Yang, 2010; Long et al., 2015; Tzeng et al., 2015) or lifelong learning (Chen & Liu, 2016; Thrun, 1998; Mitchell et al., 2015; Ruvolo & Eaton, 2013).

parse/train/BJeOioA9Y7/BJeOioA9Y7_content_list.json

@@ -0,0 +1,1724 @@
+[
+    {
+        "type": "text",
+        "text": "KNOWLEDGE FLOW: IMPROVE UPON YOUR TEACH-ERS",
+        "text_level": 1,
+        "bbox": [
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+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Iou-Jen Liu, Jian Peng, Alexander G. Schwing University of Illinois at Urbana-Champaign {iliu3, jpeng, aschwing}@illinois.edu ",
+        "bbox": [
+            183,
+            170,
+            547,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            250,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A zoo of deep nets is available these days for almost any given task, and it is increasingly unclear which net to start with when addressing a new task, or which net to use as an initialization for fine-tuning a new model. To address this issue, in this paper, we develop knowledge flow which moves ‘knowledge’ from multiple deep nets, referred to as teachers, to a new deep net model, called the student. The structure of the teachers and the student can differ arbitrarily and they can be trained on entirely different tasks with different output spaces too. Upon training with knowledge flow the student is independent of the teachers. We demonstrate our approach on a variety of supervised and reinforcement learning tasks, outperforming fine-tuning and other ‘knowledge exchange’ methods. ",
+        "bbox": [
+            233,
+            282,
+            766,
+            421
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            454,
+            336,
+            470
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Research communities have amassed a sizable number of deep net architectures for different tasks, and new ones are added almost daily. Some of those architectures are trained from scratch while others are fine-tuned, i.e., before training, their weights are initialized using a structurally similar deep net which was trained on different data. ",
+        "bbox": [
+            176,
+            478,
+            825,
+            534
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Beyond fine-tuning, particularly in reinforcement learning, teachers have also been considered in one way or another by Rusu et al. (2016b); Fernando et al. (2017); Wang et al. (2017); Li & Hoiem (2016); Bengio et al. (2009); Patel et al. (2015); Chen & Liu (2016); Teh et al. (2017); Parisotto et al. (2016). For instance, progressive neural net (Rusu et al., 2016b) keeps multiple teachers during both training and inference, and learns to extract useful features from the teachers for a new target task. PathNet (Fernando et al., 2017) uses genetic algorithms to choose pathways from a giant network for learning new tasks. ‘Growing a Brain’ (Wang et al., 2017) fine-tunes a neural network while growing the network’s capacity (wider or deeper layers). Actor-mimic (Parisotto et al., 2016) pre-trains a big model on multiple source tasks, then the big model is used as a weight initialization for a new model which will be trained on a new target task. Knowledge distillation (Hinton et al., 2015) distills knowledge from a large ensemble of models to a smaller student model. ",
+        "bbox": [
+            174,
+            541,
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+            693
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, all the aforementioned techniques have limitations. For example, progressive neural net models (Rusu et al., 2016b) grow with the number of teachers. This large number of parameters limits the number of teachers a progressive neural net can handle, and largely increases the training and testing time. In PathNet (Fernando et al., 2017), searching over a big network for pathways is computationally intensive. For fine-tuning based methods such as ‘Growing a Brain’ (Wang et al., 2017) and actor-mimic (Parisotto et al., 2016), only one pretrained model can be used at a time. Hence, their performance heavily relies on the chosen pretrained model. ",
+        "bbox": [
+            174,
+            700,
+            825,
+            797
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To address these shortcomings, we develop knowledge flow which moves ‘knowledge’ of multiple teachers when training a student. Irrespective of how many teachers we use, the student is guaranteed to become independent at the final stage of training and the size of the resulting student net remains constant. In addition, our framework makes no restrictions on the deep net size of the teacher and student, which provides flexibility in choosing teacher models. Importantly, our approach is applicable to a variety of tasks from reinforcement learning to fully-supervised training. ",
+        "bbox": [
+            174,
+            805,
+            823,
+            888
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We evaluate knowledge flow on a variety of tasks from reinforcement learning to fully-supervised learning. In particular, we follow Rusu et al. (2016b); Fernando et al. (2017) and compare on the same ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Atari games. In addition, we also observed significant top-1 error rate improvements on supervised learning datasets, i.e., CIFAR-10, and CIFAR-100. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            154,
+            326,
+            171
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Knowledge flow is applicable to a variety of settings from supervised learning to reinforcement learning, which we briefly review to introduce notation. ",
+        "bbox": [
+            173,
+            186,
+            823,
+            215
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Supervised Learning recovers the parameters $\\theta$ of a mapping $f _ { \\theta } : \\mathcal { X }  \\mathcal { Y }$ from data space $\\mathcal { X }$ to output space $\\mathcal { V }$ . To this end, a dataset $D = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }$ containing $n$ pairs $( x _ { i } , y _ { i } )$ (assumed to be sampled i.i.d.) is used, where $x _ { i } \\in { \\mathcal { X } }$ and $y _ { i } \\in \\mathcal { V }$ . Given this dataset, the parameters $\\theta$ of the mapping $f _ { \\theta }$ are learned by minimizing a loss function $\\ell _ { ( x , y ) } ( \\theta )$ composed of a regularization term $R ( \\theta )$ and an empirical risk $\\ell ( y , f _ { \\boldsymbol { \\theta } } ( x ) )$ which compares groundtruth label $y$ and prediction $f _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } )$ The parameters $\\theta$ are obtained by optimizing the following program: ",
+        "bbox": [
+            173,
+            222,
+            825,
+            309
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e097bf9b5512764728cdff3c2d133802430822f4e05b1fd88a219a1e0d448ece.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta } \\mathbb { E } _ { ( x , y ) \\sim D } [ \\ell _ { ( x , y ) } ( \\theta ) ] : = \\mathbb { E } _ { ( x , y ) \\sim D } [ \\ell ( y , f _ { \\theta } ( x ) ) ] + R ( \\theta ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            316,
+            694,
+            338
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Hereby, the mapping $f _ { \\theta }$ is obtained by maximizing the logits or a corresponding probability distribution $\\hat { f } _ { \\boldsymbol { \\theta } } ( y | \\boldsymbol { x } )$ , i.e., $f _ { \\theta } = \\arg \\operatorname* { m a x } _ { y \\in \\mathcal { V } } \\hat { f } _ { \\theta } ( y | x )$ . Here and below let the hat $( ^ { 6 } \\hat { \\cdot } \\vec { \\cdot } )$ indicate probability distributions over appropriate domains. ",
+        "bbox": [
+            174,
+            347,
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+            393
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement Learning considers an agent interacting with an environment according to a policy   \n$\\pi _ { \\theta _ { \\pi } } : \\mathcal { X }  \\mathcal { A }$ which maps a state $x _ { t } \\in \\mathcal X$ to an action $a _ { t } \\in \\mathcal A$ at time $t$ . The policy depends on   \nthe parameters a scalar rewardthe discount fa $\\theta _ { \\pi }$ . After performing action . The discounted return ar. The expected future rew $a _ { t }$ , theime d w agent observeis defined as n observing s $x _ { t + 1 }$ and rece, where wing p esiscy $r _ { t }$ $t$ $\\begin{array} { r } { R _ { t } = \\sum _ { k = 0 } ^ { \\infty } \\gamma ^ { k } r _ { t + k } } \\end{array}$ $\\gamma$ $x$   \n$\\pi _ { \\theta _ { \\pi } }$ is defined as $V ^ { \\pi _ { \\theta _ { \\pi } } } ( x _ { t } ) = \\mathbb { E } _ { \\tau \\sim \\pi _ { \\theta _ { \\pi } } } [ R _ { t } | x _ { t } ]$ , where $\\tau = \\{ ( x _ { t } , a _ { t } , r _ { t } ) , ( x _ { t + 1 } , a _ { t + 1 } , r _ { t + 1 } ) , \\ldots \\}$ is a   \ntrajectory generated by following $\\pi _ { \\theta _ { \\pi } }$ from state $x _ { t }$ . ",
+        "bbox": [
+            173,
+            398,
+            825,
+            498
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The goal of reinforcement learning is to find a policy that maximizes the expected future reward from each state $x _ { t }$ . Without loss of generality, in this paper, we follow the asynchronous advantage actor-critic (A3C) formulation (Mnih et al., 2016). In A3C, the policy mapping $\\pi _ { \\boldsymbol { \\theta } _ { \\pi } } ( x ) = \\arg \\operatorname* { m a x } _ { a \\in \\mathcal { A } } \\hat { \\pi } _ { \\boldsymbol { \\theta } _ { \\pi } } ( a \\vert x )$ is obtained from a probability distribution over states, where ${ \\hat { \\pi } } _ { \\boldsymbol { \\theta } _ { \\pi } } ( a | \\boldsymbol { x } )$ is modeled by a deep net with parameters $\\theta _ { \\pi }$ . The value function is also approximated by a deep net $V _ { \\theta _ { v } } ( x )$ , having parameters $\\theta _ { v }$ . ",
+        "bbox": [
+            173,
+            502,
+            825,
+            588
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To optimize the policy parameters $\\theta _ { \\pi }$ given a state $x _ { t }$ , a loss function based on a scaled negative log-likelihood and a negative entropy regularizer is common: ",
+        "bbox": [
+            173,
+            594,
+            821,
+            623
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3ca8c6ccd7f22bbcd82f26f74db46456d080f1dbc2f8092d8d415aeb21e383ea.jpg",
+        "text": "$$\n\\ell _ { \\pi } ^ { \\tau } ( \\theta _ { \\pi } ) = \\frac { 1 } { | \\tau | } \\sum _ { t \\in \\tau } [ - \\log \\hat { \\pi } _ { \\theta _ { \\pi } } ( a _ { t } | x _ { t } ) ( R _ { t } - V _ { \\theta _ { v } } ( x _ { t } ) ) - \\beta H ( \\hat { \\pi } _ { \\theta _ { \\pi } } ( \\cdot | x _ { t } ) ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            631,
+            733,
+            667
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Herestate , $\\begin{array} { r } { R _ { t } = \\sum _ { i = 0 } ^ { k - 1 } \\gamma ^ { i } r _ { t + i } + \\gamma ^ { k } V _ { \\theta _ { v } } ( x _ { t + k } ) } \\end{array}$ isry e empirical generated $k$ -step return following ained when s. The scalar $x _ { t }$ $| \\tau |$ $\\tau$ $\\pi _ { \\theta _ { \\pi } }$ $\\beta \\geq 0$ a user-specified constant, and $H ( \\hat { \\pi } _ { \\boldsymbol { \\theta } _ { \\pi } } ( \\cdot | \\boldsymbol { x } _ { t } ) )$ is the entropy function, which encourages exploration by favoring a uniform probability distribution ${ \\hat { \\pi } } _ { \\theta _ { \\pi } } ( a | x )$ . To optimize the value function $V _ { \\theta _ { v } }$ , it is common to use the squared loss $\\begin{array} { r } { \\ell _ { v } ^ { \\tau } ( \\theta _ { v } ) = \\frac { 1 } { 2 | \\tau | } \\sum _ { t \\in \\tau } ( R _ { t } - V _ { \\theta _ { v } } ( x _ { t } ) ) ^ { 2 } } \\end{array}$ . ",
+        "bbox": [
+            173,
+            678,
+            825,
+            753
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "By minimizing the empirical expectation of $\\ell _ { \\pi } ^ { \\tau } ( \\theta _ { \\pi } )$ and $\\ell _ { v } ^ { \\tau } ( \\theta _ { v } )$ , i.e., by addressing ",
+        "bbox": [
+            173,
+            760,
+            712,
+            776
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/725f5becc4e67d947a6e806c3aff0f5d7a4d2f75334f6f4017e2d1a44e1a89fa.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta _ { \\pi } } \\mathbb { E } _ { \\tau \\sim \\pi _ { \\theta _ { \\pi } } } [ \\ell _ { \\pi } ^ { \\tau } ( \\theta _ { \\pi } ) ] , \\quad \\mathrm { ~ a n d ~ } \\quad \\operatorname* { m i n } _ { \\theta _ { v } } \\mathbb { E } _ { \\tau \\sim \\pi _ { \\theta _ { \\pi } } } [ \\ell _ { v } ^ { \\tau } ( \\theta _ { v } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
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+            669,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "alternatingly, we learn a policy and a value function that maximize expected return. ",
+        "bbox": [
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+            832
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 KNOWLEDGE FLOW ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            845,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Instead of optimizing the programs given in Eq. (1) and Eq. (2) from scratch, the aforementioned warm-start techniques (see Sec. 5 for more) are applicable. To address their mentioned shortcomings, we propose knowledge flow, a framework that moves ‘knowledge’ from an arbitrary number of deep nets, henceforth referred to as ‘teachers’ to a deep net under training, called the ‘student.’ ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/23ea2fe0c61b350b0c7fa17bfe9b615831f0fafec1d03d55c7e243319a4ca5bc.jpg",
+        "image_caption": [
+            "Figure 1: (a) Example of a two-teacher knowledge flow. (b) Deep net transformation of knowledge flow. (c) Average normalized weights for teachers’ and the student’s layers. At the beginning of training, the student heavily relies on teacher one. As training progresses, teacher one’s weight decreases, and the student’s weight increases until the student is eventually independent. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 OVERVIEW ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            320,
+            290,
+            333
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Knowledge flow is outlined on example deep nets in Fig. 1 (a,b). We train the parameters of the student net which are randomly initialized. To this end we take advantage of teachers, whose parameters are fixed and obtained from pre-trained models on different source tasks by different algorithms. For example, for reinforcement learning, we may consider teachers trained by A3C (Mnih et al., 2016), A2C (Dhariwal et al., 2017) or DQN (Mnih et al., 2015). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "‘Knowledge’ of multiple teachers is transferred to a student by adding transformed and scaled intermediate representations from the teacher deep nets to the student net. To achieve this, we modify the student net, i.e., $f _ { \\theta }$ in the supervised setting and $\\pi _ { \\theta _ { \\pi } } ( a | x ) , V _ { \\theta _ { v } } ( x )$ in the reinforcement learning case. We add teacher representations which are transformed by multiplication with a trainable matrix Q and scaled via a weight $p _ { w }$ that is normalized to sum to one for each student layer and parameterized via trainable parameters $w$ . The normalized weights encode which of the teachers’ or the student’s representation to trust at every layer of the student net. Note that a teacher can help the student at different levels of abstraction with input from different levels of its net. ",
+        "bbox": [
+            173,
+            416,
+            825,
+            529
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Importantly, after training, the student model should perform well on the target task without relying on teachers. To achieve this, as training progresses, we increasingly encourage a high normalized weight on the student representation, which forces the student to eventually capture all the ‘knowledge.’ Due to the trainable scaling, at an early stage of training, we observe the student to rely heavily on the ‘knowledge’ of the teacher to quickly obtain better performance. However, as training proceeds, the student is encouraged to become more and more independent. During final stages of training, the student will no longer be able to rely on teachers, which ensures that the student has learned to master the desired task on its own. This is observed in Fig. 1 (c). ",
+        "bbox": [
+            173,
+            535,
+            825,
+            647
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To formally encourage this successive transfer we introduce two additional loss functions. The first, referred to as the dependency loss $\\ell _ { \\mathrm { d e p } } ( w )$ , captures how much a student relies on teachers. It depends on the weight vector $w$ which encodes the strength of the coupling. The second one ensures that a student’s behavior doesn’t change rapidly when the teachers’ influence decreases. We use loss $\\ell _ { \\mathrm { K L } } ( \\cdot , \\cdot )$ to capture the change. ",
+        "bbox": [
+            173,
+            652,
+            825,
+            723
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "By combining student net modifications and additional loss terms, for the supervised task we obtain ",
+        "bbox": [
+            173,
+            729,
+            821,
+            744
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1860ef437798fbf421a558f5f92f132f41413b102aa45fd09a1c6669f760dd01.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta , w , Q } \\mathbb { E } _ { ( x , y ) } [ \\widetilde { \\ell } _ { ( x , y ) } ( \\theta , w , Q ) + \\lambda _ { 1 } \\ell _ { \\mathrm { d e p } } ( w ) + \\lambda _ { 2 } \\ell _ { \\mathrm { K L } } ( \\widetilde { \\hat { f } } _ { \\theta } , \\widetilde { \\hat { f } } _ { \\theta _ { \\mathrm { o l d } } } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            752,
+            697,
+            782
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and for reinforcement learning the transformed program reads as follows: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
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+        "img_path": "images/91afe3f9169b3378ae94e04fec74c132b6c7287efabcb0142a3a722578254587.jpg",
+        "text": "$$\n\\left\\{ \\begin{array} { l l } { \\operatorname* { m i n } _ { \\theta _ { \\pi } , w , Q } \\mathbb { E } _ { \\tau \\sim \\tilde { \\pi } _ { \\theta _ { \\pi } } } [ \\tilde { \\ell } _ { \\pi } ^ { \\tau } ( \\theta _ { \\pi } , w , Q ) + \\lambda _ { 1 } \\ell _ { \\mathrm { d e p } } ( w ) + \\lambda _ { 2 } \\ell _ { \\mathrm { K L } } ^ { \\tau } ( \\tilde { \\hat { \\pi } } _ { \\theta _ { \\pi } } , \\tilde { \\hat { \\pi } } _ { \\theta _ { \\pi _ { \\mathrm { o l d } } } } ) ] } \\\\ { \\operatorname* { m i n } _ { \\theta _ { v } , w , Q } \\mathbb { E } _ { \\tau \\sim \\tilde { \\pi } _ { \\theta _ { \\pi } } } [ \\tilde { \\ell } _ { v } ^ { \\tau } ( \\theta _ { v } , w , Q ) ] } \\end{array} \\right. .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Loss $\\tilde { \\ell } \\colon ( \\theta , w , Q )$ originates from the original loss $\\ell \\colon ( \\theta )$ (Eqs. (1)-(2)) by transforming the deep net to include cross-connections, hence its dependence on $w , Q$ . The tilde $( ^ { 6  } )$ denotes this dependence, also for probability distribution $\\tilde { \\hat { f } }$ and policy distribution $\\tilde { \\hat { \\pi } }$ . Parameters from the current and a previous iteration are referred to via $\\theta$ and $\\theta _ { \\mathrm { o l d } }$ respectively. ",
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+        "text": "For both supervised and reinforcement learning, $\\lambda _ { 1 }$ and $\\lambda _ { 2 }$ control the strength which is used to decrease the influence of the teacher. A low $\\lambda _ { 1 }$ allows the student to rely on teachers. Close to the end of training, the student should be independent. Therefore, we set $\\lambda _ { 1 }$ to a small value at the beginning, and gradually increase its value as training progresses. ",
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+        "type": "text",
+        "text": "Note that we don’t make any assumptions about teachers and student’s objective. If a teacher’s and student’s objective differ, negative transfer may occur initially. However, the proposed method quickly decreases the weight for teacher layers to reduce this effect. Despite differences, students could potentially still benefit from the low level representation of the teachers. We do observe this low level knowledge transfer in our experiments. ",
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+        "type": "text",
+        "text": "In the following we first describe how to modify the deep nets, before we detail the loss functions $\\ell _ { \\mathrm { d e p } }$ and $\\ell _ { \\mathrm { K L } }$ , which are used to successively decrease the influence of the teachers. ",
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+        "type": "text",
+        "text": "3.2 DEEP NET TRANSFORMATION AND LOSS TERMS ",
+        "text_level": 1,
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+        "text": "Deep Net Transformation: Knowledge flow enhances the student by adding transformed and scaled intermediate representations from teacher models. To perform the transformation, intermediate representations from teachers are first multiplied by transformation matrices $Q$ . Then the transformed representations from teachers and representations from the student are linearly combined. The weights for this linear combination are determined by a weight $p _ { w }$ which is normalized to sum to one for each student layer. ",
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+        "type": "text",
+        "text": "Let index $m = 0$ denote the student model and let $\\theta ^ { ( 0 ) }$ refer to its parameters. Further, let $\\theta ^ { ( m ) }$ , $m \\in \\{ 1 , \\ldots , M \\}$ denote teacher models. We use $l _ { m } ^ { i }$ to refer to deep net layer $i$ of teacher $m$ , with $i \\in \\{ 1 , \\ldots , L _ { m } \\}$ and $L _ { m }$ the number of layers in teacher $m$ . We define layer $j$ of the student model to be $l _ { 0 } ^ { j }$ , where $j \\in \\{ 1 , \\dots , L _ { 0 } \\}$ and $L _ { 0 }$ the number of deep net layers in the student model. The output of layer $l _ { m } ^ { k }$ right before and after an activation unit is denoted $z ( l _ { m } ^ { k } )$ and $h ( l _ { m } ^ { k } )$ respectively. ",
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+        "type": "text",
+        "text": "To align a teacher’s layer $l _ { m } ^ { i }$ with a student’s layer $l _ { 0 } ^ { j }$ , we introduce a learnable transformation matrix $Q ^ { j } ( l _ { m } ^ { i } ) \\in \\mathbb { R } ^ { \\dim ( l _ { 0 } ^ { j } ) \\times \\dim ( l _ { m } ^ { i } ) }$ , where $\\dim ( \\cdot )$ gives the number of elements in the corresponding layer. The matrix multiplication $Q ^ { j } ( l _ { m } ^ { i } ) z ( l _ { m } ^ { i } )$ aligns the representation from layer $i$ of teacher $m$ with the representation of layer $j$ of the student. ",
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+        "type": "text",
+        "text": "For each layer $j$ in the student model, we define a candidate set $\\mathbb { L } ^ { j }$ , which contains $l _ { 0 } ^ { j }$ and all the teachers’ layers to be considered. For example, in Fig. 1 (a), layer one of the student model is combined with layer one of teacher one and layer two of teacher two. Therefore, the candidate set of layer one of the student model is given by $\\mathbb { L } ^ { 1 } \\dot { = } \\{ l _ { 0 } ^ { 1 } , l _ { 1 } ^ { 1 } , l _ { 2 } ^ { 2 } \\}$ . ",
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+        "type": "text",
+        "text": "To decide which teachers’ or the student’s representation to trust at every layer of the student net, we introduce a normalized weight $p _ { w } ^ { j } ( l )$ for all $j \\in \\{ 1 , \\ldots , L _ { 0 } \\}$ , where $l \\in \\mathbb { L } ^ { j }$ , summing to one for each layer $j$ in the student deep net, $i . e .$ ., ",
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+        "type": "equation",
+        "img_path": "images/a3a042925ea2a0db8db0885bd6408d54bd00a8d0b1804294ca73d48fb828c77b.jpg",
+        "text": "$$\n\\sum _ { l \\in \\mathbb { L } ^ { j } } p _ { w } ^ { j } ( l ) = 1 , \\forall j \\in \\{ 1 , \\dots , L _ { 0 } \\} .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "To obtain the combined intermediate representation of layer $j$ for the student model, we use ",
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+        "type": "equation",
+        "img_path": "images/2eb0249e23c17c105f7b3317dea52ee0f438dea0a69248ad1c8232374117e8dd.jpg",
+        "text": "$$\nh ( l _ { 0 } ^ { j } ) = \\sigma \\left( \\sum _ { l \\in \\mathbb { L } ^ { j } \\backslash l _ { 0 } ^ { j } } p _ { w } ^ { j } ( l ) Q ^ { j } ( l ) z ( l ) + p _ { w } ^ { j } ( l _ { 0 } ^ { j } ) z ( l _ { 0 } ^ { j } ) \\right) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $p _ { w } ^ { j } ( l _ { m } ^ { i } )$ determines how much the student layer $j$ relies on transformed representations of layer $i$ from the $m$ -th teacher. Intuitively, if the transformed representation of the $m$ -th teacher layer $i$ is helpful, $p _ { w } ^ { j } ( l _ { m } ^ { i } )$ will be close to one. We visualize the deep net transformation in Fig. 1 (b). ",
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+    {
+        "type": "text",
+        "text": "Note that the intermediate representations of teachers are not changed in our framework. To obtain the output of layer $l _ { m } ^ { k }$ we apply the original activation unit to the original representation $z ( l _ { m } ^ { i } )$ , i.e., $\\begin{array} { r } { h ( l _ { m } ^ { i } ) = \\sigma ( z ( l _ { m } ^ { i } ) ) , ~ \\overset {  } { \\forall } m \\in \\{ 1 , \\dots , M \\} , j \\in \\{ 1 , \\dots L _ { m } \\} } \\end{array}$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The maximal number of introduced matrices $Q$ in our framework is $\\textstyle \\sum _ { i = 1 } ^ { M } L _ { i } L _ { 0 }$ . In practice, we don’t link a student’s layer to every layer of a teacher network. Intuitively, a teachers’ bottom layer features are very likely irrelevant to a student’s top layer features. Indeed, we observed that linking a teachers’ bottom layer to a student’s top layer generally doesn’t yield improvements. Therefore, in practice, we recommend to link one teacher layer to one or two student layers, in which case we introduce on the order of $M L _ { 0 }$ matrices Q. Also note that while additional trainable parameters $Q$ and $w$ are introduced in our framework, $Q$ and $w$ are not part of the resulting student network since we ensure $p _ { w } ^ { j } ( l ) \\equiv 0 \\forall l \\in \\mathbb { L } ^ { j } \\backslash l _ { 0 } ^ { j }$ at the end of training as discussed next. Hence, the additional parameters function as auxiliary knobs that help the student learn faster. In the final stage of training, the student will be independent (see Fig. 1 (c)) and does no longer rely on $Q , w ,$ , or any transformed representations from teachers. ",
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+    {
+        "type": "table",
+        "img_path": "images/a5afdfbcd25dcd493d4d0c8648aa1c5ad5389dc1d13a420b4b2af1f930758a23.jpg",
+        "table_caption": [
+            "Table 1: Comparison with PathNet (Fernando et al., 2017) and progressive neural network (PNN) (Rusu et al., 2016b). Since PathNet and PNN don’t report exact scores we obtain their numbers from their plots and indicate that with a $\\sim$ symbol. The results of the state-of-the-art methods: A3C (Mnih et al., 2016), PPO (Schulman et al., 2017), and ACKTR (Wu et al., 2017) on Atari games are also listed for reference. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">w/ Seaquest teacher</td><td colspan=\"2\">w/ Riverraid teacher</td><td colspan=\"2\">w/ Sea. and River. teachers</td><td colspan=\"3\">No teachers</td></tr><tr><td></td><td>Ours</td><td>PathNet</td><td>Ours</td><td>PathNet</td><td>Ours</td><td>PNN</td><td>A3C</td><td>PPO</td><td>ACKTR</td></tr><tr><td>Alien</td><td>1254</td><td>~1700</td><td>1259</td><td>~1800</td><td>1911</td><td>~2000</td><td>182</td><td>1850</td><td>3197</td></tr><tr><td>Asterix</td><td>3982</td><td>~2000</td><td>3823</td><td>~2000</td><td>6012</td><td>~9000</td><td>6723</td><td>4533</td><td>31583</td></tr><tr><td>Boxing</td><td>96</td><td>~70</td><td>96</td><td>~80</td><td>99</td><td>~99</td><td>34</td><td>95</td><td>1</td></tr><tr><td>Gopher</td><td>4152</td><td>~3900</td><td>3820</td><td>~2100</td><td>5233</td><td>~4500</td><td>8443</td><td>2933</td><td>47730</td></tr><tr><td>Hero</td><td>21250</td><td>~12500</td><td>29343</td><td>~12500</td><td>30928</td><td>~30000</td><td>28766</td><td>n/a</td><td>n/a</td></tr><tr><td>James.</td><td>857</td><td>~600</td><td>832</td><td>~600</td><td>1245</td><td>~850</td><td>352</td><td>561</td><td>512</td></tr><tr><td>Krull</td><td>8193</td><td>~7800</td><td>6890</td><td>~7500</td><td>10000</td><td>~9954</td><td>8067</td><td>7942</td><td>9689</td></tr></table>",
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+        "text": "",
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+        "text": "Decreasing Teachers’ Influence: We successively decrease the influence of the teachers during training by gradually encouraging the normalized weight $p _ { w } ^ { j } ( l _ { 0 } ^ { j } )$ to increase to a value of $1 \\forall j \\in$ $\\{ 1 , 2 , \\ldots , L _ { 0 } \\}$ . To capture how much the student relies on teachers, we introduce the dependence cost as the negative log probability: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/af8e9241ad2c865499d1af93e3590ce3e1ffdda0995e02ba43d502754dc1689b.jpg",
+        "text": "$$\n\\ell _ { \\mathtt { d e p } } ( w ) = - \\frac { 1 } { L _ { 0 } } \\sum _ { j \\in \\{ 1 , 2 , . . . , L _ { 0 } \\} } \\log p _ { w } ^ { j } ( l _ { 0 } ^ { j } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "By minimizing $\\ell _ { \\mathrm { d e p } } ( w )$ , we encourage weights for the layers of the student to increase. Hence we encourage the student to become more and more independent. During the final stage of training, $p _ { w } ^ { j } ( l _ { 0 } ^ { j } )$ approaches one for all $j \\in \\{ 1 , \\ldots , L _ { 0 } \\}$ , making the student independent of the transformed representation obtained from teachers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Empirically, we found that a fast decrease of the influence of the teacher can degrade the performance. This is intuitive as it requires some time to find good transformations $Q$ . Moreover, decreasing the influence of a teacher too fast may change the output distribution over labels or actions of the student model too much, and thus lead to performance loss. To prevent changing a student’s output distribution too fast, we found a Kullback-Leibler (KL) regularizer to yield good results. More specifically, in the case of supervised learning we use ",
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+        "img_path": "images/a60a2719dfbf53e98460f9b7e868ced8d0927ca56b97cb5699597c999b125332.jpg",
+        "text": "$$\n\\ell _ { \\mathrm { K L } } \\big ( \\tilde { \\hat { f } } _ { \\boldsymbol { \\theta } } , \\tilde { \\hat { f } } _ { \\boldsymbol { \\theta } _ { \\mathrm { o l d } } } \\big ) = D _ { \\mathrm { K L } } \\big [ \\tilde { \\hat { f } } _ { \\boldsymbol { \\theta } } \\big ( \\cdot | \\boldsymbol { x } \\big ) | | \\tilde { \\hat { f } } _ { \\boldsymbol { \\theta } _ { \\mathrm { o l d } } } \\big ( \\cdot | \\boldsymbol { x } \\big ) \\big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Hereby, $\\theta$ is the set of current parameters, and $\\theta _ { \\mathrm { o l d } }$ are the previous ones. In the reinforcement learning case we use $D _ { \\mathrm { K L } } [ \\tilde { \\hat { \\pi } } _ { \\boldsymbol { \\theta } } ( \\cdot \\vert x _ { t } ) \\vert \\vert \\tilde { \\hat { \\pi } } _ { \\boldsymbol { \\theta } _ { \\mathrm { o l d } } } ^ { - } ( \\cdot \\vert x _ { t } ) ]$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In the following we evaluate knowledge flow on reinforcement and supervised learning tasks. Results are reported by using only the student model to avoid even the smallest influence from any teacher nets. ",
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+    {
+        "type": "text",
+        "text": "4.1 REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We evaluate knowledge flow on reinforcement learning using Atari games that were used by Rusu et al. (2016b); Fernando et al. (2017). Following existing work, the input to our agent are raw images from the environment. The agent learns to predict actions only based on the rewards and the input images from the environment. The agent chooses an action every four frames, and the last action is repeated on the skipped four frames. For all teacher models and the student model, we use the fully forward architecture of A3C (Mnih et al., 2016). The model has three hidden layers. The first layer is a convolutional layer with 16 filters of size $8 \\mathrm { x } 8$ and stride 4. The second layer is a convolutional layer with 32 filters of size $4 \\mathbf { x } 4$ and stride 2. The third layer is a fully connected layer with 256 hidden units. Following the third hidden layer are two sets of output. One is a softmax output that provides a probability distribution over all valid actions. The other one is a scalar output that provides the estimated value function. We use the same hyper-parameter settings as Mnih et al. (2016) except for the learning rate. Mnih et al. (2016) use RMSProp with shared statistics while we use Adam with shared statistics, which we found to give better results when training the baselines. The learning rate is set to $1 0 ^ { - 4 }$ and gradually decreased to zero for all experiments. To select $\\lambda _ { 1 }$ and $\\lambda _ { 2 }$ in our framework, we follow progressive neural net (Rusu et al., 2016b): randomly sample $\\lambda _ { 1 } \\in \\{ 0 . 0 5 , 0 . 1 , 0 . 5 \\}$ and $\\lambda _ { 2 } \\in \\{ 0 . 0 \\bar { 0 } 1 , \\bar { 0 } . 0 1 , 0 . 0 5 \\}$ . Note that $\\lambda _ { 1 }$ is set to zero at the beginning of training, and linearly increased to the sampled value at the end of training. Following Rusu et al. (2016b), we repeat each experiment 25 times with different random seeds and randomly sampled $\\lambda _ { 1 }$ and $\\lambda _ { 2 }$ . The results of the top three out of 25 runs are reported. As A3C, we run 16 agents on 16 CPU cores in parallel. ",
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+    {
+        "type": "image",
+        "img_path": "images/e6a6f99c691a883622a6e5d5f9ab027a25423a2f5ee0a355983477af4929eff9.jpg",
+        "image_caption": [
+            "Figure 2: Comparison with progressive neural network and PathNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Evaluation Metrics: We follow the evaluation procedure of Mnih et al. (2015). The trained student models are evaluated by playing each game for 30 episodes. We also follow the ‘no-op’ procedure: at the beginning of each testing episode, the agents perform up to 30 ‘no-op’ actions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Results: We first compare our framework with PathNet (Fernando et al., 2017) and progressive neural net (PNN) (Rusu et al., 2016b), which are state-of-the-art transfer reinforcement learning frameworks, using their experimental settings. The comparison is summarized in Table 1. The state-of-the-art results (Mnih et al., 2016; Schulman et al., 2017; Wu et al., 2017) on Atari games are also included in Table 1 for reference. Compared to PathNet, a student model trained using our transfer framework with one teacher achieves higher scores in 11 out of 14 experiments. Compared with PNN, for a two-teacher framework, our trained student model has only 0.7M parameters and PNN has 16M parameters. Nonetheless we observe higher scores in five out of the seven experiments. The results demonstrate that knowledge flow effectively transfers knowledge from teachers to the student. Table 1 also indicates that, in our framework, when the number of teachers increases from one to two, the student’s performance improves significantly across all experiments. The training curves for the experiments are shown in Fig. 2. The curve is the average of the top three out of 25 runs. We observe our approach to generally perform very well. ",
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+    {
+        "type": "image",
+        "img_path": "images/a5b8d0b83325960e3b46c40527e67f1a90b12eedc7a0c133b8bd1c25c6c5aded.jpg",
+        "image_caption": [
+            "Figure 3: Comparison with fine-tuning and baseline A3C on different combinations of environment/teacher settings. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To further evaluate knowledge flow, we experiment with different combinations of environment/teacher settings. These settings are not used by PathNet and progressive neural network. The results are summarized in Table 2, where “ours w/ expert” represents that one teacher is expert for the target game; “ours w/ non-expert” represents that both teachers are not experts for the target game; “Fine-tune” represents fine-tuning from a non-expert on a new target game; “A3C baseline” represents our implementation of the A3C baseline; “A3C” represents the scores reported originally (Mnih et al., 2016). Note that our A3C implementation achieves better scores than those reported by Mnih et al. (2016) for most of the games. As shown in Table 2, knowledge flow with expert teacher performs better than the baseline across all experiments, which we interpret as evidence that knowledge flow successfully transfers ‘knowledge’ from an expert teacher to the student. In addition, knowledge flow with non-expert teachers also outperforms fine-tuning on a non-expert teacher. The reasons are twofold: First, a student model in knowledge flow can learn from multiple teachers while the fine-tuning method can only start from one setting. Second, in knowledge flow, the student can avoid the negative impact from insufficiently pretrained teachers, while fine-tuning from an insufficiently pretrained model slows down the training process and may degrade the overall performance. The training curves for the experiments are shown in Fig. 3. More training curves are in the Appendix (Fig. 6). Note that in knowledge flow, the student can benefit from the intermediate representations of the teacher, even if input space, output space and objectives differ. For example, in Fig. 3 (a), the two teachers are Chopper Command and Space Invaders, which are quite different from the target game Seaquest. The student model still benefits from learning from the teachers and achieves scores ten times larger than learning without teacher and fine-tuning from a teacher. ",
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+    {
+        "type": "text",
+        "text": "4.2 SUPERVISED LEARNING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "For supervised learning, we use a variety of image classification benchmarks, including CIFAR10 (Krizhevsky, 2009), CIFAR-100 (Krizhevsky, 2009), STL-10 (Coates et al., 2011), and EMNIST (Cohen et al., 2017). The parameters $\\lambda _ { 1 }$ for the dependent cost and $\\lambda _ { 2 }$ for the KL cost are determined using the validation set of each dataset. ",
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+        "type": "text",
+        "text": "Evaluation Metrics: To evaluate the trained student model we report top-1 error rate on the test set of each dataset. All plots and reported numbers are the average of three runs obtained using different random seeds. ",
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+        "type": "table",
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+        "table_caption": [
+            "Table 3: Test Error $( \\% )$ on CIFAR-10/100. The parentheses following “Ours” indicates the teachers we use. I.e., ‘Ours (SVHN, C100)’ indicates that we use an SVHN expert and a C100 expert as teachers. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Baseline Densenet f</td><td>Fine-tune</td><td>Fine-tune t from C100 from SVHN(C100,SVHN)</td><td>Ours</td><td colspan=\"3\">Baseline Fine-tune Fine-tune Ours Densenet from C10 from SVHN(C10,SVHN)</td></tr><tr><td>C10</td><td>4.44</td><td>4.27</td><td>4.58</td><td>3.88</td><td>C100 21.64 20.83</td><td>21.02</td><td>20.78</td></tr><tr><td colspan=\"6\">(a) (b)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "CIFAR-10/CIFAR-100: CIFAR-10 and CIFAR-100 datasets consist of colored images of size $3 2 \\times 3 2$ . CIFAR-10 (C10) has 10 classes and CIFAR-100 (C100) has 100 classes. For both dataset, the training and test sets contain 50,000 and 10,000 images respectively. We perform all experiments on CIFAR-10 and CIFAR-100 with standard data augmentation (Huang et al., 2017). ",
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+    {
+        "type": "text",
+        "text": "We use Densenet (Huang et al., 2017) (depth 100, growth rate 24) as a baseline and follow their hyper-parameter settings to train our baseline, teacher and student models. For our approach, we first train teachers on CIFAR-10, CIFAR-100, and SVHN (Netzer et al., 2011). We then train the student model using a different combination of teachers. We compare our results to fine-tuning and the baseline model. As shown in Table 3 (a), for the CIFAR-10 target task, fine-tuning from the CIFAR-100 expert improves $4 \\%$ over the baseline. Fine-tuning from the SVHN expert performs worse than the baseline model. Intuitively, for the CIFAR-10 target task, the CIFAR-100 deep net is a good teacher while a deep net trained with SVHN isn’t. Presented with both good and inadequate teachers, knowledge flow improves by $1 3 \\%$ over the baseline. This demonstrates that knowledge flow can not only leverage a good teacher’s ‘knowledge,’ but it can also avoid misleading influence. As detailed in Table 3 (b), the results are similar on the CIFAR-100 dataset. ",
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+        "type": "text",
+        "text": "To further demonstrate the properties of knowledge flow, additional results are in the appendix. ",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As mentioned before, ‘knowledge’ transfer has been considered using a variety of techniques. We briefly discuss related work in contrast to our approach in the following and defer details to Sec. 8. ",
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+        "type": "text",
+        "text": "PathNet (Fernando et al., 2017) enables multiple agents to train the same deep net while reusing parameters and avoiding catastrophic forgetting. In contrast to this formulation we consider availability of multiple pre-trained teacher nets. ",
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+    {
+        "type": "text",
+        "text": "Progressive Net (Rusu et al., 2016b) leverages transfer and avoids catastrophic forgetting by introducing lateral connections to previously learned features. Our discussed method uses similar lateral connections. However, in contrast to Rusu et al. (2016b), our method ensures independence of the student upon training, addressing a limitation in (Rusu et al., 2016b) where only a fraction of the capacity of the student is eventually utilized. ",
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+    {
+        "type": "text",
+        "text": "Distral a neologism combining ‘distill & transfer learning’ (Teh et al., 2017) considers joint training of multiple tasks. Multiple tasks share a ‘distilled’ policy which encodes common behavior between different tasks. While each worker addresses its own task, a shared policy encourages consistency between the policies. Different from Distral, which is a multi-task learning framework, knowledge flow addresses a single task, while in multi-task learning, multiple tasks are addressed at the same time. Hence, common for multi-task learning and knowledge flow is a transfer of information. However, in multi-task learning, information extracted from different tasks are shared to boost performance, while, in knowledge flow, the information of multiple teachers is leveraged to help a student learn better a single, new, previously unseen task. ",
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+    {
+        "type": "text",
+        "text": "Other related work includes actor-mimic (Parisotto et al., 2016), learning without forgetting (Li & Hoiem, 2016), growing a brain (Wang et al., 2017), policy distillation (Rusu et al., 2016a), domain adaptation (Pan & Yang, 2010; Long et al., 2015; Tzeng et al., 2015), knowledge distillation (Hinton et al., 2015) or lifelong learning (Chen & Liu, 2016). A more detailed discussion on related work is provided in Sec. 8 of the supplementary material. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We developed a general knowledge flow approach that permits to train a deep net from any number of teachers. We showed results for reinforcement learning and supervised learning, demonstrating improvements compared to training from scratch and to fine-tuning. In the future we plan to learn when to use which teacher and how to actively swap teachers during training of a student. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES   \nYoshua Bengio, Jérôme Louradour, Ronan Collobert, and Jason Weston. Curriculum learning. In Proc. ICML, 2009.   \nAaron Chen. pytorch-playground. https://github.com/aaron-xichen/ pytorch-playground, 2017.   \nZ. Chen and B. Liu. Lifelong Machine Learning. Morgan & Claypool Publishers, 2016.   \nAdam Coates, Andrew Ng, and Honglak Lee. An analysis of single-layer networks in unsupervised feature learning. In Proc. AISTATS, 2011.   \nGregory Cohen, Saeed Afshar, Jonathan Tapson, and André van Schaik. EMNIST: an extension of MNIST to handwritten letters. arXiv preprint arXiv:1702.05373, 2017.   \nPrafulla Dhariwal, Christopher Hesse, Matthias Plappert, Alec Radford, John Schulman, Szymon Sidor, and Yuhuai Wu. Openai baselines, 2017.   \nChrisantha Fernando, Dylan Banarse, Charles Blundell, Yori Zwols, David Ha, Andrei A. Rusu, Alexander Pritzel, and Daan Wierstra. Pathnet: Evolution channels gradient descent in super neural networks. arXiv preprint arXiv:1701.08734, 2017.   \nTommaso Furlanello, Jiaping Zhao, Andrew M. Saxe, Laurent Itti, and Bosco S. Tjan. Active long term memory networks. arXiv preprint arXiv:1606.02355, 2016.   \nKaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proc. CVPR, 2016.   \nGeoffrey E. Hinton, Oriol Vinyals, and Jeffrey Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.   \nGao Huang, Zhuang Liu, and Kilian Q. Weinberger. Densely connected convolutional networks. In Proc. CVPR, 2017.   \nHeechul Jung, Jeongwoo Ju, Minju Jung, and Junmo Kim. Less-forgetting learning in deep neural networks. arxiv, 2016.   \nAlex Krizhevsky. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009.   \nZhizhong Li and Derek Hoiem. Learning without forgetting. In Proc. ECCV, 2016.   \nMingsheng Long, Yue Cao, Jianmin Wang, and Michael I. Jordan. Learning transferable features with deep adaptation networks. In Proc. ICML, 2015.   \nT. Mitchell, W. Cohen, E. Hruscha, P. Talukdar, J. Betteridge, A. Carlson, B. Dalvi, M. Gardner, B. Kisiel, J. Krishnamurthy, N. Lao, K. Mazaitis, T. Mohammad, N. Nakashole, E. Platanios, A. Ritter, M. Samadi, B. Settles, R. Wang, D. Wijaya, A. Gupta, X. Chen, A. Saparov, M. Greaves, and J. Welling. Never-ending learning. In Proc. AAAI, 2015.   \nVolodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A. Rusu, Joel Veness, Marc G. Bellemare, Alex Graves, Martin Riedmiller, Andreas K. Fidjeland, Georg Ostrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen King, Dharshan Kumaran, Daan Wierstra, Shane Legg, and Demis Hassabis. Human-level control through deep reinforcement learning. In Nature, 2015.   \nVolodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy P. Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In Proc. ICML, 2016.   \nYuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y Ng. Reading digits in natural images with unsupervised feature learning. 2011. ",
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+            174,
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+    },
+    {
+        "type": "table",
+        "img_path": "images/fcb055a052c4e38d48826ebad4896d96794d0811a470124df68b99cf149c2aaf.jpg",
+        "table_caption": [
+            "Table 4: Test error $( \\% )$ of distilled student net. "
+        ],
+        "table_footnote": [
+            "Table 5: Our approach on the EMNIST Letters dataset. "
+        ],
+        "table_body": "<table><tr><td></td><td>MNIST</td><td>MNIST w/o digit ‘3&#x27;</td><td>C100</td><td>Imagenet</td></tr><tr><td>Student alone</td><td>1.46</td><td>11.06</td><td>31.87</td><td>30.24</td></tr><tr><td>KD Hinton et al. (2015)</td><td>0.74</td><td>2.06</td><td>30.28</td><td>30.04</td></tr><tr><td>Ours</td><td>0.73</td><td>1.05</td><td>30.07</td><td>29.05</td></tr></table>",
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+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/6be278ca4d4aa882e9882f0ce97028b95c66ba035e4163556e7eddef30cb77d6.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model (Teacher)</td><td>Test error(%)</td></tr><tr><td>Cohen et al. (2017)</td><td>14.85</td></tr><tr><td>Fine-tune from EMNIST digits</td><td>9.04</td></tr><tr><td>Baseline</td><td>9.20</td></tr><tr><td>Ours (EMNIST letters)</td><td>7.13</td></tr><tr><td>Ours (EMNIST half letters)</td><td>8.13</td></tr><tr><td>Ours (EMNIST digit)</td><td>8.11</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "7 APPENDIX ",
+        "text_level": 1,
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "7.1 SUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Comparison with Knowledge Distillation: We follow knowledge Distillation (KD) (Hinton et al., 2015) to distill knowledge from a larger model (teacher) to a smaller model (student). The student models have $5 0 \\% - 5 \\%$ parameters of the teacher models. Following their setup, we conduct experiments on MNIST, MNIST with digit $\\cdot _ { 3 } \\cdot$ missing in the training set, CIFAR-100, and ImageNet. For MNIST and MNIST with digit $\\cdot _ { 3 } ,$ missing, following KD, the teacher model is an MLP with two hidden layers of 1200 hidden units, and the student model is an MLP with two hidden layers of 800 hidden units. For CIFAR-100, we use the model from Chen (2017) as teacher model. The student model follows the structure of the teacher, but the number of output channels of each convolutional layer is halved. For ImageNet, the teacher model is a 50-layer ResNet (He et al., 2016), and the student model is a 18-layer ResNet. The test error of the distilled student model are summarize in Table 4. Our framework has consistently better performance than KD, because the student model in our framework benefits not only from the output layer behavior of the teacher but also from intermediate layer representations of the teacher. ",
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+    {
+        "type": "text",
+        "text": "EMNIST: ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "The ‘EMNIST Letters’ dataset consists of images of size $2 8 \\times 2 8$ pixels showing handwritten letters. It has 26 balanced classes. Each class contains lower and upper case letters. The training and test sets contain 124,800 and 20,800 images respectively. The ‘EMNIST Digits’ dataset consists of images of size $2 8 \\times 2 8$ pixels showing handwritten digits. It has 10 balanced classes. The training and test sets contain 240,000 and 40,000 images respectively. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this case we use the MNIST model from Chen (2017) as a baseline, teacher and student model. We trained teachers on EMNIST Digits, EMNIST Letters, and EMNIST Letters with only 13 classes. Our target task is EMNIST Letters. The student model is trained with different teachers and the results are compared to fine-tuning, the baseline model, and the state-of-the-art results on EMNIST. The results are summarized in Table 5. Compared to the baseline and fine-tuning, student learning in our framework with expert teacher (EMNIST Letters), semi-expert teacher (Half EMNIST Letters), and non-expert teacher (EMNIST Digits) all have better performance. In Fig. 4 we illustrate the accuracy over epochs for training of different models. ",
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+    },
+    {
+        "type": "text",
+        "text": "STL-10: ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The STL-10 dataset consist of colored images of size $9 6 \\times 9 6$ pixels. It has 10 balanced classes. The training set contains 5,000 labeled images and 100,000 unlabeled images. The test set contains 8,000 images. In our experiment, we only use the 5,000 labeled images for training. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We use the STL-10 model from Chen (2017) as our baseline, teacher and student model. We trained teachers on CIFAR-10 and CIFAR-100. We compare our results to fine-tuning and the baseline in Table 6. Note that STL-10 is very similar to CIFAR-10 and CIFAR-100. Therefore, both CIFAR-10 and CIFAR-100 are very good teachers. As shown in Table 6, compared to the baseline, fine-tuning a model using weights pretrained on CIFAR-10 and CIFAR-100 reduce test errors by more than $1 0 \\%$ . Compared with fine-tuning, student model training in our framework further reduces the test error by $3 \\%$ . Note that we only train on the labeled data while other approaches use this data for testing of semi-supervised approaches. Hence our results are obtained using fewer data and may not be directly comparable. We still list their results in Table 6 for reference. In Fig. 5 we illustrate the accuracy over the epochs of training. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/fba38bd486366a25d5fa05fad89dd10a53d2fd19ce8f3f5288b35b8a48b60bde.jpg",
+        "image_caption": [
+            "Figure 4: Comparison of top-1 accuracy of our approach, fine-tuning and baseline on the EMNIST Letters test dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            346,
+            74,
+            630,
+            247
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/583f510b8a2fd04d61339ec76d77bfbc098cb7c76840bd91b98a1047e0728984.jpg",
+        "table_caption": [
+            "Table 6: Our approach on the STL-10 dataset (fully supervised). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Test error (%)</td></tr><tr><td>Zhao et al. (2015) Thoma (2017)</td><td>25.20 21.34</td></tr><tr><td>Baseline Fine-tune from C10 Fine-tune from C100</td><td>25.50 14.32</td></tr></table>",
+        "bbox": [
+            361,
+            324,
+            633,
+            460
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            488,
+            825,
+            571
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "7.2 REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            604,
+            415,
+            618
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We also compare to Distral (Teh et al., 2017), which is the state-of-the-art multi-task reinforcement learning framework. We used $\\mathrm { \\mathrm { ' K L } } + \\mathrm { e n t } \\ 1 \\ \\mathrm { c o l } ^ { \\mathrm { ? } }$ , which has a central model $( m _ { 0 } )$ , and a task model $( m _ { i } )$ for each task. We perform the experiments on Atari games. In the experiments, we have three tasks (task 1, task 2, task 3). The teachers of task 2 $\\left( m _ { 2 } \\right)$ and task 3 $( m _ { 3 } )$ are provided for our framework. Distral is trained for 120M steps (40M steps/task), and our model is trained for $4 0 \\mathbf { M }$ steps. For fair comparison, we report results of Distral’s task 1 model $( m _ { 1 } )$ , which is better than its center model $( m _ { 0 } )$ . The results are summarized in Table 7. Distral is suboptimal, because it aims to learn a multi-task agent. In addition, identical action and state space is assumed. When the target task is very different from the source tasks, Distral cannot decrease the teacher influence. In contrast, our framework can decrease a teacher’s influence, and thus reduce negative transfer. ",
+        "bbox": [
+            173,
+            636,
+            825,
+            775
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "7.3 VISUALIZATION OF NORMALIZED WEIGHTS OF TEACHERS AND STUDENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            808,
+            725,
+            821
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Following the reviewer’s suggestion, we plot the averaged normalized weight $( p _ { w } )$ for teachers and the student in the C10 experiment, where C100 and SVHN experts are teachers. Intuitively, the C100 teacher should have a higher $p _ { w }$ value than the SVHN teacher, because C100 is more relevant to C10. The plot verifies this intuition. As shown in Fig. 7, $p _ { w }$ of the C100 teacher is higher than that of the SVHN teacher over the entire training. Note, both teachers’ normalized weights approach zero at the end of training. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/20d400a30725b1ed0c1cb4dbb19eb137877a094e43446bf2d42dbd30b584dccd.jpg",
+        "image_caption": [
+            "Figure 5: Comparison of top-1 accuracy of our approach, fine-tuning and baseline on the STL-10 test dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            351,
+            82,
+            630,
+            253
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/a8440e339b32c16866c8a47fe22c32189da764eeb11a61f370f6eabe02666491.jpg",
+        "image_caption": [
+            "Figure 6: Comparison with fine-tuning and baseline A3C on different combinations of environment/teacher settings. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            101,
+            315,
+            910,
+            662
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "7.4 ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            733,
+            352,
+            747
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "7.4.1 UNTRAINED TEACHER MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            766,
+            449,
+            781
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To verify that the student really benefits from the knowledge of teachers, we conduct an ablation study suggested by a reviewer. We use teacher models that haven’t been trained at all. Intuitively, learning with untrained teachers should have worse performance than learning with knowledgeable teachers. Our experiments verify this intuition. In Fig. 8 (a), where the target task is hero, learning with untrained teachers (‘w/ untrained teachers’) achieves an average reward of 15934. Learning with knowledgeable teachers (‘Ours with seaquest and riverraid teacher’) achieves an average reward of 30928. More results are presented in Figs. 8 (b, c). The results show that knowledge flow achieves higher rewards than training with untrained teachers in different environments and teacher-student settings. ",
+        "bbox": [
+            173,
+            797,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/69fdf340b392de227ebffafe928f0045e355a07ba6303e9785a7238d767bc520.jpg",
+        "table_caption": [
+            "Table 7: Comparison with Distral on Task 1 score. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task1,Task2,Task3</td><td>Distral Teh et al. (2017)</td><td>Ours</td></tr><tr><td>KungFuMaster, Hero, Seaquest</td><td>27433</td><td>35103</td></tr><tr><td>Hero, Seaquest, Riverraid</td><td>15096</td><td>30928</td></tr><tr><td>James, Seaquest,Riverraid</td><td>550</td><td>1245</td></tr></table>",
+        "bbox": [
+            264,
+            94,
+            728,
+            169
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/cb0990d9b96c9b5636afc93e3de14ccf00f76a022af12d47c435bfd47e2f3fa3.jpg",
+        "image_caption": [
+            "Figure 7: Normalized weights for the teachers and the student in C10 experiments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            349,
+            194,
+            632,
+            366
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "7.4.2 TRAINING WITHOUT KL TERM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            452,
+            446,
+            465
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The KL term prevents the student’s output distribution over actions or labels from drastic changes when the teachers’ influence is decreasing. To investigate the importance of the KL term, we conduct an ablation study where the KL coefficient $\\left( \\lambda _ { 2 } \\right)$ is set to zero. The result is summarized in Fig. 9. Considering Fig. 9 (a), where the target task is MsPacman and the teachers are Riverraid and Seaquest experts. Without the KL term, when a teacher’s influence decreases, the rewards drop drastically. In contrast, with a KL term, we don’t observe performance drops. At the end of training, learning with the KL term achieves an average reward of 2907 and learning without the KL term achieves an average reward of 1215. More results are presented in Fig. 9 (b, c), which shows that training with the KL term achieves higher reward than training without the KL term. ",
+        "bbox": [
+            174,
+            481,
+            825,
+            607
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "7.5 TEACHERS WITH DIFFERENT ARCHITECTURE THAN STUDENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            636,
+            643,
+            650
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In additional experiments, following the suggestion of a reviewer, we use architectures for the teacher which differ from the student model. More specifically, we use the model of Mnih et al. (2015) as a teacher model. The teacher model consists of 3 convolutional layers, which have 32, 64, and 64 filters, followed by a hidden fully connected layer which has 512 ReLUs. We use the model of Mnih et al. (2016) as the student model. The student model consists of 2 convolutional layers, which have 16 and 32 filters respectively, followed by a hidden fully connected layer which has 256 ReLUs. Both models’ fully connected layers are followed by two output layers for actions and values. In the experiments, we link each teacher’s first convolutional layer to the student’s first convolutional layer. Moreover, we link each teacher’s third convolutional layer to the student’s second convolutional layer, and each teacher’s fully connected layer to the student’s fully connected layer. In the experiment, the target task is KungFu Master, and the teachers are experts for Seaquest and Riverraid. The results are summarized in Fig. 10. We observed that learning with teachers, whose architecture differs from the student, to have similar performance as learning with teachers which have the same architecture. Consider as an example Fig. 10 (a), where the target task is KungFu Master, and the teachers are experts for Seaquest and Riverraid. At the end of training, learning with teachers of different architectures achieves an average reward of 37520, and learning with teachers of the same architecture achieves an average reward of 35012. More results are shown in Fig. 10 (b, c). The results show that knowledge flow can enable higher rewards, even if the teachers and the student architectures differ. ",
+        "bbox": [
+            173,
+            662,
+            825,
+            926
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/c26da5f17450e4711fa09727be150c2d6aec6e942c2bc016144babf6a14e3253.jpg",
+        "image_caption": [
+            "Figure 8: Ablation study: using untrained teachers. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            98,
+            107,
+            913,
+            276
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/679b3e0c12470fc53fef19f217bccea704af2688fdd85988f755fce3e6617e96.jpg",
+        "image_caption": [
+            "Figure 9: Ablation study regarding KL term. Seaquest and Riverraid experts are used as teachers for all experiments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            102,
+            318,
+            911,
+            488
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "7.6 AVERAGE NETWORK AS $\\theta _ { o l d }$ ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            558,
+            411,
+            571
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For the parameters $\\theta _ { \\mathrm { o l d } }$ an average network can be used. To investigate how usage of an average network to obtain the parameters $\\theta _ { \\mathrm { o l d } }$ affects the performance, we conduct an experiment where $\\theta _ { \\mathrm { o l d } }$ is computed using the exponential running average of the model weight. More specifically, $\\theta _ { \\mathrm { o l d } }$ is updated as follows: $\\theta _ { \\mathrm { o l d } }  \\alpha \\cdot \\theta _ { \\mathrm { o l d } } + ( 1 - \\alpha ) \\cdot \\theta$ , where $\\alpha = 0 . 9$ . The results are summarized in Fig. 11. We observe that using an exponential average to compute $\\theta _ { \\mathrm { o l d } }$ results in very similar performance as using a single model. Consider Fig. 11 (a), where the target task is Boxing and the teacher is a Riverraid expert. At the end of training, using an average network to obtain $\\theta _ { \\mathrm { o l d } }$ achieves an average reward of 96.2 and using a single network to obtain $\\theta _ { \\mathrm { o l d } }$ achieves an average reward of 96.0. More results on using an average network are shown in Fig. 11 (b, c). ",
+        "bbox": [
+            173,
+            580,
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+            705
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "8 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            727,
+            344,
+            743
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "As mentioned before, variants of ‘knowledge’ transfer have been considered using a variety of techniques, for instance, fine-tuning, progressive neural nets (Rusu et al., 2016b), PathNet (Fernando et al., 2017), ‘Growing a Brain’ (Wang et al., 2017), actor-mimic (Parisotto et al., 2016), learning without forgetting (Li & Hoiem, 2016). Also related are techniques on transfer learning and lifelong learning. We discuss those methods and contrast them to our approach in the following. ",
+        "bbox": [
+            174,
+            750,
+            825,
+            819
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "PathNet (Fernando et al., 2017) enables multiple agents to train the same giant deep net while reusing parameters and avoiding catastrophic forgetting. To this end, agents embedded in the neural net discover which weights can be reused for new tasks and restrict application of gradients to those parameters. In contrast to this formulation we consider availability of multiple teacher nets, which are trained. ",
+        "bbox": [
+            174,
+            821,
+            825,
+            892
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Progressive Net (Rusu et al., 2016b) leverages transfer and avoids catastrophic forgetting by introducing lateral connections to previously learned features. Our discussed method uses similar lateral connections. However, in contrast to Rusu et al. (2016b), we introduce scaling with normalized weights. This ensures independence of the student upon training, addressing a limitation in (Rusu et al., 2016b) where only a fraction of the capacity of the student is eventually utilized. ",
+        "bbox": [
+            173,
+            895,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/52be82be150f4657c771b06c9198d4aef49934be076ba5b84bd90c6094887f41.jpg",
+        "image_caption": [
+            "Figure 10: Teachers’ architecture differs from the student’s architecture. Seaquest and Riverraid experts are used as teachers for all experiments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            98,
+            99,
+            915,
+            270
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/b1a149c7edaf6b1bdd230bb053fae6d38bdd114f2b767966162b08d69f459959.jpg",
+        "image_caption": [
+            "Figure 11: Average network to compute $\\theta _ { o l d }$ . Riverraid expert is used as teacher for all experiments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            106,
+            333,
+            913,
+            505
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            547,
+            825,
+            589
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Distral a neologism combining ‘distill & transfer learning’ (Teh et al., 2017) considers joint training of multiple tasks. Multiple tasks share a ‘distilled’ policy which encodes common behavior between different tasks. While each worker addresses its own task, a shared policy encourages consistency between the policies. Different from Distral, which is a multi-task learning framework, knowledge flow addresses a single task, while in multi-task learning, multiple tasks are addressed at the same time. Hence, common for multi-task learning and knowledge flow is a transfer of information. However, in multi-task learning, information extracted from different tasks are shared to boost performance, while, in knowledge flow, the information of multiple teachers is leveraged to help a student learn better a single, new, previously unseen task. ",
+        "bbox": [
+            173,
+            592,
+            825,
+            718
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Knowledge distillation (Hinton et al., 2015) distills information form a larger deep net into a smaller one. It assumes both nets are trained on the same dataset. In contrast, our technique allows knowledge transfer between different source and target domains. ",
+        "bbox": [
+            174,
+            722,
+            825,
+            763
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Actor-mimic (Parisotto et al., 2016) enables an agent to learn how to address multiple tasks simultaneously and generalize the extracted knowledge to new domains. A single policy net learns how to act in a set of tasks following the guidance of several expert teachers. A combination of feature regression and cross entropy loss is used to encourage the student to produce similar actions and representations. Our proposed technique differs in that we take advantage of a teachers representation at the beginning of training, ",
+        "bbox": [
+            173,
+            767,
+            825,
+            851
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Learning without forgetting (Li & Hoiem, 2016) permits to add a new task to a deep net without forgetting the original capabilities. Importantly, only data from the new task is used and the old capabilities are retained by first recording the old networks output on the new data. Similar techniques have been developed by Furlanello et al. (2016); Jung et al. (2016). In contrast, we transfer ‘knowledge’ from teacher networks more explicitly. ",
+        "bbox": [
+            174,
+            854,
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+            924
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Growing a Brain (Wang et al., 2017) analyzes the parameters which change during fine-tuning and points out that more natural model adaptation is obtained when increasing the model capacity, by either extending width or depth. Appropriate normalization is essential to significantly outperform classical fine-tuning. Since this technique is based on fine-tuning, it differs from our student-teacher based approach. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            172
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Other related work includes policy distillation (Rusu et al., 2016a), domain adaptation (Pan & Yang, 2010; Long et al., 2015; Tzeng et al., 2015) or lifelong learning (Chen & Liu, 2016; Thrun, 1998; Mitchell et al., 2015; Ruvolo & Eaton, 2013). ",
+        "bbox": [
+            174,
+            180,
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+            222
+        ],
+        "page_idx": 16
+    }
+]

parse/train/SJgn3lBtwH/SJgn3lBtwH.md

@@ -0,0 +1,392 @@
+# RE-EXAMINING LINEAR EMBEDDINGS FOR HIGH-DIMENSIONAL BAYESIAN OPTIMIZATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Bayesian optimization (BO) is a popular approach to optimize expensive-toevaluate black-box functions. A significant challenge in BO is to scale to highdimensional parameter spaces while retaining sample efficiency. A solution considered in previous literature is to embed the high-dimensional parameter space into a lower-dimensional manifold, often a random linear embedding. In this paper, we identify several crucial issues and misconceptions about the use of linear embeddings for BO. We thoroughly study and analyze the consequences of using linear embeddings and show that some of the design choices in current approaches adversely impact their performance. Based on this new theoretical understanding we propose ALEBO, a new algorithm for high-dimensional BO via linear embeddings that outperforms state-of-the-art methods on a range of problems, including learning a gait policy for robot locomotion.
+
+# 1 INTRODUCTION
+
+Bayesian optimization (BO) is a robust, sample-efficient technique for optimizing expensive-toevaluate black-box functions (Mockus, 1989; Jones, 2001). BO has been successfully applied to diverse applications, ranging from automated machine learning (Snoek et al., 2012; Hutter et al., 2011) to robotics (Lizotte et al., 2007; Calandra et al., 2015; Rai et al., 2018). One of the most active topics of research in BO is how to extend current methods to higher-dimensional spaces. A common framework to tackle this problem is to consider a high-dimensional BO (HDBO) task as a standard BO problem in a low-dimensional embedding, where the embedding can be either linear (typically a random projection) or nonlinear (e.g. via a multi-layer neural network); see Sec. 2 for a full review. An advantage of this framework is to explicitly decouple the problem of finding low-dimensional representations suitable for optimization from the actual optimization technique.
+
+In this paper we study the use of linear embeddings for HDBO, and in particular we re-examine prior efforts to use random linear projections. Random projections are attractive for BO because, by the Johnson-Lindenstrauss lemma, they can be approximately distance-preserving (Johnson & Lindenstrauss, 1984) without requiring any data to learn the embedding. Random embeddings come with several strong theoretical guarantees, but have shown mixed empirical performance for HDBO.
+
+The contributions of this paper are: 1) We provide new results that identify why linear embeddings have performed poorly in HDBO. We show that existing approaches produce representations that cannot be well-modeled by a Gaussian process (GP), or representations that likely do not contain an optimum (Sec. 4). 2) We construct a representation with better properties for BO (Sec. 5): we improve modelability by deriving a Mahalanobis kernel tailored for linear embeddings and adding polytope bounds to the embedding, and we show how to maintain a high probability that the embedding contains an optimum. 3) We show that using this representation for BO outperforms a wide range of previous approaches for HDBO, including on test functions up to $D = 1 0 0 0$ , and on realworld problems, such as gait optimization of a multi-legged robot (Sec. 6). These include the first results for HDBO with black-box constraints.
+
+# 2 RELATED WORK
+
+There are generally two approaches to extending BO into high dimensions. The first is to produce a low-dimensional embedding, do standard BO in this low-dimensional space, and then project up to the original space for function evaluations. The foundational work on embeddings for BO is REMBO (Wang et al., 2016), which creates a linear embedding by generating a random projection matrix. Sec. 3 provides a thorough description of REMBO and several subsequent approaches based on random linear embeddings (Qian et al., 2016; Binois et al., 2019; Nayebi et al., 2019). If derivatives of $f$ are available, the active subspace method can be used to recover a linear embedding (Constantine et al., 2014; Eriksson et al., 2018), or approximate gradients can be used (Djolonga et al., 2013). BO can also be done in nonlinear embeddings through VAEs (Gomez-Bombarelli et al., ´ 2018; Lu et al., 2018; Moriconi et al., 2019). An attractive aspect of random embeddings is that they can be extremely sample-efficient, since the only model to be estimated is a low-dimensional GP.
+
+The second approach to extend BO to high dimensions is to make use of surrogate models that better handle high dimensions, typically by imposing additional structure on the problem. Work along these lines include GPs with an additive kernel (Kandasamy et al., 2015; Wang et al., 2017; Gardner et al., 2017; Wang et al., 2018; Rolland et al., 2018; Mutny & Krause, 2018), cylindrical kernels (Oh ´ et al., 2018), or deep neural network kernels (Antonova et al., 2017). Random forest is used as the surrogate model in SMAC (Hutter et al., 2011). These methods produce trade-offs between sample efficiency of the model and the ability to effectively optimize the acquisition function.
+
+Here, we focus on the embedding approach and in particular the use of linear embeddings for HDBO. Without box bounds, REMBO comes with a strong guarantee: with probability 1, the embedding contains an optimum (Wang et al., 2016, Thm. 2). However, if function evaluations are limited to the box bounds, as is typical in BO problems, REMBO requires a collection of heuristics for which there are no longer guarantees on performance. While REMBO can perform well in some HDBO tasks, subsequent papers have found it can perform poorly even on tasks with a true low-dimensional linear subspace (e.g. Nayebi et al., 2019). In this paper, we analyze the properties of linear embeddings as they relate to BO, and show how to improve the representation of the function we seek to optimize.
+
+# 3 PROBLEM FRAMEWORK AND REMBO
+
+In this section we define the problem framework and notation, and then describe BO via random linear projections (REMBO)—a promising method for HDBO—along with known challenges and follow-up work that has been proposed to address these issues.
+
+Bayesian optimization We consider optimization problems of the form $\scriptstyle \operatorname* { m i n } _ { { \pmb x } \in B } f ( { \pmb x } )$ where $f$ is a black-box function and $\boldsymbol { B }$ are box bounds. We assume gradients of $f$ are unavailable. The box bounds on $_ { \textbf { \em x } }$ specify the range of values that are reasonable or physically possible to evaluate. For instance, Gramacy et al. (2016) use BO for an environmental remediation problem in which each $x _ { i }$ represents the pumping rate of a particular pump, which has physical limitations. The problem may also include nonlinear constraints $c _ { j } ( { \pmb x } ) \leq 0$ where each $c _ { j }$ is itself a black-box function. BO is a form of sequential model-based optimization, where we construct a surrogate model for $f$ and use that model to identify which parameters $_ { \textbf { \em x } }$ should be evaluated next, according to an explore-exploit strategy. The surrogate model is typically a GP, $f \sim \mathcal { G P } ( m ( \cdot ) , k ( \cdot , \cdot ) )$ , with mean function $m ( \cdot )$ and a kernel $k ( \cdot , \cdot )$ . Under the GP prior, the posterior for the value of $f ( { \pmb x } )$ at any point in the space is a normal distribution with closed-form mean and variance. Using that posterior, we construct an acquisition function $\alpha ( { \pmb x } )$ that specifies the value of a function evaluation at $_ { \textbf { \em x } }$ , such as Expected Improvement (EI) (Jones et al., 1998). We find $\pmb { x } ^ { * } \in \arg \operatorname* { m a x } _ { \pmb { x } \in B } \alpha ( \pmb { x } )$ , and evaluate $f ( { \pmb x } ^ { * } )$ .
+
+The GP is useful for BO because it provides a well-calibrated posterior in closed form. With typical kernels and acquisition functions, $\alpha ( { \pmb x } )$ is differentiable and can be effectively optimized. However, with typical kernels like the ARD RBF kernel, there are significant limitations. GPs are known to predict poorly in high dimensions, which for a GP is $D$ larger than 15–20 (Wang et al., 2016; Li et al., 2016; Nayebi et al., 2019). This prevents BO from being a useful tool in high dimensions.
+
+In HDBO, the objective $f : \mathbb { R } ^ { D }  \mathbb { R }$ operates in a high-dimensional $( D )$ space, which we call the ambient space. When using linear embeddings for HDBO, we assume there exists a low-dimensional linear subspace that captures all of the variation of $f$ . Specifically, let $f _ { d } : \mathbb { R } ^ { d }  \mathbb { R }$ , $d \ll D$ , and let $\pmb { T } \in \mathbb { R } ^ { d \times D ^ { 1 } }$ be a projection matrix from $D$ down to $d$ dimensions. The linear embedding assumption is that $f ( \pmb { x } ) = f _ { d } ( \pmb { \bar { T } } \pmb { x } ) ~ \forall \pmb { x } \in \mathbb { R } ^ { D }$ . $_ { \mathbf { T } }$ is unknown, and we only have access to $f$ , not $f _ { d }$ . We assume without loss of generality that the box bounds are $\boldsymbol { B } = [ - 1 , \mathrm { \bar { 1 } } ] ^ { D }$ ; the ambient space can always be scaled to these bounds.
+
+REMBO: Bayesian optimization via random embedding REMBO (Wang et al., 2016) generates a random projection matrix $\pmb { A } \in \mathbb { R } ^ { D \times d _ { e } }$ with each element drawn independently from $\mathcal { N } ( 0 , 1 )$ to specify a $d _ { e }$ -dimensional embedding. BO is done in the embedding to identify a point $\pmb { y } \in \mathbb { R } ^ { d _ { e } }$ to be evaluated, which is given objective value $f ( A y )$ . The embedding dimension $d _ { e }$ should satisfy $d _ { e } \geq d$ for the REMBO guarantee of containing an optimum to hold.
+
+The main challenges for using REMBO come when dealing with box bounds in the ambient space. We may select a point $\textbf {  { y } }$ in the embedding to be evaluated and find that its projection to the ambient space, $\pmb { A } \pmb { y }$ , falls outside $\boldsymbol { B }$ . The first challenge this poses is a theoretical challenge: $\mathbb { R } ^ { d _ { e } }$ is guaranteed to contain an optimum, but that optimum is not guaranteed to project up to $\boldsymbol { B }$ . When function evaluations are restricted to the box bounds, the embedding may not contain an optimum—it is not difficult to construct examples of this. REMBO has no theoretical guarantees in this setting. The second challenge posed by box bounds is the practical challenge of how function evaluations should be done for points that project up outside √ √ $\boldsymbol { B }$ . Here REMBO introduces three heuristics. First, the embedding is given box bounds $[ - \sqrt { d _ { e } } , \sqrt { d _ { e } } ] ^ { d _ { e } }$ . BO will only select points within those bounds to be projected up and evaluated. Second, if a point $\textbf {  { y } }$ in the embedding projects up outside $\boldsymbol { B }$ , then it is clipped to $\boldsymbol { B }$ . Let $p _ { B } : \mathbb { R } ^ { D }  \mathbb { R } ^ { D }$ be the $L ^ { 2 }$ projection that maps $_ { \textbf { \em x } }$ to its nearest point in $\boldsymbol { B }$ . A point $\textbf {  { y } }$ in the embedding is given objective value $f ( p _ { B } ( A y ) )$ , which can always be evaluated. Note that clipping to $\boldsymbol { B }$ renders the projection of $\textbf {  { y } }$ to the ambient space a nonlinear transformation whenever $A y \notin B$ . Third, the optimization is done with $k { = } 4$ separate projections, to improve the chances of√ √ generating an embedding that contains an optimum inside $[ - \sqrt { d _ { e } } , \sqrt { d _ { e } } ] ^ { d _ { e } }$ . Since these embeddings are independent, no data can be shared across them, which reduces sample efficiency.
+
+Extensions of REMBO Binois et al. (2015) consider the issue of non-injectivity, where the $L ^ { 2 }$ projection causes many points in the embedding to map to the same vertex of $\boldsymbol { B }$ . They define a warped kernel that reduces non-injectivity, which is called REMBO- $\phi k _ { \Psi }$ . Binois et al. (2019) consider the issue of setting bounds on the embedding. They define a projection matrix $B \in \mathbb { R } ^ { d \times D }$ that maps from the ambient space down to the embedding, and replace the $L ^ { 2 }$ projection with a projection $\gamma$ that maps $\textbf {  { y } }$ to the closest point in $\boldsymbol { B }$ that satisfies $\mathbf { \delta } _ { B x } = \mathbf { \delta } _ { y }$ . The $\gamma$ projection resolves the core challenge of REMBO related to setting bounds in the embedding: we can restrict the optimization in the embedding to points for which $\exists { \boldsymbol { x } } \in \mathbf { \boldsymbol { B } }$ s.t. $B x = y$ , and so heuristic box bounds in the embedding are no longer required. The $\gamma$ projection projects to the same points on the facets of $\boldsymbol { B }$ as the $L ^ { 2 }$ projection. Paired with the warped kernel of Binois et al. (2015), this is called REMBO- $\gamma k _ { \Psi }$ .
+
+Binois (2015) studies different choices for the projection matrix and shows that BO performance can be improved for small $d$ by sampling each row of $\pmb { A }$ from the unit hypersphere $\mathbb { S } ^ { d _ { e } - 1 }$ . If $\boldsymbol { z } \sim \mathcal { N } ( \mathbf { 0 } , I _ { d _ { e } } )$ , then $\frac { z } { | | z | | }$ is a random sample from $\mathbb { S } ^ { d _ { e } - 1 }$ , so this amounts to normalizing the rows of the usual REMBO projection matrix.
+
+HeSBO (Nayebi et al., 2019) is a recent extension of REMBO that avoids clipping to $\boldsymbol { B }$ and heuristic box bounds in the embedding by changing the projection matrix $\pmb { A }$ . In $d _ { e } = 1$ , it is easy to see that the projection matrix $\mathbf A = \mathbf 1$ , which sets every $x _ { i } = y$ , is optimal. With this projection we can set bounds of $[ - 1 , 1 ]$ on the embedding and there is no need for $L ^ { 2 }$ projections because every point in the embedding will map to a point in $\boldsymbol { B }$ . HeSBO extends this to $d _ { e } > 1$ by setting each row of $\pmb { A }$ to have a single non-zero element, which is randomly set to $\pm 1$ . The column with the non-zero value is chosen uniformly at random. Thus, each parameter in the ambient space is mapped directly to a parameter in the embedding: $x _ { i } = \pm y _ { j }$ , where $j$ is sampled uniformly from $\{ 1 , \ldots , d _ { e } \}$ and $\pm$ is chosen uniformly at random. The embedding is given box bounds of $[ - 1 , 1 ] ^ { d _ { e } }$ .
+
+# 4 CHALLENGES WITH LINEAR EMBEDDINGS
+
+Heuristics for handling box bounds when utilizing linear embeddings introduce several issues that impact HDBO performance. We highlight one recent observation from Binois et al. (2019), that most points in the embedding project up outside the box bounds, and discuss three novel observations about how existing methods can make it difficult to learn high-dimensional surrogates.
+
+Projection to the facets of $\boldsymbol { B }$ produces a nonlinear distortion in the function. The function value at any point in the embedding is measured as $f ( p \boldsymbol { { s } } ( \boldsymbol { { A } } \boldsymbol { { y } } ) )$ . For points $\textbf {  { y } }$ that project up outside of $\boldsymbol { B }$ , this will be a nonlinear mapping from the embedding to the ambient space, despite the use of a linear embedding. This has a powerful, detrimental effect on the ability to model $f$ in the embedding. Fig. 1 provides visualizations of an actual REMBO embedding for two classic test functions: the Branin $( d { = } 2 )$ and Hartmann6 $( d \small { = } 6 )$ functions, both extended to $D { = } 1 0 0$ by adding unused variables. The REMBO embedding for the Branin function contains all three optima, however there is visible distortion to the function caused by the the clipping to $\boldsymbol { B }$ . The embedding for the Hartmann6 function is even more heavily distorted.
+
+![](images/3f2a2ec93f38ddab5d9b1ade93fee60d07eded665058d933b110a0811bcabdb6.jpg)  
+Figure 1: A visualization of REMBO embeddings for two test functions. (Top left) The Branin function, $d { = } 2$ , extended to $D { = } 1 0 0$ . (Top right) A REMBO embedding of the $D { = } 1 0 0$ Branin function. (Bottom left) A center slice of the $d { = } 6$ Hartmann6 function, similarly extended to $D { = } 1 0 0$ . (Bottom right) The same slice of a REMBO embedding of that function. The embedding produces distortions in the function that render it difficult to model.
+
+Even if the function is well-modeled by a GP in the true low-dimensional space, the distortion produced by the REMBO projection transforms it into one on the embedding that is not appropriate for a GP. This can happen for any embedding strategy that cannot guarantee all points in the embedding project into $\boldsymbol { B }$ . The distortion induced by mapping to the facet depends on the relative angles of the facet and the true embedding. Projection to a facet essentially induces a non-stationarity in the kernel: each of the $2 D$ facets sits at different angles to the true subspace, and so the change in the rate of function variance will differ for each. To correct for the non-stationarity, we would have to estimate the true subspace $\mathbf { T }$ , which with $d \times D$ entries is not feasible for $D$ large.
+
+The idea behind using low-dimensional embeddings for HDBO is that it enables the use of standard BO techniques on the embedding. However, from these results we see that for the REMBO projection with box bounds we cannot expect to successfully model the function on the embedding with a regular GP. The problem is especially acute for $d _ { e } > 2$ where, as we will see next, nearly all points in the embedding map to one of the $2 D$ facets.
+
+Most points in the embedding map to the facets of $\boldsymbol { B }$ . Fig. 2 shows the probability that an interior point in the embedding projects up to the interior of √ √ $\boldsymbol { B }$ . This is measured empirically by sampling $\textbf {  { y } }$ uniformly at random from $[ - \dot { \sqrt { d _ { e } } } , \sqrt { d _ { e } } ] ^ { d _ { e } }$ , sampling $\pmb { A }$ with $\mathcal { N } ( 0 , 1 )$ entries, and then checking if $\boldsymbol { A } \boldsymbol { y } \in \mathcal { B }$ (with 1000 samples). Even for small $D$ , with $d _ { e } > 2$ practically all of the volume in the embedding projects up outside the box bounds, and is thus clipped to a facet of $\boldsymbol { B }$ .
+
+This is an issue because it means the optimization will be done primarily on the facets of $\boldsymbol { B }$ and not in the interior, which will likely not even be reached in a typical BO initialization. We saw in Fig. 1 that the function behaves very differently on points projected to the facets, and that these parts of the space can be hard to model with a GP. The problem cannot be resolved by simply shrinking the box bounds in the embedding. Binois et al. (2019) provide an excellent study of the issue of setting bounds in the embedding and show that with the REMBO strategy there is no good way to do this. The pup to $\boldsymbol { B }$ onto the embedding produces a star-shaped object called a zonotope, which has vertices (Ferrez et al., 2005). Shrinking box bounds in the embedding cuts off $2 \textstyle \sum _ { i = 0 } ^ { d - 1 } { \binom { D - 1 } { i } }$ the vertices of the zonotope and increases the chance of not containing an optimum.
+
+Linear projections do not preserve product kernels. Although less visible than that produced by the projection to the facets, there is also distortion to interior points just from the linear projection $\pmb { A }$ . The ARD kernels typically used in GP modeling are product kernels that decompose the covariance into the covariance across each dimension. Inside the embedding, moving along a single dimension will move across all dimensions of the ambient space, at rates depending on the projection matrix. Consider moving along a single dimension in the embedding, from $\mathbf { \mu } _ { \mathbf { \mu } _ { y 1 } }$ to $\mathbf { { \boldsymbol { { y } } } } _ { 2 }$ where only a single element has changed. The corresponding points in the ambient space are $\pmb { x } _ { 1 } = \pmb { A } \pmb { y } _ { 1 }$ and $\mathbf { x } _ { 2 } = A y _ { 2 }$ : even though $\mathbf { \pmb { y } } _ { 1 }$ and $\mathbf { { \boldsymbol { { y } } } } _ { 2 }$ differ in only one element, $\mathbf { \delta x } _ { 1 }$ and $\mathbf { x } _ { 2 }$ will differ in all their elements. Thus a product kernel in the true subspace will not produce a product kernel in the embedding; this is shown mathematically in Proposition 1.
+
+![](images/29d3ae0225171b3586f40e8b024e74d652a6f8321f53ae6b46b902b07c2cedce.jpg)  
+Figure 2: The probability that a randomly selected point in the REMBO embedding satisfies the ambient box bounds after being projected up. For $d _ { e } > 2$ , nearly all points in the embedding map outside the box bounds.
+
+Linear embeddings can have a low probability of containing an optimum. HeSBO avoids the challenges of REMBO related to box bounds: all interior points in the embedding map to interior points of $\boldsymbol { B }$ , and there is no need for the $L ^ { 2 }$ projection and thus the ability to model in the embedding is improved. However, for $d _ { e } > 1$ there is no guarantee that the embedding will contain an optimum, and in fact the probability of containing an optimum can be quite low. Consider the example of an axis-aligned true subspace: $f$ operates only on some set of $d$ elements of $_ { \textbf { \em x } }$ , which we denote $\mathcal { T } = \{ i _ { 1 } , \ldots , i _ { d } \}$ . For $d = 2$ and $d _ { e } \geq 2$ , there are three possible embeddings: $\boldsymbol { x } _ { i _ { 1 } }$ and $x _ { i _ { 2 } }$ map to different features in the embedding, $x _ { i _ { 1 } } = x _ { i _ { 2 } }$ , or $x _ { i _ { 1 } } = - x _ { i _ { 2 } }$ . These three embeddings are visualized in Appendix A.1. In the first case the embedding successfully captures the entire true subspace and we can expect the optimization to be successful. However, in the other two cases the embedding is only able to reach the diagonals of the true subspace, which, unless $f$ happens to have an optimum on the diagonal, will not reach the optimal value. Under a uniform prior on the location of optima, we can compute analytically the probability that the HeSBO embedding contains an optimum (see Appendix A.1). The probability is independent of $D$ , but is low for even moderate values of $d$ . For instance, with $d = 6$ , $d _ { e } = 2 0$ gives only a $44 \%$ chance of recovering an optimum.
+
+Relative to REMBO, HeSBO improves the ability to effectively model and optimize in the embedding, but reduces the likelihood of the embedding containing an optimum. Empirically, this trade-off leads to HeSBO having better BO performance than REMBO. Like HeSBO, here we wish to eliminate the $L ^ { 2 }$ projection and thus improve our ability to model and optimize in the embedding. We will show that this can be done while maintaining a much higher chance of the embedding containing an optimum, which will further improve BO performance.
+
+# 5 LEARNING AND OPTIMIZING IN LINEAR EMBEDDINGS
+
+We now show how to overcome the embedding issues described in Sec. 4. Similarly to Binois et al. (2019), we define the embedding via a matrix $\boldsymbol { B } \in \mathbb { R } ^ { d _ { e } \times D }$ that projects from the ambient space down to the embedding, and $f _ { B } ( \mathbf { \bar { y } } ) = f ( B ^ { \dagger } y )$ as the function evaluated on the embedding, where $B ^ { \dagger }$ denotes the matrix pseudo-inverse. The new techniques we develop here are applicable to any linear embedding, not just random embeddings.
+
+# 5.1 A KERNEL FOR LEARNING IN A LINEAR EMBEDDING
+
+As discussed in Sec. 4, a product kernel over dimensions of the true subspace (ARD) does not translate to a product kernel over dimensions in the embedding. However, stationarity in the true subspace does imply stationarity in the embedding, and this result gives the appropriate kernel structure.
+
+Proposition 1. Suppose the function on the true subspace is drawn from a $G P$ with an ARD RBF kernel: $f _ { d } \sim \mathcal { G P } ( m ( \cdot ) , k _ { R B F } ( \cdot , \cdot ) )$ . For any pair of points in the embedding $\textbf {  { y } }$ and $\mathbf { \Delta } _ { \mathbf { \boldsymbol { y } } ^ { \prime } }$ ,
+
+$$
+C o \nu [ f _ { B } ( \pmb { y } ) , f _ { B } ( \pmb { y } ^ { \prime } ) ] = \sigma ^ { 2 } \exp \left( - ( \pmb { y } - \pmb { y } ^ { \prime } ) ^ { \top } \pmb { \Gamma } ( \pmb { y } - \pmb { y } ^ { \prime } ) \right) ,
+$$
+
+where $\sigma ^ { 2 }$ is the kernel variance of $f _ { d } ,$ , and $\mathbf { T } \in \mathbb { R } ^ { d _ { e } \times d _ { e } }$ is symmetric and positive definite.
+
+Proof. To determine the covariance in function values of points in the embedding, we first project up to the ambient space and then project down to the true subspace
+
+$$
+f _ { B } ( \pmb { y } ) = f ( B ^ { \dagger } \pmb { y } ) = f _ { d } ( \pmb { T } B ^ { \dagger } \pmb { y } ) .
+$$
+
+Then,
+
+$$
+\begin{array} { r l } & { \mathrm { C o v } [ f _ { B } ( \pmb { y } ) , f _ { B } ( \pmb { y } ^ { \prime } ) ] = \mathrm { C o v } [ f _ { d } ( \pmb { T } \pmb { B } ^ { \dag } \pmb { y } ) , f _ { d } ( \pmb { T } \pmb { B } ^ { \dag } \pmb { y } ) ] } \\ & { \quad \quad \quad \quad = \sigma ^ { 2 } \exp \left( - ( \pmb { T } \pmb { B } ^ { \dag } \pmb { y } - \pmb { T } \pmb { B } ^ { \dag } \pmb { y } ^ { \prime } ) ^ { \top } \pmb { D } ( \pmb { T } \pmb { B } ^ { \dag } \pmb { y } - \pmb { T } \pmb { B } ^ { \dag } \pmb { y } ^ { \prime } ) \right) , } \end{array}
+$$
+
+where $\begin{array} { r } { D = \mathrm { d i a g } \left( \left[ \frac { 1 } { 2 \ell _ { 1 } ^ { 2 } } , \dots , \frac { 1 } { 2 \ell _ { d } ^ { 2 } } \right] \right) } \end{array}$ . Let $\mathbf { \Gamma } \mathbf { \Gamma } = ( T B ^ { \dagger } ) ^ { \top } D ( T B ^ { \dagger } )$ . Because $_ D$ is positive definite, it follows that $\mathbf { \delta T }$ is symmetric and positive definite.
+
+This kernel replaces the ARD Euclidean distance with a Mahalanobis distance, and so we refer to it as the Mahalanobis kernel. Similar kernels have been used for GP regression in other settings (Vivarelli & Williams, 1999; Snelson $\&$ Ghahramani, 2006). This result shows that the impact of the linear projection on the kernel can be correctly handled by fitting a $\frac { d _ { e } ( d _ { e } + 1 ) } { 2 }$ -parameter distance metric rather than the typical $d _ { e }$ -parameter ARD metric. The use of this kernel is vital for obtaining good model fits in the embedding. Appendix A.2 shows GP predictive performance on a linear embedding of the Hartmann6 function, in which an ARD RBF kernel entirely fails to predict, while the Mahalanobis kernel does not. We handle uncertainty in $\mathbf { \delta T }$ by posterior sampling from a Laplace approximation of its posterior; this is described in the appendix.
+
+# 5.2 AVOIDING NONLINEAR PROJECTIONS
+
+The most significant distortions seen in Fig. 1 result from clipping projected points to $\boldsymbol { B }$ . We can avoid this by constraining the optimization in the embedding to points that do not project up outside the bounds, that is, $B ^ { \dagger } y \in B$ . Let $\alpha ( \pmb { y } )$ be the acquisition function evaluated in the embedding that we wish to optimize. We select the next point to evaluate by solving
+
+$$
+\operatorname* { m a x } _ { \pmb { y } \in \mathbb { R } ^ { d _ { e } } } \alpha ( \pmb { y } ) \mathrm { s u b j e c t t o } - \mathbf { 1 } \leq \pmb { B } ^ { \dag } \pmb { y } \leq \mathbf { 1 } .
+$$
+
+Note that there are no box bounds on the embedding. The constraints $- \mathbf { 1 } \le B ^ { \dagger } y \le \mathbf { 1 }$ form a polytope, which is convex and can be efficiently optimized over with off-the-shelf optimization tools. Appendix A.3 provides visualizations of the embedding subject to these constraints. Within this space, the projection is entirely linear and can be effectively modeled with the GP described in Sec. 5.1.
+
+# 5.3 THE PROBABILITY THE EMBEDDING CONTAINS AN OPTIMUM
+
+Restricting the embedding with the constraints in (1) eliminates distortions from clipping to $\boldsymbol { B }$ , but it also reduces the volume of the ambient space that can be reached from the embedding and thus reduces the probability that the embedding contains an optimum. To understand the performance of BO in the linear embedding, it is critical to understand this probability, which we denote $P _ { \mathrm { o p t } }$ . Recall that even with clipping, the REMBO theoretical result does not hold when function evaluations are restricted to box bounds, and so even REMBO will generally have $P _ { \mathrm { o p t } } < 1$ .
+
+![](images/96749c96afb60a53f5b7ea6fe83f250f2b6e49176b535e51e200970b94ae49dd.jpg)  
+Figure 3: Probability the embedding contains an optimum $( P _ { \mathrm { o p t } } )$ when restricted to the constraints of (1), under a uniform prior for the location of the optima and $D = 1 0 0$ , for three embedding strategies. Setting $d _ { e } > d$ rapidly increases $P _ { \mathrm { o p t } }$ , and high probabilities can achieved with reasonable values of $d _ { e }$ . Hypersphere sampling produces the best embedding, particularly for $d$ small.
+
+$P _ { \mathrm { o p t } }$ depends on where the optima are in the ambient space—for instance, an optimum at 0 will always be contained in the embedding. Suppose the true subspace has an optimum at $z ^ { * }$ . Then, $\mathcal { O } ( T , z ^ { * } ) = \{ { \pmb x } : T { \pmb x } = z ^ { * } \}$ defines the set of optima in the ambient space. We wish to determine if any of these optima can be reached from the embedding. The points $_ { \textbf { \em x } }$ that can be reached from the embedding are those for which there exists a $\textbf {  { y } }$ in the embedding that projects up to $_ { \textbf { \em x } }$ , that is, $B ^ { \dagger } y =$ $_ { \textbf { \em x } }$ . Since the embedding itself is produced from the projection ${ \mathbf { } } _ { { \mathbf { } } } { \mathbf { } } _ { { \mathbf { } } }$ , $\bar { \mathcal { E } ( B ) } \overset { v } { = } \left\{ \pmb { x } : B ^ { \dagger } B \pmb { x } = \pmb { x } \right\}$ defines the set of points in ambient space that can be reached from the embedding. The embedding contains an optimum if and only if the intersection $\mathcal { O } ( T , z ^ { * } ) \cap \mathcal { E } ( B ) \cap B$ is non-empty. Given a prior for the locations of optima (that is, over $\mathbf { T }$ and $z ^ { * }$ ), we can compute $P _ { \mathrm { o p t } }$ as
+
+$$
+P _ { \mathrm { o p t } } = \mathbb { E } _ { B , T , z ^ { * } } \left[ \mathbf { 1 } _ { \mathcal { O } ( T , z ^ { * } ) \cap \mathcal { E } ( B ) \cap \mathcal { B } \neq \mathcal { O } } \right] .
+$$
+
+Importantly, ${ \mathcal { O } } ( T , z ^ { * } )$ , $\mathcal { E } ( B )$ , and $\boldsymbol { B }$ are all polyhedra, so their intersection can be tested by solving a linear program (see Appendix A.4). The expectation can be estimated with Monte Carlo sampling from the prior over $\mathbf { T }$ and $z ^ { * }$ and from the chosen generating distribution of $\textbf {  { B } }$ .
+
+For our analysis here, we give $_ { \mathbf { T } }$ a uniform prior over axis-aligned subspaces as described in Sec. 4, and we give $z ^ { * }$ a uniform prior in that subspace. Under these uniform priors, we can evaluate (2) to compute $P _ { \mathrm { o p t } }$ as a function of $B , D , d ,$ , and $d _ { e }$ . Fig. 3 shows these probabilities for $D = 1 0 0$ as a function of $d$ and $d _ { e }$ , for three strategies for generating the projection matrix: the REMBO strategy of $\mathcal { N } ( 0 , 1 )$ , the HeSBO projection matrix, and the unit hypersphere sampling described in Sec. 4. Increasing $d _ { e }$ above $d$ rapidly improves the probability of containing an optimum. For $d = 6$ , with $d _ { e } = 6$ the probability is nearly 0, while increasing $d _ { e }$ to 12 is sufficient to raise it to 0.5 and with $d _ { e } = 2 0$ it is nearly 1. Across all values of $d$ and $d _ { e }$ , hypersphere sampling produces the embedding with the best chance of containing an optimum. Appendix A.4 shows $P _ { \mathrm { o p t } }$ for more values of $D$ and $d$ . By using hypersphere sampling and selecting $d _ { e } > d$ , we can maintain a high $P _ { \mathrm { o p t } }$ while still avoiding clipping to $\boldsymbol { B }$ .
+
+# 5.4 A NEW METHOD FOR BO WITH LINEAR EMBEDDINGS: ALEBO
+
+We combine the results and insight gained into a new method for HDBO, which we call adaptive linear embedding BO (ALEBO), since the kernel metric and embedding bounds are adapted with the choice of $\textbf {  { B } }$ . The approach is given in algorithm form in Algorithm 1. Code is available at github.com/anonymized-for-review. In Line 1 the embedding is specified by generating a random projection matrix. We use hypersphere sampling, which gave the best $P _ { \mathrm { o p t } }$ in Fig. 3 among strategies tried here, but this could be replaced with a different projection strategy should one be more appropriate for a particular setting.
+
+# 6 BENCHMARK EXPERIMENTS
+
+We evaluate the performance of ALEBO on synthetic HDBO tasks, and compare its performance to a broad selection of HDBO methods. We include in these benchmarks: REMBO and HeSBO;
+
+Data: D, de, ninit, nBO. Result: Approximate optimizer $\pmb { x } ^ { * }$ . 1 Generate a random projection matrix $\textbf {  { B } }$ by sampling $D$ points from the hypersphere $\mathbb { S } ^ { d _ { e } - 1 }$ . 2 Generate $n _ { \mathrm { i n i t } }$ random points $y ^ { i }$ in the embedding using rejection sampling to satisfy polytope (1). 3 Let $\mathcal { D } = \{ ( \pmb { y } ^ { i } , f ( \pmb { B } ^ { \dag } \pmb { y } ^ { i } ) \} _ { i = 1 } ^ { n _ { \mathrm { i n i t } } }$ be the initial data. 4 for $j = 1 , \dots , n _ { B O }$ do 5 Fit a GP by maximizing marginal log-likelihood of $\mathcal { D }$ , with the Mahalanobis kernel. 6 Draw posterior samples of $\mathbf { \delta T }$ using a Laplace approximation. Marginalize over the posterior with moment matching. 7 Use the GP to find $\boldsymbol { y } ^ { j }$ that maximizes the acquisition function according to (1). 8 Update $\mathcal { D }$ with $( { \pmb y } ^ { j } , f ( { \pmb B } ^ { \dagger } { \pmb y } ^ { j } ) )$
+
+REMBO variants $\phi k _ { \Psi }$ (Binois et al., 2015) and $\gamma k _ { \Psi }$ (Binois et al., 2019); additive kernel methods Add-GP-UCB (Kandasamy et al., 2015) and Ensemble BO (EBO) (Wang et al., 2018); SMAC, which uses a random forest model; CMA-ES, an evolutionary strategy (Hansen et al., 2003); and quasirandom search (Sobol). For ALEBO we took $d _ { e } = 2 d$ for these experiments. In their evaluation of HeSBO, Nayebi et al. (2019) used $d _ { e } = 2 d$ when $d = 2$ but $d _ { e } = d$ on the Hartmann6 problem. Our results in Fig. 3 indicate that with $d = 6$ HeSBO will have a much higher chance of reaching an optimum with $d _ { e } = 2 d$ , so we evaluate this alongside their original choice of $d _ { e } = d$ .
+
+Fig. 4 shows optimization performance for three HDBO tasks: the Branin problem extended to $D { = } 1 0 0$ as described above; the Hartmann6 problem extended to $D { = } 1 0 0 0$ ; and the Gramacy problem extended to $D { = } 1 0 0$ . The Gramacy problem (Gramacy et al., 2016) includes two black-box constraints. The linear embedding methods (ALEBO, REMBO, and HeSBO) can naturally be extended to constrained optimization as described in Appendix A.5. The $D { = } 1 0 0$ problems were repeated with 50 runs, and the $D { = } 1 0 0 0$ problem was repeated with 25 runs. Appendix A.6 provides additional details of the benchmark methods, additional experimental results (including plots of log regret and error bars), and an extended analysis of the results.
+
+For all problems, ALEBO had the best average optimization performance. Relative to other linear embedding approaches, ALEBO also had low variance in the final best-value, which is important in real applications where one can typically only run one optimization run. For the $D { = } 1 0 0 0$ problem, REMBO- $\gamma k _ { \Psi }$ , EBO, and Add-GP-UCB did not finish a single run after 24 hours and so were terminated and not included in the results. These methods, along with SMAC and CMA-ES, also do not support blackbox constraints and so were not included in the results for the Gramacy problem.
+
+We used the Branin problem to explore the sensitivity of optimization performance to $D$ and $d _ { e }$ , by varying $d _ { e }$ from 2 to 8 and $D$ from 50 to 1000. We found that $d _ { e } = d$ performed significantly worse than larger values, but for $d _ { e } > d$ and across all values of $D$ there was little change in the BO performance. Figures with these results are in Appendix A.6.
+
+# 7 POLICY SEARCH FOR ROBOT LOCOMOTION
+
+Next, we evaluate our approach on a hexapod robot simulation for learning walking controllers. Sample efficiency is crucial in robotics as collecting data on real robots is time consuming and can cause wear-andtear on the robot. We optimize the walking gait of the simulated hexapod robot “Daisy” (Hebi Robotics, 2019). The Daisy robot is simulated in PyBullet (Coumans & McCutchan, 2008), and has 6 legs with 3 motors in each leg, as shown in Fig. 5. The goal is to learn the policy parameters that enable the robot to walk to a target location while avoiding high joint velocities and height deviations. More details about this task can be found in Appendix A.7.
+
+![](images/5fc61076f6865cf32552e7b95622d7420249fa152c1f5c3f3a20ef51ce400350.jpg)  
+Figure 5: The simulated hexapod robot Daisy.
+
+![](images/732447d5398502b9bf2d0fa3a5863eb570e1e9d8a93afe328243f19007076288.jpg)  
+Figure 4: Optimization performance on three HDBO minimization problems. For each row, the left plot shows the best value by each iteration, averaged over repeated runs. The right plot shows the distribution of the best value at the final iteration. For all three tasks, ALEBO achieved the best average performance, and had the lowest variance in final performance of the linear embedding methods.
+
+We use a Central Pattern Generator (CPG) (Crespi & Ijspeert, 2008) with $D = 7 2$ to control the robot. The CPG controller induces a cyclical motion in each joint of the robot. Different parameters of the CPG change the phase, amplitude, frequency, and offset of each joint. While the 72- dimensional controller assumes that each joint is independent of the others, one could construct a lower-dimensional embedding by coupling multiple joints. For example, the tripod gait in hexapods assumes three sets of legs synced, and out of phase with the remaining three legs. The dimensionality of the CPG controller can be reduced to 11 dimensions by restricting the movement to a tripod gait, and learning the common amplitude, offset and frequency of the joints. The existence of such low-dimensional parameterizations motivates the use of ALEBO for learning the parameters of the CPG controller, although it is not known if there is a linear low-dimensional representation. In a real robot, each motor can have different physical properties, such as friction, damping, etc. This could make a pre-defined constrained space sub-optimal, and we could benefit from learning with a flexible embedding, as in ALEBO.
+
+![](images/2d101a5a3862e201dc03ca63c48d77f7f0a873d358c40c1aed3ceb8eed5dc53a.jpg)  
+Figure 6: Optimization performance on the $D = 7 2$ hexapod locomotion task (higher is better). (Left) Mean and two standard errors (over 50 repeated runs) of the best value found by each iteration. (Right) Distribution of the best value found across repeated runs. ALEBO had the best average performance, and the lowest variance.
+
+Fig. 6 shows optimization performance for the linear embedding methods on this task, which is a maximization problem. ALEBO improves on the state-of-the-art, with both higher mean performance and a lower variance (thus, a lower chance of poor performance). Expert tuning can achieve reward values above 40, so while ALEBO is an advance in terms of linear embedding BO, there is still much room for additional work in high-dimensional BO.
+
+# 8 DISCUSSION
+
+Our work highlights the importance of two basic requirements for an embedding to be useful for optimization that are often not examined critically by the literature: 1) the function must be wellmodeled on the embedding; and 2) the embedding should contain an optimum. To the first point, we showed how polytope constraints on the embedding eliminate boundary distortions, and we derived a Mahalanobis kernel appropriate for GP modeling in a linear embedding. These two contributions allow effective modeling in the embedding space. To the second point, we developed an approach for computing the probability that the embedding contains an optimum, which we then used to construct embeddings with a higher chance of containing an optimum, via hypersphere sampling and selecting $d _ { e }$ larger than $d$ .
+
+These same two considerations are important for any embedding, not just linear. For instance, when constructing a VAE for BO it will be equally important to ensure the function remains well-modeled on the embedding and that box bounds are not handled in a way that adds distortion. We must also ensure that the VAE embedding captures enough of the ambient space to have a high chance of containing an optimum. With linear embeddings we were able to derive analytical quantities for answering these questions—more work in this area is needed for nonlinear embeddings. Here we applied linear constraints to restrict the acquisition function optimization to points that project up inside the ambient box bounds. For a VAE these constraints will be general nonlinear functions, but their gradients can be backpropped and so constrained optimization could be done in a similar way.
+
+Given $D$ and $d$ , we can solve (2) to determine the probability of containing an optimum for any $d _ { e }$ , and thus select $d _ { e }$ based on a desired target probability. We showed on test problems that BO performance was not too sensitive to the exact choice of $d _ { e }$ . In reality, such as in the robot locomotion task, we do not know $d$ , or even if the problem has low-dimensional linear structure. In this case selecting an appropriate embedding dimension remains an important open question.
+
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+
+# A APPENDIX
+
+This appendix contains a number of additional results and analyses to supplement the main text.
+
+# A.1 HESBO EMBEDDINGS
+
+We consider HeSBO embeddings in the case of a random axis-aligned true subspace, and a uniform prior on the location of the optimum within that subspace. As explained in Sec. 4, with $d = 2$ and this prior, regardless of $d _ { e }$ or $D$ there are three possible embeddings: (1) each of the active parameters are captured by a parameter in the embedding; (2) the embedding is constrained to the diagonal $x _ { i _ { 1 } } = x _ { i _ { 2 } }$ ; or (3) the embedding is constrained to the diagonal $x _ { i _ { 1 } } = - x _ { i _ { 2 } }$ . Fig. 7 shows these three embeddings for the Branin problem from the top row of Fig. 1.
+
+Within the first embedding, the optimal value of 0.398 can be reached. Within the second, the best value is 0.925 and within the third it is 17.18. Under a uniform prior on the location of the optimum within a random axis-aligned true subspace, it is easy to compute the probability that the HeSBO embedding contains an optimum:
+
+$$
+P _ { \mathrm { o p t } } ( d _ { e } ) = \frac { d _ { e } ! } { ( d _ { e } - d ) ! d _ { e } ^ { d } } .
+$$
+
+For $d = 2$ , this is exactly the probability of the first embedding shown in Fig. 7. This probability increases with $d _ { e }$ , and is exactly the probability shown in Fig. 3.
+
+# A.2 THE MAHALANOBIS KERNEL
+
+When fitting the Mahalanobis kernel derived in Proposition 1, we use an approximate Bayesian treatment of $\mathbf { \delta T }$ to improve model performance while still maintaining tractability. We propagate uncertainty in $\mathbf { \delta T }$ into the GP posterior by first constructing a posterior for $\mathbf { \delta T }$ using a Laplace approximation with a diagonal Hessian, and then drawing $m$ samples from that posterior. The marginal posterior for $f ( y )$ can then be approximated as:
+
+$$
+p ( f ( \pmb { y } ) ) \approx \frac { 1 } { m } \sum _ { i = 1 } ^ { m } p ( f ( \pmb { y } ) | \mathbf { \Gamma } ^ { i } ) .
+$$
+
+Because of the GP prior, each conditional posterior $p ( f ( \pmb { y } ) | \mathbf { \Gamma } ^ { i } )$ is a normal distribution with known mean $\mu _ { i }$ and variance $\sigma _ { i } ^ { 2 }$ . Thus the posterior $p ( f ( \pmb { y } ) )$ is a mixture of Gaussians, which we can approximate using moment matching:
+
+$$
+p ( f ( \pmb { y } ) ) \approx \mathcal { N } \left( \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \mu _ { i } , \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \sigma _ { i } ^ { 2 } + \mathrm { V a r } _ { i } [ \mu _ { i } ] \right) .
+$$
+
+![](images/965de4d5f0bcb307bcd4b5ad62afaf960302f2c26f7d0e519a8aab96b08b58b8.jpg)  
+Figure 7: Three possible HeSBO embeddings of the $d = 2$ Branin function. (Left) The first embedding fully captures the function, and thus captures all three optima. (Middle) The second is restricted to the subspace $x _ { 1 } = - x _ { 2 }$ . This subspace does not contain an optimum, but comes fairly close. (Right) The third embedding is restricted to the subspace $x _ { 1 } = x _ { 2 }$ and does not come close to any optima.
+
+![](images/2d15aa85a5abd8021601a0f2633b741a00eefc6e6f85f5a77acde56aa7252125.jpg)  
+Figure 8: Test-set model predictions for three GP kernels on the same train/test data generated by evaluating the Hartmann6 $D { = } 1 0 0$ function on a fixed linear embedding. A typical ARD kernel fails to learn and predicts the mean. The Mahalanobis kernel predicts well, and posterior sampling is important for getting reasonable predictive variance.
+
+![](images/21e250556f5bbad1670e27eed7201970f89c109c7a2973a7d7a465ea4090970c.jpg)  
+Figure 9: Average test-set log likelihood as a function of training set size, for training sets randomly sampled from a fixed linear embedding. Log marginal probabilities were averaged over a fixed test set of 1000 random points. For each training set size, 20 random training sets were drawn of that size and the figure shows the average result over those draws (with error bars for two standard errors). The ARD RBF kernel continues to predict the mean as the training set size is increased, while the Mahalanobis kernel is able to learn as the training set is expanded.
+
+We do this to maintain a Gaussian posterior, under which acquisition functions like EI have analytic form and can easily be optimized, even subject to constraints as in (1).
+
+We show the importance of the Mahalanobis kernel using models fit to data from the Hartmann6 $D { = } 1 0 0$ function, from Fig. 1. We generated a projection matrix $\textbf {  { B } }$ using hypersphere sampling to define a 6-d linear embedding. We then generated a training set (100 points) and a test set (50 points) within that embedding (that is, within the polytope given by (1)) using rejection sampling. We fit three GP models with different kernels to the training set, and then evaluated each on the test set: a typical ARD RBF kernel in 6 dimensions, the Mahalanobis kernel using a point estimate for $\mathbf { \delta T }$ , and the Mahalanobis kernel with posterior marginalization for $\mathbf { \delta T }$ as described above.
+
+Fig. 8 compares model predictions for each of these models with the actual test-set outcomes. With an ARD RBF kernel, the GP predicts the function mean everywhere, which is typical behavior of a GP that has failed to learn the function. With the same training data, the Mahalanobis kernel is able to make accurate predictions on the test set. Using a point estimate for $\mathbf { \delta T }$ significantly underestimates the predictive variance, which is rectified by using posterior sampling as described above. In BO exploration is driven by model uncertainty, so well-calibrated uncertainty intervals are especially important.
+
+![](images/1ce5a286038a7fd1a45435e7d2f36c90c4a819b193181e2fc2cf020f72adbd5c.jpg)  
+Figure 10: (Left) An embedding from a $\mathcal { N } ( 0 , 1 )$ projection matrix on the same Branin $D = 1 0 0$ problem from Fig. 1 subject to constraints of (1). (Right) The embedding from the same projection matrix after normalizing the columns to produce unit circle samples. Sampling from the unit circle increases the probability that an optimum will fall within the embedding, and polytope bounds avoid nonlinear distortions.
+
+Fig. 9 evaluates the predictive log marginal probabilities for the ARD RBF kernel and the Mahalanobis kernel with posterior sampling across a wide range of training sets with different sizes (without posterior sampling, Fig. 8 shows that the Mahalanobis point estimate significantly under covers and so has very poor predictive log marginal probabilities). We used the same linear embedding and Hartmann6 $D { = } 1 0 0$ function used in Fig. 8 to sample 1000 test points which were held fixed. For each of 8 training set sizes ranging from 40 to 200, we randomly sampled 20 training sets from the embedding. For each training set, we fit the two GPs, made predictions on the 1000 test points, and then computed the average marginal log probability of the true values. Fig. 9 shows that as we vary the training set size from 40 to 200, the ARD RBF kernel continues to predict the mean, as in Fig. 8; even 200 points in the 6-d embedding are not sufficient to learn. For small training set sizes, the Mahalanobis kernel (with sampling) has high variance in log likelihood, as it has the potential to overfit and thus under cover. But for training set sizes of 50 and greater it had better predictive log likelihood than the ARD RBF, and continued to learn as the training set size was increased. For small datasets, the Mahalanobis kernel can overfit and thus have poor predictive likelihood, but for the purposes of BO, overfitting can be better than not fitting at all (predicting the mean), even when predicting the mean has better predictive log likelihood. This can be seen in the optimization results (Figs. 4 and 12) where ALEBO shows strong performance even with less than 50 iterations.
+
+# A.3 POLYTOPE BOUNDS ON THE EMBEDDING
+
+Rather than using projections to the box bounds $\boldsymbol { B }$ , we specify polytope constraints in (1). Fig. 10 illustrates the embedding with these constraints for the same Branin $D = 1 0 0$ problem from the top row of Fig. 1. The embedding in the left figure was created with the REMBO strategy of sampling each entry from $\mathcal { N } ( 0 , 1 )$ . For the embedding in the right figure, that same projection matrix had each column normalized. This converts the projection matrix to be a sample from the unit circle, as described in Sec. 4.
+
+The $\mathcal { N } ( 0 , 1 )$ embedding does not contain any optima within the polytope bounds. Converting that projection matrix to a hypersphere sample rounds out the vertices of the polytope and expands the space to capture two of the optima. Consistent with Fig. 3, we see that hypersphere sampling significantly improves the chances of the embedding containing an optimum. Fig. 10 also shows that with the polytope bounds, we avoid the nonlinear distortions seen in Fig. 1.
+
+# A.4 EVALUATING THE PROBABILITY THE EMBEDDING CONTAINS AN OPTIMUM
+
+As in other parts of the paper, we consider a uniform prior on the location of the optimum within a random axis-aligned subspace. A random true projection matrix $_ { \mathbf { T } }$ is sampled by selecting $d$ columns at random and setting each to one of the $d$ -dimensional unit vectors. $z ^ { * }$ is then sampled uniformly at random from $[ - \bar { 1 } , 1 ] ^ { d }$ . $\textbf {  { B } }$ is sampled according to the desired strategy, which in our experiments was REMBO, HeSBO, or hypersphere. Given these three quantities, we can evaluate whether or not $\textbf {  { B } }$ contains an optimum subject to the constraints of (1) by solving the following
+
+![](images/040fc84db691d0194491e3d67ee51ffd9ba962dc1fdad4d6e17049e60a2ff3d4.jpg)  
+Figure 11: $P _ { \mathrm { o p t } }$ for hypersphere sampling, as estimated in Fig. 3 but here for a wider range of values of $d$ and $D$ . Contour color indicates $P _ { \mathrm { o p t } }$ . Doubling $D$ decreases $P _ { \mathrm { o p t } }$ for $d$ and $d _ { e }$ fixed, however even at $D = 2 0 0$ , high values of $P _ { \mathrm { o p t } }$ with reasonable values of $d _ { e }$ can be had for many values of $d$ .
+
+linear program:
+
+$$
+\begin{array} { l } { ( B ^ { \dagger } B - I ) { \pmb x } = { \bf 0 } , } \\ { { \pmb x } \geq - { \bf 1 } , } \\ { { \pmb x } \leq { \bf 1 } . } \end{array}
+$$
+
+If this problem is feasible, then the embedding produced by $\textbf {  { B } }$ contains an optimum. If it is infeasible, then it does not. Solving this over many draws of ${ \mathbf { } } T , z ^ { * }$ , and $\textbf {  { B } }$ produces an estimate of $P _ { \mathrm { o p t } }$ under that prior for the location of optima. Here we used a uniform prior, but this linear program can be taken to compute $P _ { \mathrm { o p t } }$ under any prior.
+
+Fig. 11 shows $P _ { \mathrm { o p t } }$ for a wide range of values of $d$ and $D$ , for hypersphere sampling. Across this wide range we see that for many values of $d$ we can achieve high values of $P _ { \mathrm { o p t } }$ with reasonable values of $d _ { e }$ , even for relatively high values of $D$ .
+
+# A.5 HANDLING BLACK-BOX CONSTRAINTS IN HIGH-DIMENSIONAL BAYESIAN OPTIMIZATION
+
+In many applications of BO, in addition to the black-box objective $f$ there are black-box constraints $c _ { j }$ and we seek to solve the optimization problem
+
+$$
+{ \begin{array} { r l } & { { \mathrm { m i n i m i z e ~ } } f ( { \boldsymbol { x } } ) } \\ & { { \mathrm { s u b j e c t ~ t o ~ } } c _ { j } ( { \boldsymbol { x } } ) \leq 0 , \quad j = 1 , \ldots , J , } \\ & { \qquad { \boldsymbol { x } } \in { \boldsymbol { \mathcal { B } } } . } \end{array} }
+$$
+
+In most settings the constraint functions $c _ { j }$ are evaluated simultaneously with the objective $f$ . Constraints are typically handled in BO by fitting a separate GP to each outcome (that is, to $f$ and to each $c _ { j }$ ). The acquisition function is then modified to consider not only the objective value but also whether the constraints are likely to be satisfied (e.g., Gardner et al., 2014).
+
+The extension of BO in an embedding to constrained BO is straightforward, so long as the same embedding is used for every outcome. A separate GP (in our case, using the Mahalanobis kernel) is fit to data from each outcome. Because the embedding is shared, predictions can be made for all of the outcomes at any point in the embedding. This allows us to evaluate and optimize an acquisition function for constrained BO in the embedding. Once a point is selected, it is projected up to the ambient space and evaluated on $f$ and each $c _ { j }$ as usual. Random projections are especially wellsuited for constrained BO because there is no harm in requiring the same projection for all outcomes, since it is a random projection anyway.
+
+# A.6 ADDITIONAL EXPERIMENTAL RESULTS
+
+Here we provide results from an additional problem (Hartmann6 $D { = } 1 0 0 $ ), three additional methods (LineBO variants), and provide a study of the sensitivity of ALEBO performance to $d _ { e }$ and $D$ . We also provide implementation details for the experiments.
+
+# A.6.1 METHOD IMPLEMENTATIONS AND EXPERIMENT SETUP
+
+The linear embedding methods (REMBO, HeSBO, and ALEBO) were all implemented using BoTorch, a framework for BO in PyTorch (Balandat et al., 2019), and so used the same acquisition functions and the same tooling for optimizing the acquisition function. EI was the acquisition function for the Hartmann6 and Branin benchmarks, and NEI (Letham et al., 2019) was used to handle the constraints in the Gramacy problem. ALEBO and HeSBO were given a quasirandom initialization of 10 points from a scrambled Sobol sequence. REMBO was given a Sobol initialization of 2 points for each of its 4 projections used within a run.
+
+The remaining methods used reference implementations from their authors with default settings for the package: REMBO- $\phi k _ { \Psi }$ and REMBO- $\gamma k _ { \Psi } { } ^ { 1 }$ ; $\mathrm { E B O } ^ { 2 }$ ; Add-GP-UCB 3; SMAC4; CMA-ES5; and CoordinateLineBO, RandomLineBO, and DescentLine $\mathrm { B O } ^ { 6 }$ . EBO requires an estimate of the best function value, and for each problem was given the true best function value. SMAC and CMA-ES require an initial point, and were given the point at the center of the ambient space box bounds.
+
+The function evaluations for all problems were noiseless, so the stochasticity throughout the run and in the final value all comes from stochasticity in the methods themselves. For linear embedding methods the main sources of stochasticity are in generating the random projection matrix and in the quasirandom initialization.
+
+# A.6.2 ANALYSIS OF EXPERIMENTAL RESULTS
+
+Fig. 12 provides a different view of the benchmark results of Fig. 4, showing log regret for each method, averaged over runs with error bars indicating two standard errors of the mean. This is evaluated by measuring the difference between the best point found so far, subtracting from that the optimal value for the problem, and then taking the log of that difference. The results are consistent with those seen in Fig. 4, and the standard errors show that ALEBO’s improvement in average performance over the other methods is statistically significant. We now discuss some specific aspects of these experimental results.
+
+Branin ${ \cal D } { \bf = } { \bf 1 0 0 }$ The additive kernel methods and SMAC all performed similarly on this problem, and, starting from around iteration 20, ALEBO performed the best. The distribution of final iteration values shows that in one iteration the ALEBO embedding did not contain an optimum and so achieved a final value near 10. However, across all 50 runs nearly all achieved a value very close to the optimum, leading to the best average performance.
+
+The poor performance of HeSBO on this problem (particularly in Fig. 4 without the log, where it is outperformed by all methods other than Sobol) can be attributed entirely to the embedding not containing an optimum. Recall that for this problem there are exactly three possible HeSBO embeddings, which are shown in Fig. 7. As explained in Appendix A.1, the first embedding contains the optimum of 0.398, while the best value in the other embeddings are 0.925 and 17.18. Thus, if the BO were able to find the true optimum within each embedding with the budget of 50 function evaluations given in this experiment, the expected best value found by HeSBO would be:
+
+$$
+0 . 3 9 8 P _ { \mathrm { o p t } } + 0 . 9 2 5 \left( \frac { 1 - P _ { \mathrm { o p t } } } { 2 } \right) + 1 7 . 1 8 \left( \frac { 1 - P _ { \mathrm { o p t } } } { 2 } \right) .
+$$
+
+![](images/3ab29bb999a8ebe91f56db0eef78cb827c0fb46ae29b80eac87f82ebbbc4fde2.jpg)  
+Figure 12: Log regret for the benchmark experiments of Fig. 4, plus Hartmann6 $D { = } 1 0 0$ . Each trace is the mean over repeated runs, with errors bars showing two standard errors of the mean. On the first three problems ALEBO performs significantly better than the other methods, and on Hartmann6 $D { = } 1 0 0$ it is tied with REMBO- $\gamma k _ { \Psi }$ as the best methods.
+
+This is the best average performance one can hope to achieve using the HeSBO embedding on this problem. Using (3) we can compute $P _ { \mathrm { o p t } }$ for $d _ { e } = 4$ as 0.75, and it follows that the HeSBO expected best value is 2.56. This is nearly exactly the average best-value shown in Fig. 4. The poor performance of HeSBO is thus not related to BO, but comes entirely from the $12 . 5 \%$ chance of generating an embedding whose optimal value is 17.18. The presence of these embeddings can be clearly seen in the distribution of final best values in Fig. 4.
+
+Hartmann6 ${ \cal D } { \bf = } { \bf 1 0 0 0 }$ As noted in the main text, the additive kernel methods and REMBO- $\gamma k _ { \Psi }$ could not scale up to the 1000 dimensional problem. A nice property of linear embedding approaches is that the running time is not significantly impacted by the ambient dimensionality. Table
+
+Table 1: Average running time per iteration in seconds on the Hartmann6 problem, $D { = } 1 0 0$ and $D { = } 1 0 0 0$ .   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>D=100</td><td rowspan=1 colspan=1>D=1000</td></tr><tr><td rowspan=1 colspan=1>ALEBO</td><td rowspan=1 colspan=1>29.5</td><td rowspan=1 colspan=1>32.6</td></tr><tr><td rowspan=2 colspan=1>REMBOHeSBO, de=d</td><td rowspan=1 colspan=1>1.3</td><td rowspan=1 colspan=1>1.4</td></tr><tr><td rowspan=1 colspan=1>1.0</td><td rowspan=1 colspan=1>1.6</td></tr><tr><td rowspan=1 colspan=1>HeSBO, de=2d</td><td rowspan=1 colspan=1>1.0</td><td rowspan=1 colspan=1>1.4</td></tr><tr><td rowspan=1 colspan=1>REMBO-𝜙ky</td><td rowspan=1 colspan=1>2.1</td><td rowspan=7 colspan=1>1.1404.50.00.0</td></tr><tr><td rowspan=6 colspan=1>REMBO-yk亚EBOAdd-GP-UCBSMACCMA-ESSobol</td><td rowspan=1 colspan=1>7.2</td></tr><tr><td rowspan=1 colspan=1>27.3</td></tr><tr><td rowspan=1 colspan=1>695.2</td></tr><tr><td rowspan=1 colspan=1>9.5</td></tr><tr><td rowspan=1 colspan=1>0.0</td></tr><tr><td rowspan=1 colspan=1>0.0</td></tr></table>
+
+1 gives the average running time per iteration for the various benchmark methods. Inferring the additional parameters in the Mahalanobis kernel and the added linear constraints make ALEBO slower than other linear embedding methods, but it has similar running time as EBO and is an order of magnitude faster than Add-GP-UCB, and at $D { = } 1 0 0 0$ is even an order of magnitude faster than SMAC. The average of 30s per iteration is short relative to the function evaluation time of typical resource-intensive BO applications.
+
+Hartmann6 ${ \cal D } { \bf = } { \bf 1 0 0 }$ REMBO performed worse than Sobol on this problem, despite there being a true linear subspace that satisfies the REMBO assumptions. The source of the poor performance is the poor representation of the function on the embedding illustrated in Fig. 1. The remaining methods all performed better than quasirandom. CMA-ES was competitive with all of the methods except SMAC, REMBO- $\gamma k _ { \Psi }$ , and ALEBO, which is somewhat surprising since it is not designed to have the same degree of sample efficiency as BO methods. HeSBO and Add-GP-UCB both did very well early on, but then got stuck and did not progress significantly after about iteration 50.
+
+This problem was used to test three additional methods beyond those in Fig. 4: CoordinateLineBO, RandomLineBO, and DescentLineBO (Kirschner et al., 2019). These are recent methods developed for high-dimensional safe BO, in which one must optimize subject to safety constraints that certain bounds on the functions must not be violated. The performance of these methods can be seen in the bottom panel of Fig. 12: all three LineBO variants perform much worse than Sobol, and show almost no reduction of log regret. This finding is consistent with the results of Kirschner et al. (2019), who used the Hartmann6 $D { = } 2 0$ problem as a benchmark problem. At $D { = } 2 0$ , they found that CoordinateLineBO required about 400 iterations to outperform random search, and even after 1200 iterations RandomLineBO and DescentLineBO did not perform better than random search. These methods are designed specifically for safe BO, which is a significantly harder problem than usual BO that has much worse scaling with dimensionality. The primary challenge for high-dimensional safe BO lies in optimizing the acquisition function, which is difficult even for relatively small numbers of parameters where there is no difficulty in optimizing the traditional BO acquisition function. The LineBO methods develop new techniques for acquisition function optimization, but do not consider difficulties with GP modeling in high dimensions, which is the main focus of HDBO work. LineBO methods perform very well on safe BO problems relative to other methods, but ultimately non-safe HDBO is not the problem that they were developed for, and so it is not surprising to see that they were not successful on this task.
+
+# A.6.3 SENSITIVITY OF ALEBO TO EMBEDDING AND AMBIENT DIMENSIONS
+
+We study sensitivity of ALEBO optimization performance to the embedding dimension $d _ { e }$ and the ambient dimension $D$ using the Branin function. To test dependence on $d _ { e }$ , for $D = 1 0 0$ we ran 50 optimization runs for each of $d _ { e } \in \{ 2 , 3 , 4 , 5 , 6 , 7 , 8 \}$ . To test dependence on $D$ , for $d _ { e } = 4$ we ran 50 optimization runs for each of $D \in \{ 5 0 , 1 0 0 , 2 0 0 , 5 0 0 , 1 0 0 0 \}$ . Note that the $d _ { e } = 4$ and $D = 1 0 0$ case in each of these is exactly the optimization problem of Fig. 4.
+
+![](images/d03f2feeb3b47b7967cdaf5b77251ba8d6e2bea5668067ba44e2a4c5401f1873.jpg)  
+Figure 13: ALEBO performance on the Branin problem, $( L e f t )$ as a function of embedding dimension $d _ { e }$ and $( R i g h t )$ as a function of ambient dimension $D$ . Performance shown is the average of 50 repeated runs. Optimization performance is poor with $d _ { e } = 2$ , but shows little sensitivity to $d _ { e }$ for values greater than 2. Optimization performance shows little sensitivity with $D$ , all the way up to $D = 1 0 0 0$ .
+
+![](images/39a382e2544e2a1a13b1ed4c4c9e02532caa9084c0110f10fa969831d5fd1cb9.jpg)  
+Figure 14: Final best value for the Branin problem optimizations Fig. 13, as mean with error bars showing two standard errors. With the exception of $d _ { e } = 2$ , optimization performance was good across a wide range of values of $d _ { e }$ and $D$ .
+
+The results of the optimizations are shown in Figs. 13 and 14. For $d _ { e } = d$ , optimization performance was poor. From Fig. 3 we know this is because there is a low probability of the embedding containing an optimizer. Increasing $d _ { e }$ increases that probability, but also increases the dimensionality of the embedding and thus reduces the sample efficiency of the BO in the embedding. This trade-off can be seen clearly in the figure: with $d _ { e } = 2$ there is rapid improvement that then flattens out because of the lack of good solutions in the embedding, whereas for $d _ { e } = 8$ the initial iterations are worse but then it ultimately is able to find much better solutions. Even at $d _ { e } = 8$ the average best final value was better than that of any of the comparison methods in Fig. 4.
+
+The ambient dimension $D$ will not directly impact the GP modeling in ALEBO, which depends only on $d _ { e }$ , however it will impact the probability the embedding contains an optimum as shown in Fig. 11. Consistent with the strong ALEBO performance for the Hartmann6 $D { = } 1 0 0 0$ problem, we see here that even increasing $D$ to 1000 produces only a small degradation in optimization performance. Even at $D = 1 0 0 0$ , ALEBO had better performance than the other benchmark methods had on $D = 1 0 0$ .
+
+# A.7 LOCOMOTION BENCHMARK PROBLEM
+
+The task for the final set of experiments was to learn a gait policy for a simulated robot. As a controller, we use the Central Pattern Generator (CPG) from Crespi & Ijspeert (2008). The goal in
+
+this task is for the robot to walk to a target location in a given amount of time, while reducing joint velocities, and average deviation from a desired height
+
+$$
+\boldsymbol { f } ( \boldsymbol { p } ) = \boldsymbol { C } - | | \boldsymbol { x } _ { \mathrm { f i n a l } } - \boldsymbol { x } _ { \mathrm { g o a l } } | | - \sum _ { t = 0 } ^ { T } ( w _ { 1 } | | \dot { q } _ { t } | | - w _ { 2 } | h _ { \mathrm { r o b o t } , t } - h _ { \mathrm { t a r g e t } } | ) ,
+$$
+
+where $C = 1 0$ , $w _ { 1 } = 0 . 0 0 5$ , and $w _ { 2 } = 0 . 0 1$ are constants. ${ \pmb x } _ { \mathrm { f i n a l } }$ is the location of the robot on a plane at the end of the episode, $\pmb { x } _ { \mathrm { g o a l } }$ is the target location, $\dot { \pmb q } _ { t }$ are the joint velocities at time $t$ during the trajectory, $h _ { \mathrm { r o b o t } , t }$ is the height of the robot at time $t$ , and $h _ { \mathrm { t a r g e t } }$ is a target height. $T = 3 0 0 0$ is the total length of the trajectory, leading to 30s of experiment. Cost is evaluated at the end of the trajectory.

parse/train/Sy2ogebAW/Sy2ogebAW.md

@@ -0,0 +1,214 @@
+# UNSUPERVISED NEURAL MACHINE TRANSLATION
+
+Mikel Artetxe, Gorka Labaka & Eneko Agirre IXA NLP Group University of the Basque Country (UPV/EHU) {mikel.artetxe,gorka.labaka,e.agirre}@ehu.eus
+
+Kyunghyun Cho New York University CIFAR Azrieli Global Scholar kyunghyun.cho@nyu.edu
+
+# ABSTRACT
+
+In spite of the recent success of neural machine translation (NMT) in standard benchmarks, the lack of large parallel corpora poses a major practical problem for many language pairs. There have been several proposals to alleviate this issue with, for instance, triangulation and semi-supervised learning techniques, but they still require a strong cross-lingual signal. In this work, we completely remove the need of parallel data and propose a novel method to train an NMT system in a completely unsupervised manner, relying on nothing but monolingual corpora. Our model builds upon the recent work on unsupervised embedding mappings, and consists of a slightly modified attentional encoder-decoder model that can be trained on monolingual corpora alone using a combination of denoising and backtranslation. Despite the simplicity of the approach, our system obtains 15.56 and 10.21 BLEU points in WMT 2014 French English and German English translation. The model can also profit from small parallel corpora, and attains 21.81 and 15.24 points when combined with 100,000 parallel sentences, respectively. Our implementation is released as an open source project1.
+
+# 1 INTRODUCTION
+
+Neural machine translation (NMT) has recently become the dominant paradigm to machine translation (Bahdanau et al., 2014; Sutskever et al., 2014). As opposed to the traditional statistical machine translation (SMT), NMT systems are trained end-to-end, take advantage of continuous representations that greatly alleviate the sparsity problem, and make use of much larger contexts, thus mitigating the locality problem. Thanks to this, NMT has been reported to significantly improve over SMT both in automatic metrics and human evaluation (Wu et al., 2016).
+
+Nevertheless, for the same reasons described above, NMT requires a large parallel corpus to be effective, and is known to fail when the training data is not big enough (Koehn & Knowles, 2017). Unfortunately, the lack of large parallel corpora is a practical problem for the vast majority of language pairs, including low-resource languages (e.g. Basque) as well as many combinations of major languages (e.g. German-Russian). Several authors have recently tried to address this problem using pivoting or triangulation techniques (Chen et al., 2017) as well as semi-supervised approaches (He et al., 2016), but these methods still require a strong cross-lingual signal.
+
+In this work, we eliminate the need of cross-lingual information and propose a novel method to train NMT systems in a completely unsupervised manner, relying solely on monolingual corpora. Our approach builds upon the recent work on unsupervised cross-lingual embeddings (Artetxe et al., 2017; Zhang et al., 2017). Thanks to a shared encoder for both translation directions that uses these fixed cross-lingual embeddings, the entire system can be trained, with monolingual data, to reconstruct its input. In order to learn useful structural information, noise in the form of random token swaps is introduced in this input. In addition to denoising, we also incorporate backtranslation (Sennrich et al., 2016a) into the training procedure to further improve results. Figure 1 summarizes this general schema of the proposed system.
+
+![](images/2e405cbed5e113f97798717bed1c97c3d70953d97ca38689c4963a02ca6b6995.jpg)  
+Figure 1: Architecture of the proposed system. For each sentence in language L1, the system is trained alternating two steps: denoising, which optimizes the probability of encoding a noised version of the sentence with the shared encoder and reconstructing it with the L1 decoder, and on-the-fly backtranslation, which translates the sentence in inference mode (encoding it with the shared encoder and decoding it with the L2 decoder) and then optimizes the probability of encoding this translated sentence with the shared encoder and recovering the original sentence with the L1 decoder. Training alternates between sentences in L1 and L2, with analogous steps for the latter.
+
+In spite of the simplicity of the approach, our experiments show that the proposed system can reach up to 15.56 BLEU points for French English and 10.21 BLEU points for German English in the standard WMT 2014 translation task using nothing but monolingual training data. Moreover, we show that combining this method with a small parallel corpus can further improve the results, obtaining 21.81 and 15.24 BLEU points with 100,000 parallel sentences, respectively. Our manual analysis confirms the effectiveness of the proposed approach, revealing that the system is learning non-trivial translation relations that go beyond a word-by-word substitution.
+
+The remaining of this paper is organized as follows. Section 2 analyzes the related work. Section 3 then describes the proposed method. The experimental settings are discussed in Section 4, while Section 5 presents and discusses the obtained results. Section 6 concludes the paper.
+
+# 2 RELATED WORK
+
+We will first discuss unsupervised cross-lingual embeddings, which are the basis of our proposal, in Section 2.1. Section 2.2 then addresses statistical decipherment, an SMT-inspired approach to build a machine translation system in an unsupervised manner. Finally, Section 2.3 presents previous work on training NMT systems in different low-resource scenarios.
+
+# 2.1 UNSUPERVISED CROSS-LINGUAL EMBEDDINGS
+
+Most methods for learning cross-lingual word embeddings rely on some bilingual signal at the document level, typically in the form of parallel corpora (Gouws et al., 2015; Luong et al., 2015a). Closer to our scenario, embedding mapping methods independently train the embeddings in different languages using monolingual corpora, and then learn a linear transformation that maps them to a shared space based on a bilingual dictionary (Mikolov et al., 2013a; Lazaridou et al., 2015; Artetxe et al., 2016; Smith et al., 2017). While the dictionary used in these earlier work typically contains a few thousands entries, Artetxe et al. (2017) propose a simple self-learning extension that gives comparable results with an automatically generated list of numerals, which is used as a shortcut for practical unsupervised learning. Alternatively, adversarial training has also been proposed to learn such mappings in an unsupervised manner (Miceli Barone, 2016; Zhang et al., 2017).
+
+# 2.2 STATISTICAL DECIPHERMENT FOR MACHINE TRANSLATION
+
+There is a considerable body of work in statistical decipherment techniques to induce a machine translation model from monolingual data, which follows the same noisy-channel model used by SMT (Ravi & Knight, 2011; Dou & Knight, 2012). More concretely, they treat the source language as ciphertext, and model the process by which this ciphertext is generated as a two-stage process involving the generation of the original English sequence and the probabilistic replacement of the words in it. The English generative process is modeled using a standard n-gram language model, and the channel model parameters are estimated using either expectation maximization or Bayesian inference. This approach was shown to benefit from the incorporation of syntactic knowledge of the languages involved (Dou & Knight, 2013; Dou et al., 2015). More in line with our proposal, the use of word embeddings has also been shown to bring significant improvements in statistical decipherment for machine translation (Dou et al., 2015).
+
+# 2.3 LOW-RESOURCE NEURAL MACHINE TRANSLATION
+
+There have been several proposals to exploit resources other than direct parallel corpora to train NMT systems. The scenario that is most often considered is one where two languages have little or no parallel data between them but are well connected through a third language (e.g. there might be little direct resources for German-Russian but plenty for German-English and English-Russian). The most basic approach in this scenario is to independently translate from the source language to the pivot language and from the pivot language to the target language. It has however been shown that the use of more advanced models like a teacher-student framework can bring considerable improvements over this basic baseline (Firat et al., 2016b; Chen et al., 2017). In the same line, Johnson et al. (2017) show that a multilingual extension of a standard NMT architecture performs reasonably well even for language pairs for which no direct data was given during training.
+
+In addition to that, there have been several attempts to exploit monolingual corpora for NMT in combination with the more scarce parallel corpora. A simple yet effective approach is to create a synthetic parallel corpus by backtranslating a monolingual corpus in the target language (Sennrich et al., 2016a). At the same time, Currey et al. (2017) showed that training an NMT system to directly copy target language text is also helpful and complementary with backtranslation. Finally, Ramachandran et al. (2017) pre-train the encoder and the decoder in language modeling.
+
+To the best of our knowledge, the more ambitious scenario where an NMT model is trained from monolingual corpora alone has never been explored to date, but He et al. (2016) made an important contribution in this direction. More concretely, their method trains two agents to translate in opposite directions (e.g. French English and English French), and make them teach each other through a reinforcement learning process. While promising, this approach still requires a parallel corpus of a considerable size for a warm start (1.2 million sentences in the reported experiments), whereas our work does not use any parallel data at all.
+
+# 3 PROPOSED METHOD
+
+This section describes the proposed unsupervised NMT approach. Section 3.1 first presents the architecture of the proposed system, and Section 3.2 then describes the method to train it in an unsupervised manner.
+
+# 3.1 SYSTEM ARCHITECTURE
+
+As shown in Figure 1, the proposed system follows a fairly standard encoder-decoder architecture with an attention mechanism (Bahdanau et al., 2014). More concretely, we use a two-layer bidirectional RNN in the encoder, and another two-layer RNN in the decoder. All RNNs use GRU cells with 600 hidden units (Cho et al., 2014), and the dimensionality of the embeddings is set to 300. As for the attention mechanism, we use the global attention method proposed by Luong et al. (2015b) with the general alignment function. There are, however, three important aspects in which our system differs from the standard NMT, and these are critical so the system can be trained in an unsupervised manner as described next in Section 3.2:
+
+1. Dual structure. While NMT systems are typically built for a specific translation direction (e.g. either French English or English French), we exploit the dual nature of machine translation (He et al., 2016; Firat et al., 2016a) and handle both directions together (e.g. French English).   
+2. Shared encoder. Our system makes use of one and only one encoder that is shared by both languages involved, similarly to Ha et al. (2016), Lee et al. (2017) and Johnson et al. (2017). For instance, the exact same encoder would be used for both French and English. This universal encoder is aimed to produce a language independent representation of the input text, which each decoder should then transform into its corresponding language.   
+3. Fixed embeddings in the encoder. While most NMT systems randomly initialize their embeddings and update them during training, we use pre-trained cross-lingual embeddings in the encoder that are kept fixed during training. This way, the encoder is given language independent word-level representations, and it only needs to learn how to compose them to build representations of larger phrases. As discussed in Section 2.1, there are several unsupervised methods to train these cross-lingual embeddings from monolingual corpora, so this is perfectly feasible in our scenario. Note that, even if the embeddings are crosslingual, we use separate vocabularies for each language. This way, the word chair, which exists both in French and English (meaning “flesh” in the former), would get a different vector in each language, although they would both be in a common space.
+
+# 3.2 UNSUPERVISED TRAINING
+
+As NMT systems are typically trained to predict the translations in a parallel corpus, such supervised training procedure is infeasible in our scenario, where we only have access to monolingual corpora. However, thanks to the architectural modifications proposed above, we are able to train the entire system in an unsupervised manner using the following two strategies:
+
+1. Denoising. Thanks to the use of a shared encoder, and exploiting the dual structure of machine translation, the proposed system can be directly trained to reconstruct its own input. More concretely, the whole system can be optimized to take an input sentence in a given language, encode it using the shared encoder, and reconstruct the original sentence using the decoder of that language. Given that we use pre-trained cross-lingual embeddings in the shared encoder, this encoder should learn to compose the embeddings of both languages in a language-independent fashion, and each decoder should learn to decompose this representation into their corresponding language. At inference time, we simply replace the decoder with that of the target language, so it generates the translation of the input text from the language-independent representation given by the encoder.
+
+Nevertheless, this ideal behavior is severely compromised by the fact that the resulting training procedure is essentially a trivial copying task. As such, the optimal solution for this task would not need to capture any real knowledge of the languages involved, as there would be many degenerated solutions that blindly copy all the elements in the input sequence. If this were the case, the system would at best make very literal word-by-word substitutions when used to translate from one language to another at inference time.
+
+In order to avoid such degenerated solutions and make the encoder truly learn the compositionality of its input words in a language independent manner, we propose to introduce random noise in the input sentences. The idea is to exploit the same underlying principle of denoising autoencoders (Vincent et al., 2010), where the system is trained to reconstruct the original version of a corrupted input sentence (Dai & Le, 2015; Hill et al., 2016). For that purpose, we alter the word order of the input sentence by making random swaps between contiguous words. More concretely, for a sequence of $N$ elements, we make $N / 2$ random swaps of this kind. This way, the system needs to learn about the internal structure of the languages involved to be able to recover the correct word order. At the same time, by discouraging the system to rely too much on the word order of the input sequence, we can better account for the actual word order divergences across languages. This training procedure can be seen as an instance of contrastive estimation (Smith & Eisner, 2005), where the neighborhood is defined by local swaps in our case, although other functions would also be possible.
+
+2. On-the-fly backtranslation. In spite of the denoising strategy, the training procedure above is still a copying task with some synthetic alterations that, most importantly, involves a single language at each time, without considering our final goal of translating between two languages. In order to train our system in a true translation setting without violating the constraint of using nothing but monolingual corpora, we propose to adapt the backtranslation approach proposed by Sennrich et al. (2016a) to our scenario. More concretely, given an input sentence in one language, we use the system in inference mode with greedy decoding to translate it to the other language (i.e. apply the shared encoder and the decoder of the other language). This way, we obtain a pseudo-parallel sentence pair, and train the system to predict the original sentence from this synthetic translation. Note that, contrary to standard backtranslation, which uses an independent model to backtranslate the entire corpus at one time, we take advantage of the dual structure of the proposed architecture to backtranslate each mini-batch on-the-fly using the model that is being trained itself. This way, as training progresses and the model improves, it will produce better synthetic sentence pairs through backtranslation, which will serve to further improve the model in the following iterations.
+
+During training, we alternate these different training objectives from mini-batch to mini-batch. This way, given two languages L1 and L2, each iteration would perform one mini-batch of denoising for L1, another one for L2, one mini-batch of on-the-fly backtranslation from L1 to L2, and another one from L2 to L1. Moreover, by further assuming that we have access to a small parallel corpus, the system can also be trained in a semi-supervised fashion by combining these steps with directly predicting the translations in this parallel corpus just as in standard NMT.
+
+# 4 EXPERIMENTAL SETTINGS
+
+We make our experiments comparable with previous work by using the French-English and GermanEnglish datasets from the WMT 2014 shared task2. Following common practice, the systems are evaluated on newstest2014 using tokenized BLEU scores as computed by the multi-bleu.perl script3. As for the training data, we test the proposed system under three different settings:
+
+• Unsupervised: This is the main scenario under consideration in our work, where the system has access to nothing but monolingual corpora. For that purpose, we used the News Crawl corpus with articles from 2007 to 2013.   
+• Semi-supervised: We assume that, in addition to monolingual corpora, we also have access to a small in-domain parallel corpus. This scenario has a great practical interest, as we might often have some parallel data from which we could potentially benefit, but it is insufficient to train a full traditional NMT system. For that purpose, we used the same monolingual data from the unsupervised settings together with either 10,000 or 100,000 random sentence pairs from the News Commentary parallel corpus.   
+Supervised: This is the traditional scenario in NMT where we have access to a large parallel corpus. While not the focus of our work, this setting should provide an approximate upper-bound for the proposed system. For that purpose, we used the combination of all parallel corpora provided at WMT 2014, which comprise Europarl, Common Crawl and News Commentary for both language pairs plus the UN and the Gigaword corpus for FrenchEnglish. For direct comparison with the semi-supervised scenario, we also ran separate experiments using the same subsets of News Commentary alone.
+
+Note that, to be faithful to our target scenario, we did not make use of any parallel data in these language pairs for development or tuning purposes. Instead, we used Spanish-English WMT data for our preliminary experiments, where we also decided all the hyperparameters without any rigorous exploration.
+
+As for the corpus preprocessing, we perform tokenization and truecasing using standard Moses tools.4 We then apply byte pair encoding (BPE) as proposed by Sennrich et al. (2016b) using the implementation provided by the authors5. Learning was done on the monolingual corpus of each language independently, using 50,000 operations. While BPE is known to be an effective way to overcome the rare word problem in standard NMT, it is less clear how it would perform in our more challenging unsupervised scenario, as it might be difficult to learn the translation relations between subword units. For that reason, we also run experiments at the word level in this unsupervised scenario, limiting the vocabulary to the most frequent 50,000 tokens and replacing the rest with a special token ${ \mathrm { < U N K > } }$ . We accelerate training by discarding all sentences with more than 50 elements (either BPE units or actual tokens).
+
+Given that the proposed system uses pre-trained cross-lingual embeddings in the encoder as described in Section 3.1, we use the monolingual corpora described above to independently train the embeddings for each language using word2vec (Mikolov et al., 2013b). More concretely, we use the skip-gram model with ten negative samples, a context window of ten words, 300 dimensions, a sub-sampling of $1 0 ^ { - 5 }$ , and ten training iterations. We then use the public implementation6 of the method proposed by Artetxe et al. (2017) to map these embeddings to a shared space, using the recommended configuration with numeral-based initialization. In addition to being a component of the proposed system, the resulting embeddings are also used to build a simple baseline system that translates a sentence word-by-word, replacing each word by their nearest neighbor in the other language and leaving out-of-vocabularies unchanged.
+
+The training of the proposed system itself is done using the procedure described in Section 3.2 with the cross-entropy loss function and a batch size of 50 sentences. For the unsupervised systems, we try using denoising alone as well as the combination of both denoising and backtranslation, in order to better analyze the contribution of the latter. We use Adam as our optimizer with a learning rate of $\alpha = 0 . 0 0 0 2$ (Kingma & Ba, 2015). During training, we use dropout regularization with a drop probability $p = 0 . 3$ . Given that we restrict ourselves not to use any parallel data for development purposes, we perform a fixed number of iterations (300,000) to train each variant. Using our PyTorch implementation, training each system took about 4-5 days on a single Titan X GPU for the full unsupervised variant. Although we observed that the system had not fully converged after this number of iterations in our preliminary experiments, we decide to stop training at this point in order to accelerate experimentation due to hardware constraints.
+
+As described in Section 3.2, we use greedy decoding at training time for backtranslation, but actual inference at test time was done using beam-search with a beam size of 12 following common practice (Sutskever et al., 2014; Sennrich et al., 2016a;b; He et al., 2016). We do not use any length or coverage penalty, which might further improve the reported results.
+
+# 5 RESULTS AND DISCUSSION
+
+We discuss the quantitative results in Section 5.1, and present a qualitative analysis in Section 5.2.
+
+# 5.1 QUANTITATIVE ANALYSIS
+
+The BLEU scores obtained by all the tested variants are reported in Table 1.
+
+As it can be seen, the proposed unsupervised system obtains very strong results considering that it was trained on nothing but monolingual corpora, reaching 14-15 BLEU points in French-English and 6-10 BLEU points in German-English depending on the variant and direction (rows 3 and 4). This is much stronger than the baseline system of word-by-word substitution (row 1), with improvements of at least $40 \%$ in all cases, and up to $140 \%$ in some (e.g. from 6.25 to 15.13 BLEU points in English French). This shows that the proposed system is able to go beyond very literal translations, effectively learning to use context information and account for the internal structure of the languages.
+
+The results also show that backtranslation is essential for the proposed system to work properly. In fact, the denoising technique alone is below the baseline (row 1 vs 2), while big improvements are seen when introducing backtranslation (row 2 vs 3). Test perplexities also confirm this: for instance, the proposed system with denoising alone obtains a per-word perplexity of 634.79 for French English, whereas the one with backtranslation achieves a much lower perplexity of 44.74. We emphasize, however, that the proposed training procedure would not work using backtranslation alone without denoising, as the initial translations would be meaningless sentences produced by a random NMT model, encouraging the system to completely ignore the input sentence and simply learn a language model of the target language. We thus conclude that both denoising and backtranslation play an essential role during training: denoising forces the system to capture broad word-level equivalences, while backtranslation encourages it to learn more subtle relations in an increasingly natural setting.
+
+Table 1: BLEU scores in newstest2014. Unsupervised systems are trained in the News Crawl monolingual corpus, semi-supervised systems are trained in the News Crawl monolingual corpus and a subset of the News Commentary parallel corpus, and supervised systems (provided for comparison) are trained in either these same subsets or the full parallel corpus, all from WMT 2014. For GNMT, we report the best single model scores from Wu et al. (2016).   
+
+<table><tr><td colspan="2"></td><td>FR-EN</td><td>EN-FR</td><td>DE-EN</td><td>EN-DE</td></tr><tr><td rowspan="4">Unsupervised</td><td>1.Baseline (emb. nearest neighbor)</td><td>9.98</td><td>6.25</td><td>7.07</td><td>4.39</td></tr><tr><td>2.Proposed (denoising)</td><td>7.28</td><td>5.33</td><td>3.64</td><td>2.40</td></tr><tr><td>3.Proposed (+ backtranslation)</td><td>15.56</td><td>15.13</td><td>10.21</td><td>6.55</td></tr><tr><td>4.Proposed (+ BPE)</td><td>15.56</td><td>14.36</td><td>10.16</td><td>6.89</td></tr><tr><td rowspan="2">Semi- supervised</td><td>5. Proposed (full) + 10k parallel</td><td>18.57</td><td>17.34</td><td>11.47</td><td>7.86</td></tr><tr><td>6.Proposed (full) + 100k parallel</td><td>21.81</td><td>21.74</td><td>15.24</td><td>10.95</td></tr><tr><td rowspan="4">Supervised</td><td>7. Comparable NMT (10k parallel)</td><td>1.88</td><td>1.66</td><td>1.33</td><td>0.82</td></tr><tr><td>8.Comparable NMT (100k parallel)</td><td>10.40</td><td>9.19</td><td>8.11</td><td>5.29</td></tr><tr><td>9. Comparable NMT (full parallel)</td><td>20.48</td><td>19.89</td><td>15.04</td><td>11.05</td></tr><tr><td>10. GNMT (Wu et al., 2016)</td><td>1</td><td>38.95</td><td>1</td><td>24.61</td></tr></table>
+
+As for the role of subword translation, we observe that BPE is slightly beneficial when German is the target language, detrimental when French is the target language, and practically equivalent when English is the target language (row 3 vs 4). This might be a bit surprising considering that the wordlevel system does not handle out-of-vocabularies in any way, so it always fails to translate rare words. Having a closer look, however, we observe that, while BPE manages to correctly translate some rare words, it also introduces some new errors. In particular, it sometimes happens that a subword unit from a rare word gets prefixed to a properly translated word, yielding to translations like SevAgency (split as S- ev- Agency). Moreover, we observe that BPE is of little help when translating infrequent named entities. For instance, we observed that our system translated Tymoshenko as Ebferchenko (split as Eb- fer- chenko). While standard NMT would easily learn to copy this kind of named entities using BPE, such relations are much more challenging to model under our unsupervised learning procedure. This way, we believe that a better handling of rare words and, in particular, named entities and numerals, could further improve the results in the future.
+
+In addition to that, the results of the semi-supervised system (rows 5 and 6) show that the proposed model can greatly benefit from a small parallel corpus. Note that these semi-supervised systems differ from the full unsupervised system (row 4) in the use of either 10,000 or 100,000 parallel sentences from News Crawl, so that their training alternates between denoising, backtranslation and, additionally, maximizing the translation probability of these parallel sentences as described in Section 3.2. As it can be seen, 10,000 parallel sentences alone bring an improvement of 1-3 BLEU points, while 100,000 sentences bring an improvement of 4-7 points. These results are much better than those of a comparable NMT system trained in the same parallel data (rows 7 and 8), showing the potential interest of our approach beyond the strictly unsupervised scenario. In fact, the semisupervised system trained in 100,000 parallel sentences (row 6) even surpasses the comparable NMT system trained in the full parallel corpus (row 9) in all cases but one, presumably because the domain of both the monolingual and the parallel corpora that it uses matches that of the test set.
+
+Table 2: Sample French English translations from newstest2014 by the full proposed system with BPE. See text for comments.   
+
+<table><tr><td>Source Reference</td><td></td><td>Proposed system (full)</td></tr><tr><td>Une fusillade a eu lieuä l&#x27;aéroport international de Los Angeles.</td><td>There was a shooting in Los An- geles International Airport.</td><td>A shooting occurred at Los An- geles International Airport.</td></tr><tr><td>Cette controverse croissante au- tour de l&#x27;agence a provoqué beaucoup de spéculations selon lesquelles l&#x27;incident de ce soir était le résultat d&#x27;une cyber- opération ciblée.</td><td>Such growing controversy sur- rounding the agency prompted early speculation that tonight&#x27;s incident was the result of a tar- geted cyber operation.</td><td>This growing scandal around the agency has caused much spec- ulation about how this incident was the outcome of a targeted cyber operation.</td></tr><tr><td>Le nombre total de morts en oc- tobre est le plusélevé depuis avril 2008,quand 1 073 person- nes avaient été tuées.</td><td>The total number of deaths in October is the highest since April 2008,when1,073 people were killed.</td><td>The total number of deaths in May is the highest since April 2008,when 1 064 people had been killed.</td></tr><tr><td>A l&#x27;exception de l&#x27;opéra,la province reste le parent pauvre de la culture en France.</td><td>With the exception of opera, the provinces remain the poor rela- tive of culture in France.</td><td>At an exception,opera remains of the state remains the poorest parent culture.</td></tr></table>
+
+As for the supervised system, it is remarkable that the comparable NMT model (rows 7-9), which uses the proposed architecture but trains it to predict the translations in the corresponding parallel corpus, obtains poor results compared to the state of the art in NMT (e.g. GNMT in row 10). Note that the comparable NMT system is equivalent to the semi-supervised system (rows 5 and 6), except that it does not use any monolingual corpora nor, consequently, denoising and backtranslation. As such, the comparable NMT differs from standard NMT in the use of a shared encoder with fixed embeddings (Section 3.1) and input corruption (Section 3.2).
+
+The relatively poor results of the comparable NMT model suggest that these additional constraints in our system, which were introduced to enable unsupervised learning, may also be a factor limiting its potential performance, so we believe that the system could be further improved in the future by progressively relaxing these constraints during training. For instance, using fixed cross-lingual embeddings in the encoder is necessary in the early stages of training, as it forces the encoder to use a common word representation for both languages, but it might also limit what it can ultimately learn in the process. For that reason, one could start to progressively update the weights of the encoder embeddings as training progresses. Similarly, one could also decouple the shared encoder into two independent encoders at some point during training, or progressively reduce the noise level. At the same time, note that we did not perform any rigorous hyperparameter exploration, and favored efficiency over performance in the experimental design due to hardware constraints. As such, we think that there is a considerable margin to improve these results by using larger models, longer training times, and incorporating several well-known NMT techniques (e.g. ensembling and length/coverage penalty).
+
+# 5.2 QUALITATIVE ANALYSIS
+
+In order to better understand the behavior of the proposed system, we manually analyzed some translations for French English, and present some illustrative examples in Table 2.
+
+Our analysis shows that the proposed system is able to produce high-quality translations, adequately modeling non-trivial translation relations. For instance, in the first example it translates the expression a eu lieu (literally ”has had place”) as occurred, going beyond a literal word-by-word substitution. At the same time, it correctly translates l’aeroport international de Los Angeles ´ as Los Angeles International Airport, properly modeling structural differences between the languages. As shown by the second example, the system is also capable of producing high-quality translations for considerably longer and more complex sentences.
+
+Nevertheless, our analysis also points that the proposed system has limitations and, perhaps not surprisingly, its translation quality often lags behind that of a standard supervised NMT system. In particular, we observe that the proposed model has difficulties to preserve some concrete details from source sentences. For instance, in the third example April and 2008 are properly translated, but octobre (”October”) is mistranslated as May and 1 073 as 1 064. While these clearly point to some adequacy issues, they are also understandable given the unsupervised nature of the system, and it is remarkable that the system managed to at least replace a month by another month and a number by another close number. We believe that incorporating character level information might help to mitigate some of these issues, as it could for instance favor October as the translation of octobre instead of the selected May.
+
+Finally, there are also some cases where there are both fluency and adequacy problems that severely hinders understanding the original message from the proposed translation. For instance, in the last example our system preserves most keywords in the original sentence, but it would be difficult to correctly guess its meaning just by looking at its translation. In concordance with our quantitative analysis, this suggests that there is still room for improvement, opening new research avenues for the future.
+
+# 6 CONCLUSIONS AND FUTURE WORK
+
+In this work, we propose a novel method to train an NMT system in a completely unsupervised manner. We build upon existing work on unsupervised cross-lingual embeddings (Artetxe et al., 2017; Zhang et al., 2017), and incorporate them in a modified attentional encoder-decoder model. By using a shared encoder with these fixed cross-lingual embeddings, we are able to train the system from monolingual corpora alone, combining denoising and backtranslation.
+
+The experiments show the effectiveness of our proposal, obtaining significant improvements in the BLEU score over a baseline system that performs word-by-word substitution in the standard WMT 2014 French-English and German-English benchmarks. Our manual analysis confirms the quality of the proposed system, showing that it is able to model complex cross-lingual relations and produce high-quality translations. Moreover, we show that combining our method with a small parallel corpus can bring further improvements, showing its potential interest beyond the strictly unsupervised scenario.
+
+Our work opens exciting opportunities for future research, as our analysis reveals that, in spite of the solid results, there is still a considerable room for improvement. In particular, we observe that the performance of a comparable supervised NMT system is considerably below the state of the art, which suggests that the architectural modifications introduced by our proposal (Section 3.1) are also limiting its potential performance. For that reason, we would like to explore progressively relaxing these constraints during training as discussed in Section 5.1. Additionally, we would like to incorporate character level information into the model, which we believe that could be very helpful to address some of the adequacy issues observed in our manual analysis (Section 5.2). Finally, we would like to explore other neighborhood functions for denoising, and analyze their effect in relation to the typological divergences of different language pairs.
+
+# ACKNOWLEDGMENTS
+
+This research was partially supported by a Google Faculty Award, the Spanish MINECO (TUNER TIN2015-65308-C5-1-R, MUSTER PCIN-2015-226 and TADEEP TIN2015-70214-P, cofunded by EU FEDER), the Basque Government (MODELA KK-2016/00082), the UPV/EHU (excellence research group), and the NVIDIA GPU grant program. Mikel Artetxe enjoys a doctoral grant from the Spanish MECD. Kyunghyun Cho thanks support by eBay, TenCent, Facebook, Google, NVIDIA and CIFAR, and was partly supported by Samsung Advanced Institute of Technology (Next Generation Deep Learning: from pattern recognition to AI).
+
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parse/train/Sy2ogebAW/Sy2ogebAW_content_list.json

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+        "text": "Mikel Artetxe, Gorka Labaka & Eneko Agirre IXA NLP Group University of the Basque Country (UPV/EHU) {mikel.artetxe,gorka.labaka,e.agirre}@ehu.eus ",
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+        "type": "text",
+        "text": "Kyunghyun Cho New York University CIFAR Azrieli Global Scholar kyunghyun.cho@nyu.edu ",
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+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "In spite of the recent success of neural machine translation (NMT) in standard benchmarks, the lack of large parallel corpora poses a major practical problem for many language pairs. There have been several proposals to alleviate this issue with, for instance, triangulation and semi-supervised learning techniques, but they still require a strong cross-lingual signal. In this work, we completely remove the need of parallel data and propose a novel method to train an NMT system in a completely unsupervised manner, relying on nothing but monolingual corpora. Our model builds upon the recent work on unsupervised embedding mappings, and consists of a slightly modified attentional encoder-decoder model that can be trained on monolingual corpora alone using a combination of denoising and backtranslation. Despite the simplicity of the approach, our system obtains 15.56 and 10.21 BLEU points in WMT 2014 French English and German English translation. The model can also profit from small parallel corpora, and attains 21.81 and 15.24 points when combined with 100,000 parallel sentences, respectively. Our implementation is released as an open source project1. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Neural machine translation (NMT) has recently become the dominant paradigm to machine translation (Bahdanau et al., 2014; Sutskever et al., 2014). As opposed to the traditional statistical machine translation (SMT), NMT systems are trained end-to-end, take advantage of continuous representations that greatly alleviate the sparsity problem, and make use of much larger contexts, thus mitigating the locality problem. Thanks to this, NMT has been reported to significantly improve over SMT both in automatic metrics and human evaluation (Wu et al., 2016). ",
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+        "type": "text",
+        "text": "Nevertheless, for the same reasons described above, NMT requires a large parallel corpus to be effective, and is known to fail when the training data is not big enough (Koehn & Knowles, 2017). Unfortunately, the lack of large parallel corpora is a practical problem for the vast majority of language pairs, including low-resource languages (e.g. Basque) as well as many combinations of major languages (e.g. German-Russian). Several authors have recently tried to address this problem using pivoting or triangulation techniques (Chen et al., 2017) as well as semi-supervised approaches (He et al., 2016), but these methods still require a strong cross-lingual signal. ",
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+        "type": "text",
+        "text": "In this work, we eliminate the need of cross-lingual information and propose a novel method to train NMT systems in a completely unsupervised manner, relying solely on monolingual corpora. Our approach builds upon the recent work on unsupervised cross-lingual embeddings (Artetxe et al., 2017; Zhang et al., 2017). Thanks to a shared encoder for both translation directions that uses these fixed cross-lingual embeddings, the entire system can be trained, with monolingual data, to reconstruct its input. In order to learn useful structural information, noise in the form of random token swaps is introduced in this input. In addition to denoising, we also incorporate backtranslation (Sennrich et al., 2016a) into the training procedure to further improve results. Figure 1 summarizes this general schema of the proposed system. ",
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+        "img_path": "images/2e405cbed5e113f97798717bed1c97c3d70953d97ca38689c4963a02ca6b6995.jpg",
+        "image_caption": [
+            "Figure 1: Architecture of the proposed system. For each sentence in language L1, the system is trained alternating two steps: denoising, which optimizes the probability of encoding a noised version of the sentence with the shared encoder and reconstructing it with the L1 decoder, and on-the-fly backtranslation, which translates the sentence in inference mode (encoding it with the shared encoder and decoding it with the L2 decoder) and then optimizes the probability of encoding this translated sentence with the shared encoder and recovering the original sentence with the L1 decoder. Training alternates between sentences in L1 and L2, with analogous steps for the latter. "
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+        "text": "",
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+        "type": "text",
+        "text": "In spite of the simplicity of the approach, our experiments show that the proposed system can reach up to 15.56 BLEU points for French English and 10.21 BLEU points for German English in the standard WMT 2014 translation task using nothing but monolingual training data. Moreover, we show that combining this method with a small parallel corpus can further improve the results, obtaining 21.81 and 15.24 BLEU points with 100,000 parallel sentences, respectively. Our manual analysis confirms the effectiveness of the proposed approach, revealing that the system is learning non-trivial translation relations that go beyond a word-by-word substitution. ",
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+        "text": "The remaining of this paper is organized as follows. Section 2 analyzes the related work. Section 3 then describes the proposed method. The experimental settings are discussed in Section 4, while Section 5 presents and discusses the obtained results. Section 6 concludes the paper. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "text": "We will first discuss unsupervised cross-lingual embeddings, which are the basis of our proposal, in Section 2.1. Section 2.2 then addresses statistical decipherment, an SMT-inspired approach to build a machine translation system in an unsupervised manner. Finally, Section 2.3 presents previous work on training NMT systems in different low-resource scenarios. ",
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+    {
+        "type": "text",
+        "text": "2.1 UNSUPERVISED CROSS-LINGUAL EMBEDDINGS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Most methods for learning cross-lingual word embeddings rely on some bilingual signal at the document level, typically in the form of parallel corpora (Gouws et al., 2015; Luong et al., 2015a). Closer to our scenario, embedding mapping methods independently train the embeddings in different languages using monolingual corpora, and then learn a linear transformation that maps them to a shared space based on a bilingual dictionary (Mikolov et al., 2013a; Lazaridou et al., 2015; Artetxe et al., 2016; Smith et al., 2017). While the dictionary used in these earlier work typically contains a few thousands entries, Artetxe et al. (2017) propose a simple self-learning extension that gives comparable results with an automatically generated list of numerals, which is used as a shortcut for practical unsupervised learning. Alternatively, adversarial training has also been proposed to learn such mappings in an unsupervised manner (Miceli Barone, 2016; Zhang et al., 2017). ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "2.2 STATISTICAL DECIPHERMENT FOR MACHINE TRANSLATION",
+        "text_level": 1,
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+        "type": "text",
+        "text": "There is a considerable body of work in statistical decipherment techniques to induce a machine translation model from monolingual data, which follows the same noisy-channel model used by SMT (Ravi & Knight, 2011; Dou & Knight, 2012). More concretely, they treat the source language as ciphertext, and model the process by which this ciphertext is generated as a two-stage process involving the generation of the original English sequence and the probabilistic replacement of the words in it. The English generative process is modeled using a standard n-gram language model, and the channel model parameters are estimated using either expectation maximization or Bayesian inference. This approach was shown to benefit from the incorporation of syntactic knowledge of the languages involved (Dou & Knight, 2013; Dou et al., 2015). More in line with our proposal, the use of word embeddings has also been shown to bring significant improvements in statistical decipherment for machine translation (Dou et al., 2015). ",
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+    {
+        "type": "text",
+        "text": "2.3 LOW-RESOURCE NEURAL MACHINE TRANSLATION ",
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+        "type": "text",
+        "text": "There have been several proposals to exploit resources other than direct parallel corpora to train NMT systems. The scenario that is most often considered is one where two languages have little or no parallel data between them but are well connected through a third language (e.g. there might be little direct resources for German-Russian but plenty for German-English and English-Russian). The most basic approach in this scenario is to independently translate from the source language to the pivot language and from the pivot language to the target language. It has however been shown that the use of more advanced models like a teacher-student framework can bring considerable improvements over this basic baseline (Firat et al., 2016b; Chen et al., 2017). In the same line, Johnson et al. (2017) show that a multilingual extension of a standard NMT architecture performs reasonably well even for language pairs for which no direct data was given during training. ",
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+        "type": "text",
+        "text": "In addition to that, there have been several attempts to exploit monolingual corpora for NMT in combination with the more scarce parallel corpora. A simple yet effective approach is to create a synthetic parallel corpus by backtranslating a monolingual corpus in the target language (Sennrich et al., 2016a). At the same time, Currey et al. (2017) showed that training an NMT system to directly copy target language text is also helpful and complementary with backtranslation. Finally, Ramachandran et al. (2017) pre-train the encoder and the decoder in language modeling. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To the best of our knowledge, the more ambitious scenario where an NMT model is trained from monolingual corpora alone has never been explored to date, but He et al. (2016) made an important contribution in this direction. More concretely, their method trains two agents to translate in opposite directions (e.g. French English and English French), and make them teach each other through a reinforcement learning process. While promising, this approach still requires a parallel corpus of a considerable size for a warm start (1.2 million sentences in the reported experiments), whereas our work does not use any parallel data at all. ",
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+    {
+        "type": "text",
+        "text": "3 PROPOSED METHOD ",
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+        "type": "text",
+        "text": "This section describes the proposed unsupervised NMT approach. Section 3.1 first presents the architecture of the proposed system, and Section 3.2 then describes the method to train it in an unsupervised manner. ",
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+        "type": "text",
+        "text": "3.1 SYSTEM ARCHITECTURE ",
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+        "type": "text",
+        "text": "As shown in Figure 1, the proposed system follows a fairly standard encoder-decoder architecture with an attention mechanism (Bahdanau et al., 2014). More concretely, we use a two-layer bidirectional RNN in the encoder, and another two-layer RNN in the decoder. All RNNs use GRU cells with 600 hidden units (Cho et al., 2014), and the dimensionality of the embeddings is set to 300. As for the attention mechanism, we use the global attention method proposed by Luong et al. (2015b) with the general alignment function. There are, however, three important aspects in which our system differs from the standard NMT, and these are critical so the system can be trained in an unsupervised manner as described next in Section 3.2: ",
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+        "type": "text",
+        "text": "1. Dual structure. While NMT systems are typically built for a specific translation direction (e.g. either French English or English French), we exploit the dual nature of machine translation (He et al., 2016; Firat et al., 2016a) and handle both directions together (e.g. French English).   \n2. Shared encoder. Our system makes use of one and only one encoder that is shared by both languages involved, similarly to Ha et al. (2016), Lee et al. (2017) and Johnson et al. (2017). For instance, the exact same encoder would be used for both French and English. This universal encoder is aimed to produce a language independent representation of the input text, which each decoder should then transform into its corresponding language.   \n3. Fixed embeddings in the encoder. While most NMT systems randomly initialize their embeddings and update them during training, we use pre-trained cross-lingual embeddings in the encoder that are kept fixed during training. This way, the encoder is given language independent word-level representations, and it only needs to learn how to compose them to build representations of larger phrases. As discussed in Section 2.1, there are several unsupervised methods to train these cross-lingual embeddings from monolingual corpora, so this is perfectly feasible in our scenario. Note that, even if the embeddings are crosslingual, we use separate vocabularies for each language. This way, the word chair, which exists both in French and English (meaning “flesh” in the former), would get a different vector in each language, although they would both be in a common space. ",
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+    {
+        "type": "text",
+        "text": "3.2 UNSUPERVISED TRAINING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As NMT systems are typically trained to predict the translations in a parallel corpus, such supervised training procedure is infeasible in our scenario, where we only have access to monolingual corpora. However, thanks to the architectural modifications proposed above, we are able to train the entire system in an unsupervised manner using the following two strategies: ",
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+    {
+        "type": "text",
+        "text": "1. Denoising. Thanks to the use of a shared encoder, and exploiting the dual structure of machine translation, the proposed system can be directly trained to reconstruct its own input. More concretely, the whole system can be optimized to take an input sentence in a given language, encode it using the shared encoder, and reconstruct the original sentence using the decoder of that language. Given that we use pre-trained cross-lingual embeddings in the shared encoder, this encoder should learn to compose the embeddings of both languages in a language-independent fashion, and each decoder should learn to decompose this representation into their corresponding language. At inference time, we simply replace the decoder with that of the target language, so it generates the translation of the input text from the language-independent representation given by the encoder. ",
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+        "type": "text",
+        "text": "Nevertheless, this ideal behavior is severely compromised by the fact that the resulting training procedure is essentially a trivial copying task. As such, the optimal solution for this task would not need to capture any real knowledge of the languages involved, as there would be many degenerated solutions that blindly copy all the elements in the input sequence. If this were the case, the system would at best make very literal word-by-word substitutions when used to translate from one language to another at inference time. ",
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+    {
+        "type": "text",
+        "text": "In order to avoid such degenerated solutions and make the encoder truly learn the compositionality of its input words in a language independent manner, we propose to introduce random noise in the input sentences. The idea is to exploit the same underlying principle of denoising autoencoders (Vincent et al., 2010), where the system is trained to reconstruct the original version of a corrupted input sentence (Dai & Le, 2015; Hill et al., 2016). For that purpose, we alter the word order of the input sentence by making random swaps between contiguous words. More concretely, for a sequence of $N$ elements, we make $N / 2$ random swaps of this kind. This way, the system needs to learn about the internal structure of the languages involved to be able to recover the correct word order. At the same time, by discouraging the system to rely too much on the word order of the input sequence, we can better account for the actual word order divergences across languages. This training procedure can be seen as an instance of contrastive estimation (Smith & Eisner, 2005), where the neighborhood is defined by local swaps in our case, although other functions would also be possible. ",
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+    {
+        "type": "text",
+        "text": "2. On-the-fly backtranslation. In spite of the denoising strategy, the training procedure above is still a copying task with some synthetic alterations that, most importantly, involves a single language at each time, without considering our final goal of translating between two languages. In order to train our system in a true translation setting without violating the constraint of using nothing but monolingual corpora, we propose to adapt the backtranslation approach proposed by Sennrich et al. (2016a) to our scenario. More concretely, given an input sentence in one language, we use the system in inference mode with greedy decoding to translate it to the other language (i.e. apply the shared encoder and the decoder of the other language). This way, we obtain a pseudo-parallel sentence pair, and train the system to predict the original sentence from this synthetic translation. Note that, contrary to standard backtranslation, which uses an independent model to backtranslate the entire corpus at one time, we take advantage of the dual structure of the proposed architecture to backtranslate each mini-batch on-the-fly using the model that is being trained itself. This way, as training progresses and the model improves, it will produce better synthetic sentence pairs through backtranslation, which will serve to further improve the model in the following iterations. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "During training, we alternate these different training objectives from mini-batch to mini-batch. This way, given two languages L1 and L2, each iteration would perform one mini-batch of denoising for L1, another one for L2, one mini-batch of on-the-fly backtranslation from L1 to L2, and another one from L2 to L1. Moreover, by further assuming that we have access to a small parallel corpus, the system can also be trained in a semi-supervised fashion by combining these steps with directly predicting the translations in this parallel corpus just as in standard NMT. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL SETTINGS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We make our experiments comparable with previous work by using the French-English and GermanEnglish datasets from the WMT 2014 shared task2. Following common practice, the systems are evaluated on newstest2014 using tokenized BLEU scores as computed by the multi-bleu.perl script3. As for the training data, we test the proposed system under three different settings: ",
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+        "text": "• Unsupervised: This is the main scenario under consideration in our work, where the system has access to nothing but monolingual corpora. For that purpose, we used the News Crawl corpus with articles from 2007 to 2013.   \n• Semi-supervised: We assume that, in addition to monolingual corpora, we also have access to a small in-domain parallel corpus. This scenario has a great practical interest, as we might often have some parallel data from which we could potentially benefit, but it is insufficient to train a full traditional NMT system. For that purpose, we used the same monolingual data from the unsupervised settings together with either 10,000 or 100,000 random sentence pairs from the News Commentary parallel corpus.   \nSupervised: This is the traditional scenario in NMT where we have access to a large parallel corpus. While not the focus of our work, this setting should provide an approximate upper-bound for the proposed system. For that purpose, we used the combination of all parallel corpora provided at WMT 2014, which comprise Europarl, Common Crawl and News Commentary for both language pairs plus the UN and the Gigaword corpus for FrenchEnglish. For direct comparison with the semi-supervised scenario, we also ran separate experiments using the same subsets of News Commentary alone. ",
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+    {
+        "type": "text",
+        "text": "Note that, to be faithful to our target scenario, we did not make use of any parallel data in these language pairs for development or tuning purposes. Instead, we used Spanish-English WMT data for our preliminary experiments, where we also decided all the hyperparameters without any rigorous exploration. ",
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+    {
+        "type": "text",
+        "text": "As for the corpus preprocessing, we perform tokenization and truecasing using standard Moses tools.4 We then apply byte pair encoding (BPE) as proposed by Sennrich et al. (2016b) using the implementation provided by the authors5. Learning was done on the monolingual corpus of each language independently, using 50,000 operations. While BPE is known to be an effective way to overcome the rare word problem in standard NMT, it is less clear how it would perform in our more challenging unsupervised scenario, as it might be difficult to learn the translation relations between subword units. For that reason, we also run experiments at the word level in this unsupervised scenario, limiting the vocabulary to the most frequent 50,000 tokens and replacing the rest with a special token ${ \\mathrm { < U N K > } }$ . We accelerate training by discarding all sentences with more than 50 elements (either BPE units or actual tokens). ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Given that the proposed system uses pre-trained cross-lingual embeddings in the encoder as described in Section 3.1, we use the monolingual corpora described above to independently train the embeddings for each language using word2vec (Mikolov et al., 2013b). More concretely, we use the skip-gram model with ten negative samples, a context window of ten words, 300 dimensions, a sub-sampling of $1 0 ^ { - 5 }$ , and ten training iterations. We then use the public implementation6 of the method proposed by Artetxe et al. (2017) to map these embeddings to a shared space, using the recommended configuration with numeral-based initialization. In addition to being a component of the proposed system, the resulting embeddings are also used to build a simple baseline system that translates a sentence word-by-word, replacing each word by their nearest neighbor in the other language and leaving out-of-vocabularies unchanged. ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "The training of the proposed system itself is done using the procedure described in Section 3.2 with the cross-entropy loss function and a batch size of 50 sentences. For the unsupervised systems, we try using denoising alone as well as the combination of both denoising and backtranslation, in order to better analyze the contribution of the latter. We use Adam as our optimizer with a learning rate of $\\alpha = 0 . 0 0 0 2$ (Kingma & Ba, 2015). During training, we use dropout regularization with a drop probability $p = 0 . 3$ . Given that we restrict ourselves not to use any parallel data for development purposes, we perform a fixed number of iterations (300,000) to train each variant. Using our PyTorch implementation, training each system took about 4-5 days on a single Titan X GPU for the full unsupervised variant. Although we observed that the system had not fully converged after this number of iterations in our preliminary experiments, we decide to stop training at this point in order to accelerate experimentation due to hardware constraints. ",
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+        "type": "text",
+        "text": "As described in Section 3.2, we use greedy decoding at training time for backtranslation, but actual inference at test time was done using beam-search with a beam size of 12 following common practice (Sutskever et al., 2014; Sennrich et al., 2016a;b; He et al., 2016). We do not use any length or coverage penalty, which might further improve the reported results. ",
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+        "type": "text",
+        "text": "5 RESULTS AND DISCUSSION ",
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+        "text": "We discuss the quantitative results in Section 5.1, and present a qualitative analysis in Section 5.2. ",
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+    {
+        "type": "text",
+        "text": "5.1 QUANTITATIVE ANALYSIS ",
+        "text_level": 1,
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+        "text": "The BLEU scores obtained by all the tested variants are reported in Table 1. ",
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+        "type": "text",
+        "text": "As it can be seen, the proposed unsupervised system obtains very strong results considering that it was trained on nothing but monolingual corpora, reaching 14-15 BLEU points in French-English and 6-10 BLEU points in German-English depending on the variant and direction (rows 3 and 4). This is much stronger than the baseline system of word-by-word substitution (row 1), with improvements of at least $40 \\%$ in all cases, and up to $140 \\%$ in some (e.g. from 6.25 to 15.13 BLEU points in English French). This shows that the proposed system is able to go beyond very literal translations, effectively learning to use context information and account for the internal structure of the languages. ",
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+        "text": "The results also show that backtranslation is essential for the proposed system to work properly. In fact, the denoising technique alone is below the baseline (row 1 vs 2), while big improvements are seen when introducing backtranslation (row 2 vs 3). Test perplexities also confirm this: for instance, the proposed system with denoising alone obtains a per-word perplexity of 634.79 for French English, whereas the one with backtranslation achieves a much lower perplexity of 44.74. We emphasize, however, that the proposed training procedure would not work using backtranslation alone without denoising, as the initial translations would be meaningless sentences produced by a random NMT model, encouraging the system to completely ignore the input sentence and simply learn a language model of the target language. We thus conclude that both denoising and backtranslation play an essential role during training: denoising forces the system to capture broad word-level equivalences, while backtranslation encourages it to learn more subtle relations in an increasingly natural setting. ",
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+        "table_caption": [
+            "Table 1: BLEU scores in newstest2014. Unsupervised systems are trained in the News Crawl monolingual corpus, semi-supervised systems are trained in the News Crawl monolingual corpus and a subset of the News Commentary parallel corpus, and supervised systems (provided for comparison) are trained in either these same subsets or the full parallel corpus, all from WMT 2014. For GNMT, we report the best single model scores from Wu et al. (2016). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td>FR-EN</td><td>EN-FR</td><td>DE-EN</td><td>EN-DE</td></tr><tr><td rowspan=\"4\">Unsupervised</td><td>1.Baseline (emb. nearest neighbor)</td><td>9.98</td><td>6.25</td><td>7.07</td><td>4.39</td></tr><tr><td>2.Proposed (denoising)</td><td>7.28</td><td>5.33</td><td>3.64</td><td>2.40</td></tr><tr><td>3.Proposed (+ backtranslation)</td><td>15.56</td><td>15.13</td><td>10.21</td><td>6.55</td></tr><tr><td>4.Proposed (+ BPE)</td><td>15.56</td><td>14.36</td><td>10.16</td><td>6.89</td></tr><tr><td rowspan=\"2\">Semi- supervised</td><td>5. Proposed (full) + 10k parallel</td><td>18.57</td><td>17.34</td><td>11.47</td><td>7.86</td></tr><tr><td>6.Proposed (full) + 100k parallel</td><td>21.81</td><td>21.74</td><td>15.24</td><td>10.95</td></tr><tr><td rowspan=\"4\">Supervised</td><td>7. Comparable NMT (10k parallel)</td><td>1.88</td><td>1.66</td><td>1.33</td><td>0.82</td></tr><tr><td>8.Comparable NMT (100k parallel)</td><td>10.40</td><td>9.19</td><td>8.11</td><td>5.29</td></tr><tr><td>9. Comparable NMT (full parallel)</td><td>20.48</td><td>19.89</td><td>15.04</td><td>11.05</td></tr><tr><td>10. GNMT (Wu et al., 2016)</td><td>1</td><td>38.95</td><td>1</td><td>24.61</td></tr></table>",
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+        "text": "",
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+        "type": "text",
+        "text": "As for the role of subword translation, we observe that BPE is slightly beneficial when German is the target language, detrimental when French is the target language, and practically equivalent when English is the target language (row 3 vs 4). This might be a bit surprising considering that the wordlevel system does not handle out-of-vocabularies in any way, so it always fails to translate rare words. Having a closer look, however, we observe that, while BPE manages to correctly translate some rare words, it also introduces some new errors. In particular, it sometimes happens that a subword unit from a rare word gets prefixed to a properly translated word, yielding to translations like SevAgency (split as S- ev- Agency). Moreover, we observe that BPE is of little help when translating infrequent named entities. For instance, we observed that our system translated Tymoshenko as Ebferchenko (split as Eb- fer- chenko). While standard NMT would easily learn to copy this kind of named entities using BPE, such relations are much more challenging to model under our unsupervised learning procedure. This way, we believe that a better handling of rare words and, in particular, named entities and numerals, could further improve the results in the future. ",
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+    {
+        "type": "text",
+        "text": "In addition to that, the results of the semi-supervised system (rows 5 and 6) show that the proposed model can greatly benefit from a small parallel corpus. Note that these semi-supervised systems differ from the full unsupervised system (row 4) in the use of either 10,000 or 100,000 parallel sentences from News Crawl, so that their training alternates between denoising, backtranslation and, additionally, maximizing the translation probability of these parallel sentences as described in Section 3.2. As it can be seen, 10,000 parallel sentences alone bring an improvement of 1-3 BLEU points, while 100,000 sentences bring an improvement of 4-7 points. These results are much better than those of a comparable NMT system trained in the same parallel data (rows 7 and 8), showing the potential interest of our approach beyond the strictly unsupervised scenario. In fact, the semisupervised system trained in 100,000 parallel sentences (row 6) even surpasses the comparable NMT system trained in the full parallel corpus (row 9) in all cases but one, presumably because the domain of both the monolingual and the parallel corpora that it uses matches that of the test set. ",
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+    {
+        "type": "table",
+        "img_path": "images/7bf8ec4af1c409c19e90c2d6ee353156f65439f18cae7c5da5e7f65a81a60d00.jpg",
+        "table_caption": [
+            "Table 2: Sample French English translations from newstest2014 by the full proposed system with BPE. See text for comments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Source Reference</td><td></td><td>Proposed system (full)</td></tr><tr><td>Une fusillade a eu lieuä l&#x27;aéroport international de Los Angeles.</td><td>There was a shooting in Los An- geles International Airport.</td><td>A shooting occurred at Los An- geles International Airport.</td></tr><tr><td>Cette controverse croissante au- tour de l&#x27;agence a provoqué beaucoup de spéculations selon lesquelles l&#x27;incident de ce soir était le résultat d&#x27;une cyber- opération ciblée.</td><td>Such growing controversy sur- rounding the agency prompted early speculation that tonight&#x27;s incident was the result of a tar- geted cyber operation.</td><td>This growing scandal around the agency has caused much spec- ulation about how this incident was the outcome of a targeted cyber operation.</td></tr><tr><td>Le nombre total de morts en oc- tobre est le plusélevé depuis avril 2008,quand 1 073 person- nes avaient été tuées.</td><td>The total number of deaths in October is the highest since April 2008,when1,073 people were killed.</td><td>The total number of deaths in May is the highest since April 2008,when 1 064 people had been killed.</td></tr><tr><td>A l&#x27;exception de l&#x27;opéra,la province reste le parent pauvre de la culture en France.</td><td>With the exception of opera, the provinces remain the poor rela- tive of culture in France.</td><td>At an exception,opera remains of the state remains the poorest parent culture.</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "As for the supervised system, it is remarkable that the comparable NMT model (rows 7-9), which uses the proposed architecture but trains it to predict the translations in the corresponding parallel corpus, obtains poor results compared to the state of the art in NMT (e.g. GNMT in row 10). Note that the comparable NMT system is equivalent to the semi-supervised system (rows 5 and 6), except that it does not use any monolingual corpora nor, consequently, denoising and backtranslation. As such, the comparable NMT differs from standard NMT in the use of a shared encoder with fixed embeddings (Section 3.1) and input corruption (Section 3.2). ",
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+    {
+        "type": "text",
+        "text": "The relatively poor results of the comparable NMT model suggest that these additional constraints in our system, which were introduced to enable unsupervised learning, may also be a factor limiting its potential performance, so we believe that the system could be further improved in the future by progressively relaxing these constraints during training. For instance, using fixed cross-lingual embeddings in the encoder is necessary in the early stages of training, as it forces the encoder to use a common word representation for both languages, but it might also limit what it can ultimately learn in the process. For that reason, one could start to progressively update the weights of the encoder embeddings as training progresses. Similarly, one could also decouple the shared encoder into two independent encoders at some point during training, or progressively reduce the noise level. At the same time, note that we did not perform any rigorous hyperparameter exploration, and favored efficiency over performance in the experimental design due to hardware constraints. As such, we think that there is a considerable margin to improve these results by using larger models, longer training times, and incorporating several well-known NMT techniques (e.g. ensembling and length/coverage penalty). ",
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+    {
+        "type": "text",
+        "text": "5.2 QUALITATIVE ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In order to better understand the behavior of the proposed system, we manually analyzed some translations for French English, and present some illustrative examples in Table 2. ",
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+    {
+        "type": "text",
+        "text": "Our analysis shows that the proposed system is able to produce high-quality translations, adequately modeling non-trivial translation relations. For instance, in the first example it translates the expression a eu lieu (literally ”has had place”) as occurred, going beyond a literal word-by-word substitution. At the same time, it correctly translates l’aeroport international de Los Angeles ´ as Los Angeles International Airport, properly modeling structural differences between the languages. As shown by the second example, the system is also capable of producing high-quality translations for considerably longer and more complex sentences. ",
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+    {
+        "type": "text",
+        "text": "Nevertheless, our analysis also points that the proposed system has limitations and, perhaps not surprisingly, its translation quality often lags behind that of a standard supervised NMT system. In particular, we observe that the proposed model has difficulties to preserve some concrete details from source sentences. For instance, in the third example April and 2008 are properly translated, but octobre (”October”) is mistranslated as May and 1 073 as 1 064. While these clearly point to some adequacy issues, they are also understandable given the unsupervised nature of the system, and it is remarkable that the system managed to at least replace a month by another month and a number by another close number. We believe that incorporating character level information might help to mitigate some of these issues, as it could for instance favor October as the translation of octobre instead of the selected May. ",
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+    {
+        "type": "text",
+        "text": "Finally, there are also some cases where there are both fluency and adequacy problems that severely hinders understanding the original message from the proposed translation. For instance, in the last example our system preserves most keywords in the original sentence, but it would be difficult to correctly guess its meaning just by looking at its translation. In concordance with our quantitative analysis, this suggests that there is still room for improvement, opening new research avenues for the future. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS AND FUTURE WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this work, we propose a novel method to train an NMT system in a completely unsupervised manner. We build upon existing work on unsupervised cross-lingual embeddings (Artetxe et al., 2017; Zhang et al., 2017), and incorporate them in a modified attentional encoder-decoder model. By using a shared encoder with these fixed cross-lingual embeddings, we are able to train the system from monolingual corpora alone, combining denoising and backtranslation. ",
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+    {
+        "type": "text",
+        "text": "The experiments show the effectiveness of our proposal, obtaining significant improvements in the BLEU score over a baseline system that performs word-by-word substitution in the standard WMT 2014 French-English and German-English benchmarks. Our manual analysis confirms the quality of the proposed system, showing that it is able to model complex cross-lingual relations and produce high-quality translations. Moreover, we show that combining our method with a small parallel corpus can bring further improvements, showing its potential interest beyond the strictly unsupervised scenario. ",
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+        "type": "text",
+        "text": "Our work opens exciting opportunities for future research, as our analysis reveals that, in spite of the solid results, there is still a considerable room for improvement. In particular, we observe that the performance of a comparable supervised NMT system is considerably below the state of the art, which suggests that the architectural modifications introduced by our proposal (Section 3.1) are also limiting its potential performance. For that reason, we would like to explore progressively relaxing these constraints during training as discussed in Section 5.1. Additionally, we would like to incorporate character level information into the model, which we believe that could be very helpful to address some of the adequacy issues observed in our manual analysis (Section 5.2). Finally, we would like to explore other neighborhood functions for denoising, and analyze their effect in relation to the typological divergences of different language pairs. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This research was partially supported by a Google Faculty Award, the Spanish MINECO (TUNER TIN2015-65308-C5-1-R, MUSTER PCIN-2015-226 and TADEEP TIN2015-70214-P, cofunded by EU FEDER), the Basque Government (MODELA KK-2016/00082), the UPV/EHU (excellence research group), and the NVIDIA GPU grant program. Mikel Artetxe enjoys a doctoral grant from the Spanish MECD. Kyunghyun Cho thanks support by eBay, TenCent, Facebook, Google, NVIDIA and CIFAR, and was partly supported by Samsung Advanced Institute of Technology (Next Generation Deep Learning: from pattern recognition to AI). ",
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+    {
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+# LATENT CONSTRAINTS: LEARNING TO GENERATE CONDITIONALLY FROM UNCONDITIONAL GENERATIVE MODELS
+
+Jesse Engel   
+Google Brain   
+San Francisco, CA, USA
+
+Matthew D. Hoffman Google Inc. San Francisco, CA, USA
+
+Adam Roberts Google Brain San Francisco, CA, USA
+
+# ABSTRACT
+
+Deep generative neural networks have proven effective at both conditional and unconditional modeling of complex data distributions. Conditional generation enables interactive control, but creating new controls often requires expensive retraining. In this paper, we develop a method to condition generation without retraining the model. By post-hoc learning latent constraints, value functions that identify regions in latent space that generate outputs with desired attributes, we can conditionally sample from these regions with gradient-based optimization or amortized actor functions. Combining attribute constraints with a universal “realism” constraint, which enforces similarity to the data distribution, we generate realistic conditional images from an unconditional variational autoencoder. Further, using gradient-based optimization, we demonstrate identity-preserving transformations that make the minimal adjustment in latent space to modify the attributes of an image. Finally, with discrete sequences of musical notes, we demonstrate zero-shot conditional generation, learning latent constraints in the absence of labeled data or a differentiable reward function.
+
+# 1 INTRODUCTION
+
+Generative modeling of complicated data such as images and audio is a long-standing challenge in machine learning. While unconditional sampling is an interesting technical problem, it is arguably of limited practical interest in its own right: if one needs a non-specific image (or sound, song, document, etc.), one can simply pull something at random from the unfathomably vast media databases on the web. But that naive approach may not work for conditional sampling (i.e., generating data to match a set of user-specified attributes), since as more attributes are specified, it becomes exponentially less likely that a satisfactory example can be pulled from a database. One might also want to modify some attributes of an object while preserving its core identity. These are crucial tasks in creative applications, where the typical user desires fine-grained controls (Bernardo et al., 2017).
+
+One can enforce user-specified constraints at training time, either by training on a curated subset of data or with conditioning variables. These approaches can be effective if there is enough labeled data available, but they require expensive model retraining for each new set of constraints and may not leverage commonalities between tasks. Deep latent-variable models, such as Generative Adversarial Networks (GANs; Goodfellow et al., 2014) and Variational Autoencoders (VAEs; Kingma & Welling, 2013; Rezende et al., 2014), learn to unconditionally generate realistic and varied outputs by sampling from a semantically structured latent space. One might hope to leverage that structure in creating new conditional controls for sampling and transformations (Brock et al., 2016).
+
+Here, we show that new constraints can be enforced post-hoc on pre-trained unsupervised generative models. This approach removes the need to retrain the model for each new set of constraints, allowing users to more easily define custom behavior. We separate the problem into (1) creating an unsupervised model that learns how to reconstruct data from latent embeddings, and (2) leveraging the latent structure exposed in that embedding space as a source of prior knowledge, upon which we can impose behavioral constraints.
+
+Our key contributions are as follows:
+
+![](images/9b445d4241da4a0a100e524c0e5fec9fef7838190a45a3b31264de73da209d9a.jpg)  
+Figure 1: (a) Diagram of latent constraints for a VAE. We use one critic $D _ { \mathrm { a t t r } }$ to predict which regions of the latent space will generate outputs with desired attributes, and another critic $D _ { \mathrm { r e a l i s m } }$ to predict which regions have high mass under the marginal posterior, $q ( z )$ , of the training data. (b) We begin by pretraining a standard VAE, with an emphasis on achieving good reconstructions. (c) To train the actor-critic pair we use constraint-satisfaction labels, $c .$ , to train $D$ to discriminate between encodings of actual data, $z \sim q ( z | x )$ , versus latent vectors $z \sim p ( z )$ sampled from the prior or transformed prior samples $G ( z \sim p ( z ) , y )$ . Similar to a Conditional GAN, both $G$ and $D$ operate on a concatenation of $z$ and a binary attribute vector, $y$ , allowing $G$ to learn conditional mappings in latent space. If $G$ is an optimizer, a separate attribute discriminator, $D _ { \mathrm { a t t r } }$ is trained and the latent vector is optimized to reduce the cost of both $D _ { \mathrm { a t t r } }$ and $D _ { \mathrm { r e a l i s m } }$ . (d) To sample from the intersection of these regions, we use either gradient-based optimization or an amortized generator, $G$ , to shift latent samples from either the prior $z \sim p ( z )$ , sampling) or from the data $( z \sim q ( z | x )$ , transformation).
+
+• We show that it is possible to generate conditionally from an unconditional model, learning a critic function $D ( z )$ in latent space and generating high-value samples with either gradient-based optimization or an amortized actor function $G ( z )$ , even with a nondifferentiable decoder (e.g., discrete sequences). Focusing on VAEs, we address the tradeoff between reconstruction quality and sample quality (without sacrificing diversity) by enforcing a universal “realism” constraint that requires samples in latent space to be indistinguishable from encoded data (rather than prior samples). Because we start from a VAE that can reconstruct inputs well, we are able to apply identitypreserving transformations by making the minimal adjustment in latent space needed to satisfy the desired constraints. For example, when we adjust a person’s expression or hair, the result is still clearly identifiable as the same person (see Figure 5). This contrasts with pure GAN-based transformation approaches, which often fail to preserve identity. Zero-shot conditional generation. Using samples from the VAE to generate exemplars, we can learn an actor-critic pair that satisfies user-specified rule-based constraints in the absence of any labeled data.
+
+# 2 BACKGROUND
+
+Decoder-based deep generative models such as VAEs and GANs generate samples that approximate a population distribution $p ^ { \star } ( x )$ by passing samples from some simple tractable distribution $p ( z )$ (often $p ( z ) \ \triangleq \ N ( 0 , I ) )$ through a deep neural network. GANs are trained to fool an auxiliary classifier that tries to learn to distinguish between real and synthetic samples. VAEs are fit to data using a variational approximation to maximum-likelihood estimation:
+
+$$
+\begin{array} { r } { \mathcal { L } ^ { \mathrm { E L B O } } \triangleq \frac 1 N \sum _ { n } \mathbb { E } _ { z \sim q ( z | x _ { n } ) } [ \log \pi ( x _ { n } ; g ( z ) ) ] - \mathrm { K L } ( q ( z \mid x _ { n } ) \mid | p ( z ) ) \le \frac 1 N \sum _ { n } \log p ( x _ { n } ) , } \end{array}
+$$
+
+![](images/542a8eab718a513b2c83e30bbbfd38da241907c397d572b24d2d635aa1f3da90.jpg)  
+Figure 2: Typical VAEs use a pixel-wise data likelihood, $\mathcal { N } ( \mu _ { x } ( z ) , \sigma _ { x } I )$ , with $\sigma _ { x } = 1$ to produce coherent samples at the expense of visual and conceptual blurriness (Row 3). Some reconstructions (Row 2) actually change attributes of the original data. Decreasing $\sigma _ { x }$ to 0.1 maximizes the ELBO (supplemental Table 4) and increases the fidelity of reconstructions (Row 4) at the cost of sample realism (Row 5). Using an actor to shift prior samples to satisfy the realism constraint, we achieve more realistic samples without sacrificing sharpness (Row 6). The samples are mapped to the closest point in latent space that both satisfies the realism constraint and has the same attributes as the original data.
+
+where the “encoder” distribution $q ( z \mid x )$ is an approximation to the posterior $p ( z \mid x )$ , $\pi ( x ; g ( z ) ) \triangleq$ $p ( x \mid z )$ is a tractable likelihood function that depends on some parameters output by a “decoder” function $g ( z )$ , and $q$ and $g$ are fit to maximize the evidence lower bound (ELBO) $\dot { \mathcal { L } } ^ { \mathrm { E L B O } }$ . The likelihood $\pi ( x ; g )$ is often chosen to be a product of simple distributions such as $\pi ( x ; g ) = \mathcal { N } ( x ; g , \sigma _ { x } ^ { 2 } I )$ for continuous data or $\begin{array} { r } { \pi ( x ; g ) = \tilde { \prod _ { d } \mathrm { B e r n o u l l i } } ( x _ { d } ; g _ { d } ) } \end{array}$ for binary data.
+
+GANs and VAEs have complementary strengths and weaknesses. GANs suffer from the “modecollapse” problem, where the generator assigns mass to a small subset of the support of the population distribution—that is, it may generate realistic samples, but there are many more realistic samples that it cannot generate. This is particularly problematic if we want to use GANs to manipulate data rather than generate new data; even GAN variants that include some kind of inference machinery (e.g., Donahue et al., 2016; Dumoulin et al., 2016; Perarnau et al., 2016) to determine what $z$ best matches some $x$ tend to produce reconstructions that are reminiscent of the input but do not preserve its identity.
+
+On the other hand, VAEs (especially those with simple likelihoods $\pi$ ) often exhibit a tradeoff between sharp reconstructions and sensible-looking samples (see Figure 2). That is, depending on what hyperparameters they are trained with (e.g., latent dimensionality and the scale of the likelihood term), VAEs tend to either produce blurry reconstructions and plausible (but blurry) novel samples, or bizarre samples but sharp reconstructions. It has been argued (Makhzani et al., 2016) that this is due to the “holes” problem; the decoder is trained on samples from the marginal posterior $q ( z ) \triangleq { \frac { 1 } { N } } \sum _ { n } q ( z \mid x _ { n } )$ , which may have very high KL divergence to the presupposed marginal $p ( z )$ (Hoffman & Johnson, 2016). In particular, if the decoder, $g ( z )$ , can reconstruct arbitrary values of $x$ with high accuracy (as in the case of small $\sigma _ { x }$ ) then the typical posterior $p ( z \mid x )$ will be highly concentrated. We show this experimentally in supplemental Figure 16. If $q ( z \mid x )$ underestimates the posterior variance (as it usually does), then the marginal posterior $q ( z )$ will also be highly concentrated, and samples from $\begin{array} { r } { p ( x ) \stackrel { \cdot } { = } \int _ { z } p ( z ) p ( x \mid z ) d z } \end{array}$ may produce results that are far from typical reconstructions $\mathbb { E } _ { p } [ x \mid z \sim q ( z \mid x ) ]$ . If we tune $\sigma _ { x }$ to maximize the ELBO (Bishop, 2006), we find the optimal $\sigma _ { x } \approx 0 . 1$ (supplemental Table 4). Figure 2 shows that this choice does indeed lead to good reconstructions but strange-looking samples.
+
+![](images/c35df0540875959975d667e8f870eeab6bec5d19ea90e3173421d5a5a6105159.jpg)  
+Figure 3: Contour maps of the critic value functions for the marginal posterior (“realism”) constraint. We look at the two latent dimensions that have the lowest average posterior standard deviation on the training set, taking low variance in $z$ space as a proxy for influence over the generated images. All other latent dimensions are held fixed at their original values (from a sample from $p ( z )$ on the left, and from a sample from $q ( z \mid x )$ for a held-out $x$ on the right). Gray x marks correspond to the points in latent space of the generated images to the right. The cross-section on the left, taken from a prior sample, shows contours that point towards more realistic looking digits. In the cross-section on the right, a sample from the validation set (indicated by orange squares) resides within a local maximum of the critic, as one would hope.
+
+Conditional GANs (CGAN; Mirza & Osindero, 2014) and conditional VAEs (CVAE; Sohn et al., 2015) can generate samples conditioned on attribute information when available, but they must be trained with knowledge of the attribute labels for the whole training set, and it is not clear how to adapt them to new attributes without retraining from scratch. Furthermore, CGANs and CVAEs suffer from the same problems of mode-collapse and blurriness as their unconditional cousins.
+
+We take a different approach to conditional generation and identity-preserving transformation. We begin by training an unconditional VAE with hyperparameters chosen to ensure good reconstruction (at the expense of sample quality). We then train a “realism” critic to predict whether a given $z$ maps to a high-quality sample. We also train critics to predict whether a given $z$ maps to a sample that manifests various attributes of interest. To generate samples that are both realistic and exhibit desired attributes, one option is to optimize random $z$ vectors until they satisfy both the realism and attribute critics. Alternately, we can amortize this cost by training an “actor” network to map a random set of $z$ vectors to a subregion of latent space that satisfies the constraints encoded by the critics. By encouraging these transformed $z$ vectors to remain as close as possible to where they started, we alleviate the mode-collapse problem common to GANs.
+
+Our approach is summarized visually in Figure 1. The details follow in sections 3, 4, 5, and 6.
+
+# 3 THE “REALISM” CONSTRAINT: SHARPENING VAE SAMPLES
+
+We define the realism constraint implicitly as being satisfied by samples from the marginal posterior $\begin{array} { r } { q ( z ) \triangleq \frac { 1 } { N } \sum _ { n } q ( z \mid \underline { { x } } _ { n } ) } \end{array}$ and not those from $p ( z )$ . By enforcing this constraint, we can close the gap between reconstruction quality and sample quality (without sacrificing sample diversity).
+
+As shown in Figure 1, we can train a critic $D$ to differentiate between samples from $p ( z )$ and $q ( z )$ . The critic loss, $\mathcal { L } _ { D } ( z )$ , is simply the cross-entropy, with labels $c = 1$ for $z \sim q ( z \mid x )$ and $c = 0$ for $z \sim p ( z )$ . We found that the realism critic had little trouble generalizing to unseen data; that is, it was able to recognize samples from $q ( z \mid x ^ { \mathrm { h e l d - o u t } } )$ as being “realistic” (Figure 3).
+
+Sampling from the prior is sufficient to train $D$ for models with lower KL Divergence, but if the KL Divergence between $q$ and $p$ is large, the chances of sampling a point $p ( z )$ that has high probability under $q ( z )$ becomes vanishingly small. This leads to poor sample quality and makes it difficult for $D$ to learn a tight approximation of $q ( z )$ solely by sampling from $p ( z )$ . Instead, we use an inner-loop of gradient-based optimization, $G _ { \mathrm { o p t } } ( z ) = \mathrm { G r a d i e n t D e s c e n t } ( z ; \mathcal { L } _ { D } ( z ) )$ , to move prior samples to points deemed more like $q ( z )$ by $D$ . For clarity, we introduce the shorthand $\mathcal { L } _ { c = 1 } ( z ) \triangleq - \log ( D ( z ) )$ and $\mathcal { L } _ { c = 0 } ( z ) \triangleq - ( 1 - \log ( D ( z ) ) )$ . This gives us our critic loss for the realism constraint:
+
+$$
+\mathcal { L } _ { D } ( z ) = \mathbb { E } _ { z \sim q ( z | x ) } [ \mathcal { L } _ { c = 1 } ( z ) ] + \mathbb { E } _ { z \sim p ( z ) } [ \mathcal { L } _ { c = 0 } ( z ) ] + \mathbb { E } _ { z \sim G ( p ( z ) ) } [ \mathcal { L } _ { c = 0 } ( z ) ]
+$$
+
+![](images/c11634da67cd323ff172bf8c40def51cabbde20b6ccd6132f98a3bd62dafbec6.jpg)  
+Figure 4: Conditional generation with a CGAN actor-critic pair acting in the latent space of a VAE with $\sigma _ { x } = 0 . 1$ . Each row starts from a different prior sample and maps it to a new point in latent space that satisfies both the attribute constraints and the realism constraint. The attribute constraints are changed one at a time to produce as smooth a transition as possible from left to right. The bottom CGAN is regularized during training to prefer small shifts in latent space $\lambda _ { \mathrm { d i s t } } = 0 . 1 $ , while the top is not $\lambda _ { \mathrm { d i s t } } = 0 . 0$ ). Compared to the images generated by the unregularized model, the images generated by the regularized model are much less diverse across columns, suggesting that the regularization does indeed enforce some degree of identity preservation. The regularized model produces images that are somewhat more diverse across rows, suggesting that the regularization fights mode collapse (arguably at the expense of image quality). For each column, the complete list of attributes is given in supplemental Table 3.
+
+Since this inner-loop of optimization can slow down training, we amortize the generation by using a neural network as a function approximator. There are many examples of such amortization tricks, including the encoder of a VAE, generator of a GAN, and fast neural style transfer (Ulyanov et al., 2016; Li & Wand, 2016; Johnson et al., 2016). As with a traditional GAN, the parameters of the function $G$ are updated to maximize the value $D$ ascribes to the shifted latent points. One of the challenges using a GAN in this situation is that it is prone to mode-collapse. However, an advantage of applying the GAN in latent space is that we can regularize $G$ to try and find the closest point in latent space that satisfies $D$ , thus encouraging diverse solutions. We introduce a regularization term, $\bar { \mathcal { L } _ { \mathrm { d i s t } } ( z ^ { \prime } , z ) } = 1 / \bar { \sigma _ { z } } ^ { 2 } \log ( 1 + ( z ^ { \prime } - z ) ^ { 2 } )$ to encourage nearby solutions, while allowing more exploration than a mean square error term. As a VAE utilizes only a fraction of its latent dimensions, we scale the distance penalty of each dimension by its utilization, as indicated by the squared reciprocal of the scale $\sigma _ { z } ( x ) ^ { \top }$ of the encoder distribution $\overset { \cdot } { q } ( z \mid x )$ , averaged over the training dataset, $\begin{array} { r } { \bar { \sigma } _ { z } \triangleq \frac { 1 } { N } \sum _ { n } \sigma _ { z } ( x _ { n } ) } \end{array}$ . The regularized loss is
+
+$$
+\mathcal { L } _ { G } ( z ) = \mathbb { E } _ { z \sim p ( z ) } [ \mathcal { L } _ { c = 1 } ( G ( z ) ) + \lambda _ { \mathrm { d i s t } } \mathcal { L } _ { \mathrm { d i s t } } ( G ( z ) , z ) ] .
+$$
+
+# 4 ATTRIBUTE CONSTRAINTS: CONDITIONAL GENERATION
+
+We want to generate samples that are realistic, but we also want to control what attributes they exhibit. Given binary attribute labels $y$ for a dataset, we can accomplish this by using a CGAN in the latent space, which amounts to replacing $D ( z )$ and $G ( z )$ with conditional versions $D ( z , y )$ and $G ( z , y )$ and concatenating $y$ to $z$ as input. If both the actor and critic see attribute information, $G$ must find points in latent space that could be samples from $q ( z )$ with attributes $y$ .
+
+![](images/0517b9ec5c42259496004e8c839f373c09004ea747d762da9fc2f7ba95731a14.jpg)  
+Figure 5: Identity-preserving transformations with optimization. Two separate critics are trained, one for attributes and one for the realism constraint. Starting at the latent points corresponding to the data reconstructions, we then perform gradient ascent in latent space on a weighted combination of critic values (1.0 attribute, 0.1 marginal posterior), stopping when a threshold value is passed for both critics. Images remain semantically close to the original because the pixel-wise likelihood of VAE training encourages identity-preserving reconstructions, and the dynamics of gradient ascent are naturally limited to finding solutions close in latent space. Panels are black for attributes of the original image, as the procedure just returns the original point in latent space.
+
+This procedure is computationally inexpensive relative to training a generative model from scratch. In most of our experiments, we use a relatively large CGAN actor-critic pair (4 fully connected ReLU layers of 2048 units each), which during training uses about $9 6 \times$ fewer FLOPs/iteration than the unconditional VAE. We also trained a much smaller CGAN actor-critic pair (3 fully connected ReLU layers of 256 units), which uses about $2 8 8 4 \times$ fewer FLOPs/iteration than the VAE, and achieves only slightly worse results than the larger CGAN (supplemental Figure 14 and Table 1).
+
+Figure 4 demonstrates the quality of conditional samples from a CGAN actor-critic pair and the effect of the distance penalty, which constrains generation to be closer to the prior sample, maintaining similarity between samples with different attributes. The regularized CGAN actor has less freedom to ignore modes by pushing many random $z$ vectors to the same area of the latent space, since it is penalized for moving samples from $p ( z )$ too far. The increased diversity across rows of the regularized CGAN is evidence that this regularization does fight mode-collapse (additional qualitative evidence is in supplemental Figures 7 and 8). However, without a distance penalty, samples appear more a bit realistic with more prominent attributes. This is supported by Table 1, where we use a separately trained attribute classification model to quantitatively evaluate samples. The actor with no penalty generates samples that are more accurately classified than the actor with a penalty but also shifts the samples much farther in latent space.
+
+<table><tr><td>CelebA</td><td>Accuracy</td><td>Precision</td><td>Recall</td><td>F1 Score</td><td>2ZMSE</td></tr><tr><td>(This Work) 10 Attributes</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Test Data</td><td>0.936</td><td>0.901</td><td>0.893</td><td>0.895</td><td></td></tr><tr><td>GcGAN(入dist = 0)</td><td>0.942</td><td>0.914</td><td>0.904</td><td>0.906</td><td>80.7</td></tr><tr><td>GcGAN(入dist = O) (Small Model) GcGAN(入dist = 0.1)</td><td>0.926 0.928</td><td>0.898</td><td>0.860</td><td>0.870</td><td>58.9</td></tr><tr><td>(Perarnau et al., 2016) 18 Attributes</td><td></td><td>0.903</td><td>0.863</td><td>0.874</td><td>17.0</td></tr><tr><td>Test Data</td><td>0.928</td><td></td><td></td><td></td><td></td></tr><tr><td>IcGAN</td><td>0.860</td><td></td><td></td><td>0.715 0.524</td><td></td></tr></table>
+
+Table 1: Accuracy of a separate model trained to classify attributes from images, evaluated on test data and generated images. We condition and evaluate the generated images on the same labels as the test data. For comparison, the results of a similar task using invertible CGANs for generation (Perarnau et al., 2016) are provided. However, since the full list of salient attributes was not given in the paper, we emphasize that they are not directly comparable as the two experiments use a slightly different set of attribute labels. We also measure the distance in latent space that prior samples are shifted, weighted by $1 / \bar { \sigma _ { z } } ^ { 2 }$ . Actors trained with a latent distance penalty $\lambda _ { \mathrm { d i s t } }$ have slightly worse accuracy, but find latent points much closer to the prior samples and produce a greater diversity of images (see supplemental Figures 7 and 8). Interestingly, an actor trained without a distance penalty achieves higher classification accuracy than the test set itself, possibly by generating images with more exaggerated and distinctive features than real data. A ”small model” CGAN with $8 5 \mathrm { x }$ fewer parameters (3 fully connected layers of 256 units) generates images (supplemental Figure 14) of comperable quality. Due to the smaller capacity, the model finds more local solutions (smaller $z _ { M S E } )$ that have slightly less attribute accuracy, but are more visually similar to the prior sample without an explicit regularization term.
+
+Although we used a VAE as the base generative model, our approach could also be used to generate high-quality conditional samples from pretrained classical autoencoders. We show in supplemental Figure 15 that we obtain reasonably good conditional samples (albeit with high-frequency spatial artifacts) as $\sigma _ { x }  0$ (equivalent to a classical autoencoder). Learning the decoder using VAE training encourages $\mathsf { q } ( \mathbf { z } )$ to fill up as much of the latent space as possible (without sacrificing reconstruction quality), which in turn encourages the decoder to map more of the latent space to reasonable-looking images. The prior $p ( z ) = \mathcal { N } ( \bar { 0 , } I )$ also imposes a natural scale on the latent variables.
+
+# 5 IDENTITY-PRESERVING TRANSFORMATIONS
+
+If we have a VAE that can produce good reconstructions of held-out data, we can transform the attributes of the output by gradient-based optimization. We simply need to train a critic, $D _ { a t t r } ( z )$ , to predict the attribute labels $p ( y \mid z )$ of the data embeddings $z \sim q ( z \mid x )$ , and use a cross-entropy loss to train. Then, starting from a data point, $z \sim q ( z \mid x )$ , we can perform gradient descent on the the realism constraint and attribute constraint jointly, $\mathcal { L } _ { D _ { \mathrm { r e a l } } } ( z ) + \lambda _ { \mathrm { a t t r } } \mathcal { L } _ { D _ { \mathrm { a t t r } } } ( z )$ . Note that it is helpful to maintain the realism constraint to keep the image from distorting unrealistically. Using the same procedure, we can also conditionally generate new samples (supplemental Figure 9) by starting from $z \sim p ( z )$ .
+
+Figure 5 demonstrates transformations applied to samples from the held-out evaluation dataset. Note that since the reconstructions are close to the original images, the transformed images also maintain much of their structure. This contrasts with supplemental Figure 10, where a distance-penalty-free CGAN actor produces transformations that share attributes with the original but shift identity. We could preserve identity by introducing a distance penalty, but find that it is much easier to find the correct weighting of realism cost, attribute cost, and distance penalty through optimization, as each combination does not require retraining the network.
+
+# 6 RULE-BASED CONSTRAINTS: ZERO-SHOT CONDITIONAL GENERATION
+
+So far, we have assumed access to labeled data to train attribute classifiers. We can remove the need to provide labeled examples by leveraging the structure learned by our pre-trained model, using it to generate exemplars that are scored by a user-supplied reward function. If we constrain the reward function to be bounded, $c ( x ) : \mathbb { R } ^ { N }  [ 0 , 1 ]$ , the problem becomes very similar to previous GAN settings, but now the actor, $G$ , and critic, $D$ , are working together. $D$ aims to best approximate the true value of each latent state, $\mathbb { E } _ { x \sim p ( x | z ) } c ( x )$ , and $G$ aims to shift samples from the prior to highvalue states. The critic loss is the cross-entropy from $c ( x )$ , and the actor loss is the same as $\mathcal { L } _ { G }$ in equation 3, where we again have a distance penalty to promote diversity of outputs.
+
+![](images/071568b3db247dc7bdd2ac007549f03830848063407064fd55571d86af87d845.jpg)  
+Figure 6: Transformations from a prior sample for the Melody VAE model. In each 16-bar pianoroll, time is in the horizontal direction and pitch in the vertical direction. In the prior sample, notes falling outside of the C Major scale are shown in red. After transformation by $G _ { \mathcal { P } = \mathrm { C } _ { \mathrm { M a j } } , d = 0 }$ , all sampled notes fall within the scale, without a significant change to note density. After transformation of the original $z$ by $G _ { \mathcal { P } = \mathrm { C _ { M a j } } , d = 1 9 2 }$ , all sampled notes lay within the scale and the density increases beyond 192. Synthesized audio of these samples can be heard at https://goo. ${ \mathfrak { g l } }$ /ouULt9.
+
+Note that the reward function and VAE decoder need not necessarily be differentiable, as the critic learns a value function to approximate the reward, which the actor uses for training. To highlight this, we demonstrate that the output of a recurrent VAE model can be constrained to satisfy hardcoded rule-based constraints.
+
+We first train an LSTM VAE (details in the Appendix) on melodic fragments. Each melody, $m$ , is represented as a sequence of categorical variables. In order to examine our ability to constrain the pitch classes and note density of the outputs, we define two reward functions, one that encourages notes from a set of pitches $\mathcal { P }$ , and another for that encourages melodies to have at least $d$ notes:
+
+$$
+\begin{array} { r } { c _ { \mathrm { p i t c h } } ( m , \mathcal { P } ) = \sum _ { p \in m } \mathbb { 1 } ( p \in \mathcal { P } ) / | m | \qquad c _ { \mathrm { d e n s i t y } } ( m , d ) = \operatorname* { m i n } ( 1 , | m | / d ) } \end{array}
+$$
+
+Figure 6 gives an example of controlling the pitch class and note density of generated outputs, which is quantitatively supported by the results in Table 2. During training, the actor goes through several phases of exploration and exploitation, oscillating between expanding to find new modes with high reward and then contracting to find the nearest locations of those modes, eventually settling into high value states that require only small movements in the latent space (supplemental Figure 11).
+
+# 7 RELATED WORK
+
+Conditional GANs (Mirza & Osindero, 2014) and VAEs (Sohn et al., 2015) introduce conditioning variables at training time. Sohn et al. (2015) allow these variables to affect the distribution in latent $z$ space, but still require that $p ( z \mid y )$ be a tractable distribution. Perarnau et al. (2016) use CGANs to adjust images, but because CGANs cannot usually reconstruct arbitrary inputs accurately, they must resort to image-space processing techniques to transfer effects to the original input. White (2016) propose adding “attribute vectors” to samples from $p ( z )$ as a simple and effective heuristic to perform transformations, which relies heavily on the linearity of the latent space.
+
+$$
+\begin{array} { r } { \left. \begin{array} { l l l l } { { \bf A c t o r } } & { { \bf A c t o r } } & { { \bf \Phi } } & { { \bf c } _ { \mathrm { p i t c h } } ( m , \mathcal { P } = { \bf C } _ { \mathrm { M a j } } ) } & { c _ { \mathrm { d e n s i t y } } ( m , d = 1 9 2 ) } \\ { { \bf F r i o r } } & { { \bf 0 . 5 7 9 } ( 0 . 4 3 \% ) } & { { \bf 0 . 4 1 7 } ( 0 . 0 4 \% ) } & { - } \\ { G _ { { \mathcal P } = { \bf C } _ { \mathrm { M a j } } , d = 0 } } & { { \bf 0 . 9 9 1 } ( 7 0 . 8 \% ) } & { { \bf 0 . 4 5 9 } ( 0 . 0 1 \% ) } & { 0 . 0 1 5 } \\ { G _ { { \mathcal P } = { \bf C } _ { \mathrm { M a j } } , d = 1 9 2 } } & { { \bf 0 . 9 8 2 } ( 6 2 . 4 \% ) } & { { \bf 0 . 9 8 5 } ( 8 4 . 9 \% ) } & { { \bf 0 . 0 3 9 } } \end{array} \right| \left. \begin{array} { l } { { z } _ { \mathrm { M S E } } } \\ { { \bf 0 . 0 4 9 } } \\ { { \bf 0 . 0 1 5 } } \end{array} \right. } \end{array}
+$$
+
+Table 2: Average rewards and constraint satisfaction rates (in parentheses) for unconditional (Prior) and conditional generation. Samples from the prior receive low rewards, on average, and near zero satisfaction rates from both the pitch class (C Major) and note density $\ge 1 9 2$ notes) constraints. After applying an actor optimized only for the C Major scale $\scriptstyle \left( G _ { \mathcal { P } = \mathrm { C _ { M a j } } , d = 0 } \right)$ , the pitch class constraint is fully satisfied $7 0 . 8 \%$ of the time with only a minor effect on density. The average value close to 1 also indicates that when the constraint is not satisfied, it is typically off by only a few notes. Applying an actor function optimized for the C Major scale and high density $\scriptstyle \left( G _ { \mathcal { P } = \mathrm { C _ { M a j } } , d = 1 9 2 } \right)$ causes both constraints to be satisfied at high rates, with a slightly larger shift in latent space.
+
+Some recent work has focused on applying more expressive prior constraints to VAEs (Rezende et al., 2014; Sønderby et al., 2016; Chen et al., 2017; Tomczak & Welling, 2017). The prior that maximizes the ELBO is $p ^ { \star } ( z ) = q ( z )$ (Hoffman & Johnson, 2016); one can interpret our realism constraint as trying to find an implicit distribution that is indistinguishable from $q ( z )$ . Like the adversarial autoencoder of Makhzani et al. (2016), our realism constraint relies on a discriminative model, but instead of trying to force $q ( z )$ to equal some simple $p ( z )$ , we only weakly constrain $q ( z )$ and then use a classifier to “clean up” our results.
+
+Like this work, the recently proposed adversarially regularized autoencoder (Junbo et al., 2017) uses adversarial training to generate latent codes in a latent space discovered by an autoencoder; that work focuses on unconditional generation. Gomez-Bombarelli et al. (2016) train classifiers in the latent ´ space of a VAE to predict what latent variables map to molecules with various properties, and then use iterative gradient-based optimization in the latent space to find molecules that have a desired set of properties. On molecule data, their procedure generates invalid molecules rarely enough that they can simply reject these samples, which are detected using off-the-shelf software. By contrast, the probability of generating realistic images under our pretrained VAE is astronomically small, and no simple criterion for detecting valid images exists.
+
+Jaques et al. (2017) also use a classifier to constrain generation; they use a Deep Q-network as an auxiliary loss for training an LSTM. Closest to Section 6, Nguyen et al. (2016a;b) generate very high quality conditional images by optimizing a sample from the latent space of a generative network to create an image that maximizes the class activations of a pretrained ImageNet classifier. Our work differs in that we learn an amortized generator/discriminator directly in the latent space and we achieve diversity through regularizing by the natural scale of the latent space rather than through a modified Langevin sampling algorithm.
+
+# 8 DISCUSSION AND FUTURE WORK
+
+We have demonstrated a new approach to conditional generation by constraining the latent space of an unconditional generative model. This approach could be extended in a number of ways.
+
+One possibility would be to plug in different architectures, including powerful autoregressive decoders or adversarial decoder costs, as we make no assumptions specific to independent likelihoods. While we have considered constraints based on implicit density estimation, we could also estimate the constrained distribution directly with an explicit autoregressive model or another variational autoencoder. The efficacy of autoregressive priors in VAEs is promising for this approach (Kingma et al., 2016). Conditional samples could then be obtained by ancestral sampling, and transformations by using gradient ascent to increase the likelihood under the model. Active or semisupervised learning approaches could reduce the sample complexity of learning constraints. Real-time constraint learning would also enable new applications; it might be fruitful to extend the reward approximation of Section 6 to incorporate user preferences as in (Christiano et al., 2017).
+
+# ACKNOWLEDGMENTS
+
+Many thanks to Jascha Sohl-Dickstein, Colin Raffel, and Doug Eck for their helpful brainstorming and encouragement.
+
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+
+# 9 APPENDIX
+
+# 9.1 EXPERIMENTAL DETAILS
+
+For images, we use the MNIST digits dataset (LeCun & Cortes, 2010) and the Large-scale CelebFaces Attributes (CelebA) dataset (Liu et al., 2015). MNIST images are $2 8 \times 2 8$ pixels and greyscale scaled to [0, 1]. For attributes, we use the number class label of each digit. CelebA images are centercropped to $1 2 8 \times 1 2 8$ pixels and then downsampled to $6 4 \times 6 4$ RGB pixels and scaled to [0, 1]. We find that many of the attribute labels are not strongly correlated with changes in the images, so we narrow the original 40 attributes to the 10 most visually salient: blond hair, black hair, brown hair, bald, eyeglasses, facial hair, hat, smiling, gender, and age.
+
+For melodies, we scraped the web to collect over 1.5 million publicly available MIDI files. We then extracted 16-bar melodies by sliding a window with a single bar stride over each non-percussion instrument with a $\frac { 4 } { 4 }$ time signature, keeping only the note with the highest pitch when multiple overlap. This produced over 3 million unique melodies. We represent each melody as a sequence of 256 (16 per bar) categorical variables taking one of 130 discrete states at each sixteenth note: 128 note-on pitches, a hold state, and a rest state.
+
+# 9.2 MODEL ARCHITECTURES
+
+All encoders, decoders, and classifiers are trained with the Adam optimizer (Kingma & Ba, 2015), with learning rate $= 3 \mathrm { e } { - } 4$ , $\beta _ { 1 } = 0 . 9$ , and $\beta _ { 2 } = 0 . 9 9 9$ .
+
+To train $D _ { r e a l } ( z ) , D _ { a t t r } ( z )$ and $G ( z )$ we follow the training procedure of Gulrajani et al. (2017), applying a gradient penalty of 10, training $D$ and $G$ in a 10:1 step ratio, and use the Adam optimizer with learning rate $= 3 \mathrm { e } { - } 4$ , $\beta _ { 1 } ~ = 0 . 0$ , and $\beta _ { 2 } ~ = 0 . 9$ . While not necessary to converge, we find it improves the stability of optimization. We do not apply any of the other tricks of GAN training such as batch normalization, minibatch discrimination, or one-sided label smoothing (Radford et al., 2015; Salimans et al., 2016). As samples from $p ( z )$ are easier to discriminate than samples from $G ( p ( z ) )$ , we train $D$ by sampling from $p ( z )$ at a rate 10 times less than $G ( p ( z ) )$ . For actors with inner-loop optimization, $G _ { \mathrm { o p t } }$ , 100 iterations of Adam are used with with learning rate $= 1 \mathrm { e } \mathrm { - } 1$ , $\beta _ { 1 } =$ 0.9, and $\beta _ { 2 } = 0 . 9 9 9$ .
+
+# 9.2.1 MNIST FEED-FORWARD VAE
+
+To model the MNIST data, we use a deep feed-forward neural network (Figure 13a).
+
+The encoder is a series of 3 linear layers with 1024 outputs, each followed by a ReLU, after which an additional linear layer is used to produce 2048 outputs. Half of the outputs are used as the $\mu$ and the softplus of the other half are used as the $\sigma$ to parameterize a 1024-dimension multivariate Gaussian distribution with a diagonal covariance matrix for $z$ .
+
+The decoder is a series of 3 linear layers with 1024 outputs, each followed by a ReLU, after which an additional linear layer is used to produce $2 8 \mathbf { x } 2 8$ outputs. These outputs are then passed through a sigmoid to generate the output image.
+
+# 9.2.2 CELEBA CONVOLUTIONAL VAE
+
+To model the CelebA data, we use a deep convolutional neural network (Figure 13b).
+
+The encoder is a series of $4 ~ 2 \mathrm { D }$ convolutional layers, each followed by a ReLU. The convolution kernels are of size $3 \times 3$ , $3 \times 3$ , $5 \times 5$ , and $5 \times 5$ , with 2048, 1024, 512, and 256 output channels, respectively. All convolutional layers have a stride of 2. After the final ReLU, a linear layer is used to produce 2048 outputs. Half of the outputs are used as the $\mu$ and the softplus of the other half are used as the $\sigma$ to parameterize a 1024-dimension multivariate Gaussian distribution with a diagonal covariance matrix for $z$ .
+
+The decoder passes the $z$ through a $4 \mathrm { x } 4 \mathrm { x } 2 0 4 8$ linear layer, and then a series of 4 2D transposed convolutional layers, all but the last of which are followed by a ReLU. The deconvolution kernels are of size $5 \times 5 , 5 \times 5 , 3 \times 3$ , and $3 \times 3$ , with 1024, 512, 256, and 3 output channels, respectively. All deconvolution layers have a stride of 2. The output from the final deconvolution is passed through a sigmoid to generate the output image.
+
+The classifier that is trained to predict labels from images are identical to the VAE encoders except that they end with a sigmoid cross-entropy loss.
+
+# 9.2.3 MELODY SEQUENCE VAE
+
+Music is fundamentally sequential, so we use an LSTM-based sequence VAE for modelling monophonic melodies (Figure 13c).
+
+The encoder is made up of a single-layer bidirectional LSTM, with 2048 units per cell. The final output in each direction is concatenated and passed through a linear layer to produce 1024 outputs. Half of the outputs are used as the $\mu$ and the softplus of the other half are used as a $\sigma$ to parameterize a 512-dimension multivariate Gaussian distribution with a diagonal covariance matrix for $z$ .
+
+Since musical sequences often have structure at the bar level, we use a hierarchical decoder to model long melodies. First, the $z$ goes through a linear layer to initialize the state of a 2-layer LSTM with 1024 units per layer, which outputs 16 embeddings of size 512 each, one per bar. Each of these embeddings are passed through a linear layer to produce 16 initial states for another 2-layer LSTM with 1024 units per layer. This bar-level LSTM autoregressively produces individual sixteenth note events, passing its output through a linear layer and softmax to create a distribution over the 130 classes. This categorical distribution is used to compute a cross-entropy loss during training or samples at inference time. In addition to generating the initial state at the start of each bar, the embedding for the current bar is concatenated with the previous output as the input at each time step.
+
+# 9.2.4 ACTOR FEED-FORWARD NETWORK
+
+For $G ( z )$ , we use a deep feed-forward neural network (Figure 12a) in all of our experiments.
+
+The network is a series of 4 linear layers with 2048 outputs, each followed by a ReLU, after which an additional linear layer is used to produce $2 * d i m ( z )$ outputs. Half of the outputs are used as the $\delta z$ and the sigmoid of the other half are used as gates. The transformed $z ^ { \prime }$ is the computed as $( 1 - g a t e s ) * z + g a t e s * \delta z$ . This aids in training as the network only has to then predict shifts in $z$ .
+
+When conditioning on attribute labels, $y$ , to compute $G ( z , y )$ , the labels are passed through a linear layer producing 2048 outputs which are concatenated with $z$ as the model input.
+
+# 9.2.5 CRITIC FEED-FORWARD NETWORK
+
+For $D ( z )$ , we use a deep feed-forward neural network (Figure 12b) in all of our experiments.
+
+The network is a series of 4 linear layers with 2048 outputs, each followed by a ReLU, after which an additional linear layer is used to produce a single output. This output is passed through a sigmoid to compute $D ( z )$ .
+
+When conditioning on attribute labels, $y$ , to compute $D ( z , y )$ , the labels are passed through a linear layer producing 2048 outputs which are concatenated with $z$ as the model input.
+
+# 9.3 SUPPLEMENTAL FIGURES
+
+![](images/c7271bc588426b5575cc43d68291dd43b6a2951e774d979aaf34823640dd61fa.jpg)  
+Figure 7: Additional generated CelebA faces by $G _ { \mathrm { C G A N } }$ with $\lambda _ { \mathrm { d i s t } } = 0$ . Full attribute labels are given in supplementary Table 3
+
+![](images/b77cb599d9dddb0451265adee643ada6dfeb1a26318158c1994cde29b9c5bc27.jpg)  
+Figure 8: Additional generated CelebA faces by $G _ { \mathrm { C G A N } }$ with $\lambda _ { \mathrm { d i s t } } = 0 . 1$ . Full attribute labels are given in supplementary Table 3
+
+![](images/0c9693a16a3b9e8281a0253d76908badcc7a1bf21d7403c4feac61198a6e884b.jpg)  
+Figure 9: Optimization of samples drawn from the prior to satisfy both the realism constraint and attribute constraints (drawn from the test set). The optimization takes 100 steps, and images are shown at 0, 10, 30, 50 and 100 steps. $D$ is trained with inner-loop optimization, $G _ { \mathrm { o p t } }$ , as described in Section 9.2
+
+![](images/ef19112b6592d78019a654a7c1cb0b4ff26e30ce2b4848ea671d418a9c7cfb43.jpg)  
+Figure 10: Identity-distorting transformations with CGAN actor-critic. Without a penalty to encourage small moves in latent space, the actor maps the latent vectors of the original data points to generated images that have the correct attributes, but a different identity. Panels are black for attributes of the original image, as the procedure just returns the same image as the reconstruction.
+
+![](images/d7228d1a204c64ab4cfde7920be6b879b613b90a99a551489d60c0efcddc7575.jpg)  
+Figure 11: Training curves for melody actor $( G )$ and critic $( D )$ pair for pitch class constraint $c _ { \mathrm { p i t c h } } ( m , \mathcal { P } = \mathrm { C _ { M a j } } ) ,$ ).
+
+![](images/ef83a5eac17c8d98a5925bf6d30cd03e235814d62ff8b045ce406ea347ad5889.jpg)  
+Figure 12: Architecture for the (a) actors and (b) critics used in all experiments.
+
+Table 3: Complete list of attributes for label names in Figures 4, 7, and 8   
+
+<table><tr><td>Figure Label</td><td>Bald</td><td>Black Hair</td><td>Blond Hair</td><td>Brown Hair</td><td>Eye- glasses</td><td>Male</td><td>Beard</td><td>Smiling</td><td>Hat</td><td>Young</td></tr><tr><td>Blond Hair</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Brown Hair</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Black Hair</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Male</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Facial Hair</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Eyeglasses</td><td>0</td><td>1</td><td>0</td><td>0</td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Bald</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Aged</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>1</td><td>0</td><td>0</td><td>0</td></tr></table>
+
+<table><tr><td></td><td>LL</td><td>KL</td><td>ELBO</td></tr><tr><td>1</td><td>-11360</td><td>30</td><td>-11390</td></tr><tr><td>1e-1</td><td>-11325</td><td>150</td><td>-11475</td></tr><tr><td>1e-2</td><td>15680</td><td>600</td><td>15080</td></tr><tr><td>1e-3</td><td>16090</td><td>1950</td><td>14140</td></tr><tr><td>1e-4</td><td>16150</td><td>3650</td><td>12500</td></tr></table>
+
+Table 4: Selection of $\sigma _ { x } = 0 . 1$ for the CelebA VAEs by ELBO maximization. All results are given in Nats.
+
+![](images/bc6a46f235dfa72750a10f4560f90a19b564bdee54ba80ca938988213816835f.jpg)  
+Figure 13: Architectures for the (a) feed-forward MNIST, (b) convolutional CelebA, and (c) hierarchical LSTM melody VAEs. In (b), all convolutions have a stride of 2. In (c), LSTM cells shown in the same color share weights and linear layers between levels are omitted.
+
+# Small G/D Models (256 ReLU × 3)
+
+![](images/56f7770ad96713af1e31a7e0cf8d16ef63bca62f373d9a77aee8f609762fe969.jpg)  
+Figure 14: Samples generated with smaller (3 ReLU layers of 256 units each) $G$ and $D$ models are comparable quality despite having $8 5 \mathrm { x }$ fewer parameters, $\lambda _ { \mathrm { d i s t } } = 0 . 0$ . Full attribute labels are given in supplementary Table 3.
+
+![](images/e12c823878b34da528fa8dbef4b7a3e2eb2c8d0d959f24089a66aca5f4665e02.jpg)  
+Figure 15: Latent constraints applied to a vanilla autoencoder with no latent prior. Samples are similar quality to VAEs with $\sigma _ { x } ~ = ~ 0 . 1$ , but with less diversity and more high-frequency visual artifacts. Full attribute labels are given in supplementary Table 3.
+
+![](images/389dcabc3eef37e9e3ed28351331289d6d040ac48592c4ad9d43bd89f6e92ff1.jpg)  
+Figure 16: Smaller decoder standard deviations, $\sigma _ { x }$ , lead to lower-variance posteriors, $\sigma _ { z } ( x )$ of the encoder $q ( z \mid x )$ , averaged over the training set per a dimension. The $\mathbf { X } ^ { } -$ -axis is sorted from lowest to highest variance. Tighter posteriors correspond to more utilization of the latent dimension, and we scale our distance regularization the square inverse on a per-dimension basis.

parse/train/Sy8XvGb0-/Sy8XvGb0-_content_list.json

@@ -0,0 +1,1669 @@
+[
+    {
+        "type": "text",
+        "text": "LATENT CONSTRAINTS: LEARNING TO GENERATE CONDITIONALLY FROM UNCONDITIONAL GENERATIVE MODELS ",
+        "text_level": 1,
+        "bbox": [
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+            99,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jesse Engel   \nGoogle Brain   \nSan Francisco, CA, USA ",
+        "bbox": [
+            184,
+            195,
+            348,
+            237
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Matthew D. Hoffman Google Inc. San Francisco, CA, USA ",
+        "bbox": [
+            398,
+            195,
+            565,
+            237
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Adam Roberts Google Brain San Francisco, CA, USA ",
+        "bbox": [
+            614,
+            195,
+            781,
+            237
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            273,
+            544,
+            289
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep generative neural networks have proven effective at both conditional and unconditional modeling of complex data distributions. Conditional generation enables interactive control, but creating new controls often requires expensive retraining. In this paper, we develop a method to condition generation without retraining the model. By post-hoc learning latent constraints, value functions that identify regions in latent space that generate outputs with desired attributes, we can conditionally sample from these regions with gradient-based optimization or amortized actor functions. Combining attribute constraints with a universal “realism” constraint, which enforces similarity to the data distribution, we generate realistic conditional images from an unconditional variational autoencoder. Further, using gradient-based optimization, we demonstrate identity-preserving transformations that make the minimal adjustment in latent space to modify the attributes of an image. Finally, with discrete sequences of musical notes, we demonstrate zero-shot conditional generation, learning latent constraints in the absence of labeled data or a differentiable reward function. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generative modeling of complicated data such as images and audio is a long-standing challenge in machine learning. While unconditional sampling is an interesting technical problem, it is arguably of limited practical interest in its own right: if one needs a non-specific image (or sound, song, document, etc.), one can simply pull something at random from the unfathomably vast media databases on the web. But that naive approach may not work for conditional sampling (i.e., generating data to match a set of user-specified attributes), since as more attributes are specified, it becomes exponentially less likely that a satisfactory example can be pulled from a database. One might also want to modify some attributes of an object while preserving its core identity. These are crucial tasks in creative applications, where the typical user desires fine-grained controls (Bernardo et al., 2017). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "One can enforce user-specified constraints at training time, either by training on a curated subset of data or with conditioning variables. These approaches can be effective if there is enough labeled data available, but they require expensive model retraining for each new set of constraints and may not leverage commonalities between tasks. Deep latent-variable models, such as Generative Adversarial Networks (GANs; Goodfellow et al., 2014) and Variational Autoencoders (VAEs; Kingma & Welling, 2013; Rezende et al., 2014), learn to unconditionally generate realistic and varied outputs by sampling from a semantically structured latent space. One might hope to leverage that structure in creating new conditional controls for sampling and transformations (Brock et al., 2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, we show that new constraints can be enforced post-hoc on pre-trained unsupervised generative models. This approach removes the need to retrain the model for each new set of constraints, allowing users to more easily define custom behavior. We separate the problem into (1) creating an unsupervised model that learns how to reconstruct data from latent embeddings, and (2) leveraging the latent structure exposed in that embedding space as a source of prior knowledge, upon which we can impose behavioral constraints. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our key contributions are as follows: ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/9b445d4241da4a0a100e524c0e5fec9fef7838190a45a3b31264de73da209d9a.jpg",
+        "image_caption": [
+            "Figure 1: (a) Diagram of latent constraints for a VAE. We use one critic $D _ { \\mathrm { a t t r } }$ to predict which regions of the latent space will generate outputs with desired attributes, and another critic $D _ { \\mathrm { r e a l i s m } }$ to predict which regions have high mass under the marginal posterior, $q ( z )$ , of the training data. (b) We begin by pretraining a standard VAE, with an emphasis on achieving good reconstructions. (c) To train the actor-critic pair we use constraint-satisfaction labels, $c .$ , to train $D$ to discriminate between encodings of actual data, $z \\sim q ( z | x )$ , versus latent vectors $z \\sim p ( z )$ sampled from the prior or transformed prior samples $G ( z \\sim p ( z ) , y )$ . Similar to a Conditional GAN, both $G$ and $D$ operate on a concatenation of $z$ and a binary attribute vector, $y$ , allowing $G$ to learn conditional mappings in latent space. If $G$ is an optimizer, a separate attribute discriminator, $D _ { \\mathrm { a t t r } }$ is trained and the latent vector is optimized to reduce the cost of both $D _ { \\mathrm { a t t r } }$ and $D _ { \\mathrm { r e a l i s m } }$ . (d) To sample from the intersection of these regions, we use either gradient-based optimization or an amortized generator, $G$ , to shift latent samples from either the prior $z \\sim p ( z )$ , sampling) or from the data $( z \\sim q ( z | x )$ , transformation). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "• We show that it is possible to generate conditionally from an unconditional model, learning a critic function $D ( z )$ in latent space and generating high-value samples with either gradient-based optimization or an amortized actor function $G ( z )$ , even with a nondifferentiable decoder (e.g., discrete sequences). Focusing on VAEs, we address the tradeoff between reconstruction quality and sample quality (without sacrificing diversity) by enforcing a universal “realism” constraint that requires samples in latent space to be indistinguishable from encoded data (rather than prior samples). Because we start from a VAE that can reconstruct inputs well, we are able to apply identitypreserving transformations by making the minimal adjustment in latent space needed to satisfy the desired constraints. For example, when we adjust a person’s expression or hair, the result is still clearly identifiable as the same person (see Figure 5). This contrasts with pure GAN-based transformation approaches, which often fail to preserve identity. Zero-shot conditional generation. Using samples from the VAE to generate exemplars, we can learn an actor-critic pair that satisfies user-specified rule-based constraints in the absence of any labeled data. ",
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+        "text": "2 BACKGROUND ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Decoder-based deep generative models such as VAEs and GANs generate samples that approximate a population distribution $p ^ { \\star } ( x )$ by passing samples from some simple tractable distribution $p ( z )$ (often $p ( z ) \\ \\triangleq \\ N ( 0 , I ) )$ through a deep neural network. GANs are trained to fool an auxiliary classifier that tries to learn to distinguish between real and synthetic samples. VAEs are fit to data using a variational approximation to maximum-likelihood estimation: ",
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+    {
+        "type": "equation",
+        "img_path": "images/3634b8b9de08d49b9b9960dcea96547c2e19f25a5e6cafd4435a14b746106492.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } ^ { \\mathrm { E L B O } } \\triangleq \\frac 1 N \\sum _ { n } \\mathbb { E } _ { z \\sim q ( z | x _ { n } ) } [ \\log \\pi ( x _ { n } ; g ( z ) ) ] - \\mathrm { K L } ( q ( z \\mid x _ { n } ) \\mid | p ( z ) ) \\le \\frac 1 N \\sum _ { n } \\log p ( x _ { n } ) , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "image",
+        "img_path": "images/542a8eab718a513b2c83e30bbbfd38da241907c397d572b24d2d635aa1f3da90.jpg",
+        "image_caption": [
+            "Figure 2: Typical VAEs use a pixel-wise data likelihood, $\\mathcal { N } ( \\mu _ { x } ( z ) , \\sigma _ { x } I )$ , with $\\sigma _ { x } = 1$ to produce coherent samples at the expense of visual and conceptual blurriness (Row 3). Some reconstructions (Row 2) actually change attributes of the original data. Decreasing $\\sigma _ { x }$ to 0.1 maximizes the ELBO (supplemental Table 4) and increases the fidelity of reconstructions (Row 4) at the cost of sample realism (Row 5). Using an actor to shift prior samples to satisfy the realism constraint, we achieve more realistic samples without sacrificing sharpness (Row 6). The samples are mapped to the closest point in latent space that both satisfies the realism constraint and has the same attributes as the original data. "
+        ],
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+    {
+        "type": "text",
+        "text": "where the “encoder” distribution $q ( z \\mid x )$ is an approximation to the posterior $p ( z \\mid x )$ , $\\pi ( x ; g ( z ) ) \\triangleq$ $p ( x \\mid z )$ is a tractable likelihood function that depends on some parameters output by a “decoder” function $g ( z )$ , and $q$ and $g$ are fit to maximize the evidence lower bound (ELBO) $\\dot { \\mathcal { L } } ^ { \\mathrm { E L B O } }$ . The likelihood $\\pi ( x ; g )$ is often chosen to be a product of simple distributions such as $\\pi ( x ; g ) = \\mathcal { N } ( x ; g , \\sigma _ { x } ^ { 2 } I )$ for continuous data or $\\begin{array} { r } { \\pi ( x ; g ) = \\tilde { \\prod _ { d } \\mathrm { B e r n o u l l i } } ( x _ { d } ; g _ { d } ) } \\end{array}$ for binary data. ",
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+    {
+        "type": "text",
+        "text": "GANs and VAEs have complementary strengths and weaknesses. GANs suffer from the “modecollapse” problem, where the generator assigns mass to a small subset of the support of the population distribution—that is, it may generate realistic samples, but there are many more realistic samples that it cannot generate. This is particularly problematic if we want to use GANs to manipulate data rather than generate new data; even GAN variants that include some kind of inference machinery (e.g., Donahue et al., 2016; Dumoulin et al., 2016; Perarnau et al., 2016) to determine what $z$ best matches some $x$ tend to produce reconstructions that are reminiscent of the input but do not preserve its identity. ",
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+    {
+        "type": "text",
+        "text": "On the other hand, VAEs (especially those with simple likelihoods $\\pi$ ) often exhibit a tradeoff between sharp reconstructions and sensible-looking samples (see Figure 2). That is, depending on what hyperparameters they are trained with (e.g., latent dimensionality and the scale of the likelihood term), VAEs tend to either produce blurry reconstructions and plausible (but blurry) novel samples, or bizarre samples but sharp reconstructions. It has been argued (Makhzani et al., 2016) that this is due to the “holes” problem; the decoder is trained on samples from the marginal posterior $q ( z ) \\triangleq { \\frac { 1 } { N } } \\sum _ { n } q ( z \\mid x _ { n } )$ , which may have very high KL divergence to the presupposed marginal $p ( z )$ (Hoffman & Johnson, 2016). In particular, if the decoder, $g ( z )$ , can reconstruct arbitrary values of $x$ with high accuracy (as in the case of small $\\sigma _ { x }$ ) then the typical posterior $p ( z \\mid x )$ will be highly concentrated. We show this experimentally in supplemental Figure 16. If $q ( z \\mid x )$ underestimates the posterior variance (as it usually does), then the marginal posterior $q ( z )$ will also be highly concentrated, and samples from $\\begin{array} { r } { p ( x ) \\stackrel { \\cdot } { = } \\int _ { z } p ( z ) p ( x \\mid z ) d z } \\end{array}$ may produce results that are far from typical reconstructions $\\mathbb { E } _ { p } [ x \\mid z \\sim q ( z \\mid x ) ]$ . If we tune $\\sigma _ { x }$ to maximize the ELBO (Bishop, 2006), we find the optimal $\\sigma _ { x } \\approx 0 . 1$ (supplemental Table 4). Figure 2 shows that this choice does indeed lead to good reconstructions but strange-looking samples. ",
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+        "img_path": "images/c35df0540875959975d667e8f870eeab6bec5d19ea90e3173421d5a5a6105159.jpg",
+        "image_caption": [
+            "Figure 3: Contour maps of the critic value functions for the marginal posterior (“realism”) constraint. We look at the two latent dimensions that have the lowest average posterior standard deviation on the training set, taking low variance in $z$ space as a proxy for influence over the generated images. All other latent dimensions are held fixed at their original values (from a sample from $p ( z )$ on the left, and from a sample from $q ( z \\mid x )$ for a held-out $x$ on the right). Gray x marks correspond to the points in latent space of the generated images to the right. The cross-section on the left, taken from a prior sample, shows contours that point towards more realistic looking digits. In the cross-section on the right, a sample from the validation set (indicated by orange squares) resides within a local maximum of the critic, as one would hope. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Conditional GANs (CGAN; Mirza & Osindero, 2014) and conditional VAEs (CVAE; Sohn et al., 2015) can generate samples conditioned on attribute information when available, but they must be trained with knowledge of the attribute labels for the whole training set, and it is not clear how to adapt them to new attributes without retraining from scratch. Furthermore, CGANs and CVAEs suffer from the same problems of mode-collapse and blurriness as their unconditional cousins. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We take a different approach to conditional generation and identity-preserving transformation. We begin by training an unconditional VAE with hyperparameters chosen to ensure good reconstruction (at the expense of sample quality). We then train a “realism” critic to predict whether a given $z$ maps to a high-quality sample. We also train critics to predict whether a given $z$ maps to a sample that manifests various attributes of interest. To generate samples that are both realistic and exhibit desired attributes, one option is to optimize random $z$ vectors until they satisfy both the realism and attribute critics. Alternately, we can amortize this cost by training an “actor” network to map a random set of $z$ vectors to a subregion of latent space that satisfies the constraints encoded by the critics. By encouraging these transformed $z$ vectors to remain as close as possible to where they started, we alleviate the mode-collapse problem common to GANs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our approach is summarized visually in Figure 1. The details follow in sections 3, 4, 5, and 6. ",
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+    {
+        "type": "text",
+        "text": "3 THE “REALISM” CONSTRAINT: SHARPENING VAE SAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We define the realism constraint implicitly as being satisfied by samples from the marginal posterior $\\begin{array} { r } { q ( z ) \\triangleq \\frac { 1 } { N } \\sum _ { n } q ( z \\mid \\underline { { x } } _ { n } ) } \\end{array}$ and not those from $p ( z )$ . By enforcing this constraint, we can close the gap between reconstruction quality and sample quality (without sacrificing sample diversity). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As shown in Figure 1, we can train a critic $D$ to differentiate between samples from $p ( z )$ and $q ( z )$ . The critic loss, $\\mathcal { L } _ { D } ( z )$ , is simply the cross-entropy, with labels $c = 1$ for $z \\sim q ( z \\mid x )$ and $c = 0$ for $z \\sim p ( z )$ . We found that the realism critic had little trouble generalizing to unseen data; that is, it was able to recognize samples from $q ( z \\mid x ^ { \\mathrm { h e l d - o u t } } )$ as being “realistic” (Figure 3). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Sampling from the prior is sufficient to train $D$ for models with lower KL Divergence, but if the KL Divergence between $q$ and $p$ is large, the chances of sampling a point $p ( z )$ that has high probability under $q ( z )$ becomes vanishingly small. This leads to poor sample quality and makes it difficult for $D$ to learn a tight approximation of $q ( z )$ solely by sampling from $p ( z )$ . Instead, we use an inner-loop of gradient-based optimization, $G _ { \\mathrm { o p t } } ( z ) = \\mathrm { G r a d i e n t D e s c e n t } ( z ; \\mathcal { L } _ { D } ( z ) )$ , to move prior samples to points deemed more like $q ( z )$ by $D$ . For clarity, we introduce the shorthand $\\mathcal { L } _ { c = 1 } ( z ) \\triangleq - \\log ( D ( z ) )$ and $\\mathcal { L } _ { c = 0 } ( z ) \\triangleq - ( 1 - \\log ( D ( z ) ) )$ . This gives us our critic loss for the realism constraint: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/72cccd329687ad88c9109a1357967119348d651ed0e0fddaabbb5fa09c05f98d.jpg",
+        "text": "$$\n\\mathcal { L } _ { D } ( z ) = \\mathbb { E } _ { z \\sim q ( z | x ) } [ \\mathcal { L } _ { c = 1 } ( z ) ] + \\mathbb { E } _ { z \\sim p ( z ) } [ \\mathcal { L } _ { c = 0 } ( z ) ] + \\mathbb { E } _ { z \\sim G ( p ( z ) ) } [ \\mathcal { L } _ { c = 0 } ( z ) ]\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/c11634da67cd323ff172bf8c40def51cabbde20b6ccd6132f98a3bd62dafbec6.jpg",
+        "image_caption": [
+            "Figure 4: Conditional generation with a CGAN actor-critic pair acting in the latent space of a VAE with $\\sigma _ { x } = 0 . 1$ . Each row starts from a different prior sample and maps it to a new point in latent space that satisfies both the attribute constraints and the realism constraint. The attribute constraints are changed one at a time to produce as smooth a transition as possible from left to right. The bottom CGAN is regularized during training to prefer small shifts in latent space $\\lambda _ { \\mathrm { d i s t } } = 0 . 1 $ , while the top is not $\\lambda _ { \\mathrm { d i s t } } = 0 . 0$ ). Compared to the images generated by the unregularized model, the images generated by the regularized model are much less diverse across columns, suggesting that the regularization does indeed enforce some degree of identity preservation. The regularized model produces images that are somewhat more diverse across rows, suggesting that the regularization fights mode collapse (arguably at the expense of image quality). For each column, the complete list of attributes is given in supplemental Table 3. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Since this inner-loop of optimization can slow down training, we amortize the generation by using a neural network as a function approximator. There are many examples of such amortization tricks, including the encoder of a VAE, generator of a GAN, and fast neural style transfer (Ulyanov et al., 2016; Li & Wand, 2016; Johnson et al., 2016). As with a traditional GAN, the parameters of the function $G$ are updated to maximize the value $D$ ascribes to the shifted latent points. One of the challenges using a GAN in this situation is that it is prone to mode-collapse. However, an advantage of applying the GAN in latent space is that we can regularize $G$ to try and find the closest point in latent space that satisfies $D$ , thus encouraging diverse solutions. We introduce a regularization term, $\\bar { \\mathcal { L } _ { \\mathrm { d i s t } } ( z ^ { \\prime } , z ) } = 1 / \\bar { \\sigma _ { z } } ^ { 2 } \\log ( 1 + ( z ^ { \\prime } - z ) ^ { 2 } )$ to encourage nearby solutions, while allowing more exploration than a mean square error term. As a VAE utilizes only a fraction of its latent dimensions, we scale the distance penalty of each dimension by its utilization, as indicated by the squared reciprocal of the scale $\\sigma _ { z } ( x ) ^ { \\top }$ of the encoder distribution $\\overset { \\cdot } { q } ( z \\mid x )$ , averaged over the training dataset, $\\begin{array} { r } { \\bar { \\sigma } _ { z } \\triangleq \\frac { 1 } { N } \\sum _ { n } \\sigma _ { z } ( x _ { n } ) } \\end{array}$ . The regularized loss is ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/49fd62f2e27ae2133f2e25fcd4d4c2289bdbe915ac00b648c179866e723c4206.jpg",
+        "text": "$$\n\\mathcal { L } _ { G } ( z ) = \\mathbb { E } _ { z \\sim p ( z ) } [ \\mathcal { L } _ { c = 1 } ( G ( z ) ) + \\lambda _ { \\mathrm { d i s t } } \\mathcal { L } _ { \\mathrm { d i s t } } ( G ( z ) , z ) ] .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "4 ATTRIBUTE CONSTRAINTS: CONDITIONAL GENERATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We want to generate samples that are realistic, but we also want to control what attributes they exhibit. Given binary attribute labels $y$ for a dataset, we can accomplish this by using a CGAN in the latent space, which amounts to replacing $D ( z )$ and $G ( z )$ with conditional versions $D ( z , y )$ and $G ( z , y )$ and concatenating $y$ to $z$ as input. If both the actor and critic see attribute information, $G$ must find points in latent space that could be samples from $q ( z )$ with attributes $y$ . ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/0517b9ec5c42259496004e8c839f373c09004ea747d762da9fc2f7ba95731a14.jpg",
+        "image_caption": [
+            "Figure 5: Identity-preserving transformations with optimization. Two separate critics are trained, one for attributes and one for the realism constraint. Starting at the latent points corresponding to the data reconstructions, we then perform gradient ascent in latent space on a weighted combination of critic values (1.0 attribute, 0.1 marginal posterior), stopping when a threshold value is passed for both critics. Images remain semantically close to the original because the pixel-wise likelihood of VAE training encourages identity-preserving reconstructions, and the dynamics of gradient ascent are naturally limited to finding solutions close in latent space. Panels are black for attributes of the original image, as the procedure just returns the original point in latent space. "
+        ],
+        "image_footnote": [],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "This procedure is computationally inexpensive relative to training a generative model from scratch. In most of our experiments, we use a relatively large CGAN actor-critic pair (4 fully connected ReLU layers of 2048 units each), which during training uses about $9 6 \\times$ fewer FLOPs/iteration than the unconditional VAE. We also trained a much smaller CGAN actor-critic pair (3 fully connected ReLU layers of 256 units), which uses about $2 8 8 4 \\times$ fewer FLOPs/iteration than the VAE, and achieves only slightly worse results than the larger CGAN (supplemental Figure 14 and Table 1). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 4 demonstrates the quality of conditional samples from a CGAN actor-critic pair and the effect of the distance penalty, which constrains generation to be closer to the prior sample, maintaining similarity between samples with different attributes. The regularized CGAN actor has less freedom to ignore modes by pushing many random $z$ vectors to the same area of the latent space, since it is penalized for moving samples from $p ( z )$ too far. The increased diversity across rows of the regularized CGAN is evidence that this regularization does fight mode-collapse (additional qualitative evidence is in supplemental Figures 7 and 8). However, without a distance penalty, samples appear more a bit realistic with more prominent attributes. This is supported by Table 1, where we use a separately trained attribute classification model to quantitatively evaluate samples. The actor with no penalty generates samples that are more accurately classified than the actor with a penalty but also shifts the samples much farther in latent space. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/cc7bb3a60a7f67cd3cbf29b444909629c1d49592ac3db863e181b1321510bc4f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>CelebA</td><td>Accuracy</td><td>Precision</td><td>Recall</td><td>F1 Score</td><td>2ZMSE</td></tr><tr><td>(This Work) 10 Attributes</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Test Data</td><td>0.936</td><td>0.901</td><td>0.893</td><td>0.895</td><td></td></tr><tr><td>GcGAN(入dist = 0)</td><td>0.942</td><td>0.914</td><td>0.904</td><td>0.906</td><td>80.7</td></tr><tr><td>GcGAN(入dist = O) (Small Model) GcGAN(入dist = 0.1)</td><td>0.926 0.928</td><td>0.898</td><td>0.860</td><td>0.870</td><td>58.9</td></tr><tr><td>(Perarnau et al., 2016) 18 Attributes</td><td></td><td>0.903</td><td>0.863</td><td>0.874</td><td>17.0</td></tr><tr><td>Test Data</td><td>0.928</td><td></td><td></td><td></td><td></td></tr><tr><td>IcGAN</td><td>0.860</td><td></td><td></td><td>0.715 0.524</td><td></td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Accuracy of a separate model trained to classify attributes from images, evaluated on test data and generated images. We condition and evaluate the generated images on the same labels as the test data. For comparison, the results of a similar task using invertible CGANs for generation (Perarnau et al., 2016) are provided. However, since the full list of salient attributes was not given in the paper, we emphasize that they are not directly comparable as the two experiments use a slightly different set of attribute labels. We also measure the distance in latent space that prior samples are shifted, weighted by $1 / \\bar { \\sigma _ { z } } ^ { 2 }$ . Actors trained with a latent distance penalty $\\lambda _ { \\mathrm { d i s t } }$ have slightly worse accuracy, but find latent points much closer to the prior samples and produce a greater diversity of images (see supplemental Figures 7 and 8). Interestingly, an actor trained without a distance penalty achieves higher classification accuracy than the test set itself, possibly by generating images with more exaggerated and distinctive features than real data. A ”small model” CGAN with $8 5 \\mathrm { x }$ fewer parameters (3 fully connected layers of 256 units) generates images (supplemental Figure 14) of comperable quality. Due to the smaller capacity, the model finds more local solutions (smaller $z _ { M S E } )$ that have slightly less attribute accuracy, but are more visually similar to the prior sample without an explicit regularization term. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Although we used a VAE as the base generative model, our approach could also be used to generate high-quality conditional samples from pretrained classical autoencoders. We show in supplemental Figure 15 that we obtain reasonably good conditional samples (albeit with high-frequency spatial artifacts) as $\\sigma _ { x }  0$ (equivalent to a classical autoencoder). Learning the decoder using VAE training encourages $\\mathsf { q } ( \\mathbf { z } )$ to fill up as much of the latent space as possible (without sacrificing reconstruction quality), which in turn encourages the decoder to map more of the latent space to reasonable-looking images. The prior $p ( z ) = \\mathcal { N } ( \\bar { 0 , } I )$ also imposes a natural scale on the latent variables. ",
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+    {
+        "type": "text",
+        "text": "5 IDENTITY-PRESERVING TRANSFORMATIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "If we have a VAE that can produce good reconstructions of held-out data, we can transform the attributes of the output by gradient-based optimization. We simply need to train a critic, $D _ { a t t r } ( z )$ , to predict the attribute labels $p ( y \\mid z )$ of the data embeddings $z \\sim q ( z \\mid x )$ , and use a cross-entropy loss to train. Then, starting from a data point, $z \\sim q ( z \\mid x )$ , we can perform gradient descent on the the realism constraint and attribute constraint jointly, $\\mathcal { L } _ { D _ { \\mathrm { r e a l } } } ( z ) + \\lambda _ { \\mathrm { a t t r } } \\mathcal { L } _ { D _ { \\mathrm { a t t r } } } ( z )$ . Note that it is helpful to maintain the realism constraint to keep the image from distorting unrealistically. Using the same procedure, we can also conditionally generate new samples (supplemental Figure 9) by starting from $z \\sim p ( z )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 5 demonstrates transformations applied to samples from the held-out evaluation dataset. Note that since the reconstructions are close to the original images, the transformed images also maintain much of their structure. This contrasts with supplemental Figure 10, where a distance-penalty-free CGAN actor produces transformations that share attributes with the original but shift identity. We could preserve identity by introducing a distance penalty, but find that it is much easier to find the correct weighting of realism cost, attribute cost, and distance penalty through optimization, as each combination does not require retraining the network. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 RULE-BASED CONSTRAINTS: ZERO-SHOT CONDITIONAL GENERATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "So far, we have assumed access to labeled data to train attribute classifiers. We can remove the need to provide labeled examples by leveraging the structure learned by our pre-trained model, using it to generate exemplars that are scored by a user-supplied reward function. If we constrain the reward function to be bounded, $c ( x ) : \\mathbb { R } ^ { N }  [ 0 , 1 ]$ , the problem becomes very similar to previous GAN settings, but now the actor, $G$ , and critic, $D$ , are working together. $D$ aims to best approximate the true value of each latent state, $\\mathbb { E } _ { x \\sim p ( x | z ) } c ( x )$ , and $G$ aims to shift samples from the prior to highvalue states. The critic loss is the cross-entropy from $c ( x )$ , and the actor loss is the same as $\\mathcal { L } _ { G }$ in equation 3, where we again have a distance penalty to promote diversity of outputs. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/071568b3db247dc7bdd2ac007549f03830848063407064fd55571d86af87d845.jpg",
+        "image_caption": [
+            "Figure 6: Transformations from a prior sample for the Melody VAE model. In each 16-bar pianoroll, time is in the horizontal direction and pitch in the vertical direction. In the prior sample, notes falling outside of the C Major scale are shown in red. After transformation by $G _ { \\mathcal { P } = \\mathrm { C } _ { \\mathrm { M a j } } , d = 0 }$ , all sampled notes fall within the scale, without a significant change to note density. After transformation of the original $z$ by $G _ { \\mathcal { P } = \\mathrm { C _ { M a j } } , d = 1 9 2 }$ , all sampled notes lay within the scale and the density increases beyond 192. Synthesized audio of these samples can be heard at https://goo. ${ \\mathfrak { g l } }$ /ouULt9. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Note that the reward function and VAE decoder need not necessarily be differentiable, as the critic learns a value function to approximate the reward, which the actor uses for training. To highlight this, we demonstrate that the output of a recurrent VAE model can be constrained to satisfy hardcoded rule-based constraints. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "We first train an LSTM VAE (details in the Appendix) on melodic fragments. Each melody, $m$ , is represented as a sequence of categorical variables. In order to examine our ability to constrain the pitch classes and note density of the outputs, we define two reward functions, one that encourages notes from a set of pitches $\\mathcal { P }$ , and another for that encourages melodies to have at least $d$ notes: ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fd36f8d3a3b5653568cf02d57679b0cca5d4e5e3c34d178299f3626724c5f1e0.jpg",
+        "text": "$$\n\\begin{array} { r } { c _ { \\mathrm { p i t c h } } ( m , \\mathcal { P } ) = \\sum _ { p \\in m } \\mathbb { 1 } ( p \\in \\mathcal { P } ) / | m | \\qquad c _ { \\mathrm { d e n s i t y } } ( m , d ) = \\operatorname* { m i n } ( 1 , | m | / d ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Figure 6 gives an example of controlling the pitch class and note density of generated outputs, which is quantitatively supported by the results in Table 2. During training, the actor goes through several phases of exploration and exploitation, oscillating between expanding to find new modes with high reward and then contracting to find the nearest locations of those modes, eventually settling into high value states that require only small movements in the latent space (supplemental Figure 11). ",
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+    {
+        "type": "text",
+        "text": "7 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Conditional GANs (Mirza & Osindero, 2014) and VAEs (Sohn et al., 2015) introduce conditioning variables at training time. Sohn et al. (2015) allow these variables to affect the distribution in latent $z$ space, but still require that $p ( z \\mid y )$ be a tractable distribution. Perarnau et al. (2016) use CGANs to adjust images, but because CGANs cannot usually reconstruct arbitrary inputs accurately, they must resort to image-space processing techniques to transfer effects to the original input. White (2016) propose adding “attribute vectors” to samples from $p ( z )$ as a simple and effective heuristic to perform transformations, which relies heavily on the linearity of the latent space. ",
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+    {
+        "type": "equation",
+        "img_path": "images/787d40fc747220b493b97abe1d35109fb35e7217d11b5024778f9b49e44b1d72.jpg",
+        "text": "$$\n\\begin{array} { r } { \\left. \\begin{array} { l l l l } { { \\bf A c t o r } } & { { \\bf A c t o r } } & { { \\bf \\Phi } } & { { \\bf c } _ { \\mathrm { p i t c h } } ( m , \\mathcal { P } = { \\bf C } _ { \\mathrm { M a j } } ) } & { c _ { \\mathrm { d e n s i t y } } ( m , d = 1 9 2 ) } \\\\ { { \\bf F r i o r } } & { { \\bf 0 . 5 7 9 } ( 0 . 4 3 \\% ) } & { { \\bf 0 . 4 1 7 } ( 0 . 0 4 \\% ) } & { - } \\\\ { G _ { { \\mathcal P } = { \\bf C } _ { \\mathrm { M a j } } , d = 0 } } & { { \\bf 0 . 9 9 1 } ( 7 0 . 8 \\% ) } & { { \\bf 0 . 4 5 9 } ( 0 . 0 1 \\% ) } & { 0 . 0 1 5 } \\\\ { G _ { { \\mathcal P } = { \\bf C } _ { \\mathrm { M a j } } , d = 1 9 2 } } & { { \\bf 0 . 9 8 2 } ( 6 2 . 4 \\% ) } & { { \\bf 0 . 9 8 5 } ( 8 4 . 9 \\% ) } & { { \\bf 0 . 0 3 9 } } \\end{array} \\right| \\left. \\begin{array} { l } { { z } _ { \\mathrm { M S E } } } \\\\ { { \\bf 0 . 0 4 9 } } \\\\ { { \\bf 0 . 0 1 5 } } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "table",
+        "img_path": "",
+        "table_caption": [
+            "Table 2: Average rewards and constraint satisfaction rates (in parentheses) for unconditional (Prior) and conditional generation. Samples from the prior receive low rewards, on average, and near zero satisfaction rates from both the pitch class (C Major) and note density $\\ge 1 9 2$ notes) constraints. After applying an actor optimized only for the C Major scale $\\scriptstyle \\left( G _ { \\mathcal { P } = \\mathrm { C _ { M a j } } , d = 0 } \\right)$ , the pitch class constraint is fully satisfied $7 0 . 8 \\%$ of the time with only a minor effect on density. The average value close to 1 also indicates that when the constraint is not satisfied, it is typically off by only a few notes. Applying an actor function optimized for the C Major scale and high density $\\scriptstyle \\left( G _ { \\mathcal { P } = \\mathrm { C _ { M a j } } , d = 1 9 2 } \\right)$ causes both constraints to be satisfied at high rates, with a slightly larger shift in latent space. "
+        ],
+        "table_footnote": [],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Some recent work has focused on applying more expressive prior constraints to VAEs (Rezende et al., 2014; Sønderby et al., 2016; Chen et al., 2017; Tomczak & Welling, 2017). The prior that maximizes the ELBO is $p ^ { \\star } ( z ) = q ( z )$ (Hoffman & Johnson, 2016); one can interpret our realism constraint as trying to find an implicit distribution that is indistinguishable from $q ( z )$ . Like the adversarial autoencoder of Makhzani et al. (2016), our realism constraint relies on a discriminative model, but instead of trying to force $q ( z )$ to equal some simple $p ( z )$ , we only weakly constrain $q ( z )$ and then use a classifier to “clean up” our results. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Like this work, the recently proposed adversarially regularized autoencoder (Junbo et al., 2017) uses adversarial training to generate latent codes in a latent space discovered by an autoencoder; that work focuses on unconditional generation. Gomez-Bombarelli et al. (2016) train classifiers in the latent ´ space of a VAE to predict what latent variables map to molecules with various properties, and then use iterative gradient-based optimization in the latent space to find molecules that have a desired set of properties. On molecule data, their procedure generates invalid molecules rarely enough that they can simply reject these samples, which are detected using off-the-shelf software. By contrast, the probability of generating realistic images under our pretrained VAE is astronomically small, and no simple criterion for detecting valid images exists. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jaques et al. (2017) also use a classifier to constrain generation; they use a Deep Q-network as an auxiliary loss for training an LSTM. Closest to Section 6, Nguyen et al. (2016a;b) generate very high quality conditional images by optimizing a sample from the latent space of a generative network to create an image that maximizes the class activations of a pretrained ImageNet classifier. Our work differs in that we learn an amortized generator/discriminator directly in the latent space and we achieve diversity through regularizing by the natural scale of the latent space rather than through a modified Langevin sampling algorithm. ",
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+    },
+    {
+        "type": "text",
+        "text": "8 DISCUSSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have demonstrated a new approach to conditional generation by constraining the latent space of an unconditional generative model. This approach could be extended in a number of ways. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "One possibility would be to plug in different architectures, including powerful autoregressive decoders or adversarial decoder costs, as we make no assumptions specific to independent likelihoods. While we have considered constraints based on implicit density estimation, we could also estimate the constrained distribution directly with an explicit autoregressive model or another variational autoencoder. The efficacy of autoregressive priors in VAEs is promising for this approach (Kingma et al., 2016). Conditional samples could then be obtained by ancestral sampling, and transformations by using gradient ascent to increase the likelihood under the model. Active or semisupervised learning approaches could reduce the sample complexity of learning constraints. Real-time constraint learning would also enable new applications; it might be fruitful to extend the reward approximation of Section 6 to incorporate user preferences as in (Christiano et al., 2017). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            104,
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+            117
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Many thanks to Jascha Sohl-Dickstein, Colin Raffel, and Doug Eck for their helpful brainstorming and encouragement. ",
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "text": "Jakub M. Tomczak and Max Welling. VAE with a VampPrior. CoRR, abs/1705.07120, 2017. URL http://arxiv.org/abs/1705.07120. ",
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+    },
+    {
+        "type": "text",
+        "text": "Dmitry Ulyanov, Vadim Lebedev, Andrea Vedaldi, and Victor S. Lempitsky. Texture networks: Feed-forward synthesis of textures and stylized images. In Proceedings of the 33rd International Conference on Machine Learning (ICML), 2016. URL http://arxiv.org/abs/1603. 03417. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Tom White. Sampling generative networks: Notes on a few effective techniques. arXiv preprint, 2016. URL https://arxiv.org/abs/1609.04468. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "9 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            294,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "9.1 EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            390,
+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For images, we use the MNIST digits dataset (LeCun & Cortes, 2010) and the Large-scale CelebFaces Attributes (CelebA) dataset (Liu et al., 2015). MNIST images are $2 8 \\times 2 8$ pixels and greyscale scaled to [0, 1]. For attributes, we use the number class label of each digit. CelebA images are centercropped to $1 2 8 \\times 1 2 8$ pixels and then downsampled to $6 4 \\times 6 4$ RGB pixels and scaled to [0, 1]. We find that many of the attribute labels are not strongly correlated with changes in the images, so we narrow the original 40 attributes to the 10 most visually salient: blond hair, black hair, brown hair, bald, eyeglasses, facial hair, hat, smiling, gender, and age. ",
+        "bbox": [
+            174,
+            162,
+            825,
+            260
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For melodies, we scraped the web to collect over 1.5 million publicly available MIDI files. We then extracted 16-bar melodies by sliding a window with a single bar stride over each non-percussion instrument with a $\\frac { 4 } { 4 }$ time signature, keeping only the note with the highest pitch when multiple overlap. This produced over 3 million unique melodies. We represent each melody as a sequence of 256 (16 per bar) categorical variables taking one of 130 discrete states at each sixteenth note: 128 note-on pitches, a hold state, and a rest state. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "9.2 MODEL ARCHITECTURES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            366,
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+            381
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "All encoders, decoders, and classifiers are trained with the Adam optimizer (Kingma & Ba, 2015), with learning rate $= 3 \\mathrm { e } { - } 4$ , $\\beta _ { 1 } = 0 . 9$ , and $\\beta _ { 2 } = 0 . 9 9 9$ . ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To train $D _ { r e a l } ( z ) , D _ { a t t r } ( z )$ and $G ( z )$ we follow the training procedure of Gulrajani et al. (2017), applying a gradient penalty of 10, training $D$ and $G$ in a 10:1 step ratio, and use the Adam optimizer with learning rate $= 3 \\mathrm { e } { - } 4$ , $\\beta _ { 1 } ~ = 0 . 0$ , and $\\beta _ { 2 } ~ = 0 . 9$ . While not necessary to converge, we find it improves the stability of optimization. We do not apply any of the other tricks of GAN training such as batch normalization, minibatch discrimination, or one-sided label smoothing (Radford et al., 2015; Salimans et al., 2016). As samples from $p ( z )$ are easier to discriminate than samples from $G ( p ( z ) )$ , we train $D$ by sampling from $p ( z )$ at a rate 10 times less than $G ( p ( z ) )$ . For actors with inner-loop optimization, $G _ { \\mathrm { o p t } }$ , 100 iterations of Adam are used with with learning rate $= 1 \\mathrm { e } \\mathrm { - } 1$ , $\\beta _ { 1 } =$ 0.9, and $\\beta _ { 2 } = 0 . 9 9 9$ . ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "9.2.1 MNIST FEED-FORWARD VAE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To model the MNIST data, we use a deep feed-forward neural network (Figure 13a). ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The encoder is a series of 3 linear layers with 1024 outputs, each followed by a ReLU, after which an additional linear layer is used to produce 2048 outputs. Half of the outputs are used as the $\\mu$ and the softplus of the other half are used as the $\\sigma$ to parameterize a 1024-dimension multivariate Gaussian distribution with a diagonal covariance matrix for $z$ . ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The decoder is a series of 3 linear layers with 1024 outputs, each followed by a ReLU, after which an additional linear layer is used to produce $2 8 \\mathbf { x } 2 8$ outputs. These outputs are then passed through a sigmoid to generate the output image. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "9.2.2 CELEBA CONVOLUTIONAL VAE ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To model the CelebA data, we use a deep convolutional neural network (Figure 13b). ",
+        "bbox": [
+            176,
+            768,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The encoder is a series of $4 ~ 2 \\mathrm { D }$ convolutional layers, each followed by a ReLU. The convolution kernels are of size $3 \\times 3$ , $3 \\times 3$ , $5 \\times 5$ , and $5 \\times 5$ , with 2048, 1024, 512, and 256 output channels, respectively. All convolutional layers have a stride of 2. After the final ReLU, a linear layer is used to produce 2048 outputs. Half of the outputs are used as the $\\mu$ and the softplus of the other half are used as the $\\sigma$ to parameterize a 1024-dimension multivariate Gaussian distribution with a diagonal covariance matrix for $z$ . ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The decoder passes the $z$ through a $4 \\mathrm { x } 4 \\mathrm { x } 2 0 4 8$ linear layer, and then a series of 4 2D transposed convolutional layers, all but the last of which are followed by a ReLU. The deconvolution kernels are of size $5 \\times 5 , 5 \\times 5 , 3 \\times 3$ , and $3 \\times 3$ , with 1024, 512, 256, and 3 output channels, respectively. All deconvolution layers have a stride of 2. The output from the final deconvolution is passed through a sigmoid to generate the output image. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The classifier that is trained to predict labels from images are identical to the VAE encoders except that they end with a sigmoid cross-entropy loss. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "9.2.3 MELODY SEQUENCE VAE ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Music is fundamentally sequential, so we use an LSTM-based sequence VAE for modelling monophonic melodies (Figure 13c). ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The encoder is made up of a single-layer bidirectional LSTM, with 2048 units per cell. The final output in each direction is concatenated and passed through a linear layer to produce 1024 outputs. Half of the outputs are used as the $\\mu$ and the softplus of the other half are used as a $\\sigma$ to parameterize a 512-dimension multivariate Gaussian distribution with a diagonal covariance matrix for $z$ . ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Since musical sequences often have structure at the bar level, we use a hierarchical decoder to model long melodies. First, the $z$ goes through a linear layer to initialize the state of a 2-layer LSTM with 1024 units per layer, which outputs 16 embeddings of size 512 each, one per bar. Each of these embeddings are passed through a linear layer to produce 16 initial states for another 2-layer LSTM with 1024 units per layer. This bar-level LSTM autoregressively produces individual sixteenth note events, passing its output through a linear layer and softmax to create a distribution over the 130 classes. This categorical distribution is used to compute a cross-entropy loss during training or samples at inference time. In addition to generating the initial state at the start of each bar, the embedding for the current bar is concatenated with the previous output as the input at each time step. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "9.2.4 ACTOR FEED-FORWARD NETWORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For $G ( z )$ , we use a deep feed-forward neural network (Figure 12a) in all of our experiments. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The network is a series of 4 linear layers with 2048 outputs, each followed by a ReLU, after which an additional linear layer is used to produce $2 * d i m ( z )$ outputs. Half of the outputs are used as the $\\delta z$ and the sigmoid of the other half are used as gates. The transformed $z ^ { \\prime }$ is the computed as $( 1 - g a t e s ) * z + g a t e s * \\delta z$ . This aids in training as the network only has to then predict shifts in $z$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "When conditioning on attribute labels, $y$ , to compute $G ( z , y )$ , the labels are passed through a linear layer producing 2048 outputs which are concatenated with $z$ as the model input. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "9.2.5 CRITIC FEED-FORWARD NETWORK",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For $D ( z )$ , we use a deep feed-forward neural network (Figure 12b) in all of our experiments. ",
+        "bbox": [
+            173,
+            633,
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+            648
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The network is a series of 4 linear layers with 2048 outputs, each followed by a ReLU, after which an additional linear layer is used to produce a single output. This output is passed through a sigmoid to compute $D ( z )$ . ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "When conditioning on attribute labels, $y$ , to compute $D ( z , y )$ , the labels are passed through a linear layer producing 2048 outputs which are concatenated with $z$ as the model input. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "9.3 SUPPLEMENTAL FIGURES ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/c7271bc588426b5575cc43d68291dd43b6a2951e774d979aaf34823640dd61fa.jpg",
+        "image_caption": [
+            "Figure 7: Additional generated CelebA faces by $G _ { \\mathrm { C G A N } }$ with $\\lambda _ { \\mathrm { d i s t } } = 0$ . Full attribute labels are given in supplementary Table 3 "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/b77cb599d9dddb0451265adee643ada6dfeb1a26318158c1994cde29b9c5bc27.jpg",
+        "image_caption": [
+            "Figure 8: Additional generated CelebA faces by $G _ { \\mathrm { C G A N } }$ with $\\lambda _ { \\mathrm { d i s t } } = 0 . 1$ . Full attribute labels are given in supplementary Table 3 "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/0c9693a16a3b9e8281a0253d76908badcc7a1bf21d7403c4feac61198a6e884b.jpg",
+        "image_caption": [
+            "Figure 9: Optimization of samples drawn from the prior to satisfy both the realism constraint and attribute constraints (drawn from the test set). The optimization takes 100 steps, and images are shown at 0, 10, 30, 50 and 100 steps. $D$ is trained with inner-loop optimization, $G _ { \\mathrm { o p t } }$ , as described in Section 9.2 "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/ef19112b6592d78019a654a7c1cb0b4ff26e30ce2b4848ea671d418a9c7cfb43.jpg",
+        "image_caption": [
+            "Figure 10: Identity-distorting transformations with CGAN actor-critic. Without a penalty to encourage small moves in latent space, the actor maps the latent vectors of the original data points to generated images that have the correct attributes, but a different identity. Panels are black for attributes of the original image, as the procedure just returns the same image as the reconstruction. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/d7228d1a204c64ab4cfde7920be6b879b613b90a99a551489d60c0efcddc7575.jpg",
+        "image_caption": [
+            "Figure 11: Training curves for melody actor $( G )$ and critic $( D )$ pair for pitch class constraint $c _ { \\mathrm { p i t c h } } ( m , \\mathcal { P } = \\mathrm { C _ { M a j } } ) ,$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/ef83a5eac17c8d98a5925bf6d30cd03e235814d62ff8b045ce406ea347ad5889.jpg",
+        "image_caption": [
+            "Figure 12: Architecture for the (a) actors and (b) critics used in all experiments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "table",
+        "img_path": "images/24692e02085c2ae4ca713890f1af51edff971766265b373170d7122e52c78e9e.jpg",
+        "table_caption": [
+            "Table 3: Complete list of attributes for label names in Figures 4, 7, and 8 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Figure Label</td><td>Bald</td><td>Black Hair</td><td>Blond Hair</td><td>Brown Hair</td><td>Eye- glasses</td><td>Male</td><td>Beard</td><td>Smiling</td><td>Hat</td><td>Young</td></tr><tr><td>Blond Hair</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Brown Hair</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Black Hair</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Male</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Facial Hair</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Eyeglasses</td><td>0</td><td>1</td><td>0</td><td>0</td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Bald</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td></tr><tr><td>Aged</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>1</td><td>0</td><td>0</td><td>0</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "table",
+        "img_path": "images/19e53e3b797b77aea64835a148e47d38838155d220ad6ba25b4f62cc01b052a8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>LL</td><td>KL</td><td>ELBO</td></tr><tr><td>1</td><td>-11360</td><td>30</td><td>-11390</td></tr><tr><td>1e-1</td><td>-11325</td><td>150</td><td>-11475</td></tr><tr><td>1e-2</td><td>15680</td><td>600</td><td>15080</td></tr><tr><td>1e-3</td><td>16090</td><td>1950</td><td>14140</td></tr><tr><td>1e-4</td><td>16150</td><td>3650</td><td>12500</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Selection of $\\sigma _ { x } = 0 . 1$ for the CelebA VAEs by ELBO maximization. All results are given in Nats. ",
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/bc6a46f235dfa72750a10f4560f90a19b564bdee54ba80ca938988213816835f.jpg",
+        "image_caption": [
+            "Figure 13: Architectures for the (a) feed-forward MNIST, (b) convolutional CelebA, and (c) hierarchical LSTM melody VAEs. In (b), all convolutions have a stride of 2. In (c), LSTM cells shown in the same color share weights and linear layers between levels are omitted. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Small G/D Models (256 ReLU × 3) ",
+        "text_level": 1,
+        "bbox": [
+            380,
+            108,
+            666,
+            125
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "image",
+        "img_path": "images/56f7770ad96713af1e31a7e0cf8d16ef63bca62f373d9a77aee8f609762fe969.jpg",
+        "image_caption": [
+            "Figure 14: Samples generated with smaller (3 ReLU layers of 256 units each) $G$ and $D$ models are comparable quality despite having $8 5 \\mathrm { x }$ fewer parameters, $\\lambda _ { \\mathrm { d i s t } } = 0 . 0$ . Full attribute labels are given in supplementary Table 3. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "image",
+        "img_path": "images/e12c823878b34da528fa8dbef4b7a3e2eb2c8d0d959f24089a66aca5f4665e02.jpg",
+        "image_caption": [
+            "Figure 15: Latent constraints applied to a vanilla autoencoder with no latent prior. Samples are similar quality to VAEs with $\\sigma _ { x } ~ = ~ 0 . 1$ , but with less diversity and more high-frequency visual artifacts. Full attribute labels are given in supplementary Table 3. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "image",
+        "img_path": "images/389dcabc3eef37e9e3ed28351331289d6d040ac48592c4ad9d43bd89f6e92ff1.jpg",
+        "image_caption": [
+            "Figure 16: Smaller decoder standard deviations, $\\sigma _ { x }$ , lead to lower-variance posteriors, $\\sigma _ { z } ( x )$ of the encoder $q ( z \\mid x )$ , averaged over the training set per a dimension. The $\\mathbf { X } ^ { } -$ -axis is sorted from lowest to highest variance. Tighter posteriors correspond to more utilization of the latent dimension, and we scale our distance regularization the square inverse on a per-dimension basis. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 21
+    }
+]