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+        "text": "CHARACTER LEVEL BASED DETECTION OF DGA DOMAIN NAMES ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+        "text": "ABSTRACT ",
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+        "text": "Recently several different deep learning architectures have been proposed that take a string of characters as the raw input signal and automatically derive features for text classification. Few studies are available that compare the effectiveness of these approaches for character based text classification with each other. In this paper we perform such an empirical comparison for the important cybersecurity problem of DGA detection: classifying domain names as either benign vs. produced by malware (i.e., by a Domain Generation Algorithm). Training and evaluating on a dataset with 2M domain names shows that there is surprisingly little difference between various convolutional neural network (CNN) and recurrent neural network (RNN) based architectures in terms of accuracy, prompting a preference for the simpler architectures, since they are faster to train and less prone to overfitting. ",
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+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Malware is software that infects computers in order to perform unauthorized malicious activities. In order to successfully achieve its goals, the malware needs to be able to connect to a command and control (C&C) center. To this end, both the controller behind the C&C center (hereafter called botmaster) and the malware on the infected machines can run a Domain Generation Algorithm (DGA) that generates hundreds or even thousands of domains automatically. The malware then attempts at resolving each one of these domains with its local DNS server. The botmaster will have registered one or a few of these automatically generated domains. For these domains that have been actually registered, the malware will obtain a valid IP address and will be able to communicate with the C&C center. ",
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+        "text": "The binary text classification task that we address in this paper is: given a domain name string as input, classify it as either malicious, i.e. generated by a DGA, or as benign. Deep neural networks have recently appeared in the literature on DGA detection Woodbridge et al. (2016); Saxe & Berlin (2017); Yu et al. (2017). They significantly outperform traditional machine learning methods in accuracy, at the price of increasing the complexity of training the model and requiring larger datasets. Independent of the work on deep networks for DGA detection, other deep learning approaches for character based text classification have recently been proposed, including deep neural network architectures designed for processing and classification of tweets (Dhingra et al. (2016); Vosoughi et al. (2016)) as well as general natural language text (Zhang et al. (2015)). No systematic study is available that compares the predictive accuracy of all these different character based deep learning architectures, leaving one to wonder which one works best for DGA detection. ",
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+        "text": "To answer this open question, in this paper we compare the performance of five different deep learning architectures for character based text classification (see Table 1) for the problem of detecting DGAs. They all rely on character-level embeddings, and they all use a deep learning architecture based on convolutional neural network (CNN) layers, recurrent neural network (RNN) layers, or a combination of both. Our most important finding is that for DGA detection, which can be thought of as classification of short character strings, despite of vast differences in the deep network architectures, there is remarkably little difference among the methods in terms of accuracy and false positive rates, while they all comfortably outperform a random forest trained on human engineered features. This finding is of practical value for the design of deep neural network based classifiers for short text classification in industry and academia: it provides evidence that one can select an architecture that is faster to train, without loss of accuracy. In the context of DGA detection, optimizing the training time is of particular importance, as the models need to be retrained on a regular basis to stay current with respect to new, emerging malware. ",
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+        "img_path": "images/8da383fc2555e476d25288e0fb385d05b72e28384bfa907e62e595e1a9908fd3.jpg",
+        "table_caption": [
+            "Table 1: High level overview of recent deep learning approaches for character based text classification "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>Architecture</td><td rowspan=1 colspan=1>Reference</td></tr><tr><td rowspan=1 colspan=1>Endgame</td><td rowspan=1 colspan=1>single LSTMlayer</td><td rowspan=1 colspan=1>Woodbridge et al. (2016)</td></tr><tr><td rowspan=1 colspan=1>Invincea</td><td rowspan=1 colspan=1>parallel CNN layers</td><td rowspan=1 colspan=1>Saxe&amp;Berlin (2017)</td></tr><tr><td rowspan=1 colspan=1>CMU</td><td rowspan=1 colspan=1>forward LSTM layer+ backward LSTMlayer</td><td rowspan=1 colspan=1>Dhingra et al. (2016)</td></tr><tr><td rowspan=1 colspan=1>MIT</td><td rowspan=1 colspan=1> stacked CNN layers + single LSTM layer</td><td rowspan=1 colspan=1>Vosoughi et al. (2016)</td></tr><tr><td rowspan=1 colspan=1>NYU</td><td rowspan=1 colspan=1>stacked CNNlayers</td><td rowspan=1 colspan=1>Zhang et al. (2015)</td></tr></table>",
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+        "text": "2 BACKGROUND ",
+        "text_level": 1,
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+        "text": "Malware controllers, or botmasters, use malware for all kinds of unauthorized malicious activities. These activities range from stealing information, to exploiting the victims’ computing resources to mine bitcoin. They can also include launching a distributed denial of service attack from the victims’s computers or encrypting the victims hard drive (ransomware). In order to successfully achieve its goals, it is vital that the malware be able to connect to a command and control (C&C) center. This communication can serve many purposes. The malware can use it to send stolen information (such as passwords or access credentials) to the malware designer behind the C&C center, it can use this communication channel to receive instructions or even to update itself to a newer version. ",
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+        "text": "Initially, botmasters established such a communication channel to the C&C center by hard-coding an IP address inside the malware. This approach has obvious shortcomings from the botmasters’ perspective: once the malware is reversed engineered, the IP address is discovered and shut down. Over time, malware designers came up with a much more effective strategy: Domain Generation Algorithms (DGAs). Domain Generation Algorithms work by having the malware accessing some available source of randomness and inputting it into an algorithm that generates hundreds or even thousands of domains automatically. The malware then attempts at resolving each one of these domains with its local DNS server (a DNS server runs a protocol that translates domain names into IP addresses; it is a vital piece of the Internet). The botmaster will have registered one or a few of these automatically generated domains. For these domains that have been actually registered, the malware will obtain a valid IP address and will be able to communicate with the C&C center. For all the other domains that were automatically generated but not registered, the malware obtains a message stating that these domains could not have been resolved and ignores them. ",
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+        "text": "DGAs make blacklisting of domains extremely difficult, since by changing the initial randomness (while keeping the same algorithm) the malware can potentially generate completely different domains. This technique has been used by high-profile malware such as Conficker, Stuxnet (the malware designed to attack Iran nuclear facilities) and Flame. Catching domain names generated by malware has become a central topic in information security, leading to a recent interest in detecting DGA domains using machine learning techniques. Models that classify domain names as benign or malicious based solely on the domain name string are of particular interest for their generality, as context information beyond the domain name string might be unavailable or expensive to acquire. Traditional machine learning methods for DGA detection based on the domain name string rely on extraction of predefined, human engineered lexical features, see e.g. Antonakakis et al. (2012); Schiavoni et al. (2014). Whenever human engineered features are used, it is obvious that this opens the door for an adversary to carefully craft its DGA to avoid detection by using the aforementioned features. This makes maintaining such machine learning systems labor intensive. Recently proposed deep learning techniques for detecting DGAs learn features automatically, thereby offering the potential to bypass the human effort of feature engineering (Woodbridge et al. (2016); Saxe & Berlin (2017); Yu et al. (2017)). In addition these deep learning approaches outperform the traditional machine learning techniques with human engineered features in terms of accuracy and false positive rates. The choice of deep network architecture in these recent works appears fairly arbitrary, and it is unclear whether they would perform better or worse than other deep neural networks approaches for text classification that have been introduced recently, in particular character-level methods for processing and classification of tweets (Dhingra et al. (2016); Vosoughi et al. (2016)) and for general natural language text (Zhang et al. (2015)). In this paper, we provide the first comparative study of all these different methods, showing that, despite of vast differences in the architectures, they can be easily tuned to result in the same predictive accuracy. ",
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+        "text": "The problem of adversarial examples, i.e. instances that have intentionally been designed to cause the model to make a mistake, are a well known problem in machine learning. The scenario sketched above — in which a malware designer exploits knowledge about the lexical features used by a random forest to craft his DGA to avoid detection — is a prime example of this. Deep neural networks are famously not immune to adversarial examples either, and generative adversarial networks (GANs) can be trained to generate them automatically (see e.g. Goodfellow et al. (2014a;b)). Specifically in the context of DGA detection, Anderson et al. (2016) have used a character-based generative adversarial network (GAN) to augment training sets in order to harden other machine learning models (like a random forest) against yet-to-be-observed DGAs. It is highly unlikely for attackers to use GANs themselves, because DGA algorithms must be light enough to be embedded inside malware code. Furthermore, generating domain names that look like a benign domain is not enough for an effective DGA. Ideally, every domain produced by a DGA must not have been registered yet or must have a low likelihood of being registered already – if a domain produced by a DGA has already been taken, it is useless for the botmaster. Combining all these requirements is essential for a serious study of adversarial generated domains and outside the scope of this paper. ",
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+    {
+        "type": "text",
+        "text": "3 METHODS ",
+        "text_level": 1,
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+        "text": "We compare five different deep learning methods for short string classification, when applied to the problem of DGA detection specifically. For each of the methods, we start from the original proposals as can be found in the references in Table 1 and only make modifications when they improve the predictive accuracy for the classification of domain names. Below we give an overview of the methods and the adaptations made. A Keras1 code snippet for each method is included in Appendix A. ",
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+        "text": "The strings that we give as input to all classifiers consist of a second level domain (SLD) and a top level domain (TLD), separated by a dot, as in e.g. wikipedia.org. Following Woodbridge et al. (2016), we set the maximum length at 75 characters, padded with zeros on the left for domains whose length is less than 75.2 We convert each domain name string to lower case, since domain names are case insensitive, and encode it as an ASCII code sequence of length 128, effectively representing each domain name string as a 75 by 128 matrix in which each character corresponds to a column. ",
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+        "text": "3.1 RNN BASED ARCHITECTURES ",
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+        "text": "Endgame Model Long short-term memory networks (LSTMs), a special kind of recurrent neural networks (RNNs) have recently attracted a lot of attention because of their successful application to problems that involve processing of sequences (Hochreiter & Schmidhuber (1997)). Since domain names can be thought of as sequences of characters, LSTMs are a natural kind of classifiers to apply. The LSTM network proposed by Woodbridge et al. (2016) was designed specifically for DGA detection, so we stay very close to the original model. The network is comprised of an embedding layer, an LSTM layer (128 LSTM cells with default Tanh activation), and a single node output layer with sigmoid activation. Instead of using RMSProp as the optimization algorithm, as was done in Woodbridge et al. (2016), we switched to Adam (Kingma & Ba (2014)) because it resulted in better loss convergence results (see Section 4). The Endgame model includes dropout, a technique to improve model performance and overcome over-fitting by randomly excluding nodes during training, which serves to break up complex co-adaptations in the network Srivastava et al. (2014). This is confined to the training phase; all nodes are active during testing and deployment. ",
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+        "text": "The role of the embedding layer is to learn to represent each character that can occur in a domain name by a 128-dimensional numerical vector. This vector is different from the original 128- dimensional ASCII encoding. The embedding maps semantically similar characters to similar vectors, where the notion of similarity is implicitly derived (learned) based on the classification task at hand. As will become clear in the remainder of this section, all five deep neural network architectures under study start with such an embedding layer. To allow for a fair comparison, we have made the parameter choices for the embedding layer, such as the dimensionality of the embedding space, identical for all five models. In addition, for comparison purposes, in Section 4 we also present the results of a “baseline neural network model” consisting only of an embedding layer as its hidden layer. ",
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+        "text": "CMU Model Bidirectional RNNs extend regular RNNs by processing the input string in two ways. In a forward layer, the input sequence is processed from the left to the right, as in a traditional RNN, while in a backward layer, the processing happens from the right to the left. The output from the forward and the backward layer is then combined and passed on to further layers. Bidirectional LSTMs for character level text processing have been proposed in Ling et al. (2015), and, following up on that, very similar bidirectional GRUs (gated recurrent units) have been applied in a “Tweet2Vec” model for tweet classification (predicting hashtags of tweets) by Dhingra et al. (2016). We use an adaptation of the latter; see Listing 2 in Appendix A. Including dropout or replacing LSTM by GRU did not cause a significant change in predictive accuracy, although the latter did result in a decrease of training runtime. ",
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+        "type": "text",
+        "text": "3.2 CNN BASED ARCHITECTURES ",
+        "text_level": 1,
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+        "text": "NYU Model Convolutional neural networks (CNNs) are known for their ability to process input data with a grid like topology, such as images consisting of a grid of pixels. To the best of our knowledge, Zhang et al. (2015) were the first to apply 1-dimensional or “temporal” CNNs successfully to text classification at character level. Their proposed deep network architecture, which is intended to process full-blown natural language text such as news articles or reviews, includes 6 stacked CNN layers, with each subsequent layer consuming the output from the previous layer. In contrast to natural language text, domain names are very short and they do not have an internal grammatical structure, naturally resulting the original architectures from Zhang et al. (2015) to overfit on our data. We therefore reduced the number of stacked CNN layers to two, and decreased the size and the number of filters on the CNN layers (see Listing 3). ",
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+        "text": "Invincea Model Saxe & Berlin (2017) proposed a CNN based classifier that takes generic short character strings as its input and learns to detect whether they are indicators of malicious behavior. The short character strings can be e.g. URLs, file paths, or registry keys. The fundamental difference between the Invincea model versus the NYU model described above, is that in the Invincea model the CNN layers are parallel instead of stacked, and that pooling always happens over the entire domain name instead of within a small pooling window. That means that the Invincea model is only detecting the presence or absence of patterns in the domain names, and does not retain any information on where exactly in the domain name string these patterns occur. In the Invincea model, the embedding layer is followed by a convolutional layer with 1024 filters, namely 256 filters for each of the sizes 2, 3, 4, and 5. Each of these filters learns to detect the soft presence of an interesting soft $n$ -gram (with $n = 2 , 3 , 4 , 5$ ). The output of the convolutional layer is consumed by two dense hidden layers, each with 1024 nodes, before reaching a single node output layer with sigmoid activation. Out of all the models that we compared, this one has the most extensive architecture. ",
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+    {
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+        "text": "3.3 HYBRID CNN/RNN BASED ARCHITECTURE ",
+        "text_level": 1,
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+        "text": "MIT Model The MIT model proposed by Vosoughi et al. (2016) is an extension of the NYU model, where the stacked CNN layers are followed by an LSTM layer. Similarly as with the NYU model, the use of multiple stacked CNN layers (which worked well for tweets in Vosoughi et al. (2016)) resulted in the models to overfit on our data. For this reason, we reduced the MIT model architecture to the minimum that preserves its spirit: one CNN layer followed by one LSTM layer. ",
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+        "img_path": "images/85063668b9a10fc7a667cd524993b7ec538cda6f99609e6ed9697650223202bf.jpg",
+        "image_caption": [
+            "Figure 1: Training and validation loss curves. The vertical axis in Figure (f) has a different scale than the other figures, related to the fact that the “Embedding only” baseline model has a much higher loss than the other models. "
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+        "text": "4 RESULTS AND LEARNED REPRESENTATIONS ",
+        "text_level": 1,
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+        "text": "We trained and evaluated the models on a dataset with 1 million DGA domain names from Bambenek3 (positive examples) and the top 1 million domain names from Alexa4 (negative examples). Alexa ranks websites based on their popularity in terms of number of page views and number of unique visitors. It only retains the websites’ SLD and TLD, aggregating across any subdomains. For example, according to Alexa, the five highest ranked domain names in terms of popularity on 2017-10-26 are google.com, youtube.com, facebook.com baidu.com, and wikipedia.org. For our experiments, we assume that the top 1 million domain names in this ranking are benign domain names, although it is possible that the bottom of the ranking may contain some noise. ",
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+        "text": "In addition to these benign domain names, we collected 1 million DGA domain names from the Bambenek Consulting feeds for 3 different days, namely Jun 24, Jul 22, Jul 23, 2017. These feeds contain DGA domain names from specific malware families that were observed in real traffic on those days. Such domain names can be collected by reverse engineering a known malware family, generating lists of domain names with the reverse engineered malware, and checking which of these domain names also occur in real traffic. Note that our goal in this paper is the development of a neural network classifier that can detect DGAs without the need to reverse engineer malware families. An important advantage of such a classifier is that it can also be used against new and previously unknown malware families. ",
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+        "img_path": "images/540acc38e362ba110e32a05379b9dcf448ade6507cedf43f7461b3e481bb3394.jpg",
+        "table_caption": [
+            "Table 2: Results on test data from July 2017. Accuracy, TPR, FPR are w.r.t. a threshold that gives a FPR of 0.001 on the validation data. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=2>Architecture</td><td rowspan=1 colspan=1>Acc</td><td rowspan=1 colspan=1>TPR</td><td rowspan=1 colspan=1>FPR</td><td rowspan=1 colspan=1>AUC@1%</td></tr><tr><td rowspan=1 colspan=1>RF</td><td rowspan=1 colspan=2>Lexical features</td><td rowspan=1 colspan=1>91.51%</td><td rowspan=1 colspan=1>83.15%</td><td rowspan=1 colspan=1>0.00128</td><td rowspan=1 colspan=1>84.77%</td></tr><tr><td rowspan=1 colspan=1>MLP</td><td rowspan=1 colspan=2>Lexical features</td><td rowspan=1 colspan=1>73.74%</td><td rowspan=1 colspan=1>47.61%</td><td rowspan=1 colspan=1>0.00091</td><td rowspan=1 colspan=1>58.81%</td></tr><tr><td rowspan=1 colspan=1>Embedding</td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>84.29%</td><td rowspan=1 colspan=1>68.69%</td><td rowspan=1 colspan=1>0.00108</td><td rowspan=1 colspan=1>80.88%</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>LSTM</td><td rowspan=1 colspan=1>CNN</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Endgame</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>98.72%</td><td rowspan=1 colspan=1>97.55%</td><td rowspan=1 colspan=1>0.00102</td><td rowspan=1 colspan=1>98.03%</td></tr><tr><td rowspan=1 colspan=1>Invincea</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>98.95%</td><td rowspan=1 colspan=1>98.01%</td><td rowspan=1 colspan=1>0.00109</td><td rowspan=1 colspan=1>97.47%</td></tr><tr><td rowspan=1 colspan=1>CMU</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>98.54%</td><td rowspan=1 colspan=1>97.18%</td><td rowspan=1 colspan=1>0.00108</td><td rowspan=1 colspan=1>98.25%</td></tr><tr><td rowspan=1 colspan=1>MIT</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>98.70%</td><td rowspan=1 colspan=1>97.49%</td><td rowspan=1 colspan=1>0.00099</td><td rowspan=1 colspan=1>97.55%</td></tr><tr><td rowspan=1 colspan=1>NYU</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>98.58%</td><td rowspan=1 colspan=1>97.27%</td><td rowspan=1 colspan=1>0.00116</td><td rowspan=1 colspan=1>97.93%</td></tr></table>",
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+        "text": "We randomly split the data into $80 \\%$ for training, $10 \\%$ for validation, and $10 \\%$ for testing. Figure 1 shows the training and validation loss curves for each of the models described in Section 3. The displayed epochs indicate where we stopped the training to obtain the models used to produce the final results in Table 2 and 5. The training loss is higher than the validation loss in the pictures in Figure 1 because the loss against the training data is computed in an average way across batches (the batch size is 100) while dropout is being applied, whereas performance on the validation set is determined at the end of each epoch with dropout disabled. Figure 1(f) displays the loss curves for training a simple neural network consisting of only an embedding layer. We include the performance of this network in our results as a baseline. ",
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+        "text": "The accuracy of each of the trained models when applied to the test data is recorded in Table 2. In addition to accuracy, this table includes the true positive rate (TPR) and false positive rate (FPR) for each of the models. Recall that $\\mathrm { T P R } = \\mathrm { T P } / ( \\mathrm { T P } { + } \\mathrm { F N } )$ and $\\mathrm { F P R } = \\mathrm { F P } / ( \\mathrm { F P + T N } )$ where TP, FP, TN, and FN are the number of true positives, false positives, true negatives, and false negatives respectively. A low false positive rate is very important in deployed DGA detection systems, because blocking legitimate traffic is highly undesirable. All classifiers in Table 2 output a probability that a given instance belongs to the positive class, so we can tune a threshold probability at which to consider a prediction positive. For each model, we choose this threshold such that the model trained over the training data has a 0.001 FPR over the validation data. Then we report the accuracy, TPR and FPR obtained with this classification threshold over the test data. Finally, we also report AUC $@1 \\%$ FPR, which is the integral of the ROC curve from $\\mathrm { { F P R } = 0 }$ to $\\mathrm { { F P R } = 0 . 0 1 }$ on the test data. ",
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+        "type": "text",
+        "text": "For comparison purposes, Table 2 also contains results for a Random Forest (RF) and a Multilayer Perceptron (MLP) trained on the following 11 features, extracted from each domain name string (see Yu et al. (2014; 2016)): ent (normalized entropy of characters); nl2 (median of 2-gram); nl3 (median of 3-gram); naz (symbol character ratio); hex (hex character ratio); vwl (vowel character ratio); len (domain label length); gni (gini index of characters); cer (classification error of characters); tld (top level domain hash); dgt (first character digit). The Random Forest consists of 100 trees. The MLP has a single hidden layer with 128 nodes; see Listing 7 in Appendix A. Like the deep networks in this paper, this MLP was trained with batch size 100. The values of the 11 features are normalized so that they are all on the same scale before presenting them to the MLP. ",
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+    {
+        "type": "text",
+        "text": "As expected, the FPR of all classifiers is around 0.001. There is a clear variation in the TPR that the classifiers achieve against that small FPR. While the Random Forest is only able to “catch” $83 \\%$ of the malicious domain names, all the deep neural network architectures achieve a recall of 97- $98 \\%$ . The baseline neural network consisting of only an embedding layer as its hidden layer clearly performs the worst with a TPR of less than $69 \\%$ , highlighting that it is advantageous to extend the network architecture with one or more LSTM or CNN layers. Interestingly, there is little to no variation among the five deep neural network architectures in terms of TPR. ",
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+    {
+        "type": "text",
+        "text": "Table 3 contains examples of domain names that were randomly selected among those misclassified by either the Random Forest or by all five deep neural networks. Inspecting the column of the benign domain names, i.e. the Alexa domain names, it is interesting to note that most of those misclassified by the deep neural networks (bottom left in the table) come across as gibberish that a human annotator would likely also classify as malicious. As also evident from the top right of Table 3, the deep neural networks have become very good at considering such gibberish-looking domain names to be malicious, even though they were never explicitly told to do so (unlike the Random Forest, which explicitly includes a normalized entropy of characters feature). The fact that the malicious domain names at the bottom right of Table 3 were missed by the deep neural networks might be due to our deliberate choice to tune the classification threshold to achieve a very low FPR. This makes all the classifiers hold back from labeling a domain name as malicious if they are not almost completely certain. As explained above, a low FPR is very important in deployed DGA detection systems, as blocking legitimate traffic is highly undesirable. Note that if a deployed DGA detection system would rely on the Random Forest classifier, it would block all domain names from the first row in Table 3, whereas, if it would rely on any of the deep neural network classifiers, it would block all domain names from the second row in Table 3. The domain names from the first column would have been unjustly blocked. For those negatively affected by this, it would be easier to “understand” (and perhaps forgive) the decisions made by the deep neural network classifiers, as they are more in line with decisions that a human would make when confronted with these domain name strings. ",
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+        "type": "table",
+        "img_path": "images/0231fd63a9749dda745cc66d28ef815052c7c3ff357fd7fef7b7bfa10070eaf2.jpg",
+        "table_caption": [
+            "Table 3: Examples of domain names that were either misclassified by the Random Forest or by the deep neural networks. For the malicious domain names, the name of the malware family is shown between parentheses. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>benign</td><td rowspan=1 colspan=1>malicious</td></tr><tr><td rowspan=1 colspan=1>misclassifiedby RF&amp;correctlyclassifiedby all fivedeep networks</td><td rowspan=1 colspan=1>kosmetikosdnr.ltjobrankingcommittee.comnaturalandhealthytips.compokemonrubysapphire.comturkcehdpornoizle.combollywoodparksdubai.com</td><td rowspan=1 colspan=1>mowvcssclilpomqi.com (murofet)ntearasildeafeninguvuc.com (banjori)raklloblmuppono.info (cryptolocker)5alolch3wvn5olcc.org (chinad)ldjucxqhivnaperisusb.ga (necurs)daeontibyxgask.cc (ranbyus)</td></tr><tr><td rowspan=1 colspan=1>misclassifiedby all fivedeep networks &amp;correctlyclassified by RF</td><td rowspan=1 colspan=1>rfembassy.kz 4553t5pugtt1qslvsnmpc0tpfz5fo.xyz9odyefocculgririlemjijbab.topa5rtngpo9840oyd.commydwnldsghtfv.com</td><td rowspan=1 colspan=1>doycsnramt.com (qakbot)gypjuytopleh.com (ramnit)zamdazhocs.com (nymaim)mxdsbbnxmogo.online (tinba)pelkbazgro.info (pykspa)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "The results in Table 2 are for DGA domain names that appeared at the same time in real traffic as the training data used to construct the classifiers, namely July 2017. To evaluate how well the trained models hold up against DGA domain names that appear at a later point, we collected an additional 100K DGA domain names from the Bambenek Consulting feeds for Dec 6, Dec 9, and Dec 10, 2017. Table 4 contains results for this “prospective” dataset. The only difference between the experimental setup for Table 2 and Table 4 are the 100K DGA domain names used in the test set. The training dataset, the validation dataset, and the 100K Alexa domain names used in the test set are kept the same. As can be seen when comparing the results in Table 2 and Table 4, the to be expected drop in TPR is minor. ",
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+    },
+    {
+        "type": "text",
+        "text": "As Table 5 shows there is a substantial distinction among the different models in terms of complexity (number of parameters that have to be learned during the training process) and required training time per epoch. The platform used for training is an AWS virtual machine with access to multiple GPUs. Both the number of epochs needed to train a network, and the number of seconds required per epoch, are contributing factors to the overall training runtime. The NYU model took less than 8 minutes to train, while the CMU model took as much as 10 hours. The last column in Table 5 shows the time needed to classify 200K domain names with an already trained model. The ranking of the deep networks in terms of this “scoring time” coincides with the ranking in terms of training time. Given that all deep networks achieve a similar accuracy (TPR), the NYU model with it short training and scoring time comes out as the winner. ",
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+    {
+        "type": "table",
+        "img_path": "images/68701f5daa51c054e68525005fa9f09385e5bb0ff81de4a39a6a79965ef494ca.jpg",
+        "table_caption": [
+            "Table 4: Results on test data from December 2017. Accuracy, TPR, FPR are w.r.t. a threshold that gives a FPR of 0.001 on the validation data. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=2>Architecture</td><td rowspan=1 colspan=1>Acc</td><td rowspan=1 colspan=1>TPR</td><td rowspan=1 colspan=1>FPR</td><td rowspan=1 colspan=1>AUC@1%</td></tr><tr><td rowspan=1 colspan=1>RF</td><td rowspan=1 colspan=2>Lexical features</td><td rowspan=1 colspan=1>91.57%</td><td rowspan=1 colspan=1>83.26%</td><td rowspan=1 colspan=1>0.00128</td><td rowspan=1 colspan=1>84.47%</td></tr><tr><td rowspan=1 colspan=1>MLP</td><td rowspan=1 colspan=2>Lexical features</td><td rowspan=1 colspan=1>78.87%</td><td rowspan=1 colspan=1>57.86%</td><td rowspan=1 colspan=1>0.00091</td><td rowspan=1 colspan=1>67.23%</td></tr><tr><td rowspan=1 colspan=1>Embedding</td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>86.75%</td><td rowspan=1 colspan=1>73.61%</td><td rowspan=1 colspan=1>0.00108</td><td rowspan=1 colspan=1>82.93%</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>LSTM</td><td rowspan=1 colspan=1>CNN</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Endgame</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>98.22%</td><td rowspan=1 colspan=1>96.54%</td><td rowspan=1 colspan=1>0.00102</td><td rowspan=1 colspan=1>97.53%</td></tr><tr><td rowspan=1 colspan=1>Invincea</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>98.44%</td><td rowspan=1 colspan=1>97.00%</td><td rowspan=1 colspan=1>0.00109</td><td rowspan=1 colspan=1>97.07%</td></tr><tr><td rowspan=1 colspan=1>CMU</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>98.06%</td><td rowspan=1 colspan=1>96.23%</td><td rowspan=1 colspan=1>0.00108</td><td rowspan=1 colspan=1>97.46%</td></tr><tr><td rowspan=1 colspan=1>MIT</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>98.21%</td><td rowspan=1 colspan=1>96.52%</td><td rowspan=1 colspan=1>0.00099</td><td rowspan=1 colspan=1>97.52%</td></tr><tr><td rowspan=1 colspan=1>NYU</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>98.12%</td><td rowspan=1 colspan=1>96.35%</td><td rowspan=1 colspan=1>0.00116</td><td rowspan=1 colspan=1>97.10%</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6aa1d644a880781ff2eff04c193104acbcded649738d518172396569f02b1e74.jpg",
+        "table_caption": [
+            "Table 5: Comparison of complexity and efficiency of classifiers for DGA detection. The complexity refers to the number of parameters that have to be learned in the deep learning architectures. The training time is reported in terms of seconds needed for an epoch times the number of epochs. The scoring time is the time in seconds needed to label 200K domain names by an already trained model. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=2>Architecture</td><td rowspan=1 colspan=1>Complexity</td><td rowspan=1 colspan=1>Training&#x27;Time</td><td rowspan=1 colspan=1>Scoring time</td></tr><tr><td rowspan=1 colspan=1>RF</td><td rowspan=1 colspan=2>Lexical features</td><td rowspan=1 colspan=1>100 trees</td><td rowspan=1 colspan=1>1,800s</td><td rowspan=1 colspan=1>5s</td></tr><tr><td rowspan=1 colspan=1>MLP</td><td rowspan=1 colspan=2>Lexical features</td><td rowspan=1 colspan=1>1,665 par</td><td rowspan=1 colspan=1>10s× 40=  400s</td><td rowspan=1 colspan=1>1s</td></tr><tr><td rowspan=1 colspan=1>Embedding</td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>25,985 par</td><td rowspan=1 colspan=1>15s × 40=  600s</td><td rowspan=1 colspan=1>3s</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>LSTM</td><td rowspan=1 colspan=1>CNN</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Endgame</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>148,097 par</td><td rowspan=1 colspan=1>430s × 10= 4,300s</td><td rowspan=1 colspan=1>13s</td></tr><tr><td rowspan=1 colspan=1>Invincea</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>2,576,385 par</td><td rowspan=1 colspan=1>105s × 40= 4,200s</td><td rowspan=1 colspan=1>7s</td></tr><tr><td rowspan=1 colspan=1>CMU</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>115,329 par</td><td rowspan=1 colspan=1>1200s ×30= 36,000s</td><td rowspan=1 colspan=1>26s</td></tr><tr><td rowspan=1 colspan=1>MIT</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>115,137 par</td><td rowspan=1 colspan=1>800s ×15=  12,000s</td><td rowspan=1 colspan=1>10s</td></tr><tr><td rowspan=1 colspan=1>NYU</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>254,337 par</td><td rowspan=1 colspan=1>45s ×10= 450s</td><td rowspan=1 colspan=1>5s</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "DGA detection, i.e. the classification task of distinguishing between benign domain names and those generated by malware (Domain Generation Algorithms), has become a central topic in information security. In this paper we have compared five different deep neural network architectures that perform this classification task based purely on the domain name string, given as a raw input signal at character level. All five models, i.e. two RNN based architectures, two CNN based architectures, and one hybrid RNN/CNN architecture perform equally well, catching around $9 7 . 9 8 \\%$ of malicious domain names against a false positive rate of 0.001. This roughly means that for every 970 malicious domain names that the deep networks catch, they flag only one benign domain name erroneously as malicious. A Random Forest based on human defined linguistic features achieves a recall of only $83 \\%$ against the same 0.001 false positive rate when trained and tested on the same data that was used for the deep networks. The use of a deep neural network that automatically learns features is attractive in a cybersecurity setting because it is a lot harder to craft malware to avoid detection by a system that relies on automatically learned features instead of on human engineered features. An interesting direction for future work is to test the trained deep networks more extensively on domain names generated by new and previously unseen malware families. ",
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+    {
+        "type": "text",
+        "text": "main input $=$ Input (shape $: =$ (75, ) , dtype $z '$ int32 ’, name $\\mathbf { \\equiv }$ ’main input’)   \nembedding $=$ Embedding(input dim ${ \\tt 1 2 8 }$ , output dim ${ \\tt 1 2 8 }$ , input length $= 7 5$ )(main input )   \nlstm $=$ LSTM(128, return sequence $\\vDash$ False) (embedding)   \ndrop $=$ Dropout(0.5) ( lstm )   \noutput $=$ Dense(1, activation $= ^ { \\mathrm { i } }$ ’sigmoid’) (drop)   \nmodel $=$ Model(inputs $\\mathbf { \\sigma } =$ main input, outputs $=$ output)   \nmodel.compile( ${ \\mathrm { l o s s } } = { \\ ' }$ binary crossentropy ’, optimizer $= ^ { \\ ' }$ adam’)   \nmain input $=$ Input (shape=(75, ) , dtype=’int32 ’, name $u = { \\mathrm { i } }$ ’main input’)   \nembedding $=$ Embedding(input dim ${ \\tt 1 2 8 }$ , output dim ${ \\tt 1 2 8 }$ , input length $= 7 5$ )(main input )   \nbi lstm $=$ Bidirectional ( layer =LSTM(64, return sequence $\\mathrm {  ~ \\sigma ~ } \\mathrm {  ~ \\sigma ~ } \\mathrm {  ~ \\sigma ~ } \\mathrm {  ~ \\sigma ~ }$ False) , merge mode=’concat’)(embedding)   \noutput $=$ Dense(1, activation $= ^ { \\overrightarrow { { \\mathbf { \\Gamma } } } }$ ’sigmoid’) ( bi lstm )   \nmodel $=$ Model(inputs main input, outputs $=$ output)   \nmodel.compile( ${ \\mathrm { l o s s } } = { \\ ' }$ binary crossentropy ’, optimizer $= ^ { \\ ' }$ adam’)   \nmain input $=$ Input (shape $: = 1$ (75, ) , dtype $= ^ { \\ ' }$ int32 ’, name $z '$ ’main input’)   \nembedding $=$ Embedding(input dim ${ \\tt 1 2 8 }$ , output dim ${ \\tt 1 2 8 }$ , input length $= 7 5$ )(main input )   \nconv1 $=$ Conv1D( filters ${ } = 1 2 8$ , kernel size $^ { = 3 }$ , padding $= ^ { \\ast }$ ’same’, strides $^ { = 1 }$ )(embedding)   \nthresh1 $=$ ThresholdedReLU(1e−6)(conv1)   \nmax pool1 $=$ MaxPooling1D(pool size $^ { = 2 }$ , padding=’same’)(thresh1 )   \nconv $\\ ? =$ Conv1D( filters ${ } = 1 2 8$ , kernel size $^ { = 2 }$ , padding $= ^ { \\overrightarrow { } }$ ’same’, strides $^ { = 1 }$ )(max pool1)   \nthresh2 $=$ ThresholdedReLU(1e−6)(conv2)   \nmax pool2 $=$ MaxPooling1D(pool size $^ { = 2 }$ , padding=’same’)(thresh2 )   \nflatten $=$ Flatten () (max pool2)   \nfc $=$ Dense(64)( flatten )   \nthresh fc $=$ ThresholdedReLU(1e−6)(fc)   \ndrop $=$ Dropout(0.5) ( thresh fc )   \noutput $=$ Dense(1, activation $= ^ { \\overrightarrow { { \\mathbf { \\Gamma } } } }$ ’sigmoid’) (drop)   \nmodel $=$ Model(inputs=main input, outputs $=$ output)   \nmodel.compile( ${ \\mathrm { l o s s } } = { \\ ' }$ binary crossentropy ’, optimizer $= ^ { \\ ' }$ adam’) ",
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+        "type": "text",
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+        ],
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+    {
+        "type": "text",
+        "text": "def getconvmodel( self , kernel size , filters ) : model $=$ Sequential () model.add( Conv1D( filters $=$ filters , input shape $=$ (128, 128), kernel size $=$ kernel size , padding=’same’, activation $= ^ { \\ ' }$ relu ’, strides $\\mathbf { \\tau } = 1 \\dot { }$ )) model.add(Lambda(lambda x: K.sum(x, axi $^ { - 1 }$ ), output shape $=$ ( filters , ) ) ) model.add(Dropout(0.5) ) return model ",
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "middle $=$ Dropout(0.5) (middle) ",
+        "bbox": [
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+            161
+        ],
+        "page_idx": 10
+    },
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+        "img_path": "images/c7ecb41720d72649079b0460a01d40d9e9c1b525e946b706dca106d8bfd7adc9.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>output = Dense(1,activation =&#x27;sigmoid&#x27;)(middle)model = Model(inputs=main_input, outputs =output)model.compile(loss=&#x27;binary_crossentropy’，optimizer=&#x27;adam&#x27;)</td></tr><tr><td rowspan=1 colspan=1>Listing 4: Invincea CNN model with parallel CNN layers,adapted from Saxe &amp; Berlin (2017)</td></tr><tr><td rowspan=1 colspan=1>main_input = Input(shape=(75,），dtype=&#x27; int32’，name=&#x27;main_input&#x27;)embedding =Embedding(input_dim=128,output_dim=128,input_length=75)(main_input)conv = Conv1D(filters =128,kernel_size =3，padding=&#x27;same&#x27;，activation =&#x27;relu’,strides=1)(embedding)max_pool = MaxPooling1D(pool_size=2, padding=&#x27;same&#x27;)(conv)encode = LSTM(64, return_sequences=False)(max-pool)output = Dense(1， activation =&#x27;sigmoid&#x27;)(encode)model= Model(inputs=main_input， outputs =output)model.compile(loss=’ binary_crossentropy’， optimizer=&#x27;adam&#x27;)</td></tr><tr><td rowspan=1 colspan=1>Listing 5: MIT model with a stacked CNN and LSTM layer,adapted from Vosoughi et al. (2016)</td></tr><tr><td rowspan=1 colspan=1>main_input = Input(shape=(75,)，dtype=&#x27; int32’，name=&#x27;main_input&#x27;)embedding =Embedding(input_dim=128,output_dim=128,input_length=75)(main_input)flatten = Flatten ((embedding)output = Dense(1， activation =&#x27;sigmoid&#x27;)( flatten ）model = Model(inputs=main_input， outputs =output)print (model.summaryO)model.compile(loss=&#x27; binary_crossentropy’，optimizer=&#x27;adam&#x27;)</td></tr><tr><td rowspan=1 colspan=1>Listing 6: Baseline Model with only Embedding Layer</td></tr><tr><td rowspan=1 colspan=1>main_input = Input(shape=(11,)，name=&#x27;main_input&#x27;)dense = Dense(128,activation =&#x27;relu &#x27;)(main_input)output = Dense(1， activation =&#x27;sigmoid&#x27;)(dense)model = Model(inputs=main_input， outputs =output)print (model.summaryO)model.compile(loss=&#x27;binary_crossentropy’，optimizer=&#x27;adam&#x27;)</td></tr></table>",
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+]

parse/train/GrB-y0SNkQ/GrB-y0SNkQ.md

@@ -0,0 +1,636 @@
+# CobBO: Coordinate Backoff Bayesian Optimization with Two-Stage Kernels
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Bayesian optimization is a popular method for optimizing expensive black-box   
+2 functions. Yet it oftentimes struggles in high dimensions where the computation   
+3 could be prohibitively expensive and a sufficient estimation of the global landscape   
+4 requires more observations. We introduce Coordinate backoff Bayesian optimiza  
+5 tion (CobBO) with two-stage kernels to alleviate this problem. In each iteration, a   
+6 promising subset of coordinates is selected in the first stage, as past observed points   
+7 in the full space are projected to the selected subspace adopting a simple kernel   
+8 that sacrifices the approximation accuracy for computational efficiency. Then in   
+9 the second stage of the same iteration a more sophisticated kernel is applied for   
+10 estimating the landscape in the selected low dimensional subspace where the com  
+11 putational cost becomes affordable. Effectively, this second stage kernel refines the   
+12 approximation of the global landscape estimated by the first stage kernel through   
+13 a sequence of observations in the local subspace. This refinement lasts until a   
+14 stopping rule is met determining when to back off from a certain subspace and   
+15 switch to another coordinate subset. This decoupling significantly reduces the com  
+16 putational burden in high dimensions, while the two-stage kernels of the Gaussian   
+17 process regressions fully leverage the observations in the whole space rather than   
+18 only relying on observations in each coordinate subspace. Extensive evaluations   
+19 show that CobBO finds solutions comparable to or better than other state-of-the-art   
+20 methods for dimensions ranging from tens to hundreds, while reducing the trial   
+21 complexity and computational costs.
+
+# 22 1 Introduction
+
+Bayesian optimization (BO) has emerged as an effective zero-order paradigm for optimizing expensive black-box functions. The entire sequence of iterations rely only on the function values of the already queried points without information on their derivatives. Though highly competitive in low dimensions (e.g., the dimension $D \leq 2 0$ [15]), Bayesian optimization based on Gaussian Process (GP) regression has obstacles that impede its effectiveness, especially in high dimensions.
+
+Approximation accuracy: GP regression assumes a class of random functions in a probability space as surrogates that iteratively yield posterior distributions by conditioning on the queried points. When suggesting new query points, for complex functions with numerous local optima and saddle points due to local fluctuations, always exactly using the values on the queried points as the conditional events may mismatch the function’s local landscape by overemphasizing the approximation accuracy of the global landscape.
+
+Curse of dimensionality: As a sample efficient method, Bayesian optimization often suffers from high dimensions. Fitting the GP model (estimating the parameters, e.g., length_scales [14]), computing the Gaussian process posterior and optimizing the acquisition function in high dimensions all
+
+![](images/28f04cf1e21181f3ffc763b3653d72a5cda43565631298f98897787083f12e5f.jpg)  
+Figure 1: Minimize the fluctuated Rastrigin function on $[ - 5 , 1 0 ] ^ { 5 0 }$ with 20 initial samples. [Left] Computation times for training the GP regression model and maximizing the acquisition function at each iteration. CobBO significantly reduces the execution time compared with a vanilla BO, e.g. $\times 1 3$ faster in this case. [Right] The average error between the GP predictions before making queries and the true function values at the queried points (solid curves, the higher the better) and the best observed function value (dashed curves, the lower the better) at iteration $t$ . CobBO captures the global landscape less accurately using the RBF kernel, and then explores selected subspaces $\Omega _ { t }$ more accurately using the Matern kernel. This eventually better exploits the promising subspaces.
+
+39 incur large computational costs. It also results in statistical insufficiency of exploration [11, 65]. As   
+40 the GP regression’s error grows with dimensions [8], more samples are required to balance that in   
+41 high dimensions, which could cubically increase the computational costs in the worst case [45].   
+42 To alleviate these issues, we design coordinate backoff Bayesian optimization (CobBO) with two  
+43 stage kernels, by challenging a seemingly natural intuition stating that it is always better for Bayesian   
+44 optimization to have a more accurate approximation of the objective function at all times. We   
+45 demonstrate that this is not necessarily true, by showing that smoothing out local fluctuations and   
+46 using the estimated function values instead of the true observations to serve as the conditional events   
+47 in selected subspaces can not only significantly reduce the computation time due to the curse of   
+48 dimensionality but also help in capturing the large-scale properties of the objective function $f ( x )$ .   
+49 Specifically, CobBO introduces the two-stage kernels with a stopping rule. The first stage of each   
+50 iteration adopts a simple kernel that sacrifices the approximation accuracy of $f ( x )$ for computational   
+51 efficiency. For example, by using a universal radial basis function (RBF) approximation without   
+52 learnable parameters [50], CobBO can eliminate the model fitting time in the full space. It captures   
+53 a smooth approximation ${ \hat { f } } ( x )$ of the global landscape by interpolating the values of queried points   
+54 projected to selected promising subspaces. These projected points serve as the conditional events   
+55 for GP regression. In a selected coordinate subspace, the second stage of the same iteration applies   
+56 a sophisticated kernel that can tolerate high computational cost in low dimensions. For example,   
+57 CobBO uses the Automatic Relevance Determination (ARD) Matérn 5/2 kernel [40]. It refines the   
+58 approximation of the local landscape by a sequence of observations determined by a stopping rule that   
+59 backs off from a certain subspace and switches to another coordinate subset. In addition, computing   
+60 the Gaussian process posterior and optimizing the acquisition function are both efficiently conducted   
+61 in the low dimensional subspaces, bypassing the curse of dimensionality.   
+62 For iteration $t$ , instead of directly computing the Gaussian process posterior distribution   
+63 $\left\{ \hat { f } ( x ) \middle | \mathcal { H } _ { t } = \left\{ ( x _ { i } , y _ { i } ) \right\} _ { i = 1 } ^ { t } , x \in \Omega \right\}$ by conditioning on the observations $y _ { i } = f ( x _ { i } )$ at queried   
+64 points $x _ { i }$ in the full space $\Omega \subset \mathbb { R } ^ { D }$ for $i = 1 , \ldots , t$ , we change the conditional events, and consider
+
+$$
+\left\{ \hat { f } ( x ) \Big | R \left( P _ { \Omega _ { t } } ( x _ { 1 } , \dots , x _ { t } ) , \mathcal { H } _ { t } \right) , x \in \Omega _ { t } , \Omega _ { t } \subset \Omega \right\}
+$$
+
+65 for a projection function $P _ { \Omega _ { t } } ( \cdot )$ to a random subspace $\Omega _ { t }$ and an interpolation function $R ( \cdot , \cdot )$ , e.g.,   
+66 using a RBF approximation without learnable parameters [50] as the simple kernel for the first   
+67 stage. The projection $P _ { \Omega _ { t } } ( \cdot )$ maps the queried points to virtual points on a subspace $\Omega _ { t }$ of a lower   
+68 dimension [51]. The interpolation function $R ( \cdot , \cdot )$ estimates the objective values at the virtual points   
+69 using the queried points and their values as specified by $\mathcal { H } _ { t }$ . The second stage within the subspace $\Omega _ { t }$   
+70 uses the more sophisticated kernel, e.g., Matérn $5 / 2$ kernel [40], which has a number of parameters   
+71 that otherwise would be expensive to be learned in high dimensions.   
+72 This method can be viewed as a variant of block coordinate ascent tailored to Bayesian optimization by   
+73 applying backoff stopping rules for switching coordinate blocks. While similar work exists [43, 48],   
+74 CobBO differs by introducing the two-stage kernels and addressing the following three issues:
+
+1. Selecting a block of coordinates for ascending requires determining the block size as well as the coordinates therein. CobBO selects the coordinate subsets by a multiplicative weights update method [2] to the preference probability associated with each coordinate. Thus, it samples more promising subspaces with higher probabilities.   
+2. A coordinate subspace requires a sufficient number of query points acting as the conditional events for the GP regression. CobBO leverages all observations in the whole space by interpolating the values of queried points projected to selected promising subspaces, rather than simply starting from scratch in each subspace.   
+3. Querying a certain subspace, under some trial budget, comes at the expense of exploring other coordinate blocks. Yet prematurely shifting to different subspaces does not fully exploit the full potential of a given subspace. Hence determining the number of consecutive function queries within a subspace makes a trade-off between exploration and exploitation. CobBO uses a stopping rule in each subspace to switch the selected coordinates. When consecutively querying data points in the same subspace, CobBO does not need to conduct the first-stage function approximation in the full space, which is far more efficient.
+
+90 Through comprehensive evaluations, CobBO demonstrates appealing performance for dimensions   
+91 ranging from tens to hundreds. It obtains comparable or better solutions with fewer queries, in   
+92 comparison with the state-of-the-art methods, for most of the problems tested in Section 4.2.
+
+# 2 Related work
+
+94 Certain assumptions are often imposed on the latent structure in high dimensions. Typical assumptions   
+95 include low dimensional structures and additive structures. Their advantages manifest on problems   
+96 with a low dimension or a low effective dimension. However, these assumptions do not necessarily   
+97 hold for non-separable functions with no redundant dimensions.   
+98 Low dimensional structure: The black-box function $f$ is assumed to have a low effective dimen  
+99 sion [30, 58], e.g., $f ( x ) = g ( \Phi x )$ with some function $g ( \cdot )$ and a matrix $\Phi$ of $d \times D , d < < D$ . A   
+100 number of different methods have been developed, including random embedding [66, 11, 63, 36,   
+101 44, 70, 5, 32], low-rank matrix recovery [11, 58], and learning subspaces by derivative informa  
+102 tion [11, 13]. In contrast to existing work on subspace selections, e.g., Hashing-enhanced Subspace   
+103 BO (HeSBO) [44], Mahalanobis kernel for linear embeddings [33], DROPOUT [35] and LineBO [29]   
+104 (which receives a special treatment in Appendix F), CobBO efficiently leverages all the observations   
+105 in the whole space using the two-stage kernels and the stopping rule in each subspace for consecutive   
+106 observations, rather than only relying on limited observations in each coordinate subspace. It exploits   
+107 subspace structure from a perspective of block coordinate ascent, independent of the dimensions,   
+108 different from some algorithms that are more suitable for low dimensions, e.g., BADS [1].   
+109 Additive structure: A decomposition assumption is often made by $\begin{array} { r } { f ( x ) = \sum _ { i = 1 } ^ { k } f ^ { ( i ) } \left( x _ { i } \right) } \end{array}$ , with $x _ { i }$   
+110 defined over low-dimensional components. In this case, the effective dimensionality of the model is   
+111 the largest dimension among all additive groups [45], which is usually small. The Gaussian process   
+112 is structured as an additive model [17, 28], e.g., projected-additive functions [36], ensemble Bayesian   
+113 optimization (EBO) [61], latent additive structural kernel learning (HDBBO) [65] and group additive   
+114 models [28, 36]. However, learning the unknown structure incurs a considerable computational   
+115 cost [44], and is not applicable for non-separable functions, for which CobBO can still be applied.   
+116 Trust regions and space partitions: Trust region BO has been proven effective for high-dimensional   
+117 problems. A typical pattern is to alternate between global and local search regions. In the local   
+118 trust regions, many efficient methods have been applied, e.g., local Gaussian models (TurBO [14]),   
+119 adaptive search on a mesh grid (BADS [1]) or quasi-Newton local optimization (BLOSSOM [41]).   
+120 TurBO [14] uses Thompson sampling to allocate samples across multiple regions. A related method is   
+121 to use space partitions, e.g., LA-MCTS[60] on a Monte Carlo tree search algorithm to learn efficient   
+122 partitions. CobBO differs by selecting low dimensional subspaces. It can also incorporate trust   
+123 regions in the first-stage global approximation, as shown in the Appendix.   
+125 Without loss of generality, suppose that the goal is to solve a maximization problem $\boldsymbol { x } ^ { * } =$   
+126 $\mathrm { { r g m a x } } _ { x \in \Omega } f ( x )$ for a black-box function $f : \Omega  \mathbb { R }$ . The domain is normalized $\Omega = [ 0 , 1 ] ^ { D }$   
+127 with the coordinates indexed by $I = \{ 1 , 2 , \cdots , D \}$ .   
+128 For a sequence of points $\mathcal { X } _ { t } = \{ x _ { 1 } , x _ { 2 } , \cdot \cdot \cdot , x _ { t } \}$ with $t$ indexing the most recent iteration, we observe   
+129 $\mathcal { H } _ { t } = \{ ( x _ { i } , y _ { i } = f ( x _ { i } ) ) \} _ { i = 1 } ^ { t }$ . A random subset $C _ { t } \subseteq I$ of the coordinates is selected, forming a   
+130 subspace $\Omega _ { t } \subseteq \Omega$ at iteration $t$ . As a variant of coordinate ascent, the subspace $\Omega _ { t }$ contains a pivot   
+131 point $V _ { t }$ , which presumably is the maximum point $x _ { t } ^ { M } = \operatorname { a r g m a x } _ { x \in \mathcal { X } _ { t } } { f ( x ) }$ with $M _ { t } = f \left( { { \dot { x } } _ { t } } ^ { M } \right)$ .   
+132 CobBO may set $V _ { t }$ different from $x _ { t } ^ { M }$ to escape local optima. Then, BO is conducted within $\Omega _ { t }$ while   
+133 fixing all the other coordinates $C _ { t } ^ { c } \bar { = } I \setminus C _ { t }$ , i.e., the complement of $C _ { t }$ .
+
+# $\overline { { \mathbf { A l g o r i t h m 1 : C o b B O ( f , \boldsymbol { \tau } , T ) } } }$
+
+1 $\mathcal { H } _ { \tau } \gets \mathrm { s a m p l e } ~ $ $\tau$ initial points and evaluate their values   
+2 $V _ { \tau } , M _ { \tau } \gets$ Find the tuple with the maximal objective value in $\mathcal { H } _ { \tau }$   
+3 $q _ { \tau } \gets 0$ Initialize the number of consecutive failed queries   
+4 $\pi _ { \tau } $ Initialize a uniform preference distribution on the coordinates   
+5 for $t  \tau$ to $T$ do 6 if switch $\Omega _ { t - 1 }$ by the backoff stopping rule (Section 3.2) then 7 $C _ { t } \gets$ Sample a promising coordinate block according to $\pi _ { t }$ (Section 3.1) 8 $\Omega _ { t } \gets$ Take the subspace of $\Omega _ { t }$ over the coordinate block $C _ { t }$ , such that $V _ { t } \in \Omega _ { t }$ 9 else   
+10 $\Omega _ { t } \gets \Omega _ { t - 1 }$   
+11 $\hat { \mathcal { X } } _ { t } \gets P _ { \Omega _ { t } } ( \mathcal { X } _ { t } )$ hProject $\mathcal { X } _ { t }$ onto $\Omega _ { t }$ to obtain a set of virtual points (Eq. 1)i   
+12 $\hat { \mathcal { H } } _ { t } \gets R \left( \hat { \mathcal { X } } _ { t } , \bar { \mathcal { H } } _ { t } \right)$ hSmooth function values on $\hat { \mathcal { X } } _ { t }$ by interpolation using $\mathcal { H } _ { t }$ i   
+13 $p \left[ \hat { f } _ { \Omega _ { t } } ( x ) | \hat { \mathcal { H } } _ { t } \right] \gets \mathbf { C } \mathbf { \tau }$ ompute the posterior distribution of the Gaussian process in $\Omega _ { t }$ conditional on $\hat { \mathcal { H } } _ { t }$   
+14 $\begin{array} { r } { x _ { t + 1 } \gets \mathrm { a r g m a x } _ { x \in \Omega _ { t } } Q _ { \hat { f } \sim p ( \hat { f } | \hat { \mathcal { H } } _ { t } ) } ( x | \hat { \mathcal { H } } _ { t } ) \ \Big [ } \end{array}$ Suggest the next query in $\Omega _ { t }$ (Section 3)i   
+15 $y _ { t + 1 } \gets$ Evaluate the black-box function $y _ { t + 1 } = f ( x _ { t + 1 } )$   
+16 if $y _ { t + 1 } > M _ { t }$ then   
+17 $| \quad V _ { t + 1 } \gets x _ { t + 1 } , M _ { t + 1 } \gets y _ { t + 1 } , q _ { t + 1 } \gets 0$   
+18 else   
+19 $V _ { t + 1 }  V _ { t }$ , $M _ { t + 1 } \gets M _ { t }$ , $q _ { t + 1 } \gets q _ { t } + 1$   
+20 $\pi _ { t + 1 } $ Update $\pi _ { t }$ by a multiplicative weights update method (Eq. 2)   
+21 $\mathcal { H } _ { t + 1 }  \bar { \mathcal { H } } _ { t } \bigcup \{ ( x _ { t + 1 } , y _ { t + 1 } ) \} , \mathcal { X } _ { t + 1 }  \bar { \mathcal { X } } _ { t } \bigcup \{ x _ { t + 1 } \}$   
+22 end
+
+For BO in 134 $\Omega _ { t }$ , we use Gaussian processes as the random surrogates $\hat { f } = \hat { f } _ { \Omega _ { t } } ( x )$ to describe the 135 Bayesian statistics of $f ( x )$ for $x \in \Omega _ { t }$ . At each iteration, the next query point is generated by solving
+
+$$
+\begin{array} { r } { x _ { t + 1 } = \mathrm { a r g m a x } _ { x \in \Omega _ { t } , V _ { t } \in \Omega _ { t } } Q _ { \hat { f } _ { \Omega _ { t } } ( x ) \sim p ( \hat { f } | \mathcal { H } _ { t } ) } ( x | \mathcal { H } _ { t } ) , } \end{array}
+$$
+
+136 where the acquisition function $Q ( x | \mathcal { H } _ { t } )$ incorporates the posterior distribution of the Gaussian   
+137 processes $p ( \hat { f } | \mathcal { H } _ { t } )$ . Typical acquisition functions include the expected improvement (EI) [42, 27],   
+138 the upper confidence bound (UCB) [3, 54, 55], the entropy search [24, 25, 64], and the knowledge   
+139 gradient [16, 53, 69].
+
+Instead of directly computing the posterior distribution 140 $p ( \hat { f } | \mathcal { H } _ { t } )$ , we replace the conditional events 141 $\mathcal { H } _ { t }$ by
+
+$$
+\hat { \mathcal { H } } _ { t } : = R \left( P _ { \Omega _ { t } } \left( \mathcal { X } _ { t } \right) , \mathcal { H } _ { t } \right) = \left\{ \left( \hat { x } _ { i } , \hat { y } _ { i } \right) \right\} _ { i = 1 } ^ { t }
+$$
+
+with an interpolation function 142 $R ( \cdot , \cdot )$ and a projection function $P _ { \Omega _ { t } } ( \cdot )$ ,
+
+$$
+P _ { \Omega _ { t } } ( x ) ^ { ( j ) } = { \left\{ \begin{array} { l l } { x ^ { ( j ) } } & { { \mathrm { i f ~ } } j \in C _ { t } } \\ { V _ { t } ^ { ( j ) } } & { { \mathrm { i f ~ } } j \not \in C _ { t } } \end{array} \right. }
+$$
+
+143 at coordinate $j$ . It simply keeps the values of $x$ whose corresponding coordinates are in $C _ { t }$ and   
+144 replaces the rest by the corresponding values of $V _ { t }$ , as illustrated in Fig. 2.   
+145 Applying $P _ { \Omega _ { t } } ( \cdot )$ on $\mathcal { X } _ { t }$ and discarding duplicates generate a new set of distinct virtual points $\hat { \mathcal { X } } _ { t } =$   
+146 $\{ \hat { x } _ { 1 } , \hat { x } _ { 2 } , \hat { x } _ { 3 } , \cdot \cdot \cdot , \hat { x } _ { \hat { t } } \}$ , $\hat { x } _ { i } \in \Omega _ { t } \forall 1 \leq i \leq \hat { t } \leq t$ . The function values at $\hat { x } _ { i } \in \hat { \mathcal X } _ { t }$ are interpolated as   
+147 $\hat { y } _ { i } = R ( \hat { x } _ { i } , \mathcal { H } _ { t } )$ using the standard radial basis function [6, 7] and the observed points in $\mathcal { H } _ { t }$ . It not   
+148 only significantly reduces the GP regression time due to the efficiency of RBF [6] and the acquisition   
+149 function optimization in low dimensions [11], but also eventually improves the model accuracy using   
+150 the more sophisticated kernel applied on $\Omega _ { t }$ .   
+151 Note that only a fraction of the points in $\hat { \mathcal X } _ { t } \cap \mathcal X _ { t }$   
+152 directly observe the exact function values. The   
+153 function values on the rest ones in $\hat { \mathcal { X } } _ { t } \backslash \mathcal { X } _ { t }$ are   
+154 estimated by interpolation, which captures the   
+155 landscape of $f ( x )$ by smoothing out the local   
+156 fluctuations. To control the trade-off between   
+157 the inaccurate estimations and the exact obser  
+158 vations in $\Omega _ { t }$ , we design a stopping rule that   
+159 optimizes the number of consistent queries in   
+160 $\Omega _ { t }$ . The more consistent queries conducted in a   
+161 given subspace, the more accurate observations
+
+![](images/8bfff9cb31fe8fb61f7be5662a1ff0bff267a509571d27f5354dbe4b270083af.jpg)  
+Figure 2: Two-stage kernels: subspace projection and function value interpolation
+
+62 could be obtained, albeit at the expense of a smaller remaining budget for exploring other regions.
+
+163 The key features of CobBO are listed in Algorithm 1, with more details in the following sections.   
+164 Several auxiliary components are utilized and presented in Appendix C to deal with a larger variety   
+165 of problems and corner cases.
+
+# 166 3.1 Block coordinate ascent and subspace selection
+
+167 For Bayesian optimization, consider an infeasible assumption that each iteration can exactly maximize   
+168 the function $f ( x )$ in $\Omega _ { t }$ . This is not possible for one iteration but only if one can consistently query   
+169 in $\Omega _ { t }$ , since the points converge to the maximum, e.g., under the expected improvement acquisition   
+170 function with fixed priors [59] and the convergence rate can be characterized for smooth functions   
+171 in the reproducing kernel Hilbert space [8]. However, even with this infeasible assumption, it is   
+172 known that coordinate ascent with fixed blocks can cause stagnation at a non-critical point, e.g., for   
+173 non-differentiable [67] or non-convex functions [49]. This motivates us to select a subspace with a   
+174 variable-size coordinate block $C _ { t }$ for each query. A good coordinate block can help the iterations   
+175 to escape the trapped non-critical points. For example, one condition can be based on the result   
+176 in [21] that assumes $f ( x )$ to be differentiable and strictly quasi-convex over a collection of blocks. In   
+177 practice, we do not restrict ourselves to these assumptions.   
+178 We induce a preference distribution $\pi _ { t }$ over the coordinate set $I$ , and sample a variable-size coordinate   
+179 block $C _ { t }$ accordingly. This distribution is updated at iteration $t$ through a multiplicative weights   
+180 update method [2]. Specifically, the values of $\pi _ { t }$ at coordinates in $C _ { t }$ starts off uniform and increase   
+181 in face of an improvement or decrease otherwise according to different multiplicative ratios $\alpha > 1$   
+182 and $\beta > 1$ , respectively,
+
+$$
+w _ { t , j } = w _ { t - 1 , j } \cdot \left\{ \begin{array} { l l } { \alpha } & { \mathrm { i f ~ } j \in C _ { t } \mathrm { ~ a n d ~ } y _ { t } > M _ { t - 1 } } \\ { 1 / \beta } & { \mathrm { i f ~ } j \in C _ { t } \mathrm { ~ a n d ~ } y _ { t } \leq M _ { t - 1 } } \\ { 1 } & { \mathrm { i f ~ } j \notin C _ { t } } \end{array} \right. ; \quad w _ { 0 , j } = \frac { 1 } { D } \quad ; \quad \pi _ { t , j } = \frac { w _ { t , j } } { \sum _ { j = 1 } ^ { D } w _ { t , j } }
+$$
+
+183 This update characterizes how likely a coordinate block can generate a promising search subspace.   
+184 The multiplicative ratio $\alpha$ is chosen to be relatively large, e.g., $\alpha = 2 . 0$ , and $\beta$ relatively small, e.g.,   
+185 $\beta = 1 . 1$ , since the queries that improve the best observations $y _ { t } > M _ { t - 1 }$ happen more rarely than   
+186 the opposite $y _ { t } \le M _ { t - 1 }$ .   
+187 How to dynamically select the size $| C _ { t } | \cdot \cdot$ It is known that Bayesian optimization works well for low   
+188 dimensions [15]. Thus, we specify an upper bound for the dimension of the subspace (e.g. $| C _ { t } | \le 3 0 \}$ .   
+189 In principle, $| C _ { t } |$ can be any random number in a finite set of possible block sizes $\mathcal { C }$ . This is different   
+190 from the method that partitions the coordinates into fixed blocks and selects one according to, e.g.,   
+191 cyclic order [68], random sampling or Gauss-Southwell [46].
+
+# 192 3.2 Backoff stopping rule for consistent queries
+
+Applying BO on $\Omega _ { t }$ requires a strategy to determine the number of consecutive queries for making a sufficient progress. This strategy is based on previous observations, thus forming a stopping rule. In principle, there are two different scenarios, exemplifying exploration and exploitation, respectively. Persistently querying a given subspace refrains from opportunistically exploring other coordinate combinations. Abruptly shifting to different subspaces does not fully exploit the potential of a given subspace.
+
+CobBO designs a heuristic stopping rule in compromise. It takes the above two scenarios into joint consideration, by considering not only the number of consecutive queries that fail to improve the objective function but also other factors including the improved difference $M _ { t } - M _ { t - 1 }$ , the point distance $\vert \vert x _ { t } - x _ { t - 1 } \vert \vert$ , the query budget $T$ and the problem dimension $D$ . On the one hand, switching to another subspace $\Omega _ { t + 1 } ~ ( \neq ~ \Omega _ { t } )$ prematurely without fully exploiting $\Omega _ { t }$ incurs an additional approximation error associated with the interpolation of observations in $\Omega _ { t }$ projected to $\Omega _ { t + 1 }$ . On the other hand, it is also possible to over-exploit a subspace, spending high query budget on marginal improvements around local optima. In order to mitigate this, even when a query leads to an improvement, other factors are considered for sampling a new subspace.
+
+# 3.3 Theoretical Analysis
+
+209 One can view our block coordinate selection approach in section 3.1 as a combinatorial mixture of   
+210 experts problem [10], where each coordinate is a single expert and the forecaster aims at choosing   
+211 the best combination of experts in each step. Under this view, we bound the regret of our selection   
+212 method with respect to the policy of selecting the best (unknown) block of coordinates at each step.   
+213 Assume that there is a fixed optimal choice $\mathcal { T } ^ { * }$ for the block of coordinates to pick at all steps. This   
+214 block is characterized by improving the objective function for the largest number of times among   
+215 all the possible coordinate blocks when performing Bayesian optimization over the corresponding   
+216 subspaces. The following particular design of losses expresses this cause:
+
+$$
+\ell _ { t , i } = \left\{ \begin{array} { l l } { - \log ( \tilde { \alpha } ) } & { \mathrm { i f ~ } i \in C _ { t } \mathrm { ~ a n d ~ } y _ { t } > M _ { t - 1 } } \\ { \log ( \tilde { \beta } ) } & { \mathrm { i f ~ } i \in C _ { t } \mathrm { ~ a n d ~ } y _ { t } \leq M _ { t - 1 } } \\ { 0 } & { \mathrm { i f ~ } i \not \in C _ { t } } \end{array} \right. ~ ; \quad \tilde { \alpha } , \tilde { \beta } > 1
+$$
+
+217 as all the coordinates participating in the selected block incur the same loss that effectively rewards   
+218 these coordinates for improving the objective and penalizes these for failing to improve the objective.   
+219 All other coordinates that are not selected receive a zero loss and remain untouched.
+
+Note that $\tilde { \alpha }$ and $\tilde { \beta }$ express the extent of reward and penalty, e.g. for $\tilde { \alpha } = \tilde { \beta } = e$ we have losses of $\ell _ { t , i } \in \{ - 1 , 1 , 0 \}$ . Yet, $\tilde { \alpha }$ is better chosen to be larger than $\tilde { \beta }$ , since the frequency of improving the objective is expected to be smaller.
+
+23 The loss received by the forecaster is to reflect the same motivation. This is done by averaging   
+24 the losses of the individual coordinates in the selected block, so that the size of the block does not   
+25 matter explicitly, i.e. a bigger block should not incur more loss just due to its size but only due to its   
+26 performance. Such that for each coordinate block $\mathcal { T } _ { t } \subset \mathcal { T } = \{ 1 , \cdots , D \}$ selected at time step $t$ , the   
+27 loss incurred by the forecaster is Lt,It = 1|I | $\begin{array} { r } { L _ { t , \mathcal { T } _ { t } } = \frac { 1 } { | \mathcal { T } _ { t } | } \sum _ { i \in \mathcal { T } _ { t } } \ell _ { t , i } } \end{array}$ . This is also the common loss incurred by   
+28 all the coordinates participating in that block.
+
+229 In each step we have the following multiplicative update rule of the weights associated with each coordinate (setting 230 $\alpha = \tilde { \alpha } ^ { \eta }$ and $\beta = \widetilde { \beta } ^ { \eta }$ yields the update rule in Eq. 2):
+
+$$
+w _ { t , i } = w _ { t - 1 , i } \cdot e ^ { - \eta \ell _ { t , i } } = w _ { t - 1 , i } \cdot \left\{ \begin{array} { l l } { \tilde { \alpha } ^ { \eta } } & { \mathrm { i f ~ } i \in C _ { t } \mathrm { ~ a n d ~ } y _ { t } > M _ { t - 1 } } \\ { 1 / \tilde { \beta } ^ { \eta } } & { \mathrm { i f ~ } i \in C _ { t } \mathrm { ~ a n d ~ } y _ { t } \le M _ { t - 1 } } \\ { 1 } & { \mathrm { i f ~ } i \notin C _ { t } } \end{array} \right.
+$$
+
+The probability $\tilde { \pi } _ { t , \mathcal { T } _ { t } }$ of selecting a certain coordinate block $\mathcal { T } _ { t }$ is induced by $\pi _ { t }$ as specified next. Thus the expected cumulative loss of the foreceaster is:
+
+$$
+L _ { T } = \sum _ { t = 1 } ^ { T } \sum _ { c \in \mathcal { C } } \sum _ { \mathbb { Z } _ { t } \in \mathcal { S } _ { c } } \tilde { \pi } _ { t , \mathbb { Z } _ { t } } \cdot \frac { 1 } { | \mathbb { Z } _ { t } | } \sum _ { i \in \mathcal { T } _ { t } } \ell _ { t , i }
+$$
+
+Assume the best coordinate block is $\mathcal { T } ^ { * }$ , then the corresponding cumulative loss is:
+
+$$
+L _ { T } ^ { * } = \sum _ { t = 1 } ^ { T } { L _ { t , \mathbb { Z } ^ { * } } } = \sum _ { t = 1 } ^ { T } \frac { 1 } { | \mathscr { T } ^ { * } | } \sum _ { i \in \mathscr { T } ^ { * } } \ell _ { t , i }
+$$
+
+We hence aim at bounding the regret 231 $R e g r e t _ { T } = L _ { T } - L _ { T } ^ { * }$ .
+
+232 Theorem 1. Sample from the combinatorial space of all possible coordinate blocks $\textstyle { \mathcal { T } } _ { t } \in \bigcup _ { c \in { \mathcal { C } } } S _ { c }$ with probability 233 $\begin{array} { r } { \tilde { \pi } _ { t , \mathcal { T } _ { t } } ~ = ~ \prod _ { i \in \mathcal { T } _ { t } } \tilde { w } _ { t , \mathcal { T } _ { t } } \Big / \sum _ { c \in \mathcal { C } } \sum _ { \hat { \mathcal { T } } \in \mathcal { S } _ { c } } \prod _ { j \in \hat { \mathcal { T } } } \tilde { w } _ { t , \hat { \mathcal { T } } } . } \end{array}$ . Then the update rule in Eq. 2 with 234 $\alpha = \tilde { \alpha } ^ { \eta }$ , $\beta = \widetilde { \beta } ^ { \eta }$ and $\eta = \log ( \tilde { \alpha } \tilde { \beta } ) ^ { - 1 } \sqrt { T ^ { - 1 } | \mathcal { C } | D \log ( D ) }$ yields:
+
+$$
+R e g r e t _ { t } \le \mathcal { O } \left( ( \log ( \tilde { \alpha } \tilde { \beta } ) \cdot \sqrt { T | \mathcal { C } | D \log ( D ) } ) \right)
+$$
+
+where 235 $\begin{array} { r } { \tilde { w } _ { t , \mathcal { T } _ { t } } = \prod _ { i \in \mathcal { T } _ { t } } w _ { t , i } ^ { 1 / | \mathcal { T } _ { t } | } } \end{array}$ w1/|It|t,i is the geometric mean of weights in block It. The upper bound in Eq. 5 is tight, as the lower bound can be shown to be of 236 $\Omega ( { \sqrt { T \log ( N ) } } )$ [23] where the number of experts is 237 $\begin{array} { r } { N = \sum _ { c \in \mathcal { C } } S _ { c } \leq D ^ { | \mathcal { C } | D } } \end{array}$ in our combinatorial setup, as typically $| \mathcal { C } | \ll D$ .
+
+238 In practice, the direct sampling policy introduced in Theorem 1 involves high computational costs due   
+239 to the exponential growth of combinations in $D$ . Thus CobBO suggests an alternative computationally   
+240 efficient sampling policy with a linear growth in $D$ .
+
+Theorem 2. Sample a block size $c \in { \mathcal { C } }$ with probability $p _ { c }$ and c coordinates without replacement according to $\pi _ { t }$ . Assume $c \supset \{ 1 \}$ , then the update rule in Eq. 2, with $\alpha = \tilde { \alpha } ^ { \eta }$ , $\beta = { \tilde { \beta } } ^ { \eta }$ and $\begin{array} { r } { \eta = \sqrt { \frac { \log ( D ) } { T ( \log ( \tilde { \alpha } \tilde { \beta } ) ^ { 2 } - \log ( p _ { 1 } ) ) } } \geq 1 } \end{array}$ ≥ 1 yields:
+
+$$
+R e g r e t _ { t } \le \mathcal { O } \left( \sqrt { ( \log ( \tilde { \alpha } \tilde { \beta } ) ^ { 2 } - \log ( p _ { 1 } ) ) } \cdot \sqrt { T \log ( D ) } ) \right)
+$$
+
+where $p _ { c } > 0$ for all $c \in { \mathcal { C } }$ and $\textstyle \sum _ { c \in c } p _ { c } = 1$ , e.g., uniformly set $p _ { c } \equiv | \mathcal { C } | ^ { - 1 }$ .The proof and detailed sampling policy are in Appendix A. The regret upper bound in Eq. 6 is tight, as the lower bound for an easier setup can be shown to be of $\Omega ( { \sqrt { T \log ( D ) } } )$ [23]. The implication on $\eta$ is valid only for settings of a very high dimensionality and low query budget. In particular, CobBO is designed for this kind of problems.
+
+Remark: Similar analysis and results follow when incorporating consistent queries from Section 3.2 and sampling a new coordinate block once every several steps. This is done by effectively performing less steps of aggregated temporal losses, as shown in Appendix A.
+
+# 4 Numerical Experiments
+
+This section presents detailed ablation studies of the key components presented in Section 3 and comparisons with other algorithms.
+
+# 4.1 Empirical analysis and ablation study
+
+Ablation studies are designed to study the contributions of the key components in Algorithm 1 by experimenting with the Rastrigin function on $[ - 5 , 1 0 ] ^ { 5 0 }$ with 20 initial points. The best performing run out of 5 experiments for each configuration is presented in Figure 3.
+
+![](images/2c0ffc937e0888a03bbad1004c2f441b8be3c88d37408aa85eb32eedc2e170da.jpg)  
+Figure 3: Ablation study using Rastrigin on $[ - 5 , 1 0 ] ^ { 5 0 }$ with 20 initial random samples
+
+259 RBF interpolation: RBF calculation is time efficient. Specifically, this is much beneficial in high   
+260 dimensions. Figure 1 (left) shows the computation time of plain Bayesian optimization compared to   
+261 CobBO’s. While the former applies GP regression using the Matérn kernel in the high dimensional   
+262 space directly, the later applies RBF interpolation in the high dimensional space and GP regression   
+263 with the Matérn kernel in the low dimensional subspace. This two-step composite kernel leads to a   
+264 significant speed-up. Other time efficient alternatives are, e.g., the inverse distance weighting [26]   
+265 and the simple approach of assigning the value of the observed nearest neighbour. Figure 3 (a) shows   
+266 that RBF is the most favorable.
+
+Backoff stopping rule: CobBO applies a stopping rule to query a variable number of points in subspace $\Omega _ { t }$ (Section 3.2). To validate its effectiveness, we compare it with schemes that use a fixed budget of queries for $\Omega _ { t }$ . Figure 3 (b) shows that the stopping rule yields superior results.
+
+70 Coordinate blocks of a varying size: CobBO selects a block of coordinates of a varying size $C _ { t }$   
+71 (Section 3.1). Figure 3 (c) shows that a varying size is better than fixed.
+
+Preference probability over coordinates: For demonstrating the effectiveness of coordinate selection (Section 3.1), we artificially let the function value only depend on the first 25 coordinates of its input and ignore the rest. It forms two separate sets of active and inactive coordinates, respectively. We expect CobBO to refrain from selecting inactive coordinates. Figure 4 shows the entropy of this preference probability $\pi _ { t }$ over coordinates and the overall probability for picking active and inactive coordinate at each iteration. We see that the entropy decreases, as the preference distribution concentrates on the significant active coordinates.
+
+![](images/fa5e65bd92bc8a4d02b662eaf20271bce2f5ab18293cd1e8217d2815058a697b.jpg)
+
+![](images/719eee06ccf8f9e1437c867aa75bcf2b969c5348077aca5399ea7ac1ede5e2f0.jpg)  
+Figure 4: The preference probability focuses on active coordinates as the entropy decreases   
+Figure 5: Performance over low (left) medium (middle) and high (right) dimensional problems
+
+# 286 4.2 Comparisons with other methods
+
+The default configuration for CobBO is specified in the supplementary materials. CobBO performs on par or outperforms a collection of state-of-the-art methods across the following experiments. Most of the experiments are conducted using the same settings as in TurBO [14], where it is compared with a comprehensive list of baselines, including BFGS, BOCK [47], BOHAMIANN, CMA-ES [22], BOBYQA, EBO [61], GP-TS, HeSBO [44], Nelder-Mead and random search. To avoid repetitions, we only show TuRBO and CMA-ES that achieve the best performance among this list, and additionally compare CobBO with BADS [1], REMBO [63], Differetial Evolution (Diff-Evo) [56], Tree Parzen Estimator (TPE) [4] and Adaptive TPE (ATPE) [12].
+
+# 4.2.1 Low dimensional tests
+
+To evaluate CobBO on low dimensional problems, we use the lunar landing [38, 14] and robot pushing [62], by following the setup in [14]. Confidence intervals $( 9 5 \% )$ over 30 independent experiments for each problem are shown in Fig. 5.
+
+Lunar landing (maximization): This controller learning problem (12 dimensions) is provided by the OpenAI gym [38] and evaluated in [14]. Each algorithm has 50 initial points and a budget of 1, 500 trials. TuRBO is configured with 5 trust regions and a batch size of 50 as in [14]. Fig. 5 (upper left) shows that, among the 30 independent tests, CobBO quickly exceeds 300 along some good sample paths.
+
+Robot pushing (maximization): This control problem (14 dimensions) is introduced in [62] and extensively tested in [14]. We follow the setting in [14], where TuRBO is configured with a batch size of 50 and 15 trust regions, each of which has 30 initial points. Each experiment has a budget of 10, 000 evaluations. On average CobBO exceeds 10 within 5500 trials, as shown in Fig. 5 (lower left).
+
+# 4.2.2 High dimensional tests
+
+Since the duration of each experiment in this section is long, confidence intervals $( 9 5 \% )$ ) over repeated 10 independent experiments for each problem are presented.
+
+Additive latent structure (minimization): As mentioned in Section 2, additive latent structures have been exploited in high dimensions. We construct an additive function of 56 dimensions, defined as $f _ { 5 6 } ( x ) \ = \ \mathrm { { A c k l e y } ( x _ { 1 } ) + \mathrm { { L e v y } ( x _ { 2 } ) + \mathrm { { R a s t r i g i n } ( x _ { 3 } ) + \mathrm { { H a r t m a n n } ( x _ { 4 } ) + \mathrm { { R o s e n b r o c k } ( x _ { 5 } ) + \mathrm { { A c k l e y } ( x _ { 6 } ) + \mathrm { { A c k l e y } ( x _ { 7 } ) } } } } } } } $ Schwefel $\left( \mathrm { x } _ { 6 } \right)$ , where the first three terms express the exact functions and domains described in Section 4.2.1, the Hartmann function on $[ 0 , \bar { 1 } ] ^ { 6 }$ and the Rosenbrock and Schwefel functions on $[ - 5 , 1 0 ] ^ { 1 0 }$ and $[ - 5 0 0 , 5 0 0 ] ^ { 1 0 }$ , respectively.
+
+We compare CobBO with TPE, BADS, CMA-ES and TuRBO, each with 100 initial points. Specifically, TuRBO is configured with 15 trust regions and a batch size 100. ATPE is excluded as it takes more than 24 hours per run to finish. The results are shown in Fig. 5 (upper middle), where CobBO quickly finds the best solution among the algorithms tested.
+
+Rover trajectory planning (maximization): This problem (60 dimensions) is introduced in [62]. The objective is to find a collision-avoiding trajectory of a sequence consisting of 30 positions in a 2-D plane. We compare CobBO with TuRBO, TPE and CMA-ES, each with a budget of 20, 000 evaluations and 200 initial points. TuRBO is configured with 15 trust regions and a batch size of 100, as in [14]. ATPE, BADS and REMBO are excluded for this problem and the following ones, as they all take more than 24 hours per run. Fig. 5 (lower middle) shows that CobBO has a good performance.
+
+The 200-dimensional Levy and Ackley functions (minimization): We minimize the Levy and Ackley functions over $[ - 5 , \mathsf { \bar { 1 0 } } ] ^ { 2 0 0 }$ with 500 initial points. TuRBO is configured with 15 trust regions and a batch size of 100. These two problems are challenging and have no redundant dimensions. For Levy, in Fig. 5 (upper right), CobBO reaches 100.0 within 2, 000 trials, while CMA-ES and TuRBO obtain 200.0 after 8, 000 trials. TPE cannot find a comparable solution within 10, 000 trials in this case. For Ackley, in Fig. 5 (lower right), CobBO reaches the best solution among all of the algorithms tested.
+
+Regarding running times, for Ackley, CobBO runs for 12.8 CPU hours and TuRBO-1 run for more than 80 CPU hours or $9 . 6 \ : G P U$ hours. Most other methods either cannot make any progress or find far worse solutions.
+
+# 5 Conclusion
+
+CobBO is a variant of coordinate ascent tailored for Bayesian optimization with a stopping rule to switch coordinate subspaces. The sampling policy of subspaces is proven to have tight regret bounds with respect to the best subspace in hindsight. Combining this projection on random subspaces with a two-stage kernels for function value interpolation and GP regression, we provide a practical Bayesian optimization method of affordable computational costs in high dimensions. Empirically, CobBO consistently finds comparable or better solutions with reduced trial complexity in comparison with the state-of-the-art methods across a variety of benchmarks.
+
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+
+# 556 Broader Impact
+
+As stated in [32], Bayesian optimization is a powerful optimization technique used in a wide range of industries and applications, such as robotics [37, 9, 52], internet tech companies [18, 34], designing novel molecules for pharmaceutics [19], material design for increasing efficiency of solar cells [71], and aerospace engineering [31]. All of these settings have high-dimensional optimization problems, and advances in BO will reflect on improved capabilities on these fields as well. We have fully opensourced our code for CobBO using the MIT license to be available for researchers and practitioners in these fields, and many others. The ability to optimize a larger number of parameters than has previously been possible will bring further improvements to and further accelerate work in these areas.
+
+# Checklist
+
+67 The checklist follows the references. Please read the checklist guidelines carefully for information on   
+68 how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or
+
+[N/A] . You are strongly encouraged to include a justification to your answer, either by referencing 0 the appropriate section of your paper or providing a brief inline description. For example:
+
+• Did you include the license to the code and datasets? [Yes] See Section ??.   
+• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   
+• Did you include the license to the code and datasets? [N/A]
+
+Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Section 3.   
+(b) Did you describe the limitations of your work? [No]   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Broader Impact.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 3.3.   
+(b) Did you include complete proofs of all theoretical results? [Yes] See the Appendix A.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] In the supplemental material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Table 2 in the appendix, which contains the default hyperparameters.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Fig.5 and Fig.8-10.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See page 9, line 324.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [N/A]   
+(b) Did you mention the license of the assets? [Yes] MIT license; see the appendix.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] See Section 4.2.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] It does not contain personal identifiable information or offensive content.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+        "text": "CobBO: Coordinate Backoff Bayesian Optimization with Two-Stage Kernels ",
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+        "text": "1 Bayesian optimization is a popular method for optimizing expensive black-box   \n2 functions. Yet it oftentimes struggles in high dimensions where the computation   \n3 could be prohibitively expensive and a sufficient estimation of the global landscape   \n4 requires more observations. We introduce Coordinate backoff Bayesian optimiza  \n5 tion (CobBO) with two-stage kernels to alleviate this problem. In each iteration, a   \n6 promising subset of coordinates is selected in the first stage, as past observed points   \n7 in the full space are projected to the selected subspace adopting a simple kernel   \n8 that sacrifices the approximation accuracy for computational efficiency. Then in   \n9 the second stage of the same iteration a more sophisticated kernel is applied for   \n10 estimating the landscape in the selected low dimensional subspace where the com  \n11 putational cost becomes affordable. Effectively, this second stage kernel refines the   \n12 approximation of the global landscape estimated by the first stage kernel through   \n13 a sequence of observations in the local subspace. This refinement lasts until a   \n14 stopping rule is met determining when to back off from a certain subspace and   \n15 switch to another coordinate subset. This decoupling significantly reduces the com  \n16 putational burden in high dimensions, while the two-stage kernels of the Gaussian   \n17 process regressions fully leverage the observations in the whole space rather than   \n18 only relying on observations in each coordinate subspace. Extensive evaluations   \n19 show that CobBO finds solutions comparable to or better than other state-of-the-art   \n20 methods for dimensions ranging from tens to hundreds, while reducing the trial   \n21 complexity and computational costs. ",
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+        "text": "22 1 Introduction ",
+        "text_level": 1,
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+        "text": "Bayesian optimization (BO) has emerged as an effective zero-order paradigm for optimizing expensive black-box functions. The entire sequence of iterations rely only on the function values of the already queried points without information on their derivatives. Though highly competitive in low dimensions (e.g., the dimension $D \\leq 2 0$ [15]), Bayesian optimization based on Gaussian Process (GP) regression has obstacles that impede its effectiveness, especially in high dimensions. ",
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+        "text": "Approximation accuracy: GP regression assumes a class of random functions in a probability space as surrogates that iteratively yield posterior distributions by conditioning on the queried points. When suggesting new query points, for complex functions with numerous local optima and saddle points due to local fluctuations, always exactly using the values on the queried points as the conditional events may mismatch the function’s local landscape by overemphasizing the approximation accuracy of the global landscape. ",
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+        "text": "Curse of dimensionality: As a sample efficient method, Bayesian optimization often suffers from high dimensions. Fitting the GP model (estimating the parameters, e.g., length_scales [14]), computing the Gaussian process posterior and optimizing the acquisition function in high dimensions all ",
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+            "Figure 1: Minimize the fluctuated Rastrigin function on $[ - 5 , 1 0 ] ^ { 5 0 }$ with 20 initial samples. [Left] Computation times for training the GP regression model and maximizing the acquisition function at each iteration. CobBO significantly reduces the execution time compared with a vanilla BO, e.g. $\\times 1 3$ faster in this case. [Right] The average error between the GP predictions before making queries and the true function values at the queried points (solid curves, the higher the better) and the best observed function value (dashed curves, the lower the better) at iteration $t$ . CobBO captures the global landscape less accurately using the RBF kernel, and then explores selected subspaces $\\Omega _ { t }$ more accurately using the Matern kernel. This eventually better exploits the promising subspaces. "
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+        "text": "39 incur large computational costs. It also results in statistical insufficiency of exploration [11, 65]. As   \n40 the GP regression’s error grows with dimensions [8], more samples are required to balance that in   \n41 high dimensions, which could cubically increase the computational costs in the worst case [45].   \n42 To alleviate these issues, we design coordinate backoff Bayesian optimization (CobBO) with two  \n43 stage kernels, by challenging a seemingly natural intuition stating that it is always better for Bayesian   \n44 optimization to have a more accurate approximation of the objective function at all times. We   \n45 demonstrate that this is not necessarily true, by showing that smoothing out local fluctuations and   \n46 using the estimated function values instead of the true observations to serve as the conditional events   \n47 in selected subspaces can not only significantly reduce the computation time due to the curse of   \n48 dimensionality but also help in capturing the large-scale properties of the objective function $f ( x )$ .   \n49 Specifically, CobBO introduces the two-stage kernels with a stopping rule. The first stage of each   \n50 iteration adopts a simple kernel that sacrifices the approximation accuracy of $f ( x )$ for computational   \n51 efficiency. For example, by using a universal radial basis function (RBF) approximation without   \n52 learnable parameters [50], CobBO can eliminate the model fitting time in the full space. It captures   \n53 a smooth approximation ${ \\hat { f } } ( x )$ of the global landscape by interpolating the values of queried points   \n54 projected to selected promising subspaces. These projected points serve as the conditional events   \n55 for GP regression. In a selected coordinate subspace, the second stage of the same iteration applies   \n56 a sophisticated kernel that can tolerate high computational cost in low dimensions. For example,   \n57 CobBO uses the Automatic Relevance Determination (ARD) Matérn 5/2 kernel [40]. It refines the   \n58 approximation of the local landscape by a sequence of observations determined by a stopping rule that   \n59 backs off from a certain subspace and switches to another coordinate subset. In addition, computing   \n60 the Gaussian process posterior and optimizing the acquisition function are both efficiently conducted   \n61 in the low dimensional subspaces, bypassing the curse of dimensionality.   \n62 For iteration $t$ , instead of directly computing the Gaussian process posterior distribution   \n63 $\\left\\{ \\hat { f } ( x ) \\middle | \\mathcal { H } _ { t } = \\left\\{ ( x _ { i } , y _ { i } ) \\right\\} _ { i = 1 } ^ { t } , x \\in \\Omega \\right\\}$ by conditioning on the observations $y _ { i } = f ( x _ { i } )$ at queried   \n64 points $x _ { i }$ in the full space $\\Omega \\subset \\mathbb { R } ^ { D }$ for $i = 1 , \\ldots , t$ , we change the conditional events, and consider ",
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+        "text": "$$\n\\left\\{ \\hat { f } ( x ) \\Big | R \\left( P _ { \\Omega _ { t } } ( x _ { 1 } , \\dots , x _ { t } ) , \\mathcal { H } _ { t } \\right) , x \\in \\Omega _ { t } , \\Omega _ { t } \\subset \\Omega \\right\\}\n$$",
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+        "text": "65 for a projection function $P _ { \\Omega _ { t } } ( \\cdot )$ to a random subspace $\\Omega _ { t }$ and an interpolation function $R ( \\cdot , \\cdot )$ , e.g.,   \n66 using a RBF approximation without learnable parameters [50] as the simple kernel for the first   \n67 stage. The projection $P _ { \\Omega _ { t } } ( \\cdot )$ maps the queried points to virtual points on a subspace $\\Omega _ { t }$ of a lower   \n68 dimension [51]. The interpolation function $R ( \\cdot , \\cdot )$ estimates the objective values at the virtual points   \n69 using the queried points and their values as specified by $\\mathcal { H } _ { t }$ . The second stage within the subspace $\\Omega _ { t }$   \n70 uses the more sophisticated kernel, e.g., Matérn $5 / 2$ kernel [40], which has a number of parameters   \n71 that otherwise would be expensive to be learned in high dimensions.   \n72 This method can be viewed as a variant of block coordinate ascent tailored to Bayesian optimization by   \n73 applying backoff stopping rules for switching coordinate blocks. While similar work exists [43, 48],   \n74 CobBO differs by introducing the two-stage kernels and addressing the following three issues: ",
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+        "text": "1. Selecting a block of coordinates for ascending requires determining the block size as well as the coordinates therein. CobBO selects the coordinate subsets by a multiplicative weights update method [2] to the preference probability associated with each coordinate. Thus, it samples more promising subspaces with higher probabilities.   \n2. A coordinate subspace requires a sufficient number of query points acting as the conditional events for the GP regression. CobBO leverages all observations in the whole space by interpolating the values of queried points projected to selected promising subspaces, rather than simply starting from scratch in each subspace.   \n3. Querying a certain subspace, under some trial budget, comes at the expense of exploring other coordinate blocks. Yet prematurely shifting to different subspaces does not fully exploit the full potential of a given subspace. Hence determining the number of consecutive function queries within a subspace makes a trade-off between exploration and exploitation. CobBO uses a stopping rule in each subspace to switch the selected coordinates. When consecutively querying data points in the same subspace, CobBO does not need to conduct the first-stage function approximation in the full space, which is far more efficient. ",
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+        "text": "90 Through comprehensive evaluations, CobBO demonstrates appealing performance for dimensions   \n91 ranging from tens to hundreds. It obtains comparable or better solutions with fewer queries, in   \n92 comparison with the state-of-the-art methods, for most of the problems tested in Section 4.2. ",
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+        "text": "2 Related work ",
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+        "text": "94 Certain assumptions are often imposed on the latent structure in high dimensions. Typical assumptions   \n95 include low dimensional structures and additive structures. Their advantages manifest on problems   \n96 with a low dimension or a low effective dimension. However, these assumptions do not necessarily   \n97 hold for non-separable functions with no redundant dimensions.   \n98 Low dimensional structure: The black-box function $f$ is assumed to have a low effective dimen  \n99 sion [30, 58], e.g., $f ( x ) = g ( \\Phi x )$ with some function $g ( \\cdot )$ and a matrix $\\Phi$ of $d \\times D , d < < D$ . A   \n100 number of different methods have been developed, including random embedding [66, 11, 63, 36,   \n101 44, 70, 5, 32], low-rank matrix recovery [11, 58], and learning subspaces by derivative informa  \n102 tion [11, 13]. In contrast to existing work on subspace selections, e.g., Hashing-enhanced Subspace   \n103 BO (HeSBO) [44], Mahalanobis kernel for linear embeddings [33], DROPOUT [35] and LineBO [29]   \n104 (which receives a special treatment in Appendix F), CobBO efficiently leverages all the observations   \n105 in the whole space using the two-stage kernels and the stopping rule in each subspace for consecutive   \n106 observations, rather than only relying on limited observations in each coordinate subspace. It exploits   \n107 subspace structure from a perspective of block coordinate ascent, independent of the dimensions,   \n108 different from some algorithms that are more suitable for low dimensions, e.g., BADS [1].   \n109 Additive structure: A decomposition assumption is often made by $\\begin{array} { r } { f ( x ) = \\sum _ { i = 1 } ^ { k } f ^ { ( i ) } \\left( x _ { i } \\right) } \\end{array}$ , with $x _ { i }$   \n110 defined over low-dimensional components. In this case, the effective dimensionality of the model is   \n111 the largest dimension among all additive groups [45], which is usually small. The Gaussian process   \n112 is structured as an additive model [17, 28], e.g., projected-additive functions [36], ensemble Bayesian   \n113 optimization (EBO) [61], latent additive structural kernel learning (HDBBO) [65] and group additive   \n114 models [28, 36]. However, learning the unknown structure incurs a considerable computational   \n115 cost [44], and is not applicable for non-separable functions, for which CobBO can still be applied.   \n116 Trust regions and space partitions: Trust region BO has been proven effective for high-dimensional   \n117 problems. A typical pattern is to alternate between global and local search regions. In the local   \n118 trust regions, many efficient methods have been applied, e.g., local Gaussian models (TurBO [14]),   \n119 adaptive search on a mesh grid (BADS [1]) or quasi-Newton local optimization (BLOSSOM [41]).   \n120 TurBO [14] uses Thompson sampling to allocate samples across multiple regions. A related method is   \n121 to use space partitions, e.g., LA-MCTS[60] on a Monte Carlo tree search algorithm to learn efficient   \n122 partitions. CobBO differs by selecting low dimensional subspaces. It can also incorporate trust   \n123 regions in the first-stage global approximation, as shown in the Appendix.   \n125 Without loss of generality, suppose that the goal is to solve a maximization problem $\\boldsymbol { x } ^ { * } =$   \n126 $\\mathrm { { r g m a x } } _ { x \\in \\Omega } f ( x )$ for a black-box function $f : \\Omega  \\mathbb { R }$ . The domain is normalized $\\Omega = [ 0 , 1 ] ^ { D }$   \n127 with the coordinates indexed by $I = \\{ 1 , 2 , \\cdots , D \\}$ .   \n128 For a sequence of points $\\mathcal { X } _ { t } = \\{ x _ { 1 } , x _ { 2 } , \\cdot \\cdot \\cdot , x _ { t } \\}$ with $t$ indexing the most recent iteration, we observe   \n129 $\\mathcal { H } _ { t } = \\{ ( x _ { i } , y _ { i } = f ( x _ { i } ) ) \\} _ { i = 1 } ^ { t }$ . A random subset $C _ { t } \\subseteq I$ of the coordinates is selected, forming a   \n130 subspace $\\Omega _ { t } \\subseteq \\Omega$ at iteration $t$ . As a variant of coordinate ascent, the subspace $\\Omega _ { t }$ contains a pivot   \n131 point $V _ { t }$ , which presumably is the maximum point $x _ { t } ^ { M } = \\operatorname { a r g m a x } _ { x \\in \\mathcal { X } _ { t } } { f ( x ) }$ with $M _ { t } = f \\left( { { \\dot { x } } _ { t } } ^ { M } \\right)$ .   \n132 CobBO may set $V _ { t }$ different from $x _ { t } ^ { M }$ to escape local optima. Then, BO is conducted within $\\Omega _ { t }$ while   \n133 fixing all the other coordinates $C _ { t } ^ { c } \\bar { = } I \\setminus C _ { t }$ , i.e., the complement of $C _ { t }$ . ",
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+        "text": "$\\overline { { \\mathbf { A l g o r i t h m 1 : C o b B O ( f , \\boldsymbol { \\tau } , T ) } } }$ ",
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+        "text": "1 $\\mathcal { H } _ { \\tau } \\gets \\mathrm { s a m p l e } ~ $ $\\tau$ initial points and evaluate their values   \n2 $V _ { \\tau } , M _ { \\tau } \\gets$ Find the tuple with the maximal objective value in $\\mathcal { H } _ { \\tau }$   \n3 $q _ { \\tau } \\gets 0$ Initialize the number of consecutive failed queries   \n4 $\\pi _ { \\tau } $ Initialize a uniform preference distribution on the coordinates   \n5 for $t  \\tau$ to $T$ do 6 if switch $\\Omega _ { t - 1 }$ by the backoff stopping rule (Section 3.2) then 7 $C _ { t } \\gets$ Sample a promising coordinate block according to $\\pi _ { t }$ (Section 3.1) 8 $\\Omega _ { t } \\gets$ Take the subspace of $\\Omega _ { t }$ over the coordinate block $C _ { t }$ , such that $V _ { t } \\in \\Omega _ { t }$ 9 else   \n10 $\\Omega _ { t } \\gets \\Omega _ { t - 1 }$   \n11 $\\hat { \\mathcal { X } } _ { t } \\gets P _ { \\Omega _ { t } } ( \\mathcal { X } _ { t } )$ hProject $\\mathcal { X } _ { t }$ onto $\\Omega _ { t }$ to obtain a set of virtual points (Eq. 1)i   \n12 $\\hat { \\mathcal { H } } _ { t } \\gets R \\left( \\hat { \\mathcal { X } } _ { t } , \\bar { \\mathcal { H } } _ { t } \\right)$ hSmooth function values on $\\hat { \\mathcal { X } } _ { t }$ by interpolation using $\\mathcal { H } _ { t }$ i   \n13 $p \\left[ \\hat { f } _ { \\Omega _ { t } } ( x ) | \\hat { \\mathcal { H } } _ { t } \\right] \\gets \\mathbf { C } \\mathbf { \\tau }$ ompute the posterior distribution of the Gaussian process in $\\Omega _ { t }$ conditional on $\\hat { \\mathcal { H } } _ { t }$   \n14 $\\begin{array} { r } { x _ { t + 1 } \\gets \\mathrm { a r g m a x } _ { x \\in \\Omega _ { t } } Q _ { \\hat { f } \\sim p ( \\hat { f } | \\hat { \\mathcal { H } } _ { t } ) } ( x | \\hat { \\mathcal { H } } _ { t } ) \\ \\Big [ } \\end{array}$ Suggest the next query in $\\Omega _ { t }$ (Section 3)i   \n15 $y _ { t + 1 } \\gets$ Evaluate the black-box function $y _ { t + 1 } = f ( x _ { t + 1 } )$   \n16 if $y _ { t + 1 } > M _ { t }$ then   \n17 $| \\quad V _ { t + 1 } \\gets x _ { t + 1 } , M _ { t + 1 } \\gets y _ { t + 1 } , q _ { t + 1 } \\gets 0$   \n18 else   \n19 $V _ { t + 1 }  V _ { t }$ , $M _ { t + 1 } \\gets M _ { t }$ , $q _ { t + 1 } \\gets q _ { t } + 1$   \n20 $\\pi _ { t + 1 } $ Update $\\pi _ { t }$ by a multiplicative weights update method (Eq. 2)   \n21 $\\mathcal { H } _ { t + 1 }  \\bar { \\mathcal { H } } _ { t } \\bigcup \\{ ( x _ { t + 1 } , y _ { t + 1 } ) \\} , \\mathcal { X } _ { t + 1 }  \\bar { \\mathcal { X } } _ { t } \\bigcup \\{ x _ { t + 1 } \\}$   \n22 end ",
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+        "text": "For BO in 134 $\\Omega _ { t }$ , we use Gaussian processes as the random surrogates $\\hat { f } = \\hat { f } _ { \\Omega _ { t } } ( x )$ to describe the 135 Bayesian statistics of $f ( x )$ for $x \\in \\Omega _ { t }$ . At each iteration, the next query point is generated by solving ",
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+        "text": "$$\n\\begin{array} { r } { x _ { t + 1 } = \\mathrm { a r g m a x } _ { x \\in \\Omega _ { t } , V _ { t } \\in \\Omega _ { t } } Q _ { \\hat { f } _ { \\Omega _ { t } } ( x ) \\sim p ( \\hat { f } | \\mathcal { H } _ { t } ) } ( x | \\mathcal { H } _ { t } ) , } \\end{array}\n$$",
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+        "type": "text",
+        "text": "136 where the acquisition function $Q ( x | \\mathcal { H } _ { t } )$ incorporates the posterior distribution of the Gaussian   \n137 processes $p ( \\hat { f } | \\mathcal { H } _ { t } )$ . Typical acquisition functions include the expected improvement (EI) [42, 27],   \n138 the upper confidence bound (UCB) [3, 54, 55], the entropy search [24, 25, 64], and the knowledge   \n139 gradient [16, 53, 69]. ",
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+        "type": "text",
+        "text": "Instead of directly computing the posterior distribution 140 $p ( \\hat { f } | \\mathcal { H } _ { t } )$ , we replace the conditional events 141 $\\mathcal { H } _ { t }$ by ",
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+        "text": "$$\n\\hat { \\mathcal { H } } _ { t } : = R \\left( P _ { \\Omega _ { t } } \\left( \\mathcal { X } _ { t } \\right) , \\mathcal { H } _ { t } \\right) = \\left\\{ \\left( \\hat { x } _ { i } , \\hat { y } _ { i } \\right) \\right\\} _ { i = 1 } ^ { t }\n$$",
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+        "type": "text",
+        "text": "with an interpolation function 142 $R ( \\cdot , \\cdot )$ and a projection function $P _ { \\Omega _ { t } } ( \\cdot )$ , ",
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+        "text": "$$\nP _ { \\Omega _ { t } } ( x ) ^ { ( j ) } = { \\left\\{ \\begin{array} { l l } { x ^ { ( j ) } } & { { \\mathrm { i f ~ } } j \\in C _ { t } } \\\\ { V _ { t } ^ { ( j ) } } & { { \\mathrm { i f ~ } } j \\not \\in C _ { t } } \\end{array} \\right. }\n$$",
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+        "text": "143 at coordinate $j$ . It simply keeps the values of $x$ whose corresponding coordinates are in $C _ { t }$ and   \n144 replaces the rest by the corresponding values of $V _ { t }$ , as illustrated in Fig. 2.   \n145 Applying $P _ { \\Omega _ { t } } ( \\cdot )$ on $\\mathcal { X } _ { t }$ and discarding duplicates generate a new set of distinct virtual points $\\hat { \\mathcal { X } } _ { t } =$   \n146 $\\{ \\hat { x } _ { 1 } , \\hat { x } _ { 2 } , \\hat { x } _ { 3 } , \\cdot \\cdot \\cdot , \\hat { x } _ { \\hat { t } } \\}$ , $\\hat { x } _ { i } \\in \\Omega _ { t } \\forall 1 \\leq i \\leq \\hat { t } \\leq t$ . The function values at $\\hat { x } _ { i } \\in \\hat { \\mathcal X } _ { t }$ are interpolated as   \n147 $\\hat { y } _ { i } = R ( \\hat { x } _ { i } , \\mathcal { H } _ { t } )$ using the standard radial basis function [6, 7] and the observed points in $\\mathcal { H } _ { t }$ . It not   \n148 only significantly reduces the GP regression time due to the efficiency of RBF [6] and the acquisition   \n149 function optimization in low dimensions [11], but also eventually improves the model accuracy using   \n150 the more sophisticated kernel applied on $\\Omega _ { t }$ .   \n151 Note that only a fraction of the points in $\\hat { \\mathcal X } _ { t } \\cap \\mathcal X _ { t }$   \n152 directly observe the exact function values. The   \n153 function values on the rest ones in $\\hat { \\mathcal { X } } _ { t } \\backslash \\mathcal { X } _ { t }$ are   \n154 estimated by interpolation, which captures the   \n155 landscape of $f ( x )$ by smoothing out the local   \n156 fluctuations. To control the trade-off between   \n157 the inaccurate estimations and the exact obser  \n158 vations in $\\Omega _ { t }$ , we design a stopping rule that   \n159 optimizes the number of consistent queries in   \n160 $\\Omega _ { t }$ . The more consistent queries conducted in a   \n161 given subspace, the more accurate observations ",
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+        "image_caption": [
+            "Figure 2: Two-stage kernels: subspace projection and function value interpolation "
+        ],
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+        "type": "text",
+        "text": "62 could be obtained, albeit at the expense of a smaller remaining budget for exploring other regions. ",
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+        "text": "163 The key features of CobBO are listed in Algorithm 1, with more details in the following sections.   \n164 Several auxiliary components are utilized and presented in Appendix C to deal with a larger variety   \n165 of problems and corner cases. ",
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+        "type": "text",
+        "text": "166 3.1 Block coordinate ascent and subspace selection ",
+        "text_level": 1,
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+        "text": "167 For Bayesian optimization, consider an infeasible assumption that each iteration can exactly maximize   \n168 the function $f ( x )$ in $\\Omega _ { t }$ . This is not possible for one iteration but only if one can consistently query   \n169 in $\\Omega _ { t }$ , since the points converge to the maximum, e.g., under the expected improvement acquisition   \n170 function with fixed priors [59] and the convergence rate can be characterized for smooth functions   \n171 in the reproducing kernel Hilbert space [8]. However, even with this infeasible assumption, it is   \n172 known that coordinate ascent with fixed blocks can cause stagnation at a non-critical point, e.g., for   \n173 non-differentiable [67] or non-convex functions [49]. This motivates us to select a subspace with a   \n174 variable-size coordinate block $C _ { t }$ for each query. A good coordinate block can help the iterations   \n175 to escape the trapped non-critical points. For example, one condition can be based on the result   \n176 in [21] that assumes $f ( x )$ to be differentiable and strictly quasi-convex over a collection of blocks. In   \n177 practice, we do not restrict ourselves to these assumptions.   \n178 We induce a preference distribution $\\pi _ { t }$ over the coordinate set $I$ , and sample a variable-size coordinate   \n179 block $C _ { t }$ accordingly. This distribution is updated at iteration $t$ through a multiplicative weights   \n180 update method [2]. Specifically, the values of $\\pi _ { t }$ at coordinates in $C _ { t }$ starts off uniform and increase   \n181 in face of an improvement or decrease otherwise according to different multiplicative ratios $\\alpha > 1$   \n182 and $\\beta > 1$ , respectively, ",
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+        "text": "$$\nw _ { t , j } = w _ { t - 1 , j } \\cdot \\left\\{ \\begin{array} { l l } { \\alpha } & { \\mathrm { i f ~ } j \\in C _ { t } \\mathrm { ~ a n d ~ } y _ { t } > M _ { t - 1 } } \\\\ { 1 / \\beta } & { \\mathrm { i f ~ } j \\in C _ { t } \\mathrm { ~ a n d ~ } y _ { t } \\leq M _ { t - 1 } } \\\\ { 1 } & { \\mathrm { i f ~ } j \\notin C _ { t } } \\end{array} \\right. ; \\quad w _ { 0 , j } = \\frac { 1 } { D } \\quad ; \\quad \\pi _ { t , j } = \\frac { w _ { t , j } } { \\sum _ { j = 1 } ^ { D } w _ { t , j } }\n$$",
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+        "type": "text",
+        "text": "183 This update characterizes how likely a coordinate block can generate a promising search subspace.   \n184 The multiplicative ratio $\\alpha$ is chosen to be relatively large, e.g., $\\alpha = 2 . 0$ , and $\\beta$ relatively small, e.g.,   \n185 $\\beta = 1 . 1$ , since the queries that improve the best observations $y _ { t } > M _ { t - 1 }$ happen more rarely than   \n186 the opposite $y _ { t } \\le M _ { t - 1 }$ .   \n187 How to dynamically select the size $| C _ { t } | \\cdot \\cdot$ It is known that Bayesian optimization works well for low   \n188 dimensions [15]. Thus, we specify an upper bound for the dimension of the subspace (e.g. $| C _ { t } | \\le 3 0 \\}$ .   \n189 In principle, $| C _ { t } |$ can be any random number in a finite set of possible block sizes $\\mathcal { C }$ . This is different   \n190 from the method that partitions the coordinates into fixed blocks and selects one according to, e.g.,   \n191 cyclic order [68], random sampling or Gauss-Southwell [46]. ",
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+        "text": "192 3.2 Backoff stopping rule for consistent queries ",
+        "text_level": 1,
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+        "text": "Applying BO on $\\Omega _ { t }$ requires a strategy to determine the number of consecutive queries for making a sufficient progress. This strategy is based on previous observations, thus forming a stopping rule. In principle, there are two different scenarios, exemplifying exploration and exploitation, respectively. Persistently querying a given subspace refrains from opportunistically exploring other coordinate combinations. Abruptly shifting to different subspaces does not fully exploit the potential of a given subspace. ",
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+        "type": "text",
+        "text": "CobBO designs a heuristic stopping rule in compromise. It takes the above two scenarios into joint consideration, by considering not only the number of consecutive queries that fail to improve the objective function but also other factors including the improved difference $M _ { t } - M _ { t - 1 }$ , the point distance $\\vert \\vert x _ { t } - x _ { t - 1 } \\vert \\vert$ , the query budget $T$ and the problem dimension $D$ . On the one hand, switching to another subspace $\\Omega _ { t + 1 } ~ ( \\neq ~ \\Omega _ { t } )$ prematurely without fully exploiting $\\Omega _ { t }$ incurs an additional approximation error associated with the interpolation of observations in $\\Omega _ { t }$ projected to $\\Omega _ { t + 1 }$ . On the other hand, it is also possible to over-exploit a subspace, spending high query budget on marginal improvements around local optima. In order to mitigate this, even when a query leads to an improvement, other factors are considered for sampling a new subspace. ",
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+        "type": "text",
+        "text": "3.3 Theoretical Analysis ",
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+        "text": "209 One can view our block coordinate selection approach in section 3.1 as a combinatorial mixture of   \n210 experts problem [10], where each coordinate is a single expert and the forecaster aims at choosing   \n211 the best combination of experts in each step. Under this view, we bound the regret of our selection   \n212 method with respect to the policy of selecting the best (unknown) block of coordinates at each step.   \n213 Assume that there is a fixed optimal choice $\\mathcal { T } ^ { * }$ for the block of coordinates to pick at all steps. This   \n214 block is characterized by improving the objective function for the largest number of times among   \n215 all the possible coordinate blocks when performing Bayesian optimization over the corresponding   \n216 subspaces. The following particular design of losses expresses this cause: ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "type": "equation",
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+        "text": "$$\n\\ell _ { t , i } = \\left\\{ \\begin{array} { l l } { - \\log ( \\tilde { \\alpha } ) } & { \\mathrm { i f ~ } i \\in C _ { t } \\mathrm { ~ a n d ~ } y _ { t } > M _ { t - 1 } } \\\\ { \\log ( \\tilde { \\beta } ) } & { \\mathrm { i f ~ } i \\in C _ { t } \\mathrm { ~ a n d ~ } y _ { t } \\leq M _ { t - 1 } } \\\\ { 0 } & { \\mathrm { i f ~ } i \\not \\in C _ { t } } \\end{array} \\right. ~ ; \\quad \\tilde { \\alpha } , \\tilde { \\beta } > 1\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "217 as all the coordinates participating in the selected block incur the same loss that effectively rewards   \n218 these coordinates for improving the objective and penalizes these for failing to improve the objective.   \n219 All other coordinates that are not selected receive a zero loss and remain untouched. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Note that $\\tilde { \\alpha }$ and $\\tilde { \\beta }$ express the extent of reward and penalty, e.g. for $\\tilde { \\alpha } = \\tilde { \\beta } = e$ we have losses of $\\ell _ { t , i } \\in \\{ - 1 , 1 , 0 \\}$ . Yet, $\\tilde { \\alpha }$ is better chosen to be larger than $\\tilde { \\beta }$ , since the frequency of improving the objective is expected to be smaller. ",
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+    {
+        "type": "text",
+        "text": "23 The loss received by the forecaster is to reflect the same motivation. This is done by averaging   \n24 the losses of the individual coordinates in the selected block, so that the size of the block does not   \n25 matter explicitly, i.e. a bigger block should not incur more loss just due to its size but only due to its   \n26 performance. Such that for each coordinate block $\\mathcal { T } _ { t } \\subset \\mathcal { T } = \\{ 1 , \\cdots , D \\}$ selected at time step $t$ , the   \n27 loss incurred by the forecaster is Lt,It = 1|I | $\\begin{array} { r } { L _ { t , \\mathcal { T } _ { t } } = \\frac { 1 } { | \\mathcal { T } _ { t } | } \\sum _ { i \\in \\mathcal { T } _ { t } } \\ell _ { t , i } } \\end{array}$ . This is also the common loss incurred by   \n28 all the coordinates participating in that block. ",
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+        "type": "text",
+        "text": "229 In each step we have the following multiplicative update rule of the weights associated with each coordinate (setting 230 $\\alpha = \\tilde { \\alpha } ^ { \\eta }$ and $\\beta = \\widetilde { \\beta } ^ { \\eta }$ yields the update rule in Eq. 2): ",
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+        "type": "equation",
+        "img_path": "images/80f4e40891b1bc1e98f9c54715a876886588616f6917aa92b7a69ede1076afa0.jpg",
+        "text": "$$\nw _ { t , i } = w _ { t - 1 , i } \\cdot e ^ { - \\eta \\ell _ { t , i } } = w _ { t - 1 , i } \\cdot \\left\\{ \\begin{array} { l l } { \\tilde { \\alpha } ^ { \\eta } } & { \\mathrm { i f ~ } i \\in C _ { t } \\mathrm { ~ a n d ~ } y _ { t } > M _ { t - 1 } } \\\\ { 1 / \\tilde { \\beta } ^ { \\eta } } & { \\mathrm { i f ~ } i \\in C _ { t } \\mathrm { ~ a n d ~ } y _ { t } \\le M _ { t - 1 } } \\\\ { 1 } & { \\mathrm { i f ~ } i \\notin C _ { t } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The probability $\\tilde { \\pi } _ { t , \\mathcal { T } _ { t } }$ of selecting a certain coordinate block $\\mathcal { T } _ { t }$ is induced by $\\pi _ { t }$ as specified next. Thus the expected cumulative loss of the foreceaster is: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/faaa5f6c65049764e360c1c2a4483192230aa9dbb825b4edb8b5bce50f57dfc2.jpg",
+        "text": "$$\nL _ { T } = \\sum _ { t = 1 } ^ { T } \\sum _ { c \\in \\mathcal { C } } \\sum _ { \\mathbb { Z } _ { t } \\in \\mathcal { S } _ { c } } \\tilde { \\pi } _ { t , \\mathbb { Z } _ { t } } \\cdot \\frac { 1 } { | \\mathbb { Z } _ { t } | } \\sum _ { i \\in \\mathcal { T } _ { t } } \\ell _ { t , i }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Assume the best coordinate block is $\\mathcal { T } ^ { * }$ , then the corresponding cumulative loss is: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4f5bbff65480ba246858e3fb3570fb59af8a4f4355861b962b6b0949ec77902e.jpg",
+        "text": "$$\nL _ { T } ^ { * } = \\sum _ { t = 1 } ^ { T } { L _ { t , \\mathbb { Z } ^ { * } } } = \\sum _ { t = 1 } ^ { T } \\frac { 1 } { | \\mathscr { T } ^ { * } | } \\sum _ { i \\in \\mathscr { T } ^ { * } } \\ell _ { t , i }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We hence aim at bounding the regret 231 $R e g r e t _ { T } = L _ { T } - L _ { T } ^ { * }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "232 Theorem 1. Sample from the combinatorial space of all possible coordinate blocks $\\textstyle { \\mathcal { T } } _ { t } \\in \\bigcup _ { c \\in { \\mathcal { C } } } S _ { c }$ with probability 233 $\\begin{array} { r } { \\tilde { \\pi } _ { t , \\mathcal { T } _ { t } } ~ = ~ \\prod _ { i \\in \\mathcal { T } _ { t } } \\tilde { w } _ { t , \\mathcal { T } _ { t } } \\Big / \\sum _ { c \\in \\mathcal { C } } \\sum _ { \\hat { \\mathcal { T } } \\in \\mathcal { S } _ { c } } \\prod _ { j \\in \\hat { \\mathcal { T } } } \\tilde { w } _ { t , \\hat { \\mathcal { T } } } . } \\end{array}$ . Then the update rule in Eq. 2 with 234 $\\alpha = \\tilde { \\alpha } ^ { \\eta }$ , $\\beta = \\widetilde { \\beta } ^ { \\eta }$ and $\\eta = \\log ( \\tilde { \\alpha } \\tilde { \\beta } ) ^ { - 1 } \\sqrt { T ^ { - 1 } | \\mathcal { C } | D \\log ( D ) }$ yields: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ca9b611d51d94efbbdd41f2622cdeb19d264916f0f35f863fcb5dfeb0e7ec1d4.jpg",
+        "text": "$$\nR e g r e t _ { t } \\le \\mathcal { O } \\left( ( \\log ( \\tilde { \\alpha } \\tilde { \\beta } ) \\cdot \\sqrt { T | \\mathcal { C } | D \\log ( D ) } ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where 235 $\\begin{array} { r } { \\tilde { w } _ { t , \\mathcal { T } _ { t } } = \\prod _ { i \\in \\mathcal { T } _ { t } } w _ { t , i } ^ { 1 / | \\mathcal { T } _ { t } | } } \\end{array}$ w1/|It|t,i is the geometric mean of weights in block It. The upper bound in Eq. 5 is tight, as the lower bound can be shown to be of 236 $\\Omega ( { \\sqrt { T \\log ( N ) } } )$ [23] where the number of experts is 237 $\\begin{array} { r } { N = \\sum _ { c \\in \\mathcal { C } } S _ { c } \\leq D ^ { | \\mathcal { C } | D } } \\end{array}$ in our combinatorial setup, as typically $| \\mathcal { C } | \\ll D$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "238 In practice, the direct sampling policy introduced in Theorem 1 involves high computational costs due   \n239 to the exponential growth of combinations in $D$ . Thus CobBO suggests an alternative computationally   \n240 efficient sampling policy with a linear growth in $D$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. Sample a block size $c \\in { \\mathcal { C } }$ with probability $p _ { c }$ and c coordinates without replacement according to $\\pi _ { t }$ . Assume $c \\supset \\{ 1 \\}$ , then the update rule in Eq. 2, with $\\alpha = \\tilde { \\alpha } ^ { \\eta }$ , $\\beta = { \\tilde { \\beta } } ^ { \\eta }$ and $\\begin{array} { r } { \\eta = \\sqrt { \\frac { \\log ( D ) } { T ( \\log ( \\tilde { \\alpha } \\tilde { \\beta } ) ^ { 2 } - \\log ( p _ { 1 } ) ) } } \\geq 1 } \\end{array}$ ≥ 1 yields: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/84a557876f337729336cd8d4cd689065c2729874ff721b5d9b9e0709875b3db4.jpg",
+        "text": "$$\nR e g r e t _ { t } \\le \\mathcal { O } \\left( \\sqrt { ( \\log ( \\tilde { \\alpha } \\tilde { \\beta } ) ^ { 2 } - \\log ( p _ { 1 } ) ) } \\cdot \\sqrt { T \\log ( D ) } ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $p _ { c } > 0$ for all $c \\in { \\mathcal { C } }$ and $\\textstyle \\sum _ { c \\in c } p _ { c } = 1$ , e.g., uniformly set $p _ { c } \\equiv | \\mathcal { C } | ^ { - 1 }$ .The proof and detailed sampling policy are in Appendix A. The regret upper bound in Eq. 6 is tight, as the lower bound for an easier setup can be shown to be of $\\Omega ( { \\sqrt { T \\log ( D ) } } )$ [23]. The implication on $\\eta$ is valid only for settings of a very high dimensionality and low query budget. In particular, CobBO is designed for this kind of problems. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Remark: Similar analysis and results follow when incorporating consistent queries from Section 3.2 and sampling a new coordinate block once every several steps. This is done by effectively performing less steps of aggregated temporal losses, as shown in Appendix A. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 Numerical Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This section presents detailed ablation studies of the key components presented in Section 3 and comparisons with other algorithms. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Empirical analysis and ablation study ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Ablation studies are designed to study the contributions of the key components in Algorithm 1 by experimenting with the Rastrigin function on $[ - 5 , 1 0 ] ^ { 5 0 }$ with 20 initial points. The best performing run out of 5 experiments for each configuration is presented in Figure 3. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/2c0ffc937e0888a03bbad1004c2f441b8be3c88d37408aa85eb32eedc2e170da.jpg",
+        "image_caption": [
+            "Figure 3: Ablation study using Rastrigin on $[ - 5 , 1 0 ] ^ { 5 0 }$ with 20 initial random samples "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "259 RBF interpolation: RBF calculation is time efficient. Specifically, this is much beneficial in high   \n260 dimensions. Figure 1 (left) shows the computation time of plain Bayesian optimization compared to   \n261 CobBO’s. While the former applies GP regression using the Matérn kernel in the high dimensional   \n262 space directly, the later applies RBF interpolation in the high dimensional space and GP regression   \n263 with the Matérn kernel in the low dimensional subspace. This two-step composite kernel leads to a   \n264 significant speed-up. Other time efficient alternatives are, e.g., the inverse distance weighting [26]   \n265 and the simple approach of assigning the value of the observed nearest neighbour. Figure 3 (a) shows   \n266 that RBF is the most favorable. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Backoff stopping rule: CobBO applies a stopping rule to query a variable number of points in subspace $\\Omega _ { t }$ (Section 3.2). To validate its effectiveness, we compare it with schemes that use a fixed budget of queries for $\\Omega _ { t }$ . Figure 3 (b) shows that the stopping rule yields superior results. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "70 Coordinate blocks of a varying size: CobBO selects a block of coordinates of a varying size $C _ { t }$   \n71 (Section 3.1). Figure 3 (c) shows that a varying size is better than fixed. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Preference probability over coordinates: For demonstrating the effectiveness of coordinate selection (Section 3.1), we artificially let the function value only depend on the first 25 coordinates of its input and ignore the rest. It forms two separate sets of active and inactive coordinates, respectively. We expect CobBO to refrain from selecting inactive coordinates. Figure 4 shows the entropy of this preference probability $\\pi _ { t }$ over coordinates and the overall probability for picking active and inactive coordinate at each iteration. We see that the entropy decreases, as the preference distribution concentrates on the significant active coordinates. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/fa5e65bd92bc8a4d02b662eaf20271bce2f5ab18293cd1e8217d2815058a697b.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/719eee06ccf8f9e1437c867aa75bcf2b969c5348077aca5399ea7ac1ede5e2f0.jpg",
+        "image_caption": [
+            "Figure 4: The preference probability focuses on active coordinates as the entropy decreases ",
+            "Figure 5: Performance over low (left) medium (middle) and high (right) dimensional problems "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "286 4.2 Comparisons with other methods ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "The default configuration for CobBO is specified in the supplementary materials. CobBO performs on par or outperforms a collection of state-of-the-art methods across the following experiments. Most of the experiments are conducted using the same settings as in TurBO [14], where it is compared with a comprehensive list of baselines, including BFGS, BOCK [47], BOHAMIANN, CMA-ES [22], BOBYQA, EBO [61], GP-TS, HeSBO [44], Nelder-Mead and random search. To avoid repetitions, we only show TuRBO and CMA-ES that achieve the best performance among this list, and additionally compare CobBO with BADS [1], REMBO [63], Differetial Evolution (Diff-Evo) [56], Tree Parzen Estimator (TPE) [4] and Adaptive TPE (ATPE) [12]. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2.1 Low dimensional tests ",
+        "text_level": 1,
+        "bbox": [
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+            92,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To evaluate CobBO on low dimensional problems, we use the lunar landing [38, 14] and robot pushing [62], by following the setup in [14]. Confidence intervals $( 9 5 \\% )$ over 30 independent experiments for each problem are shown in Fig. 5. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Lunar landing (maximization): This controller learning problem (12 dimensions) is provided by the OpenAI gym [38] and evaluated in [14]. Each algorithm has 50 initial points and a budget of 1, 500 trials. TuRBO is configured with 5 trust regions and a batch size of 50 as in [14]. Fig. 5 (upper left) shows that, among the 30 independent tests, CobBO quickly exceeds 300 along some good sample paths. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Robot pushing (maximization): This control problem (14 dimensions) is introduced in [62] and extensively tested in [14]. We follow the setting in [14], where TuRBO is configured with a batch size of 50 and 15 trust regions, each of which has 30 initial points. Each experiment has a budget of 10, 000 evaluations. On average CobBO exceeds 10 within 5500 trials, as shown in Fig. 5 (lower left). ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.2.2 High dimensional tests ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            315,
+            383,
+            330
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Since the duration of each experiment in this section is long, confidence intervals $( 9 5 \\% )$ ) over repeated 10 independent experiments for each problem are presented. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Additive latent structure (minimization): As mentioned in Section 2, additive latent structures have been exploited in high dimensions. We construct an additive function of 56 dimensions, defined as $f _ { 5 6 } ( x ) \\ = \\ \\mathrm { { A c k l e y } ( x _ { 1 } ) + \\mathrm { { L e v y } ( x _ { 2 } ) + \\mathrm { { R a s t r i g i n } ( x _ { 3 } ) + \\mathrm { { H a r t m a n n } ( x _ { 4 } ) + \\mathrm { { R o s e n b r o c k } ( x _ { 5 } ) + \\mathrm { { A c k l e y } ( x _ { 6 } ) + \\mathrm { { A c k l e y } ( x _ { 7 } ) } } } } } } } $ Schwefel $\\left( \\mathrm { x } _ { 6 } \\right)$ , where the first three terms express the exact functions and domains described in Section 4.2.1, the Hartmann function on $[ 0 , \\bar { 1 } ] ^ { 6 }$ and the Rosenbrock and Schwefel functions on $[ - 5 , 1 0 ] ^ { 1 0 }$ and $[ - 5 0 0 , 5 0 0 ] ^ { 1 0 }$ , respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We compare CobBO with TPE, BADS, CMA-ES and TuRBO, each with 100 initial points. Specifically, TuRBO is configured with 15 trust regions and a batch size 100. ATPE is excluded as it takes more than 24 hours per run to finish. The results are shown in Fig. 5 (upper middle), where CobBO quickly finds the best solution among the algorithms tested. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Rover trajectory planning (maximization): This problem (60 dimensions) is introduced in [62]. The objective is to find a collision-avoiding trajectory of a sequence consisting of 30 positions in a 2-D plane. We compare CobBO with TuRBO, TPE and CMA-ES, each with a budget of 20, 000 evaluations and 200 initial points. TuRBO is configured with 15 trust regions and a batch size of 100, as in [14]. ATPE, BADS and REMBO are excluded for this problem and the following ones, as they all take more than 24 hours per run. Fig. 5 (lower middle) shows that CobBO has a good performance. ",
+        "bbox": [
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+        "page_idx": 8
+    },
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+        "text": "The 200-dimensional Levy and Ackley functions (minimization): We minimize the Levy and Ackley functions over $[ - 5 , \\mathsf { \\bar { 1 0 } } ] ^ { 2 0 0 }$ with 500 initial points. TuRBO is configured with 15 trust regions and a batch size of 100. These two problems are challenging and have no redundant dimensions. For Levy, in Fig. 5 (upper right), CobBO reaches 100.0 within 2, 000 trials, while CMA-ES and TuRBO obtain 200.0 after 8, 000 trials. TPE cannot find a comparable solution within 10, 000 trials in this case. For Ackley, in Fig. 5 (lower right), CobBO reaches the best solution among all of the algorithms tested. ",
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+        "text": "Regarding running times, for Ackley, CobBO runs for 12.8 CPU hours and TuRBO-1 run for more than 80 CPU hours or $9 . 6 \\ : G P U$ hours. Most other methods either cannot make any progress or find far worse solutions. ",
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+        "text": "CobBO is a variant of coordinate ascent tailored for Bayesian optimization with a stopping rule to switch coordinate subspaces. The sampling policy of subspaces is proven to have tight regret bounds with respect to the best subspace in hindsight. Combining this projection on random subspaces with a two-stage kernels for function value interpolation and GP regression, we provide a practical Bayesian optimization method of affordable computational costs in high dimensions. Empirically, CobBO consistently finds comparable or better solutions with reduced trial complexity in comparison with the state-of-the-art methods across a variety of benchmarks. ",
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+        "text": "348 [1] Luigi Acerbi and Wei Ji Ma. Practical bayesian optimization for model fitting with bayesian   \n349 adaptive direct search. In Proceedings of the 31st International Conference on Neural Infor  \n350 mation Processing Systems, NIPS’17, page 1834–1844, Red Hook, NY, USA, 2017. Curran   \n351 Associates Inc.   \n352 [2] Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a   \n353 meta-algorithm and applications. Theory of Computing, 8(6):121–164, 2012.   \n354 [3] Peter Auer. Using confidence bounds for exploitation-exploration trade-offs. J. Mach. Learn.   \n355 Res., 3(null):397–422, Mar. 2003.   \n356 [4] James S. Bergstra, Rémi Bardenet, Yoshua Bengio, and Balázs Kégl. Algorithms for hyper  \n357 parameter optimization. In J. Shawe-Taylor, R. S. Zemel, P. L. Bartlett, F. Pereira, and K. Q.   \n358 Weinberger, editors, Advances in Neural Information Processing Systems 24, pages 2546–2554.   \n359 Curran Associates, Inc., 2011.   \n360 [5] Mickaël Binois, David Ginsbourger, and Olivier Roustant. On the choice of the low-dimensional   \n361 domain for global optimization via random embeddings. Journal of Global Optimization,   \n362 76(1):69–90, January 2020.   \n363 [6] Martin D. Buhmann. Radial Basis Functions: Theory and Implementations. Cambridge   \n364 Monographs on Applied and Computational Mathematics. Cambridge University Press, 2003.   \n365 [7] Martin D. Buhmann and M. D. Buhmann. Radial Basis Functions. Cambridge University Press,   \n366 USA, 2003.   \n367 [8] Adam D. Bull. Convergence rates of efficient global optimization algorithms. Journal of   \n368 Machine Learning Research, 12(88):2879–2904, 2011.   \n369 [9] Roberto Calandra, André Seyfarth, Jan Peters, and Marc Peter Deisenroth. Bayesian optimiza  \n370 tion for learning gaits under uncertainty. 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+        "text": "As stated in [32], Bayesian optimization is a powerful optimization technique used in a wide range of industries and applications, such as robotics [37, 9, 52], internet tech companies [18, 34], designing novel molecules for pharmaceutics [19], material design for increasing efficiency of solar cells [71], and aerospace engineering [31]. All of these settings have high-dimensional optimization problems, and advances in BO will reflect on improved capabilities on these fields as well. We have fully opensourced our code for CobBO using the MIT license to be available for researchers and practitioners in these fields, and many others. The ability to optimize a larger number of parameters than has previously been possible will bring further improvements to and further accelerate work in these areas. ",
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parse/train/H1eCw3EKvH/H1eCw3EKvH.md

@@ -0,0 +1,354 @@
+# ON THE WEAKNESSES OF REINFORCEMENT LEARNING FOR NEURAL MACHINE TRANSLATION
+
+Leshem Choshen1, Lior $\mathbf { F o x } ^ { 2 }$ , Zohar Aizenbud1, Omri Abend1,3
+
+1 School of Computer Science and Engineering, 2 The Edmond and Lily Safra Center for Brain Sciences   
+3 Department of Cognitive Sciences   
+The Hebrew University of Jerusalem   
+first.last@mail.huji.ac.il, oabend@cs.huji.ac.il
+
+# ABSTRACT
+
+Reinforcement learning (RL) is frequently used to increase performance in text generation tasks, including machine translation (MT), notably through the use of Minimum Risk Training (MRT) and Generative Adversarial Networks (GAN). However, little is known about what and how these methods learn in the context of MT. We prove that one of the most common RL methods for MT does not optimize the expected reward, as well as show that other methods take an infeasibly long time to converge. In fact, our results suggest that RL practices in MT are likely to improve performance only where the pre-trained parameters are already close to yielding the correct translation. Our findings further suggest that observed gains may be due to effects unrelated to the training signal, concretely, changes in the shape of the distribution curve.
+
+# 1 INTRODUCTION
+
+Reinforcement learning (RL) is an appealing path for advancement in Machine Translation (MT), as it allows training systems to optimize non-differentiable score functions, common in MT evaluation, as well as tackling the “exposure bias” (Ranzato et al., 2015) in standard training, namely that the model is not exposed during training to incorrectly generated tokens, and is thus unlikely to recover from generating such tokens at test time. These motivations have led to much interest in RL for text generation in general and MT in particular (see $\ S 2$ ). Various policy gradient methods have been used, notably REINFORCE (Williams, 1992) and variants thereof (e.g., Ranzato et al., 2015; Edunov et al., 2018) and Minimum Risk Training (MRT; e.g., Och, 2003; Shen et al., 2016). Another popular use of RL is for training GANs (Yang et al., 2018; Tevet et al., 2018). Nevertheless, despite increasing interest and strong results, little is known about what accounts for these performance gains, and the training dynamics involved.
+
+We present the following contributions. First, our theoretical analysis shows that commonly used approximation methods are theoretically ill-founded, and may converge to parameter values that do not minimize the risk, nor are local minima thereof (§2.2).
+
+Second, using both naturalistic experiments and carefully constructed simulations, we show that performance gains observed in the literature likely stem not from making target tokens the most probable, but from unrelated effects, such as increasing the peakiness of the output distribution (i.e., the probability mass of the most probable tokens). We do so by comparing a setting where the reward is informative, vs. one where it is constant. In $\ S 4$ we discuss this peakiness effect (PKE).
+
+Third, we show that promoting the target token to be the mode is likely to take a prohibitively long time. The only case we find, where improvements are likely, is where the target token is among the first 2-3 most probable tokens according to the pretrained model. These findings suggest that REINFORCE (§5) and CMRT $( \ S 6 )$ are likely to improve over the pre-trained model only under the best possible conditions, i.e., where the pre-trained model is “nearly” correct.
+
+We conclude by discussing other RL practices in MT which should be avoided for practical and theoretical reasons, and briefly discuss alternative RL approaches that will allow RL to tackle a larger class of errors in pre-trained models (§7).
+
+# 2 RL IN MACHINE TRANSLATION
+
+An MT system generates tokens $y = ( y _ { 1 } , . . . , y _ { n } )$ from a vocabulary $V$ one token at a time. The probability of generating $y _ { i }$ given preceding tokens $y _ { < i }$ is given by $P _ { \theta } ( \cdot | x , y _ { < i } )$ , where $x$ is the source sentence and $\theta$ are the model parameters. For each generated token $y _ { i }$ , we denote with $r ( y _ { i } ; y _ { < i } , x , y ^ { ( r e f ) } )$ the score, or reward, for generating $y _ { i }$ given $y _ { < i } , x$ , and the reference sentence $y ^ { ( r e f ) }$ . For brevity, we omit parameters where they are fixed within context. For simplicity, we assume $r$ does not depend on following tokens $y _ { > i }$ .
+
+We also assume there is exactly one valid target token, as de facto, training is done against a single reference (Schulz et al., 2018). In practice, either a token-level reward is approximated using MonteCarlo methods (e.g., Yang et al., 2018), or a sentence-level (sparse) reward is given at the end of the episode (sentence). The latter is equivalent to a uniform token-level reward.
+
+$r$ is often the negative log-likelihood, or a standard MT metric, e.g., BLEU (Papineni et al., 2002). RL’s goal is to maximize the expected episode reward (denoted with $R$ ); i.e., to find
+
+$$
+\theta ^ { * } = \arg \operatorname* { m a x } _ { \theta } \ R ( \theta ) = \arg \operatorname* { m a x } _ { \theta } \ \mathbb { E } _ { y \sim P _ { \theta } } [ r ( y ) ]
+$$
+
+# 2.1 REINFORCE
+
+For a given source sentence, and past predictions $y _ { < i }$ , REINFORCE (Williams, 1992) samples $k$ tokens $k$ is a hyperparameter) $S = \bar { ( { y ^ { ( 1 ) } } , . . . , y ^ { ( k ) } ) }$ from $P _ { \theta }$ and updates $\theta$ according to this rule:
+
+$$
+\Delta \theta \propto \frac { 1 } { k } \sum _ { i = 1 } ^ { k } r ( y _ { i } ) \nabla \log ( P _ { \theta } ( y _ { i } ) )
+$$
+
+The right-hand side of equation 2 is an unbiased estimator of the gradient of the objective function, i.e., $\mathbb { E } \left[ \Delta \theta \right] \propto \nabla _ { \theta } R \left( \theta \right)$ . Therefore, REINFORCE is performing a form of stochastic gradient ascent on $R$ , and has similar formal guarantees. From here follows that if $R$ is constant with respect to $\theta$ , then the expected $\Delta \theta$ prescribed by REINFORCE is zero. We note that $r$ may be shifted by a constant term (called a “baseline”), without affecting the optimal value for $\theta$ .
+
+REINFORCE is used in MT, text generation, and image-to-text tasks (Liu et al., 2016; Wu et al., 2018; Rennie et al., 2017; Shetty et al., 2017; Hendricks et al., 2016) – in isolation, or as a part of training (Ranzato et al., 2015). Lately, an especially prominent use for REINFORCE is adversarial training with discrete data, where another network predicts the reward (GAN). For some recent work on RL for NMT, see (Zhang et al., 2016; Li et al., 2017; Wu et al., 2017; Yu et al., 2017; Yang et al., 2018).
+
+# 2.2 MINIMUM RISK TRAINING
+
+The term Minimum Risk Training (MRT) is used ambiguously in MT to refer either to the application of REINFORCE to minimizing the risk (equivalently, to maximizing the expected reward, the negative loss), or more commonly to a somewhat different estimation method, which we term Contrastive MRT (CMRT) and turn now to analyzing. CMRT was proposed by Och (2003), adapted to NMT by Shen et al. (2016), and often used since (Ayana et al., 2016; Neubig, 2016; Shen et al., 2017; Edunov et al., 2018; Makarov & Clematide, 2018; Neubig et al., 2018).
+
+The method works as follows: at each iteration, sample $k$ tokens $S = \{ y _ { 1 } , \dots , y _ { k } \}$ from $P _ { \theta }$ , and update $\theta$ according to the gradient of
+
+$$
+\widetilde { R } ( \theta , S ) = \sum _ { i = 1 } ^ { k } Q _ { \theta , S } ( y _ { i } ) r ( y _ { i } ) = \mathbb { E } _ { y \sim Q } \left[ r ( y ) \right]
+$$
+
+where
+
+$$
+Q _ { \theta , S } ( y _ { i } ) = \frac { P ( y _ { i } ) ^ { \alpha } } { \sum _ { y _ { j } \in S } P ( y _ { j } ) ^ { \alpha } }
+$$
+
+Commonly (but not universally), deduplication is performed, so $\widetilde { R }$ sums over a set of unique values (Sennrich et al., 2017). This changes little in our empirical results and theoretical analysis.
+
+Despite the resemblance in definitions of $R$ (equation 1) and $\widetilde { R }$ (indeed, $\widetilde { R }$ is sometimes presented as an approximation of $R$ ), they differ in two important aspects. First, $Q$ ’s support is $S$ , so increasing
+
+$Q ( y _ { i } )$ for some $y _ { i }$ necessarily comes at the expense of $Q ( y )$ for some $y \in S$ . In contrast, increasing $P ( y _ { i } )$ , as in REINFORCE, may come at the expense of $P ( y )$ for any $y \in V$ . Second, $\alpha$ is a smoothness parameter: the closer $\alpha$ is to 0, the closer $Q$ is to be uniform.
+
+We show in Appendix A.1 that despite its name, CMRT does not optimize $R$ , nor does it optimize $\mathbb { E } [ { \widetilde { R } } ]$ . That is, it may well converge to values that are not local maxima of $R$ , making it theoretically ill-founded.1 However, given CMRT popularity, the strong results it yielded and the absence of theory for explaining it, we discuss it here. Given a sample $S$ , the gradient of $\widetilde { R }$ is given by
+
+$$
+\nabla \widetilde { R } = \alpha \sum _ { i = 1 } ^ { k } \Big ( Q ( y _ { i } ) \cdot r ( y _ { i } ) \cdot \nabla \log P ( y _ { i } ) \Big ) - \mathbb { E } _ { Q } [ r ] \nabla \log Z ( S )
+$$
+
+where $\begin{array} { r } { Z ( S ) = \sum _ { i } P ( y _ { i } ) ^ { \alpha } } \end{array}$ . See Appendix A.2.
+
+Comparing Equations 2 and 3, the differences between REINFORCE and CMRT are reflected again. First, $\nabla \widetilde R$ has an additional term, proportional to $\nabla \log Z ( S )$ , which yields the contrastive effect. This contrast may improve the rate of convergence since it counters the decrease of probability mass for non-sampled tokens.
+
+Second, given $S$ , the relative weighting of the gradients $\nabla \log P ( y _ { i } )$ is proportional to $r ( y _ { i } ) Q ( y _ { i } )$ , or equivalently to $r ( y _ { i } ) P ( y _ { i } ) ^ { \alpha }$ . CMRT with deduplication sums over distinct values in $S$ (equation 3), while REINFORCE sums over all values. This means that the relative weight of the unique value $y _ { i }$ is $\frac { r ( y _ { i } ) | \{ y _ { i } \in S \} | } { k }$ in REINFORCE. For $\alpha = 1$ the expected value of these relative weights is the same, and so for $\alpha < 1$ (as is commonly used), more weight is given to improbable tokens, which could also have a positive effect on the convergence rate.2 However, if $\alpha$ is too close to 0, $\nabla \widetilde R$ vanishes, as it is not affected by $\theta$ . This tradeoff explains the importance of tuning $\alpha$ reported in the literature. In $\ S 6$ we present simulations with CMRT, showing very similar trends as presented by REINFORCE.
+
+# 3 MOTIVATING DISCUSSION
+
+Implementing a stochastic gradient ascent, REINFORCE is guaranteed to converge to a stationary point of $R$ under broad conditions. However, not much is known about its convergence rate under the prevailing conditions in NMT.
+
+We begin with a qualitative, motivating analysis of these questions. As work on language generation empirically showed, RNNs quickly learn to output very peaky distributions (Press et al., 2017). This tendency is advantageous for generating fluent sentences with high probability, but may also entail slower convergence rates when using RL to fine-tune the model, because RL methods used in text generation sample from the (pretrained) policy distribution, which means they mostly sample what the pretrained model deems to be likely. Since the pretrained model (or policy) is peaky, exploration of other potentially more rewarding tokens will be limited, hampering convergence.
+
+Intuitively, REINFORCE increases the probabilities of successful (positively rewarding) observations, weighing updates by how rewarding they were. When sampling a handful of tokens in each context (source sentence $x$ and generated prefix $y _ { < i , }$ ), and where the number of epochs is not large, it is unlikely that more than a few unique tokens will be sampled from $P _ { \theta } ( \cdot | x , y _ { < i } )$ . (In practice, $k$ is typically between 1 and 20, and the number of epochs between 1 and 100.) It is thus unlikely that anything but the initially most probable candidates will be observed. Consequently, REINFORCE initially raises their probabilities, even if more rewarding tokens can be found down the list.
+
+We thus hypothesize the peakiness of the distribution, i.e., the probability mass allocated to the most probable tokens, will increase, at least in the first phase. We call this the peakiness-effect (PKE), and show it occurs both in simulations (§4.1) and in full-scale NMT experiments (§4.2).
+
+With more iterations, the most-rewarding tokens will be eventually sampled, and gradually gain probability mass. This discussion suggests that training will be extremely sample-inefficient. We assess the rate of convergence empirically in $\ S 5$ , finding this to be indeed the case.
+
+![](images/a8e0772d8762f399cf44fa7afc438edb7e4aada745a8847bb2b5a17ee8ef2ab1.jpg)  
+Figure 1: A histogram of the update size (x-axis) to the total predicted probability of the 10 most probable tokens (left) or the most probable token (right) in the Constant Reward setting. An update is overwhelmingly more probable to increase this probability than to decrease it.
+
+# 4 THE PEAKINESS EFFECT
+
+We turn to demonstrate that the initially most probable tokens will initially gain probability mass, even if they are not the most rewarding, yielding a PKE.
+
+Caccia et al. (2018) recently observed in the context of language modeling using GANs that performance gains similar to those GAN yield can be achieved by decreasing the temperature for the prediction softmax (i.e., making it peakier). However, they proposed no causes for this effect. Our findings propose an underlying mechanism leading to this trend. We return to this point in $^ { \ S 7 }$ . Furthermore, given their findings, it is reasonable to assume that our results are relevant for RL use in other generation tasks, whose output space too is discrete, high-dimensional and concentrated.
+
+# 4.1 CONTROLLED SIMULATIONS
+
+We experiment with a 1-layer softmax model, that predicts a single token $i \in V$ with probability $\frac { e ^ { \theta _ { i } } } { \sum _ { j } e ^ { \theta _ { j } } }$ . $\theta = \{ \theta _ { j } \} _ { j \in V }$ are the model’s parameters. This model simulates the top of any MT decoder that ends with a softmax layer, as essentially all NMT decoders do. To make experiments realistic, we use similar parameters as those reported in the influential Transformer NMT system (Vaswani et al., 2017). Specifically, the size of $V$ (distinct BPE tokens) is 30,715, and the initial values for $\theta$ were sampled from 1,000 sets of logits taken from decoding the standard newstest2013 development set, using a pretrained Transformer model. The model was pretrained on WMT2015 training data (Bojar et al., 2015). Hyperparameters are reported in Appendix A.3. We define one of the tokens in $V$ to be the target token and denote it with $y _ { b e s t }$ . We assign deterministic token reward, this makes learning easier than when relying on approximations and our predictions optimistic. We experiment with two reward functions:
+
+1. Simulated Reward: $r ( y ) = 2$ for $y = y _ { b e s t }$ , $r ( y ) = 1$ if $y$ is one of the 10 initially highest scoring tokens, and $r ( y ) = 0$ otherwise. This simulates a condition where the pretrained model is of decent but sub-optimal quality. $r$ here is at the scale of popular rewards used in MT, such as GAN-based rewards or BLEU (which are between 0 and 1). 2. Constant Reward: $r$ is constantly equal to 1, for all tokens. This setting is aimed to confirm that PKE is not a result of the signal carried by the reward.
+
+Experiments with the first setting were run 100 times, each time for 50K steps, updating $\theta$ after each step. With the second setting, it is sufficient to take a single step at a time, as the expected update after each step is zero, and so any $\mathrm { P K E }$ seen in a single step is only accentuated in the next. It is, therefore, more telling to run more repetitions rather than more steps per initialization. We, therefore, sample 10,000 pretrained distributions, and perform a single REINFORCE step.
+
+As RL training lasts about 30 epochs before stopping, samples about $1 0 0 \mathrm { K }$ tokens per epoch, and as the network already predicts $y _ { b e s t }$ in about two thirds of the contexts,3 we estimate the number of steps used in practice to be in the order of magnitude of 1M. For visual clarity, we present figures for 50K-100K steps. However, full experiments (with 1M steps) exhibit similar trends: where REINFORCE was not close to converging after 50K steps, the same was true after 1M steps.
+
+We evaluate the peakiness of a distribution in terms of the probability of the most probable token (the mode), the total probability of the ten most probable tokens, and the entropy of the distribution (lower entropy indicates more peakiness).
+
+![](images/80b2427a869afcc4d92132462366b3e75868a834bc4126520558d3941359d3a7.jpg)  
+Figure 2: Token probabilities through REINFORCE training, in the controlled simulations in the Simulated Reward setting. The left/center/right figures correspond to simulations where the target token $( y _ { b e s t } )$ was initially the second/third/fourth most probable token. The green line corresponds to the target token, yellow lines to medium-reward tokens and red lines to no-reward tokens.
+
+Results. The distributions become peakier in terms of all three measures: on average, the mode’s probability and the 10 most probable tokens increases, and the entropy decreases. Figure 1a presents the histogram of the update size, the difference in the probability of the $1 0 \mathrm { m o s t }$ probable tokens in the Constant Reward setting, after a single step. Figure 1b depicts similar statistics for the mode. The average entropy in the pretrained model is 2.9 is reduced to 2.85 after one REINFORCE step.
+
+Simulated Reward setting shows similar trends. For example, entropy decreases from 3 to about 0.001 in 100K steps. This extreme decrease suggests it is effectively a deterministic policy. $\mathrm { P K E }$ is achieved in a few hundred steps, usually before other effects become prominent (see Figure 2), and is stronger than for Constant Reward.
+
+# 4.2 NMT EXPERIMENTS
+
+We turn to analyzing a real-world application of REINFORCE to NMT. Important differences between this and the previous simulations are: (1) it is rare in NMT for REINFORCE to sample from the same conditional distribution more than a handful of times, given the number of source sentences $x$ and sentence prefixes $y _ { < i }$ (contexts); and (2) in NMT $P _ { \theta } ( \cdot | x , y _ { < i } )$ shares parameters between contexts, which means that updating $P _ { \theta }$ for one context may influence $P _ { \theta }$ for another.
+
+We follow the same pretraining as in $\ S 4 . 1$ . We then follow Yang et al. (2018) in defining the reward function based on the expected BLEU score. Expected BLEU is computed by sampling suffixes for the sentence, and averaging the BLEU score of the sampled sentences against the reference.
+
+We use early stopping with a patience of 10 epochs, where each epoch consists of 5,000 sentences sampled from the WMT2015 (Bojar et al., 2015) German-English training data. We use $k = 1$ . We retuned the learning-rate, and positive baseline settings against the development set. Other hyper-parameters were an exact replication of the experiments reported in (Yang et al., 2018).
+
+![](images/e97b7eecb797bb490493534fc6b6cb4cd578b2f2ee48ba45bcccf79a3a2e0728.jpg)  
+Figure 3: The cumulative distribution of the probability of the most likely token in the NMT experiments. The green distribution corresponds to the pretrained model, and the blue corresponds to the reinforced model. The yaxis is the proportion of conditional probabilities with a mode of value $\leq x$ (the $\mathbf { X }$ -axis). Note that a lower cumulative percentage means a more peaked output distribution. A lower cumulative percentage means a more peaked output distribution.
+
+Results. Results indicate an increase in the peakiness of the conditional distributions. Our results are based on a sample of 1,000 contexts from the pretrained model, and another (independent) sample from the reinforced model.
+
+The modes of the conditional distributions tend to increase. Figure 3 presents the distribution of the modes’ probability in the reinforced conditional distributions compared with the pretrained model, showing a shift of probability mass towards higher probabilities for the mode, following RL. Another indication of the increased peakiness is the decrease in the average entropy of $P _ { \theta }$ , which was reduced from 3.45 in the pretrained model to an average of 2.82 following RL. This more modest reduction in entropy (compared to $\ S 4 . 1$ ) might also suggest that the procedure did not converge to the optimal value for $\theta$ , as then we would have expected the entropy to substantially drop if not to 0 (overfit), then to the average entropy of valid next tokens (given the source and a prefix of the sentence).
+
+# 5 PERFORMANCE FOLLOWING REINFORCE
+
+We now turn to assessing under what conditions it is likely that REINFORCE will lead to an improvement in the performance of an NMT system. As in the previous section, we use both controlled simulations and NMT experiments.
+
+# 5.1 CONTROLLED SIMULATIONS
+
+We use the same model and experimental setup described in Section 4.1, this time only exploring the Simulated Reward setting, as a Constant Reward is not expected to converge to any meaningful $\theta$ . Results are averaged over 100 conditional distributions sampled from the pretrained model.
+
+Caution should be exercised when determining the learning rate (LR). Common LRs used in the NMT literature are of the scale of $1 0 ^ { - 4 }$ . However, in our simulations, no LR smaller than 0.1 yielded any improvement in $R$ . We thus set the LR to be 0.1. We note that in our simulations, a higher learning rate means faster convergence as our reward is noise-free: it is always highest for the best option. In practice, increasing the learning rate may deteriorate results, as it may cause the system to overfit to the sampled instances. Indeed, when increasing the learning rate in our NMT experiments (see below) by an order of magnitude, early stopping caused the RL procedure to stop without any parameter updates.
+
+Figure 2 shows the change in $P _ { \theta }$ over the first 50K REINFORCE steps (probabilities are averaged over 100 repetitions), for a case where $y _ { b e s t }$ was initially the second, third and fourth most probable. Although these are the easiest settings, and despite the high learning rate, it fails to make $y _ { b e s t }$ the mode of the distribution within $1 0 0 \mathrm { K }$ steps, unless $y _ { b e s t }$ was initially the second most probable. In cases where $y _ { b e s t }$ is initially of a lower rank than four, it is hard to see any increase in its probability, even after 1M steps.
+
+![](images/e7ee8c44bf8457067841e284920b801a1f4cbe676f0f9f4bfe4fcc3b96ffaf27.jpg)  
+Figure 4: Cumulative percentage of contexts where the pretrained model ranks $y _ { b e s t }$ in rank $_ x$ or below and where it does not rank $y _ { b e s t }$ first $( x = 0$ ). In about half the cases it is ranked fourth or below.
+
+# 5.2 NMT EXPERIMENTS
+
+We trained an NMT system, using the same procedure as in Section 4.2, and report BLEU scores over the news2014 test set. After training with an expected BLEU reward, we indeed see a minor improvement which is consistent between trials and pretrained models. While the pretrain BLEU score is 30.31, the reinforced one is 30.73.
+
+Analyzing what words were influenced by the RL procedure, we begin by computing the cumulative probability of the target token $y _ { b e s t }$ to be ranked lower than a given rank according to the pretrained model. Results (Figure 4) show that in about half of the cases, $y _ { b e s t }$ is not among the top three choices of the pretrained model, and we thus expect it not to gain substantial probability following REINFORCE, according to our simulations.
+
+We next turn to compare the ranks the reinforced model assigns to the target tokens, and their ranks according to the pretrained model. Figure 6 presents the difference in the probability that $y _ { b e s t }$ is ranked at a given rank following RL and the probability it is ranked there initially. Results indicate that indeed more target tokens are ranked first, and less second, but little consistent shift of probability mass occurs otherwise across the ten first ranks. It is possible that RL has managed to push $y _ { b e s t }$ in some cases between very low ranks $( < 1 , 0 0 0 )$ to medium-low ranks (between 10 and 1,000). However, token probabilities in these ranks are so low that it is unlikely to affect the system outputs in any way. This fits well with the results of our simulations that predicted that only the initially top-ranked tokens are likely to change.
+
+In an attempt to explain the improved BLEU score following RL with PKE, we repeat the NMT experiment this time using a constant reward of 1. Our results present a nearly identical improvement in BLEU, achieving 30.72, and a similar pattern in the change of the target tokens’ ranks (see Appendix 8). Therefore, there is room to suspect that even in cases where RL yields an improvement in BLEU, it may partially result from reward-independent factors, such as PKE.4
+
+![](images/5dcb22db4003b6d8b1e130bb7b8a032341116c25c187d78aba1343df058a0aad.jpg)  
+Figure 5: The probability of different tokens following CMRT, in the controlled simulations in the Simulated Reward setting. The left/right figures correspond to simulations where the target token $( y _ { b e s t } )$ was initially the second/third most probable token. The green line corresponds to the target token, yellow lines to medium-reward tokens and red lines to tokens with $r ( y ) = 0$ .
+
+# 6 EXPERIMENTS WITH CONTRASTIVE MRT
+
+In $\ S 2 . 2$ we showed that CMRT does not, in fact, maximize $R$ , and so does not enjoy the same theoretical guarantees as REINFORCE and similar policy gradient methods. However, being the RL procedure of choice in much recent work we repeat the simulations described in $\ S 4$ and $\ S 5$ , assessing CMRT’s performance in these conditions. We experiment with $\alpha = 0 . 0 0 5$ and $k = 2 0$ , common settings in the literature, and average over 100 trials.
+
+Figure 5 shows how the distribution $P _ { \theta }$ changes over the course of 50K update steps to $\theta$ , where $y _ { b e s t }$ is taken to be the second and third initially most probable token (Simulated Reward setting). Results are similar in trends to those obtained with REINFORCE: MRT succeeds in pushing $y _ { b e s t }$ to be the highest ranked token if it was initially second, but struggles where it was initially ranked third or below. We only observe a small $\mathrm { P K E }$ in MRT. This is probably due to the contrastive effect, which means that tokens that were not sampled do not lose probability mass.
+
+All graphs we present here allow sampling the same token more than once in each batch (i.e., $S$ is a sample with replacements). Simulations with deduplication show similar results.
+
+![](images/03bf2a5a7bc9cbd3fc79305ae00d3e2c99452fdba0637e6445dc57ebc0b7cda2.jpg)  
+Figure 6: Difference between the ranks of $y _ { b e s t }$ in the reinforced and the pretrained model. Each column $_ x$ corresponds to the difference in the probability that $y _ { b e s t }$ is ranked in rank $x$ in the reinforced model, and the same probability in the pretrained model.
+
+# 7 DISCUSSION
+
+In this paper, we showed that the type of distributions used in NMT entail that promoting the target token to be the mode is likely to take a prohibitively long times for existing RL practices, except under the best conditions (where the pretrained model is “nearly” correct). This leads us to conclude that observed improvements from using RL for NMT are likely due either to fine-tuning the most probable tokens in the pretrained model (an effect which may be more easily achieved using reranking methods, and uses but little of the power of RL methods), or to effects unrelated to the signal carried by the reward, such as PKE. Another contribution of this paper is in showing that CMRT does not optimize the expected reward and is thus theoretically unmotivated.
+
+A number of reasons lead us to believe that in our NMT experiments, improvements are not due to the reward function, but to artefacts such as PKE. First, reducing a constant baseline from $r$ , so as to make the expected reward zero, disallows learning. This is surprising, as REINFORCE, generally and in our simulations, converges faster where the reward is centered around zero, and so the fact that this procedure here disallows learning hints that other factors are in play. As PKE can be observed even where the reward is constant (if the expected reward is positive; see $\ S 4 . 1 \ r _ { \cdot }$ ), this suggests PKE may play a role here. Second, we observe more peakiness in the reinforced model and in such cases, we expect improvements in BLEU (Caccia et al., 2018). Third, we achieve similar results with a constant reward in our NMT experiments (§5.2). Fourth, our controlled simulations show that asymptotic convergence is not reached in any but the easiest conditions (§5.1).
+
+Our analysis further suggests that gradient clipping, sometimes used in NMT (Zhang et al., 2016; Wieting et al., 2019), is expected to hinder convergence further. It should be avoided when using REINFORCE as it violates REINFORCE’s assumptions.
+
+The per-token sampling as done in our experiments is more exploratory than beam search (Wu et al., 2018), reducing PKE. Furthermore, the latter does not sample from the behavior policy, but does not properly account for being off-policy in the parameter updates.
+
+Adding the reference to the sample $S$ , which some implementations allow (Sennrich et al., 2017) may help reduce the problems of never sampling the target tokens. However, as Edunov et al. (2018) point out, this practice may lower results, as it may destabilize training by leading the model to improve over outputs it cannot generalize over, as they are very different from anything the model assigns a high probability to, at the cost of other outputs.
+
+# 8 CONCLUSION
+
+The standard MT scenario poses several uncommon challenges for RL. First, the action space in MT problems is a high-dimensional discrete space (generally in the size of the vocabulary of the target language or the product thereof for sentences). This contrasts with the more common scenario studied by contemporary RL methods, which focuses mostly on much smaller discrete action spaces (e.g., video games (Mnih et al., 2015; 2016)), or continuous action spaces of relatively low dimensions (e.g., simulation of robotic control tasks (Lillicrap et al., 2015)). Second, reward for MT is naturally very sparse – almost all possible sentences are “wrong” (hence, not rewarding) in a given context. Finally, it is common in MT to use RL for tuning a pretrained model. Using a pretrained model ameliorates the last problem. But then, these pretrained models are in general quite peaky, and because training is done on-policy – that is, actions are being sampled from the same model being optimized – exploration is inherently limited.
+
+Here we argued that, taken together, these challenges result in significant weaknesses for current RL practices for NMT, that may ultimately prevent them from being truly useful. At least some of these challenges have been widely studied in the RL literature, with numerous techniques developed to address them, but were not yet adopted in NLP. We turn to discuss some of them.
+
+Off-policy methods, in which observations are sampled from a different policy than the one being currently optimized, are prominent in RL (Watkins & Dayan, 1992; Sutton & Barto, 1998), and were also studied in the context of policy gradient methods (Degris et al., 2012; Silver et al., 2014). In principle, such methods allow learning from a more “exploratory” policy. Moreover, a key motivation for using $\alpha$ in CMRT is smoothing; off-policy sampling allows smoothing while keeping convergence guarantees.
+
+In its basic form, exploration in REINFORCE relies on stochasticity in the action-selection (in MT, this is due to sampling). More sophisticated exploration methods have been extensively studied, for example using measures for the exploratory usefulness of states or actions (Fox et al., 2018), or relying on parameter-space noise rather than action-space noise (Plappert et al., 2017).
+
+For MT, an additional challenge is that even effective exploration (sampling diverse sets of observations), may not be enough, since the state-action space is too large to be effectively covered, with almost all sentences being not rewarding. Recently, diversity-based and multi-goal methods for RL were proposed to tackle similar challenges (Andrychowicz et al., 2017; Ghosh et al., 2018; Eysenbach et al., 2019). We believe the adoption of such methods is a promising path forward for the application of RL in NLP.
+
+# 9 ACKNOWLEDGMENTS
+
+This work was supported by the Israel Science Foundation (grant no. 929/17) and by the HUJI Cyber Security Research Center in conjunction with the Israel National Cyber Bureau in the Prime Minister’s Office.
+
+# REFERENCES
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+
+# A APPENDIX
+
+A.1 CONTRASTIVE MRT DOES NOT MAXIMIZE THE EXPECTED REWARD
+
+We hereby detail a simple example where following the Contrastive MRT method (see $\ S 2 . 2$ ) does not converge to the parameter value that maximizes $R$ .
+
+Let $\theta$ be a real number in $[ 0 , 0 . 5 ]$ , and let $P _ { \theta }$ be a family of distributions over three values $a , b , c$ such that:
+
+$$
+P _ { \theta } ( x ) = { \left\{ \begin{array} { l l } { \theta } & { x = a } \\ { 2 \theta ^ { 2 } } & { x = b } \\ { 1 - \theta - 2 \theta ^ { 2 } } & { x = c } \end{array} \right. }
+$$
+
+Let $r ( a ) = 1 , r ( b ) = 0 , r ( c ) = 0 . 5$ . The expected reward as a function of $\theta$ is:
+
+$$
+R ( \theta ) = \theta + 0 . 5 ( 1 - \theta - 2 \theta ^ { 2 } )
+$$
+
+$R ( \theta )$ is uniquely maximized by $\theta ^ { * } = 0 . 2 5$ .
+
+Table 1 details the possible samples of size $k = 2$ , their probabilities, the corresponding $\widetilde { R }$ and its gradient. Standard numerical methods show that $\mathbb { E } [ \nabla \widetilde { R } ]$ over possible samples $S$ is positive for $\theta \in ( 0 , \gamma )$ and negative for $\theta \in ( \gamma , 0 . 5 ]$ , where $\gamma \approx 0 . 2 9 5$ . This means that for any initialization of $\theta \in ( 0 , 0 . 5 ]$ , Contrastive MRT will converge to $\gamma$ if the learning rate is sufficiently small. For $\theta = 0$ , $\widetilde { R } \equiv 0 . 5$ , and there will be no gradient updates, so the method will converge to $\theta = 0$ . Neither of these values maximizes $R ( \theta )$ .
+
+We note that by using some $g \left( \theta \right)$ the $\gamma$ could be arbitrarily far from $\theta ^ { * }$ . $g$ could also map to $( - i n f , i n f )$ more often used in neural networks parameters.
+
+We further note that resorting to maximizing $\mathbb { E } [ { \widetilde { R } } ]$ instead, does not maximize $R ( \theta )$ either. Indeed, plotting $\mathbb { E } [ { \widetilde { R } } ]$ as a function of $\theta$ for this example, yields a maximum at $\theta \approx 0 . 3 2$ .
+
+<table><tr><td rowspan=1 colspan=1>S</td><td rowspan=1 colspan=1>P(S)</td><td rowspan=1 colspan=1>R</td><td rowspan=1 colspan=1>VR</td></tr><tr><td rowspan=1 colspan=1>{a，b}</td><td rowspan=1 colspan=1>403</td><td rowspan=1 colspan=1>11+20</td><td rowspan=1 colspan=1>-2(1+20)2</td></tr><tr><td rowspan=1 colspan=1>{a,c}</td><td rowspan=1 colspan=1>20(1-0-202)</td><td rowspan=1 colspan=1>0.5 +  θ2-402</td><td rowspan=1 colspan=1>2x²+12(1-202)2</td></tr><tr><td rowspan=1 colspan=1>{b,c}</td><td rowspan=1 colspan=1>402(1-0-202)</td><td rowspan=1 colspan=1>1-0-2022-20</td><td rowspan=1 colspan=1>0²-20(1-0)2</td></tr><tr><td rowspan=1 colspan=1>a,a</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>b,b</td><td rowspan=1 colspan=1>404</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>c,c</td><td rowspan=1 colspan=1>(1-0-202)2</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0</td></tr></table>
+
+Table 1: The gradients of $\widetilde { R }$ for each possible sample $S$ . The batch size is $k = 2$ . Rows correspond to different sampled outcomes. $\nabla \widetilde R$ is the gradient of $\widetilde { R }$ given the corresponding value for $S$ .
+
+# A.2 DERIVING THE GRADIENT OF $\widetilde { R }$
+
+Given $S$ , recall the definition of $\widetilde { R }$ :
+
+$$
+\widetilde { R } ( \theta , S ) = \sum _ { i = 1 } ^ { k } Q _ { \theta , S } ( y _ { i } ) r ( y _ { i } )
+$$
+
+Taking the deriviative w.r.t. $\theta$ :
+
+![](images/5c2012b67867d533776a33f3e9587557daec2c0a652af82c8bac4330da4fc23f.jpg)  
+Figure 7: The probability of different tokens following REINFORCE, in the controlled simulations in the Constant Reward setting. The left/center/right figures correspond to simulations where the target token $( y _ { b e s t } )$ was initially the second/third/fourth most probable token. The green line corresponds to the target token, yellow lines to medium-reward tokens and red lines to tokens with $\bar { \boldsymbol { r } } ( y ) = 0$ .
+
+# A.3 NMT IMPLEMENTATION DETAILS
+
+True casing and tokenization were used (Koehn et al., 2007), including escaping html symbols and "-" that represents a compound was changed into a separate token of $=$ . Some preprocessing used before us converted the latter to ##AT##-##AT## but standard tokenizers in use process that into 11 different tokens, which over-represents the significance of that character when BLEU is calculated. BPE (Sennrich et al., 2016) extracted 30,715 tokens. For the MT experiments we used 6 layers in the encoder and the decoder. The size of the embeddings was 512. Gradient clipping was used with size of 5 for pre-training (see Discussion on why not to use it in training). We did not use attention dropout, but 0.1 residual dropout rate was used. In pretraining and training sentences of more than 50 tokens were discarded. Pretraining and training were considered finished when BLEU did not increase in the development set for 10 consecutive evaluations, and evaluation was done every 1,000 and 5,000 for batches of size 100 and 256 for pretraining and training respectively. Learning rate used for rmsprop (Tieleman & Hinton, 2012) was 0.01 in pretraining and for adam (Kingma & Ba, 2015) with decay was 0.005 for training. 4,000 learning rate warm up steps were used. Pretraining took about 7 days with 4 GPUs, afterwards, training took roughly the same time. Monte Carlo used 20 sentence rolls per word.
+
+# A.4 DETAILED RESULTS FOR CONSTANT REWARD SETTING
+
+We present graphs for the constant reward setting in Figures 8 and 7. Trends are similar to the ones obtained for the Simulated Reward setting.
+
+![](images/f8626e73f517af6d8522a5bf6d4508414c5c3f4934d6cdcdc1290323a0e87186.jpg)  
+Figure 8: Difference between the ranks of $y _ { b e s t }$ in the reinforced with constant reward and the pretrained model. Each column $_ x$ corresponds to the difference in the probability that $y _ { b e s t }$ is ranked in rank $x$ in the reinforced model, and the same probability in the pretrained model.

parse/train/H1eCw3EKvH/H1eCw3EKvH_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "ON THE WEAKNESSES OF REINFORCEMENT LEARNING FOR NEURAL MACHINE TRANSLATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            823,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Leshem Choshen1, Lior $\\mathbf { F o x } ^ { 2 }$ , Zohar Aizenbud1, Omri Abend1,3 ",
+        "bbox": [
+            186,
+            169,
+            629,
+            184
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 School of Computer Science and Engineering, 2 The Edmond and Lily Safra Center for Brain Sciences   \n3 Department of Cognitive Sciences   \nThe Hebrew University of Jerusalem   \nfirst.last@mail.huji.ac.il, oabend@cs.huji.ac.il ",
+        "bbox": [
+            184,
+            185,
+            867,
+            242
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            279,
+            544,
+            292
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning (RL) is frequently used to increase performance in text generation tasks, including machine translation (MT), notably through the use of Minimum Risk Training (MRT) and Generative Adversarial Networks (GAN). However, little is known about what and how these methods learn in the context of MT. We prove that one of the most common RL methods for MT does not optimize the expected reward, as well as show that other methods take an infeasibly long time to converge. In fact, our results suggest that RL practices in MT are likely to improve performance only where the pre-trained parameters are already close to yielding the correct translation. Our findings further suggest that observed gains may be due to effects unrelated to the training signal, concretely, changes in the shape of the distribution curve. ",
+        "bbox": [
+            233,
+            309,
+            764,
+            462
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            488,
+            336,
+            503
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning (RL) is an appealing path for advancement in Machine Translation (MT), as it allows training systems to optimize non-differentiable score functions, common in MT evaluation, as well as tackling the “exposure bias” (Ranzato et al., 2015) in standard training, namely that the model is not exposed during training to incorrectly generated tokens, and is thus unlikely to recover from generating such tokens at test time. These motivations have led to much interest in RL for text generation in general and MT in particular (see $\\ S 2$ ). Various policy gradient methods have been used, notably REINFORCE (Williams, 1992) and variants thereof (e.g., Ranzato et al., 2015; Edunov et al., 2018) and Minimum Risk Training (MRT; e.g., Och, 2003; Shen et al., 2016). Another popular use of RL is for training GANs (Yang et al., 2018; Tevet et al., 2018). Nevertheless, despite increasing interest and strong results, little is known about what accounts for these performance gains, and the training dynamics involved. ",
+        "bbox": [
+            174,
+            518,
+            825,
+            671
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present the following contributions. First, our theoretical analysis shows that commonly used approximation methods are theoretically ill-founded, and may converge to parameter values that do not minimize the risk, nor are local minima thereof (§2.2). ",
+        "bbox": [
+            176,
+            679,
+            821,
+            722
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Second, using both naturalistic experiments and carefully constructed simulations, we show that performance gains observed in the literature likely stem not from making target tokens the most probable, but from unrelated effects, such as increasing the peakiness of the output distribution (i.e., the probability mass of the most probable tokens). We do so by comparing a setting where the reward is informative, vs. one where it is constant. In $\\ S 4$ we discuss this peakiness effect (PKE). ",
+        "bbox": [
+            174,
+            728,
+            825,
+            797
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Third, we show that promoting the target token to be the mode is likely to take a prohibitively long time. The only case we find, where improvements are likely, is where the target token is among the first 2-3 most probable tokens according to the pretrained model. These findings suggest that REINFORCE (§5) and CMRT $( \\ S 6 )$ are likely to improve over the pre-trained model only under the best possible conditions, i.e., where the pre-trained model is “nearly” correct. ",
+        "bbox": [
+            176,
+            805,
+            825,
+            875
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We conclude by discussing other RL practices in MT which should be avoided for practical and theoretical reasons, and briefly discuss alternative RL approaches that will allow RL to tackle a larger class of errors in pre-trained models (§7). ",
+        "bbox": [
+            176,
+            882,
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+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2 RL IN MACHINE TRANSLATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            468,
+            118
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "An MT system generates tokens $y = ( y _ { 1 } , . . . , y _ { n } )$ from a vocabulary $V$ one token at a time. The probability of generating $y _ { i }$ given preceding tokens $y _ { < i }$ is given by $P _ { \\theta } ( \\cdot | x , y _ { < i } )$ , where $x$ is the source sentence and $\\theta$ are the model parameters. For each generated token $y _ { i }$ , we denote with $r ( y _ { i } ; y _ { < i } , x , y ^ { ( r e f ) } )$ the score, or reward, for generating $y _ { i }$ given $y _ { < i } , x$ , and the reference sentence $y ^ { ( r e f ) }$ . For brevity, we omit parameters where they are fixed within context. For simplicity, we assume $r$ does not depend on following tokens $y _ { > i }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We also assume there is exactly one valid target token, as de facto, training is done against a single reference (Schulz et al., 2018). In practice, either a token-level reward is approximated using MonteCarlo methods (e.g., Yang et al., 2018), or a sentence-level (sparse) reward is given at the end of the episode (sentence). The latter is equivalent to a uniform token-level reward. ",
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+    {
+        "type": "text",
+        "text": "$r$ is often the negative log-likelihood, or a standard MT metric, e.g., BLEU (Papineni et al., 2002). RL’s goal is to maximize the expected episode reward (denoted with $R$ ); i.e., to find ",
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+    {
+        "type": "equation",
+        "img_path": "images/92f3bf2b0c72acf50f38312684819b7c0a13ab1fd1ce36786dc7f85276eb1e1a.jpg",
+        "text": "$$\n\\theta ^ { * } = \\arg \\operatorname* { m a x } _ { \\theta } \\ R ( \\theta ) = \\arg \\operatorname* { m a x } _ { \\theta } \\ \\mathbb { E } _ { y \\sim P _ { \\theta } } [ r ( y ) ]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "2.1 REINFORCE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For a given source sentence, and past predictions $y _ { < i }$ , REINFORCE (Williams, 1992) samples $k$ tokens $k$ is a hyperparameter) $S = \\bar { ( { y ^ { ( 1 ) } } , . . . , y ^ { ( k ) } ) }$ from $P _ { \\theta }$ and updates $\\theta$ according to this rule: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/0ddded5b311dd76ab647338dc6d8b159a478e6f9a9f144b85fb50d884eb1b870.jpg",
+        "text": "$$\n\\Delta \\theta \\propto \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } r ( y _ { i } ) \\nabla \\log ( P _ { \\theta } ( y _ { i } ) )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The right-hand side of equation 2 is an unbiased estimator of the gradient of the objective function, i.e., $\\mathbb { E } \\left[ \\Delta \\theta \\right] \\propto \\nabla _ { \\theta } R \\left( \\theta \\right)$ . Therefore, REINFORCE is performing a form of stochastic gradient ascent on $R$ , and has similar formal guarantees. From here follows that if $R$ is constant with respect to $\\theta$ , then the expected $\\Delta \\theta$ prescribed by REINFORCE is zero. We note that $r$ may be shifted by a constant term (called a “baseline”), without affecting the optimal value for $\\theta$ . ",
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+        "type": "text",
+        "text": "REINFORCE is used in MT, text generation, and image-to-text tasks (Liu et al., 2016; Wu et al., 2018; Rennie et al., 2017; Shetty et al., 2017; Hendricks et al., 2016) – in isolation, or as a part of training (Ranzato et al., 2015). Lately, an especially prominent use for REINFORCE is adversarial training with discrete data, where another network predicts the reward (GAN). For some recent work on RL for NMT, see (Zhang et al., 2016; Li et al., 2017; Wu et al., 2017; Yu et al., 2017; Yang et al., 2018). ",
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+    {
+        "type": "text",
+        "text": "2.2 MINIMUM RISK TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The term Minimum Risk Training (MRT) is used ambiguously in MT to refer either to the application of REINFORCE to minimizing the risk (equivalently, to maximizing the expected reward, the negative loss), or more commonly to a somewhat different estimation method, which we term Contrastive MRT (CMRT) and turn now to analyzing. CMRT was proposed by Och (2003), adapted to NMT by Shen et al. (2016), and often used since (Ayana et al., 2016; Neubig, 2016; Shen et al., 2017; Edunov et al., 2018; Makarov & Clematide, 2018; Neubig et al., 2018). ",
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+    {
+        "type": "text",
+        "text": "The method works as follows: at each iteration, sample $k$ tokens $S = \\{ y _ { 1 } , \\dots , y _ { k } \\}$ from $P _ { \\theta }$ , and update $\\theta$ according to the gradient of ",
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+    {
+        "type": "equation",
+        "img_path": "images/75d3690c76ebf01dcba6beeb14fa47d4341465106831b63b79ba231e588d2ec8.jpg",
+        "text": "$$\n\\widetilde { R } ( \\theta , S ) = \\sum _ { i = 1 } ^ { k } Q _ { \\theta , S } ( y _ { i } ) r ( y _ { i } ) = \\mathbb { E } _ { y \\sim Q } \\left[ r ( y ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/7bab940a15a59f2a4e325da639524c63665344ad507dd8674bbb04dfa52eb87c.jpg",
+        "text": "$$\nQ _ { \\theta , S } ( y _ { i } ) = \\frac { P ( y _ { i } ) ^ { \\alpha } } { \\sum _ { y _ { j } \\in S } P ( y _ { j } ) ^ { \\alpha } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Commonly (but not universally), deduplication is performed, so $\\widetilde { R }$ sums over a set of unique values (Sennrich et al., 2017). This changes little in our empirical results and theoretical analysis. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Despite the resemblance in definitions of $R$ (equation 1) and $\\widetilde { R }$ (indeed, $\\widetilde { R }$ is sometimes presented as an approximation of $R$ ), they differ in two important aspects. First, $Q$ ’s support is $S$ , so increasing ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "$Q ( y _ { i } )$ for some $y _ { i }$ necessarily comes at the expense of $Q ( y )$ for some $y \\in S$ . In contrast, increasing $P ( y _ { i } )$ , as in REINFORCE, may come at the expense of $P ( y )$ for any $y \\in V$ . Second, $\\alpha$ is a smoothness parameter: the closer $\\alpha$ is to 0, the closer $Q$ is to be uniform. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "We show in Appendix A.1 that despite its name, CMRT does not optimize $R$ , nor does it optimize $\\mathbb { E } [ { \\widetilde { R } } ]$ . That is, it may well converge to values that are not local maxima of $R$ , making it theoretically ill-founded.1 However, given CMRT popularity, the strong results it yielded and the absence of theory for explaining it, we discuss it here. Given a sample $S$ , the gradient of $\\widetilde { R }$ is given by ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8b0354f6ce8c603331f588cf49db61fcc863dadc082b3f226fee9cf82bb8f630.jpg",
+        "text": "$$\n\\nabla \\widetilde { R } = \\alpha \\sum _ { i = 1 } ^ { k } \\Big ( Q ( y _ { i } ) \\cdot r ( y _ { i } ) \\cdot \\nabla \\log P ( y _ { i } ) \\Big ) - \\mathbb { E } _ { Q } [ r ] \\nabla \\log Z ( S )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { Z ( S ) = \\sum _ { i } P ( y _ { i } ) ^ { \\alpha } } \\end{array}$ . See Appendix A.2. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Comparing Equations 2 and 3, the differences between REINFORCE and CMRT are reflected again. First, $\\nabla \\widetilde R$ has an additional term, proportional to $\\nabla \\log Z ( S )$ , which yields the contrastive effect. This contrast may improve the rate of convergence since it counters the decrease of probability mass for non-sampled tokens. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Second, given $S$ , the relative weighting of the gradients $\\nabla \\log P ( y _ { i } )$ is proportional to $r ( y _ { i } ) Q ( y _ { i } )$ , or equivalently to $r ( y _ { i } ) P ( y _ { i } ) ^ { \\alpha }$ . CMRT with deduplication sums over distinct values in $S$ (equation 3), while REINFORCE sums over all values. This means that the relative weight of the unique value $y _ { i }$ is $\\frac { r ( y _ { i } ) | \\{ y _ { i } \\in S \\} | } { k }$ in REINFORCE. For $\\alpha = 1$ the expected value of these relative weights is the same, and so for $\\alpha < 1$ (as is commonly used), more weight is given to improbable tokens, which could also have a positive effect on the convergence rate.2 However, if $\\alpha$ is too close to 0, $\\nabla \\widetilde R$ vanishes, as it is not affected by $\\theta$ . This tradeoff explains the importance of tuning $\\alpha$ reported in the literature. In $\\ S 6$ we present simulations with CMRT, showing very similar trends as presented by REINFORCE. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 MOTIVATING DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Implementing a stochastic gradient ascent, REINFORCE is guaranteed to converge to a stationary point of $R$ under broad conditions. However, not much is known about its convergence rate under the prevailing conditions in NMT. ",
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+    {
+        "type": "text",
+        "text": "We begin with a qualitative, motivating analysis of these questions. As work on language generation empirically showed, RNNs quickly learn to output very peaky distributions (Press et al., 2017). This tendency is advantageous for generating fluent sentences with high probability, but may also entail slower convergence rates when using RL to fine-tune the model, because RL methods used in text generation sample from the (pretrained) policy distribution, which means they mostly sample what the pretrained model deems to be likely. Since the pretrained model (or policy) is peaky, exploration of other potentially more rewarding tokens will be limited, hampering convergence. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Intuitively, REINFORCE increases the probabilities of successful (positively rewarding) observations, weighing updates by how rewarding they were. When sampling a handful of tokens in each context (source sentence $x$ and generated prefix $y _ { < i , }$ ), and where the number of epochs is not large, it is unlikely that more than a few unique tokens will be sampled from $P _ { \\theta } ( \\cdot | x , y _ { < i } )$ . (In practice, $k$ is typically between 1 and 20, and the number of epochs between 1 and 100.) It is thus unlikely that anything but the initially most probable candidates will be observed. Consequently, REINFORCE initially raises their probabilities, even if more rewarding tokens can be found down the list. ",
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+    {
+        "type": "text",
+        "text": "We thus hypothesize the peakiness of the distribution, i.e., the probability mass allocated to the most probable tokens, will increase, at least in the first phase. We call this the peakiness-effect (PKE), and show it occurs both in simulations (§4.1) and in full-scale NMT experiments (§4.2). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "With more iterations, the most-rewarding tokens will be eventually sampled, and gradually gain probability mass. This discussion suggests that training will be extremely sample-inefficient. We assess the rate of convergence empirically in $\\ S 5$ , finding this to be indeed the case. ",
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+    {
+        "type": "image",
+        "img_path": "images/a8e0772d8762f399cf44fa7afc438edb7e4aada745a8847bb2b5a17ee8ef2ab1.jpg",
+        "image_caption": [
+            "Figure 1: A histogram of the update size (x-axis) to the total predicted probability of the 10 most probable tokens (left) or the most probable token (right) in the Constant Reward setting. An update is overwhelmingly more probable to increase this probability than to decrease it. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 THE PEAKINESS EFFECT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We turn to demonstrate that the initially most probable tokens will initially gain probability mass, even if they are not the most rewarding, yielding a PKE. ",
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+    {
+        "type": "text",
+        "text": "Caccia et al. (2018) recently observed in the context of language modeling using GANs that performance gains similar to those GAN yield can be achieved by decreasing the temperature for the prediction softmax (i.e., making it peakier). However, they proposed no causes for this effect. Our findings propose an underlying mechanism leading to this trend. We return to this point in $^ { \\ S 7 }$ . Furthermore, given their findings, it is reasonable to assume that our results are relevant for RL use in other generation tasks, whose output space too is discrete, high-dimensional and concentrated. ",
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+    {
+        "type": "text",
+        "text": "4.1 CONTROLLED SIMULATIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We experiment with a 1-layer softmax model, that predicts a single token $i \\in V$ with probability $\\frac { e ^ { \\theta _ { i } } } { \\sum _ { j } e ^ { \\theta _ { j } } }$ . $\\theta = \\{ \\theta _ { j } \\} _ { j \\in V }$ are the model’s parameters. This model simulates the top of any MT decoder that ends with a softmax layer, as essentially all NMT decoders do. To make experiments realistic, we use similar parameters as those reported in the influential Transformer NMT system (Vaswani et al., 2017). Specifically, the size of $V$ (distinct BPE tokens) is 30,715, and the initial values for $\\theta$ were sampled from 1,000 sets of logits taken from decoding the standard newstest2013 development set, using a pretrained Transformer model. The model was pretrained on WMT2015 training data (Bojar et al., 2015). Hyperparameters are reported in Appendix A.3. We define one of the tokens in $V$ to be the target token and denote it with $y _ { b e s t }$ . We assign deterministic token reward, this makes learning easier than when relying on approximations and our predictions optimistic. We experiment with two reward functions: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1. Simulated Reward: $r ( y ) = 2$ for $y = y _ { b e s t }$ , $r ( y ) = 1$ if $y$ is one of the 10 initially highest scoring tokens, and $r ( y ) = 0$ otherwise. This simulates a condition where the pretrained model is of decent but sub-optimal quality. $r$ here is at the scale of popular rewards used in MT, such as GAN-based rewards or BLEU (which are between 0 and 1). 2. Constant Reward: $r$ is constantly equal to 1, for all tokens. This setting is aimed to confirm that PKE is not a result of the signal carried by the reward. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Experiments with the first setting were run 100 times, each time for 50K steps, updating $\\theta$ after each step. With the second setting, it is sufficient to take a single step at a time, as the expected update after each step is zero, and so any $\\mathrm { P K E }$ seen in a single step is only accentuated in the next. It is, therefore, more telling to run more repetitions rather than more steps per initialization. We, therefore, sample 10,000 pretrained distributions, and perform a single REINFORCE step. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As RL training lasts about 30 epochs before stopping, samples about $1 0 0 \\mathrm { K }$ tokens per epoch, and as the network already predicts $y _ { b e s t }$ in about two thirds of the contexts,3 we estimate the number of steps used in practice to be in the order of magnitude of 1M. For visual clarity, we present figures for 50K-100K steps. However, full experiments (with 1M steps) exhibit similar trends: where REINFORCE was not close to converging after 50K steps, the same was true after 1M steps. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate the peakiness of a distribution in terms of the probability of the most probable token (the mode), the total probability of the ten most probable tokens, and the entropy of the distribution (lower entropy indicates more peakiness). ",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/80b2427a869afcc4d92132462366b3e75868a834bc4126520558d3941359d3a7.jpg",
+        "image_caption": [
+            "Figure 2: Token probabilities through REINFORCE training, in the controlled simulations in the Simulated Reward setting. The left/center/right figures correspond to simulations where the target token $( y _ { b e s t } )$ was initially the second/third/fourth most probable token. The green line corresponds to the target token, yellow lines to medium-reward tokens and red lines to no-reward tokens. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Results. The distributions become peakier in terms of all three measures: on average, the mode’s probability and the 10 most probable tokens increases, and the entropy decreases. Figure 1a presents the histogram of the update size, the difference in the probability of the $1 0 \\mathrm { m o s t }$ probable tokens in the Constant Reward setting, after a single step. Figure 1b depicts similar statistics for the mode. The average entropy in the pretrained model is 2.9 is reduced to 2.85 after one REINFORCE step. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Simulated Reward setting shows similar trends. For example, entropy decreases from 3 to about 0.001 in 100K steps. This extreme decrease suggests it is effectively a deterministic policy. $\\mathrm { P K E }$ is achieved in a few hundred steps, usually before other effects become prominent (see Figure 2), and is stronger than for Constant Reward. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "4.2 NMT EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "We turn to analyzing a real-world application of REINFORCE to NMT. Important differences between this and the previous simulations are: (1) it is rare in NMT for REINFORCE to sample from the same conditional distribution more than a handful of times, given the number of source sentences $x$ and sentence prefixes $y _ { < i }$ (contexts); and (2) in NMT $P _ { \\theta } ( \\cdot | x , y _ { < i } )$ shares parameters between contexts, which means that updating $P _ { \\theta }$ for one context may influence $P _ { \\theta }$ for another. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We follow the same pretraining as in $\\ S 4 . 1$ . We then follow Yang et al. (2018) in defining the reward function based on the expected BLEU score. Expected BLEU is computed by sampling suffixes for the sentence, and averaging the BLEU score of the sampled sentences against the reference. ",
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+    {
+        "type": "text",
+        "text": "We use early stopping with a patience of 10 epochs, where each epoch consists of 5,000 sentences sampled from the WMT2015 (Bojar et al., 2015) German-English training data. We use $k = 1$ . We retuned the learning-rate, and positive baseline settings against the development set. Other hyper-parameters were an exact replication of the experiments reported in (Yang et al., 2018). ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/e97b7eecb797bb490493534fc6b6cb4cd578b2f2ee48ba45bcccf79a3a2e0728.jpg",
+        "image_caption": [
+            "Figure 3: The cumulative distribution of the probability of the most likely token in the NMT experiments. The green distribution corresponds to the pretrained model, and the blue corresponds to the reinforced model. The yaxis is the proportion of conditional probabilities with a mode of value $\\leq x$ (the $\\mathbf { X }$ -axis). Note that a lower cumulative percentage means a more peaked output distribution. A lower cumulative percentage means a more peaked output distribution. "
+        ],
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+    {
+        "type": "text",
+        "text": "Results. Results indicate an increase in the peakiness of the conditional distributions. Our results are based on a sample of 1,000 contexts from the pretrained model, and another (independent) sample from the reinforced model. ",
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+    {
+        "type": "text",
+        "text": "The modes of the conditional distributions tend to increase. Figure 3 presents the distribution of the modes’ probability in the reinforced conditional distributions compared with the pretrained model, showing a shift of probability mass towards higher probabilities for the mode, following RL. Another indication of the increased peakiness is the decrease in the average entropy of $P _ { \\theta }$ , which was reduced from 3.45 in the pretrained model to an average of 2.82 following RL. This more modest reduction in entropy (compared to $\\ S 4 . 1$ ) might also suggest that the procedure did not converge to the optimal value for $\\theta$ , as then we would have expected the entropy to substantially drop if not to 0 (overfit), then to the average entropy of valid next tokens (given the source and a prefix of the sentence). ",
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+        "type": "text",
+        "text": "5 PERFORMANCE FOLLOWING REINFORCE ",
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+        "type": "text",
+        "text": "We now turn to assessing under what conditions it is likely that REINFORCE will lead to an improvement in the performance of an NMT system. As in the previous section, we use both controlled simulations and NMT experiments. ",
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+        "type": "text",
+        "text": "5.1 CONTROLLED SIMULATIONS ",
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+        "type": "text",
+        "text": "We use the same model and experimental setup described in Section 4.1, this time only exploring the Simulated Reward setting, as a Constant Reward is not expected to converge to any meaningful $\\theta$ . Results are averaged over 100 conditional distributions sampled from the pretrained model. ",
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+        "type": "text",
+        "text": "Caution should be exercised when determining the learning rate (LR). Common LRs used in the NMT literature are of the scale of $1 0 ^ { - 4 }$ . However, in our simulations, no LR smaller than 0.1 yielded any improvement in $R$ . We thus set the LR to be 0.1. We note that in our simulations, a higher learning rate means faster convergence as our reward is noise-free: it is always highest for the best option. In practice, increasing the learning rate may deteriorate results, as it may cause the system to overfit to the sampled instances. Indeed, when increasing the learning rate in our NMT experiments (see below) by an order of magnitude, early stopping caused the RL procedure to stop without any parameter updates. ",
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+    {
+        "type": "text",
+        "text": "Figure 2 shows the change in $P _ { \\theta }$ over the first 50K REINFORCE steps (probabilities are averaged over 100 repetitions), for a case where $y _ { b e s t }$ was initially the second, third and fourth most probable. Although these are the easiest settings, and despite the high learning rate, it fails to make $y _ { b e s t }$ the mode of the distribution within $1 0 0 \\mathrm { K }$ steps, unless $y _ { b e s t }$ was initially the second most probable. In cases where $y _ { b e s t }$ is initially of a lower rank than four, it is hard to see any increase in its probability, even after 1M steps. ",
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+        "image_caption": [
+            "Figure 4: Cumulative percentage of contexts where the pretrained model ranks $y _ { b e s t }$ in rank $_ x$ or below and where it does not rank $y _ { b e s t }$ first $( x = 0$ ). In about half the cases it is ranked fourth or below. "
+        ],
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+    {
+        "type": "text",
+        "text": "5.2 NMT EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We trained an NMT system, using the same procedure as in Section 4.2, and report BLEU scores over the news2014 test set. After training with an expected BLEU reward, we indeed see a minor improvement which is consistent between trials and pretrained models. While the pretrain BLEU score is 30.31, the reinforced one is 30.73. ",
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+    {
+        "type": "text",
+        "text": "Analyzing what words were influenced by the RL procedure, we begin by computing the cumulative probability of the target token $y _ { b e s t }$ to be ranked lower than a given rank according to the pretrained model. Results (Figure 4) show that in about half of the cases, $y _ { b e s t }$ is not among the top three choices of the pretrained model, and we thus expect it not to gain substantial probability following REINFORCE, according to our simulations. ",
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+    {
+        "type": "text",
+        "text": "We next turn to compare the ranks the reinforced model assigns to the target tokens, and their ranks according to the pretrained model. Figure 6 presents the difference in the probability that $y _ { b e s t }$ is ranked at a given rank following RL and the probability it is ranked there initially. Results indicate that indeed more target tokens are ranked first, and less second, but little consistent shift of probability mass occurs otherwise across the ten first ranks. It is possible that RL has managed to push $y _ { b e s t }$ in some cases between very low ranks $( < 1 , 0 0 0 )$ to medium-low ranks (between 10 and 1,000). However, token probabilities in these ranks are so low that it is unlikely to affect the system outputs in any way. This fits well with the results of our simulations that predicted that only the initially top-ranked tokens are likely to change. ",
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+    {
+        "type": "text",
+        "text": "In an attempt to explain the improved BLEU score following RL with PKE, we repeat the NMT experiment this time using a constant reward of 1. Our results present a nearly identical improvement in BLEU, achieving 30.72, and a similar pattern in the change of the target tokens’ ranks (see Appendix 8). Therefore, there is room to suspect that even in cases where RL yields an improvement in BLEU, it may partially result from reward-independent factors, such as PKE.4 ",
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+    {
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+        "img_path": "images/5dcb22db4003b6d8b1e130bb7b8a032341116c25c187d78aba1343df058a0aad.jpg",
+        "image_caption": [
+            "Figure 5: The probability of different tokens following CMRT, in the controlled simulations in the Simulated Reward setting. The left/right figures correspond to simulations where the target token $( y _ { b e s t } )$ was initially the second/third most probable token. The green line corresponds to the target token, yellow lines to medium-reward tokens and red lines to tokens with $r ( y ) = 0$ . "
+        ],
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+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS WITH CONTRASTIVE MRT ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In $\\ S 2 . 2$ we showed that CMRT does not, in fact, maximize $R$ , and so does not enjoy the same theoretical guarantees as REINFORCE and similar policy gradient methods. However, being the RL procedure of choice in much recent work we repeat the simulations described in $\\ S 4$ and $\\ S 5$ , assessing CMRT’s performance in these conditions. We experiment with $\\alpha = 0 . 0 0 5$ and $k = 2 0$ , common settings in the literature, and average over 100 trials. ",
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+    {
+        "type": "text",
+        "text": "Figure 5 shows how the distribution $P _ { \\theta }$ changes over the course of 50K update steps to $\\theta$ , where $y _ { b e s t }$ is taken to be the second and third initially most probable token (Simulated Reward setting). Results are similar in trends to those obtained with REINFORCE: MRT succeeds in pushing $y _ { b e s t }$ to be the highest ranked token if it was initially second, but struggles where it was initially ranked third or below. We only observe a small $\\mathrm { P K E }$ in MRT. This is probably due to the contrastive effect, which means that tokens that were not sampled do not lose probability mass. ",
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+    {
+        "type": "text",
+        "text": "All graphs we present here allow sampling the same token more than once in each batch (i.e., $S$ is a sample with replacements). Simulations with deduplication show similar results. ",
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+    {
+        "type": "image",
+        "img_path": "images/03bf2a5a7bc9cbd3fc79305ae00d3e2c99452fdba0637e6445dc57ebc0b7cda2.jpg",
+        "image_caption": [
+            "Figure 6: Difference between the ranks of $y _ { b e s t }$ in the reinforced and the pretrained model. Each column $_ x$ corresponds to the difference in the probability that $y _ { b e s t }$ is ranked in rank $x$ in the reinforced model, and the same probability in the pretrained model. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "7 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we showed that the type of distributions used in NMT entail that promoting the target token to be the mode is likely to take a prohibitively long times for existing RL practices, except under the best conditions (where the pretrained model is “nearly” correct). This leads us to conclude that observed improvements from using RL for NMT are likely due either to fine-tuning the most probable tokens in the pretrained model (an effect which may be more easily achieved using reranking methods, and uses but little of the power of RL methods), or to effects unrelated to the signal carried by the reward, such as PKE. Another contribution of this paper is in showing that CMRT does not optimize the expected reward and is thus theoretically unmotivated. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A number of reasons lead us to believe that in our NMT experiments, improvements are not due to the reward function, but to artefacts such as PKE. First, reducing a constant baseline from $r$ , so as to make the expected reward zero, disallows learning. This is surprising, as REINFORCE, generally and in our simulations, converges faster where the reward is centered around zero, and so the fact that this procedure here disallows learning hints that other factors are in play. As PKE can be observed even where the reward is constant (if the expected reward is positive; see $\\ S 4 . 1 \\ r _ { \\cdot }$ ), this suggests PKE may play a role here. Second, we observe more peakiness in the reinforced model and in such cases, we expect improvements in BLEU (Caccia et al., 2018). Third, we achieve similar results with a constant reward in our NMT experiments (§5.2). Fourth, our controlled simulations show that asymptotic convergence is not reached in any but the easiest conditions (§5.1). ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Our analysis further suggests that gradient clipping, sometimes used in NMT (Zhang et al., 2016; Wieting et al., 2019), is expected to hinder convergence further. It should be avoided when using REINFORCE as it violates REINFORCE’s assumptions. ",
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+    {
+        "type": "text",
+        "text": "The per-token sampling as done in our experiments is more exploratory than beam search (Wu et al., 2018), reducing PKE. Furthermore, the latter does not sample from the behavior policy, but does not properly account for being off-policy in the parameter updates. ",
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+    {
+        "type": "text",
+        "text": "Adding the reference to the sample $S$ , which some implementations allow (Sennrich et al., 2017) may help reduce the problems of never sampling the target tokens. However, as Edunov et al. (2018) point out, this practice may lower results, as it may destabilize training by leading the model to improve over outputs it cannot generalize over, as they are very different from anything the model assigns a high probability to, at the cost of other outputs. ",
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+    {
+        "type": "text",
+        "text": "8 CONCLUSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The standard MT scenario poses several uncommon challenges for RL. First, the action space in MT problems is a high-dimensional discrete space (generally in the size of the vocabulary of the target language or the product thereof for sentences). This contrasts with the more common scenario studied by contemporary RL methods, which focuses mostly on much smaller discrete action spaces (e.g., video games (Mnih et al., 2015; 2016)), or continuous action spaces of relatively low dimensions (e.g., simulation of robotic control tasks (Lillicrap et al., 2015)). Second, reward for MT is naturally very sparse – almost all possible sentences are “wrong” (hence, not rewarding) in a given context. Finally, it is common in MT to use RL for tuning a pretrained model. Using a pretrained model ameliorates the last problem. But then, these pretrained models are in general quite peaky, and because training is done on-policy – that is, actions are being sampled from the same model being optimized – exploration is inherently limited. ",
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+    {
+        "type": "text",
+        "text": "Here we argued that, taken together, these challenges result in significant weaknesses for current RL practices for NMT, that may ultimately prevent them from being truly useful. At least some of these challenges have been widely studied in the RL literature, with numerous techniques developed to address them, but were not yet adopted in NLP. We turn to discuss some of them. ",
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+    {
+        "type": "text",
+        "text": "Off-policy methods, in which observations are sampled from a different policy than the one being currently optimized, are prominent in RL (Watkins & Dayan, 1992; Sutton & Barto, 1998), and were also studied in the context of policy gradient methods (Degris et al., 2012; Silver et al., 2014). In principle, such methods allow learning from a more “exploratory” policy. Moreover, a key motivation for using $\\alpha$ in CMRT is smoothing; off-policy sampling allows smoothing while keeping convergence guarantees. ",
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+    },
+    {
+        "type": "text",
+        "text": "In its basic form, exploration in REINFORCE relies on stochasticity in the action-selection (in MT, this is due to sampling). More sophisticated exploration methods have been extensively studied, for example using measures for the exploratory usefulness of states or actions (Fox et al., 2018), or relying on parameter-space noise rather than action-space noise (Plappert et al., 2017). ",
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+    },
+    {
+        "type": "text",
+        "text": "For MT, an additional challenge is that even effective exploration (sampling diverse sets of observations), may not be enough, since the state-action space is too large to be effectively covered, with almost all sentences being not rewarding. Recently, diversity-based and multi-goal methods for RL were proposed to tackle similar challenges (Andrychowicz et al., 2017; Ghosh et al., 2018; Eysenbach et al., 2019). We believe the adoption of such methods is a promising path forward for the application of RL in NLP. ",
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+    {
+        "type": "text",
+        "text": "9 ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This work was supported by the Israel Science Foundation (grant no. 929/17) and by the HUJI Cyber Security Research Center in conjunction with the Israel National Cyber Bureau in the Prime Minister’s Office. ",
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+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            299,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 CONTRASTIVE MRT DOES NOT MAXIMIZE THE EXPECTED REWARD ",
+        "bbox": [
+            173,
+            136,
+            692,
+            152
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We hereby detail a simple example where following the Contrastive MRT method (see $\\ S 2 . 2$ ) does not converge to the parameter value that maximizes $R$ . ",
+        "bbox": [
+            174,
+            162,
+            826,
+            193
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Let $\\theta$ be a real number in $[ 0 , 0 . 5 ]$ , and let $P _ { \\theta }$ be a family of distributions over three values $a , b , c$ such that: ",
+        "bbox": [
+            173,
+            198,
+            826,
+            227
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fe9f8508c9b814ea3989163a21bd214ee3af0d6b91060cf4905dfc30d63616a1.jpg",
+        "text": "$$\nP _ { \\theta } ( x ) = { \\left\\{ \\begin{array} { l l } { \\theta } & { x = a } \\\\ { 2 \\theta ^ { 2 } } & { x = b } \\\\ { 1 - \\theta - 2 \\theta ^ { 2 } } & { x = c } \\end{array} \\right. }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            248,
+            607,
+            300
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Let $r ( a ) = 1 , r ( b ) = 0 , r ( c ) = 0 . 5$ . The expected reward as a function of $\\theta$ is: ",
+        "bbox": [
+            171,
+            314,
+            684,
+            330
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/81f5ff5c9ccbf8655a471f4fd6976f945fd0b33db940259dd06793c0e6a0655f.jpg",
+        "text": "$$\nR ( \\theta ) = \\theta + 0 . 5 ( 1 - \\theta - 2 \\theta ^ { 2 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            352,
+            599,
+            369
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "$R ( \\theta )$ is uniquely maximized by $\\theta ^ { * } = 0 . 2 5$ . ",
+        "bbox": [
+            178,
+            383,
+            452,
+            400
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 1 details the possible samples of size $k = 2$ , their probabilities, the corresponding $\\widetilde { R }$ and its gradient. Standard numerical methods show that $\\mathbb { E } [ \\nabla \\widetilde { R } ]$ over possible samples $S$ is positive for $\\theta \\in ( 0 , \\gamma )$ and negative for $\\theta \\in ( \\gamma , 0 . 5 ]$ , where $\\gamma \\approx 0 . 2 9 5$ . This means that for any initialization of $\\theta \\in ( 0 , 0 . 5 ]$ , Contrastive MRT will converge to $\\gamma$ if the learning rate is sufficiently small. For $\\theta = 0$ , $\\widetilde { R } \\equiv 0 . 5$ , and there will be no gradient updates, so the method will converge to $\\theta = 0$ . Neither of these values maximizes $R ( \\theta )$ . ",
+        "bbox": [
+            173,
+            406,
+            826,
+            498
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We note that by using some $g \\left( \\theta \\right)$ the $\\gamma$ could be arbitrarily far from $\\theta ^ { * }$ . $g$ could also map to $( - i n f , i n f )$ more often used in neural networks parameters. ",
+        "bbox": [
+            171,
+            503,
+            825,
+            532
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We further note that resorting to maximizing $\\mathbb { E } [ { \\widetilde { R } } ]$ instead, does not maximize $R ( \\theta )$ either. Indeed, plotting $\\mathbb { E } [ { \\widetilde { R } } ]$ as a function of $\\theta$ for this example, yields a maximum at $\\theta \\approx 0 . 3 2$ . ",
+        "bbox": [
+            176,
+            540,
+            821,
+            574
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/156c51d553a179f445afdae5e8eb93518834bf88cf861e7e2c884f6adaa964f2.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>S</td><td rowspan=1 colspan=1>P(S)</td><td rowspan=1 colspan=1>R</td><td rowspan=1 colspan=1>VR</td></tr><tr><td rowspan=1 colspan=1>{a，b}</td><td rowspan=1 colspan=1>403</td><td rowspan=1 colspan=1>11+20</td><td rowspan=1 colspan=1>-2(1+20)2</td></tr><tr><td rowspan=1 colspan=1>{a,c}</td><td rowspan=1 colspan=1>20(1-0-202)</td><td rowspan=1 colspan=1>0.5 +  θ2-402</td><td rowspan=1 colspan=1>2x²+12(1-202)2</td></tr><tr><td rowspan=1 colspan=1>{b,c}</td><td rowspan=1 colspan=1>402(1-0-202)</td><td rowspan=1 colspan=1>1-0-2022-20</td><td rowspan=1 colspan=1>0²-20(1-0)2</td></tr><tr><td rowspan=1 colspan=1>a,a</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>b,b</td><td rowspan=1 colspan=1>404</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>c,c</td><td rowspan=1 colspan=1>(1-0-202)2</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0</td></tr></table>",
+        "bbox": [
+            173,
+            587,
+            524,
+            714
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 1: The gradients of $\\widetilde { R }$ for each possible sample $S$ . The batch size is $k = 2$ . Rows correspond to different sampled outcomes. $\\nabla \\widetilde R$ is the gradient of $\\widetilde { R }$ given the corresponding value for $S$ . ",
+        "bbox": [
+            173,
+            726,
+            823,
+            756
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 DERIVING THE GRADIENT OF $\\widetilde { R }$ ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            785,
+            442,
+            801
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Given $S$ , recall the definition of $\\widetilde { R }$ : ",
+        "bbox": [
+            174,
+            815,
+            405,
+            833
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/52636b85327050e053504170a4d8bc370bb6f1e680921074ddc47d0210cccb86.jpg",
+        "text": "$$\n\\widetilde { R } ( \\theta , S ) = \\sum _ { i = 1 } ^ { k } Q _ { \\theta , S } ( y _ { i } ) r ( y _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            853,
+            596,
+            895
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Taking the deriviative w.r.t. $\\theta$ : ",
+        "bbox": [
+            174,
+            909,
+            372,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/5c2012b67867d533776a33f3e9587557daec2c0a652af82c8bac4330da4fc23f.jpg",
+        "image_caption": [
+            "Figure 7: The probability of different tokens following REINFORCE, in the controlled simulations in the Constant Reward setting. The left/center/right figures correspond to simulations where the target token $( y _ { b e s t } )$ was initially the second/third/fourth most probable token. The green line corresponds to the target token, yellow lines to medium-reward tokens and red lines to tokens with $\\bar { \\boldsymbol { r } } ( y ) = 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            111,
+            818,
+            467
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 NMT IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            558,
+            452,
+            571
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "True casing and tokenization were used (Koehn et al., 2007), including escaping html symbols and \"-\" that represents a compound was changed into a separate token of $=$ . Some preprocessing used before us converted the latter to ##AT##-##AT## but standard tokenizers in use process that into 11 different tokens, which over-represents the significance of that character when BLEU is calculated. BPE (Sennrich et al., 2016) extracted 30,715 tokens. For the MT experiments we used 6 layers in the encoder and the decoder. The size of the embeddings was 512. Gradient clipping was used with size of 5 for pre-training (see Discussion on why not to use it in training). We did not use attention dropout, but 0.1 residual dropout rate was used. In pretraining and training sentences of more than 50 tokens were discarded. Pretraining and training were considered finished when BLEU did not increase in the development set for 10 consecutive evaluations, and evaluation was done every 1,000 and 5,000 for batches of size 100 and 256 for pretraining and training respectively. Learning rate used for rmsprop (Tieleman & Hinton, 2012) was 0.01 in pretraining and for adam (Kingma & Ba, 2015) with decay was 0.005 for training. 4,000 learning rate warm up steps were used. Pretraining took about 7 days with 4 GPUs, afterwards, training took roughly the same time. Monte Carlo used 20 sentence rolls per word. ",
+        "bbox": [
+            174,
+            583,
+            825,
+            790
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4 DETAILED RESULTS FOR CONSTANT REWARD SETTING",
+        "text_level": 1,
+        "bbox": [
+            176,
+            808,
+            602,
+            821
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We present graphs for the constant reward setting in Figures 8 and 7. Trends are similar to the ones obtained for the Simulated Reward setting. ",
+        "bbox": [
+            176,
+            833,
+            823,
+            862
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/f8626e73f517af6d8522a5bf6d4508414c5c3f4934d6cdcdc1290323a0e87186.jpg",
+        "image_caption": [
+            "Figure 8: Difference between the ranks of $y _ { b e s t }$ in the reinforced with constant reward and the pretrained model. Each column $_ x$ corresponds to the difference in the probability that $y _ { b e s t }$ is ranked in rank $x$ in the reinforced model, and the same probability in the pretrained model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            197,
+            318,
+            790,
+            655
+        ],
+        "page_idx": 13
+    }
+]

parse/train/Hyx6Bi0qYm/Hyx6Bi0qYm.md

@@ -0,0 +1,188 @@
+# ADVERSARIAL DOMAIN ADAPTATION FOR STABLE BRAIN-MACHINE INTERFACES
+
+Ali Farshchian, Juan A. Gallego, Lee E. Miller & Sara A. Solla Northwestern University, Evanston, IL, USA {a-farshchiansadegh,juan.gallego,lm,solla}@northwestern.edu
+
+Joseph P. Cohen & Yoshua Bengio   
+University of Montreal, Montreal, Canada   
+cohenjos@iro.umontreal.ca,yoshua.bengio@umontreal.ca
+
+# ABSTRACT
+
+Brain-Machine Interfaces (BMIs) have recently emerged as a clinically viable option to restore voluntary movements after paralysis. These devices are based on the ability to extract information about movement intent from neural signals recorded using multi-electrode arrays chronically implanted in the motor cortices of the brain. However, the inherent loss and turnover of recorded neurons requires repeated recalibrations of the interface, which can potentially alter the day-to-day user experience. The resulting need for continued user adaptation interferes with the natural, subconscious use of the BMI. Here, we introduce a new computational approach that decodes movement intent from a low-dimensional latent representation of the neural data. We implement various domain adaptation methods to stabilize the interface over significantly long times. This includes Canonical Correlation Analysis used to align the latent variables across days; this method requires prior point-to-point correspondence of the time series across domains. Alternatively, we match the empirical probability distributions of the latent variables across days through the minimization of their Kullback-Leibler divergence. These two methods provide a significant and comparable improvement in the performance of the interface. However, implementation of an Adversarial Domain Adaptation Network trained to match the empirical probability distribution of the residuals of the reconstructed neural signals outperforms the two methods based on latent variables, while requiring remarkably few data points to solve the domain adaptation problem.
+
+# 1 INTRODUCTION
+
+Individuals with tetraplegia due to spinal cord injury identify restoration of hand function as their highest priority (Anderson, 2004). Over $50 \%$ of respondents with a C1-C4 injury would be willing to undergo brain surgery to restore grasp (Blabe et al., 2015). Brain-Machine Interfaces (BMIs) aim to restore motor function by extracting movement intent from neural signals. Despite its great promise, current BMI technology has significant limitations. A BMI that maps neural activity in primary motor cortex (M1) onto motor intent commands should ideally provide a stable day-to-day user experience. However, the gradual alterations of the activity recorded by chronically implanted multi-electrode arrays, due to neuron turnover or electrode movement and failure (Barrese et al., 2013), causes considerable variation in the actions produced by the BMI. This turnover may occur within a single day (Downey et al., 2018), and is estimated to be on the order of $40 \%$ over two weeks (Dickey et al., 2009). In the face of changing neural signals, performance can be maintained by daily retraining the interface, but this is not a viable solution as it requires the user to keep on adapting to a new interface (Ajiboye et al., 2017).
+
+There is a high degree of correlation across the M1 neural signals. This redundancy implies that the dimensionality of the underlying motor command is much lower than the number of M1 neurons, and even lower than the number of recorded M1 neurons (Gallego et al., 2017). The use of dimensionality reduction methods is thus a common practice in BMI design, as it provides a more compact and denoised representation of neural activity, and a low-dimensional predictor of movement intent. Most of the earlier work used linear dimensionality reduction methods such as Principal Components Analysis (PCA) and Factor Analysis (FA) (Yu et al., 2009; Shenoy et al., 2013; Sadtler et al., 2014; Gallego et al., 2017); more recently, autoencoders (AEs) have been used for the nonlinear dimensionality reduction of neural signals (Pandarinath et al., 2018). Here we develop a deep learning architecture to extract a low-dimensional representation of recorded M1 activity constrained to include features related to movement intent. This is achieved by the simultaneous training of a deep, nonlinear autoencoder network based on neural signals from M1, and a network that predicts movement intent from the inferred low-dimensional signals. We show that this architecture significantly improves predictions over the standard sequential approach of first extracting a low-dimensional, latent representation of neural signals, followed by training a movement predictor based on the latent signals.
+
+To stabilize the resulting BMI against continuous changes in the neural recordings, we introduce a novel approach based on the Generative Adversarial Network (GAN) architecture (Goodfellow et al., 2014). This new approach, the Adversarial Domain Adaptation Network (ADAN), focuses on the probability distribution function (PDF) of the residuals of the reconstructed neural signals to align the residual’s PDF at a later day to the PDF of the first day the BMI was calculated. The alignment of residual PDFs results in the alignment of the PDFs of the neural data and of their latent representation across multiple days. We show that this method results in a significantly more stable performance of the BMI over time than the stability achieved using several other domain adaptation methods. The use of an ADAN thus results in a BMI that remains stable and consistent to the user over long periods of time. A successful domain adaptation of the neural data eliminates the need for frequent recalibration of the BMI, which remains fixed. This strategy is expected to alleviate the cognitive burden on the user, who would no longer need to learn novel strategies to compensate for a changing interface.
+
+# 2 RELATED WORK
+
+Current approaches to solving the stability problem for BMIs based on spiking activity recorded using chronically implanted multi-electrode arrays include gradually updating interface parameters using an exponentially weighted sliding average (Orsborn et al., 2012; Dangi et al., 2013), adjusting interface parameters by tracking recording nonstationarities (Zhang & Chase, 2013; Bishop et al., 2014) or by retrospectively inferring the user intention among a set of fixed targets (Jarosiewicz et al., 2015), cancelling out neural fluctuations by projecting the recorded activity onto a very lowdimensional space (Nuyujukian et al., 2014), training the interface with large data volumes collected over a period of several months to achieve robustness against future changes in neural recordings (Sussillo et al., 2016), and a semi-unsupervised approach based on aligning the PDF of newly predicted movements to a previously established PDF of typical movements (Dyer et al., 2017).
+
+Other approaches, more similar to ours, are based on the assumption that the relationship between latent dynamics and movement intent will remain stable despite changes in the recorded neural signals. Recent studies reveal the potential of latent dynamics for BMI stability. Kao et al. (2017) use past information about population dynamics to partially stabilize a BMI even under severe loss of recorded neurons, by aligning the remaining neurons to previously learned dynamics. Pandarinath et al. (2018) extract a single latent space from concatenating neural recordings over five months, and show that a predictor of movement kinematics based on these latent signals is reliable across all the recorded sessions.
+
+# 3 EXPERIMENTAL SETUP
+
+A male rhesus monkey (Macaca mulatta) sat in a primate chair with the forearm restrained and its hand secured into a padded, custom fit box. A torque cell with six degrees of freedom was mounted onto the box. The monkey was trained to generate isometric torques that controlled a computer cursor displayed on a screen placed at eye-level, and performed a 2D center-out task in which the cursor moved from a central target to one of eight peripheral targets equally spaced along a circle centered on the central target (Figure 1A) (Oby et al., 2012).
+
+To record neural activity, we implanted a 96-channel microelectrode array (Blackrock Microsystems, Salt Lake City, Utah) into the hand area of primary motor cortex (M1). Prior to implanting the array, we intraoperatively identified the hand area of M1 through sulcal landmarks, and by stimulating the surface of the cortex to elicit twitches of the wrist and hand muscles. We also implanted electrodes in 14 muscles of the forearm and hand, allowing us to record the electromyograms (EMGs) that quantify the level of activity in each of the muscles. Data was collected in five experimental sessions spanning 16 days. All methods were approved by Northwestern Universitys IACUC and carried out in accordance with the Guide for the Care and Use of Laboratory Animals.
+
+# 4 METHODS
+
+# 4.1 COMPUTATIONAL INTERFACE
+
+Our goal is to reliably predict the actual patterns of muscle activity during task execution, based on the recorded neural signals. Similar real-time predictions of kinematics are the basis of BMIs used to provide control of a computer cursor or a robotic limb to a paralyzed person (Taylor et al., 2002; Hochberg et al., 2006; Collinger et al., 2013). Predictions of muscle activity have been used to control the intensity of electrical stimulation of muscles that are temporarily paralyzed by a pharmacological peripheral nerve block (Ethier et al., 2012), a procedure that effectively bypasses the spinal cord to restore voluntary control of the paralyzed muscles. Similar methods have been attempted recently in humans (Bouton et al., 2016; Ajiboye et al., 2017).
+
+The BMI is a computational interface that transforms neural signals into command signals for movement control, in this case the EMG patterns. Here we propose an interface that consists of two components, a neural autoencoder (AE) and an EMG predictor (Figure 1B). The AE is a fully connected multilayer network consisting of an input layer, five layers of hidden units, and an output layer. The reconstruction aims at minimizing the mean square error (MSE) between input and output signals. Units in the latent and output layers implement linear readouts of the activity of the preceding layer. Units in the remaining hidden layers implement a linear readout followed by an exponential nonlinearity. To provide inputs to the AE, we start with neural data $\mathbf { } _  \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf { }$ consisting of the spike trains recorded from $n$ electrodes at time $t$ . We bin neural spikes at $5 0 ~ \mathrm { m s }$ intervals and apply a Gaussian filter with a standard deviation of $1 2 5 ~ \mathrm { m s }$ to the binned spike counts to obtain $n$ -dimensional smoothed firing rates $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ . The output layer provides $n$ -dimensional estimates $\widehat { \mathbf { x } } _ { t }$ of the inputs $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ . The latent layer activity ${ \boldsymbol { z } } _ { t }$ consist of $l$ latent variables, with $l < n$ .
+
+The EMG data ${ \mathbf { } } _ { \pmb { y } _ { t } }$ is the envelope of the muscle activity recorded from $m$ muscles, with $m < n$ . The $l$ -dimensional latent activity ${ \boldsymbol { z } } _ { t }$ is mapped onto the $m$ -dimensional EMGs through a long-short term memory (LSTM) (Hochreiter & Schmidhuber, 1997) layer with $m$ units, followed by a linear layer $\widehat { \pmb { y } } _ { t } = \dot { \pmb { W } } ^ { T } L S T M ( \pmb { z } _ { t } )$ . To train the model, we minimize a loss function $\mathcal { L }$ that simultaneously baccounts for two losses: $\mathcal { L } ^ { \dot { \pmb { x } } } : \mathbb { R } ^ { n }  \mathbb { R } ^ { + }$ is the MSE of the reconstruction of the smooth firing rates, and $\mathcal { L } ^ { y } : \mathbb { R } ^ { m }  \mathbb { R } ^ { + }$ is the MSE of the EMG predictions:
+
+$$
+\mathcal { L } = \lambda \mathcal { L } ^ { \pmb { x } } + \mathcal { L } ^ { \pmb { y } } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \left( \lambda | | \widehat { \pmb x } _ { t } - { \pmb x } _ { t } | | ^ { 2 } + | | \widehat { \pmb y } _ { t } - { \pmb y } _ { t } | | ^ { 2 } \right)
+$$
+
+Here $T$ is the number of time samples, and $t$ labels the time-ordered data points that constitute the training set. The factor $\lambda$ that multiplies the AE loss adjusts for the different units and different value ranges of firing rates and EMGs; it equalizes the contributions of the two terms in the loss function so that the learning algorithm does not prioritize one over the other. The value of $\lambda$ is updated for each new training iteration; it is computed as the ratio $\begin{array} { r } { \lambda = \frac { \mathcal { L } ^ { y } } { \mathcal { L } ^ { x } } } \end{array}$ of the respective losses at the end of the preceding iteration. Once the neural AE and the EMG predictor networks have been trained on the data acquired on the first recording session, indicated as day-0, their weights remain fixed.
+
+# 4.2 DOMAIN ADAPTATION
+
+To stabilize a fixed BMI, we need to align the latent space of later days to that of the first day, when the fixed interface was initially built. This step is necessary to provide statistically stationary inputs to the EMG predictor. We first use two different approaches to align latent variables across days: Canonical Correlation Analysis (CCA) between latent trajectories and Kullback-Leibler (KL)
+
+![](images/cfbcb111a80d57cb5ed1486983e8932baa2c61db93b5efacaab21f3a690f7d2b.jpg)  
+Figure 1: Experimental setup and methods. A. The isometric wrist center-out task with its eight targets, color coded. BMI schematics: recorded neural activity predicts muscle activity. B. The BMI consists of two networks: a neural AE and an EMG predictor. Recorded neural activity is binned and smoothed to provide an input to the AE. The activity of the low-dimensional latent space provides an input to the predictor of muscle activity. C. The ADAN architecture that aligns the firing rates of day- $k$ to those of day-0, when the BMI was built.
+
+divergence minimization between latent distributions. We then use an ADAN to align the distributions of the residuals of the reconstructed neural data. This procedure results in the alignment of the distributions of the neural data and of their latent representation across days.
+
+# 4.2.1 CANONICAL CORRELATION ANALYSIS (CCA)
+
+Consider the latent activities $Z _ { 0 }$ corresponding to day-0 and $Z _ { k }$ corresponding to a later day- $k$ ; the AE is fixed after being trained with day-0 data. Both $Z _ { 0 }$ and $Z _ { k }$ are matrices of dimension $l$ by $8 \tau$ , where $l$ is the dimensionality of the latent space and $\tau$ is the average time duration of each trial; the factor of 8 arises from concatenating the averaged latent activities for each of the eight targets. The goal of CCA is to find a linear transformation of the latent variables $Z _ { k }$ so that they are maximally correlated with a linear transformation of the latent variables $Z _ { 0 }$ (Bach & Jordan, 2002). This well established approach, involving only linear algebra, has been successfully applied to the analysis of M1 neural data (Sussillo et al., 2015; Gallego et al., 2018; Russo et al., 2018). In summary, the analysis starts with a $Q R$ decomposition of the transposed latent activity matrices, $Z _ { 0 } ^ { T } = \dot { Q _ { 0 } } R _ { 0 }$ , $Z _ { k } ^ { T } = Q _ { k } R _ { k }$ . Next, we construct the inner product matrix of $Q _ { 0 }$ and $Q _ { k }$ , and perform a singular value decomposition to obtain $Q _ { 0 } ^ { T } Q _ { k } = U S V ^ { T }$ . The new latent space directions along which correlations are maximized are given by $M _ { 0 } = R _ { 0 } ^ { - 1 } U$ , and $M _ { k } = R _ { k } ^ { - 1 } V$ , respectively.
+
+The implementation of CCA requires a one-to-one correspondence between data points in the two sets; this restricts its application to neural data that can be matched in time across days. Matching is achieved here through the repeated execution of highly stereotypic movements; the correspondence is then established by pairing average trials to a given target across different days. In a real-life scenario, motor behaviors are not structured and moment-to-moment movement intent is less clear, interfering with the possibility of establishing such a correspondence. Alignment using CCA requires a supervised calibration through the repetition of stereotyped tasks, but ideally the alignment would be achieved based on data obtained during natural, voluntary movements. A successful unsupervised approach to the alignment problem is thus highly desirable.
+
+# 4.2.2 KULLBACK-LEIBLER DIVERGENCE MINIMIZATION (KLDM)
+
+For the unsupervised approach, we seek to match the probability distribution of the latent variables of day- $k$ to that of day-0, without a need for the point-to-point correspondence provided in CCA by their time sequence. We use the fixed AE trained on day-0 data to map the neural data of day-0 and day- $k$ onto two sets of $l$ -dimensional latent variables, $\{ z _ { 0 } \}$ and $\left\{ z _ { k } \right\}$ , respectively. We then compute the mean and covariance matrices for each of these two empirical distributions, and capture their first and second order statistics by approximating these two distributions by multivariate Gaussians: $p _ { 0 } ( z _ { 0 } ) \sim \mathcal { N } ( z _ { 0 } ; \pmb { \mu } _ { 0 } , \pmb { \Sigma } _ { 0 } )$ and $\dot { p _ { k } } ( \bar { z } _ { k } ^ { - } ) \sim \mathcal { N } ( z _ { k } ; \mu _ { k } , \Sigma _ { k } )$ . We then minimize the KL divergence between them,
+
+$$
+D _ { \mathrm { K L } } ( p _ { k } ( z _ { k } ) | | p _ { 0 } ( z _ { 0 } ) ) = { \frac { 1 } { 2 } } { \Big ( } t r ( \Sigma _ { 0 } ^ { - 1 } \Sigma _ { k } ) + ( \mu _ { 0 } - \mu _ { k } ) ^ { T } \Sigma _ { 0 } ^ { - 1 } ( \mu _ { 0 } - \mu _ { k } ) ) - l + l n { \frac { | \Sigma _ { 0 } | } { | \Sigma _ { k } | } } { \Big ) }
+$$
+
+To minimize the KL divergence, we implemented a map from neural activity to latent activity using a network with the same architecture as the encoder section of the BMI’s AE. This network was initialized with the weights obtained after training the BMI’s AE on the day-0 data. Subsequent training was driven by the gradient of the cost function of equation 2, evaluated on a training set provided by day- $k$ recordings of neural activity. This process aligns the day- $k$ latent PDF to that of day-0 through two global linear operations: a translation through the match of the means, and a rotation through the match of the eigenvectors of the covariance matrices; a nonuniform scaling follows from the match of the eigenvalues of the covariance matrices.
+
+To improve on the Gaussian assumption for the distribution of latent variables, we have trained an alternative BMI in which the AE (Figure 1B) is replaced by a Variational AE (Kingma & Welling, 2013). We train the VAE by adding to the interface loss function (equation 1) a regularizer term: the Kullback-Leibler (KL) divergence $D _ { \mathrm { K L } } ( p _ { 0 } ( z _ { 0 } ) \vert \vert q ( z _ { 0 } ) )$ between the probability distribution $p _ { 0 } ( z _ { 0 } )$ of the latent activity on day-0 and $q ( z _ { 0 } ) = \mathcal { N } ( z _ { 0 } ; 0 , I )$ . The latent variables of the VAE are thus subject to the additional soft constraint of conforming to a normal distribution.
+
+# 4.2.3 ADVERSARIAL DOMAIN ADAPTATION NETWORK (ADAN)
+
+In addition to matching the probability distributions of latent variables of day- $k$ to those of day-0, we seek an alternative approach: to match the probability distributions of the residuals of the reconstructed firing rates (Zhao et al., 2016), as a proxy for matching the distributions of the neural recordings and their corresponding latent variables. To this end, we train an ADAN whose architecture is very similar to that of a Generative Adversarial Network (GAN): it consists of two deep neural networks, a distribution alignment module and a discriminator module (Figure 1C).
+
+The discriminator is an AE (Zhao et al., 2016) with the same architecture as the one used for the BMI (Figure 1B). The discriminator parameters $\theta _ { D }$ are initialized with the weights of the BMI AE, trained on the day-0 neural data. The goal of the discriminator is to maximize the difference between the neural reconstruction losses of day- $k$ and day-0. The great dissimilarity between the probability distribution of day-0 residuals and that of day- $k$ residuals obtained with the discriminator in its initialized state results in a strong signal that facilitates subsequent discriminator training.
+
+The distribution alignment module works as an adversary to the discriminator by minimizing the neural reconstruction losses of day- $k$ (Warde-Farley & Bengio, 2017). It consists of a hidden layer with exponential units and a linear readout layer, each with $n$ fully connected units. The aligner parameters $\theta _ { A }$ , the weights of the $n$ by $n$ connectivity matrices from input to hidden and from hidden to output, are initialized as the corresponding identity matrices. The aligner module receives as inputs the firing rates $X _ { k }$ of day- $k$ . During training, the gradients through the discriminator bring the output $A ( X _ { k } )$ of the aligner closer to $X _ { 0 }$ . The adversarial mechanism provided by the discriminator allows us to achieve this alignment in an unsupervised manner.
+
+To train the ADAN, we need to quantify the reconstruction losses. Given input data $\boldsymbol { X }$ , the discriminator outputs $\hat { \pmb X } = \hat { \pmb X } ( \pmb X , \bar { \theta } _ { D } )$ , with residuals $R ( X , \theta _ { D } ) = \left( X - \hat { X } ( X , \theta _ { D } ) \right)$ . Consider the scalar reconstruction losses $\mathbf { r }$ obtained by taking the $L ^ { 1 }$ norm of each column of $\pmb { R }$ . Let $\rho _ { 0 }$ and $\rho _ { k }$ be the distributions of the scalar losses for day-0 and day- $k$ , respectively, and let $\mu _ { 0 }$ and $\mu _ { k }$ be their corresponding means. We measure the dissimilarity between these two distributions by a lower bound to the Wasserstein distance (Arjovsky et al., 2017), provided by the absolute value of the difference between the means: $W ( \rho _ { 0 } , \rho _ { k } ) \geq | \mu _ { 0 } - \mu _ { k } |$ (Berthelot et al., 2017). The discriminator is trained to learn features that discriminate between the two distributions by maximizing the corresponding Wasserstein distance. The discriminator initialization implies $\mu _ { k } > \mu _ { 0 }$ when the ADAN training begins. By maximizing $\left( \mu _ { k } - \mu _ { 0 } \right)$ , equivalent to minimizing $\left( \mu _ { 0 } - \mu _ { k } \right)$ , this relation is maintained during training. Since scalar residuals and their means are nonnegative, the maximization of $W ( \rho _ { 0 } , \rho _ { k } )$ is achieved by decreasing $\mu _ { 0 }$ while increasing $\mu _ { k }$ .
+
+Given discriminator and aligner parameters $\theta _ { D }$ and $\theta _ { A }$ , respectively, the discriminator and aligner loss functions $\mathcal { L } _ { D }$ and $\mathcal { L } _ { A }$ to be minimized can be expressed as
+
+$$
+\left\{ \begin{array} { l l } { \mathcal { L } _ { D } = \mu _ { 0 } ( X _ { 0 } ; \theta _ { D } ) - \mu _ { k } ( A ( X _ { k } ; \theta _ { A } ) ; \theta _ { D } ) } & { \mathrm { ~ f o r } \theta _ { D } } \\ { \mathcal { L } _ { A } = \mu _ { k } ( A ( X _ { k } ; \theta _ { A } ) ; \theta _ { D } ) } & { \mathrm { ~ f o r } \theta _ { A } } \end{array} \right.
+$$
+
+# 5 RESULTS
+
+Figure 2A illustrates the firing rates $n = 9 6$ ), the latent variables $( l = 1 0 )$ ), the reconstructed firing rates, and the actual and predicted muscle activity for two representative muscles, a wrist flexor and an extensor, over a set of eight trials (one trial per target location) of a test data set randomly selected from day-0 data. The overall performance of the interface is summarized in Figure 2B, quantified using the percentage of the variance accounted for $( \% \mathrm { { V A F } ) }$ for five-fold cross-validated data. The blue bar shows the accuracy of EMG predictions directly obtained from the $n$ -dimensional firing rates, without the dimensionality reduction step; this EMG predictor consists of an LSTM layer with $n$ units, followed by a linear layer. The green bar shows the accuracy of EMG predictions when the interface is trained sequentially: first the AE is trained in an unsupervised manner, and then the EMG predictor is trained with the resulting $l$ -dimensional latent variables as input. The performance of the sequentially trained interface is worse than that of an EMG predictor trained directly on neural activity as input; the difference is small but significant (paired t-test, $\scriptstyle \mathrm { p = 0 . 0 0 6 } )$ ). In contrast, when the EMG predictor is trained simultaneously with the AE (red bar), there is no significant difference (paired t-test, $\mathrm { p { = } 0 . 9 7 1 }$ ) in performance between EMG predictions based on the $n$ -dimensional neural activity and the $l$ -dimensional latent variables. In simultaneous training, the AE is trained using the joint loss function of equation 1 that includes not only the unsupervised neural reconstruction loss but also a supervised regression loss that quantifies the quality of EMG predictions. Therefore, the supervision of the dimensionality reduction step through the integration of relevant movement information leads to a latent representation that better captures neural variability related to movement intent.
+
+For the implementation of the domain adaptation techniques, the interface was trained using only the data of day-0 and remained fixed afterward. Both CCA and KLDM were designed to match latent variables across days. Therefore, when using these methods, we first use the encoder part of the fixed AE to map neural activity of subsequent days onto latent activity, we then apply CCA or KLDM to align day- $k$ latent activity to that of day-0, and finally use the fixed EMG predictor to predict EMGs from the aligned latent activity. While KLDM explicitly seeks to match first and second order statistics of the latent variables through a Gaussian approximation, CCA aligns the latent variables using a point-to point correspondence across days provided by the latent trajectories. The effect of CCA alignment is illustrated in Figure 3A, where we show 2D t-SNE visualizations of 10D latent trajectories. Each trajectory is an average over all trials for a given target. The differences between day-16 and day-0 latent trajectories reflect the impact of turnover in the recorded neurons. Comparison between these two sets of trajectories reveals a variety of transformations, including nonuniform rotation, scaling, and skewing. In spite of the complexity of these transformations, the available point-to-point correspondence along these trajectories allows CCA to achieve a good alignment. The mechanism underlying KLDM alignment is illustrated in 3B, where we show the empirical probability distribution of the latent variables along a randomly chosen, representative direction within the 10D latent space. Results are shown for day-0 (blue), for day-16 (red), and for day-16 after alignment with KLDM (yellow). The effects of using a BMI based on a VAE instead of the AE are shown in Supplementary Figure S1.
+
+![](images/00ee1987bc8b1cd5138a59515ec4b8f0b0f5780b00d2a81e39f0a42f74026cd3.jpg)  
+Figure 2: Neural to muscle BMI. A. Example firing rates recorded from the hand area of primary motor cortex while the monkey performed the isometric wrist task; we also show latent variables and reconstructed firing rates. The quality of EMG predictions is illustrated by comparison to actual EMGs for two representative muscles, for each of the eight target directions. B. Performance comparison between EMG predictions from $n$ -dimensional firing rates (blue) and EMG predictions from $l$ -dimensional latent variables, obtained either by training the predictor sequentially (green) or simultaneously (red) with the neural AE. Error bars represent standard deviation of the mean.
+
+In contrast to CCA and KLDM, ADAN is designed to match the high-dimensional neural recordings across days via the $L ^ { 1 }$ norm of their residuals. In the ADAN architecture, the discriminator module is an AE that receives as inputs the neural activity of day-0 or day- $k$ and outputs their reconstructions. The residuals of the reconstruction of neural signals follow from the difference between the discriminator’s inputs and outputs. The ADAN aligns the day- $k$ residual statistics to those of day-0 by focusing on the $L ^ { 1 }$ norm of the residuals and minimizing the distance between these scalar PDFs across days; this procedure results in the alignment of the neural recordings and consequently their latent representation. The aligner module of ADAN aligns day- $k$ neural activity to that of day-0; this aligned neural activity is then used as input to the fixed BMI. Figure 3C shows the 1D distribution of the $L ^ { 1 }$ norm of the vector residuals, and in Figure 3D a 2D t-SNE (Maaten & Hinton, 2008) visualization of the vector residuals based on 1000 randomly sampled data points. Residuals correspond to the errors in firing rate reconstructions using the day-0 fixed AE for both day-0 data (blue) and day-16 data (red). Residuals for the day-16 data after alignment with ADAN are shown in yellow. Figure 3E shows the empirical probability distribution of the latent variables along the same representative, randomly chosen dimension within the 10D latent space used in Figure 3B. Results are shown for latent variables on day-0 using the fixed AE (blue), for the latent variables on day16 along the same dimension using the same, fixed AE (red), and for day-16 latent variables after alignment with ADAN (yellow). A 2D t-SNE visualization of latent variables aligned with ADAN is shown in comparison to the results of a simple center-and-scale alignment in Supplementary Figure S2.
+
+The performance of the BMI before and after domain adaptation with CCA, KLDM, and ADAN is summarized in Figure 4A and quantified using the $\% \mathrm { V A F }$ in EMG predictions. We report mean and standard deviation for five-fold cross-validated data. Blue symbols indicate the performance of an interface that is updated on each day; this provides an upper bound for the potential benefits of neural domain adaptation. Red symbols illustrate the natural deterioration in the performance of a fixed interface due to the gradual deterioration of neural recordings. Green, orange, and purple symbols indicate the extent to which the performance of a fixed interface improves after alignment using CCA, KLDM, and ADAN, respectively. The comparable performance of CCA and KLDM reflects that both methods achieve alignment based on latent statistics; the use of ADAN directly for latent space alignment does not produce better results than these two methods. In contrast, when ADAN is used for alignment based on residual statistics, interface stability improves. This ADAN provides a better alignment because the residuals amplify the mismatch that results when a fixed day-0 AE is applied to later-day data (see Figures 3C and D). Although the improvement achieved for day-16 with ADAN over its competitors is small, about $6 \%$ , it is statistically significant (one-way ANOVA with Tukey’s test, $p < 0 . 0 1 $ ). We have been unable to achieve this degree of improvement with any of the many other domain adaptation approaches we tried. This improvement is even more remarkable given that domain adaptation with ADAN requires a surprisingly small amount of data. Figure 4B shows the percentage improvement in EMG predictions as a function of the amount of training data. Subsequent symbols are obtained by adding 6s of data (120 samples) to the training set, and computing the average percentage improvement for the entire day $2 0 \mathrm { m i n }$ recordings), for all days after day-0. EMG prediction accuracy saturates at ${ \sim } 1 \mathrm { m i n }$ ; this need for a small training set is ideal for practical applications.
+
+![](images/e9b38601d4715d3ef1ecc5cd481252a0f21e2fd6c1bb9de7da21e12e49302a2a.jpg)  
+Figure 3: Domain adaptation. A. 2D t-SNE visualization of the averaged 10D latent neural trajectories as the monkey performed the isometric wrist task for day-0, day-16 before alignment, and day-16 after alignment with CCA. B. Probability distribution of the 10D latent variables along a randomly chosen, representative direction. We show the distribution at day-0, and the distribution at day-16 before and after alignment using KLDM. C. Probability distribution of the $L ^ { 1 }$ norm of the vector residuals of the reconstructed firing rates for day-0 and day-16, before and after adversarial alignment using ADAN. D. 2D t-SNE visualization of the vector residuals of the reconstructed firing rates for day-0 and day-16, before and after adversarial alignment using ADAN. E. Same as $\mathbf { B }$ , but for alignment using ADAN.
+
+![](images/f9ec77dbb1b97dbbacc163c530a6470f7cafb2507eb877dceb142bd58fac66de.jpg)  
+Figure 4: A. EMG prediction performance using a fixed BMI decoder. Blue symbols represent the sustained performance of interfaces retrained on a daily basis. Red symbols illustrate the deterioration in performance of a fixed interface without domain adaptation. The performance of a fixed interface after domain adaptation is shown for CCA (green), KLDM (orange), and ADAN (purple). Error bars represent standard deviation of the mean. B. Average improvements in EMG prediction performance for alignment using ADAN as a function of the amount of training data needed for domain adaptation at the beginning of each day, averaged over all days after day-0. Shading represents standard deviation of the mean.
+
+# 6 CONCLUSION
+
+We address the problem of stabilizing a fixed Brain-Machine Interface against performance deterioration due to the loss and turnover of recorded neural signals. We introduce a new approach to extracting a low-dimensional latent representation of the neural signals while simultaneously inferring movement intent. We then implement various domain adaption methods to stabilize the latent representation over time, including Canonical Correlation Analysis and the minimization of a Kullback-Leibler divergence. These two methods provide comparable improvement in the performance of the interface. We find that an Adversarial Domain Adaptation Network trained to match the empirical probability distribution of the residuals of the reconstructed neural recordings restores the latent representation of neural trajectories and outperforms the two methods based on latent variables, while requiring remarkably little data to solve the domain adaptation problem. In addition, ADAN solves the domain adaptation problem in a manner that is not task specific, and thus is potentially applicable to unconstrained movements.
+
+Here we report on improvements in interface stability obtained offline, without a user in the loop. Online, closed-loop performance is not particularly well correlated with offline accuracy; in an online evaluation of performance, the user’s ability to adapt at an unknown rate and to an unknown extent to an imperfect BMI obscures the performance improvements obtained with domain adaptation. Although the open-loop performance improvement demonstrated here is encouraging, additional experiments, both open and closed-loop, with additional animals and involving additional tasks, are required to fully validate our results and to establish that the improvements demonstrated here facilitate the sustained use of a brain-machine interface.
+
+# ACKNOWLEDGMENTS
+
+These results are based upon work supported the National Institute of Neurological Disorders and Stroke (NINDS) under Grant No. 5R01NS053603-12. We thank F. Mussa-Ivaldi, E. Perreault, X. Ma, K. Bodkin, E. Altan, and M. Medina for discussions and comments.
+
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+![](images/03baa277479065713f9a7a4d714f893b880ac9b717338f0dc002497dfd73e82a.jpg)  
+Figure S1: Variational autoencoder. A. EMG prediction performance using a different BMI, based on a VAE decoder, in comparison to the performance of a BMI based on the traditional autoencoder. Blue symbols represent the sustained performance of an interface retrained on a daily basis. Red symbols illustrate the deterioration in the performance of a fixed interface without domain adaptation. Orange symbols represent the performance of a fixed interface when the latent variables are aligned using KLDM. For each of these three cases, solid lines represent the performance of an AE-based BMI, and dashed lines that of a VAE-based BMI. Error bars represent standard deviation of the mean. B. Probability distribution of the 10D latent variables along the same dimension used in Figure 3B, now obtained with the fixed VAE trained on day-0. We show the distribution at day-0, and the distribution at day-16 before and after alignment using KLDM. In comparison to Figure 3B, the use of a VAE greatly improves the Gaussian nature of the latent variables’ distribution. However, this additional constraint in the autoencoder results in a slight deterioration of the BMI’s ability to predict EMGs, as shown in A.
+
+![](images/cee8d3124130855899869d5be03769982b0fd853864307f640e2b8d733f7e836.jpg)  
+Figure S2: Visualization of the probability distribution of latent variables in 2D using t-SNE. A. Latent variables on day-0. B. Latent variables on day-16, before alignment. C. Latent variables on day-16, after alignment to those of day-0 using T&S: a global translation to match the respective means followed by a global scaling to match the respective variances (yellow). Also shown, latent variables on day-0 (blue) on the same projection. D. Latent variables on day-16, after alignment to those of day-0 using ADAN (yellow). Also shown, latent variables on day-0 (blue) on the same projection.

parse/train/Hyx6Bi0qYm/Hyx6Bi0qYm_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "ADVERSARIAL DOMAIN ADAPTATION FOR STABLE BRAIN-MACHINE INTERFACES ",
+        "text_level": 1,
+        "bbox": [
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+            99,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ali Farshchian, Juan A. Gallego, Lee E. Miller & Sara A. Solla Northwestern University, Evanston, IL, USA {a-farshchiansadegh,juan.gallego,lm,solla}@northwestern.edu ",
+        "bbox": [
+            183,
+            170,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Joseph P. Cohen & Yoshua Bengio   \nUniversity of Montreal, Montreal, Canada   \ncohenjos@iro.umontreal.ca,yoshua.bengio@umontreal.ca ",
+        "bbox": [
+            183,
+            233,
+            692,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            313,
+            544,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Brain-Machine Interfaces (BMIs) have recently emerged as a clinically viable option to restore voluntary movements after paralysis. These devices are based on the ability to extract information about movement intent from neural signals recorded using multi-electrode arrays chronically implanted in the motor cortices of the brain. However, the inherent loss and turnover of recorded neurons requires repeated recalibrations of the interface, which can potentially alter the day-to-day user experience. The resulting need for continued user adaptation interferes with the natural, subconscious use of the BMI. Here, we introduce a new computational approach that decodes movement intent from a low-dimensional latent representation of the neural data. We implement various domain adaptation methods to stabilize the interface over significantly long times. This includes Canonical Correlation Analysis used to align the latent variables across days; this method requires prior point-to-point correspondence of the time series across domains. Alternatively, we match the empirical probability distributions of the latent variables across days through the minimization of their Kullback-Leibler divergence. These two methods provide a significant and comparable improvement in the performance of the interface. However, implementation of an Adversarial Domain Adaptation Network trained to match the empirical probability distribution of the residuals of the reconstructed neural signals outperforms the two methods based on latent variables, while requiring remarkably few data points to solve the domain adaptation problem. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            662,
+            336,
+            678
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Individuals with tetraplegia due to spinal cord injury identify restoration of hand function as their highest priority (Anderson, 2004). Over $50 \\%$ of respondents with a C1-C4 injury would be willing to undergo brain surgery to restore grasp (Blabe et al., 2015). Brain-Machine Interfaces (BMIs) aim to restore motor function by extracting movement intent from neural signals. Despite its great promise, current BMI technology has significant limitations. A BMI that maps neural activity in primary motor cortex (M1) onto motor intent commands should ideally provide a stable day-to-day user experience. However, the gradual alterations of the activity recorded by chronically implanted multi-electrode arrays, due to neuron turnover or electrode movement and failure (Barrese et al., 2013), causes considerable variation in the actions produced by the BMI. This turnover may occur within a single day (Downey et al., 2018), and is estimated to be on the order of $40 \\%$ over two weeks (Dickey et al., 2009). In the face of changing neural signals, performance can be maintained by daily retraining the interface, but this is not a viable solution as it requires the user to keep on adapting to a new interface (Ajiboye et al., 2017). ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "There is a high degree of correlation across the M1 neural signals. This redundancy implies that the dimensionality of the underlying motor command is much lower than the number of M1 neurons, and even lower than the number of recorded M1 neurons (Gallego et al., 2017). The use of dimensionality reduction methods is thus a common practice in BMI design, as it provides a more compact and denoised representation of neural activity, and a low-dimensional predictor of movement intent. Most of the earlier work used linear dimensionality reduction methods such as Principal Components Analysis (PCA) and Factor Analysis (FA) (Yu et al., 2009; Shenoy et al., 2013; Sadtler et al., 2014; Gallego et al., 2017); more recently, autoencoders (AEs) have been used for the nonlinear dimensionality reduction of neural signals (Pandarinath et al., 2018). Here we develop a deep learning architecture to extract a low-dimensional representation of recorded M1 activity constrained to include features related to movement intent. This is achieved by the simultaneous training of a deep, nonlinear autoencoder network based on neural signals from M1, and a network that predicts movement intent from the inferred low-dimensional signals. We show that this architecture significantly improves predictions over the standard sequential approach of first extracting a low-dimensional, latent representation of neural signals, followed by training a movement predictor based on the latent signals. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To stabilize the resulting BMI against continuous changes in the neural recordings, we introduce a novel approach based on the Generative Adversarial Network (GAN) architecture (Goodfellow et al., 2014). This new approach, the Adversarial Domain Adaptation Network (ADAN), focuses on the probability distribution function (PDF) of the residuals of the reconstructed neural signals to align the residual’s PDF at a later day to the PDF of the first day the BMI was calculated. The alignment of residual PDFs results in the alignment of the PDFs of the neural data and of their latent representation across multiple days. We show that this method results in a significantly more stable performance of the BMI over time than the stability achieved using several other domain adaptation methods. The use of an ADAN thus results in a BMI that remains stable and consistent to the user over long periods of time. A successful domain adaptation of the neural data eliminates the need for frequent recalibration of the BMI, which remains fixed. This strategy is expected to alleviate the cognitive burden on the user, who would no longer need to learn novel strategies to compensate for a changing interface. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            493,
+            344,
+            510
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Current approaches to solving the stability problem for BMIs based on spiking activity recorded using chronically implanted multi-electrode arrays include gradually updating interface parameters using an exponentially weighted sliding average (Orsborn et al., 2012; Dangi et al., 2013), adjusting interface parameters by tracking recording nonstationarities (Zhang & Chase, 2013; Bishop et al., 2014) or by retrospectively inferring the user intention among a set of fixed targets (Jarosiewicz et al., 2015), cancelling out neural fluctuations by projecting the recorded activity onto a very lowdimensional space (Nuyujukian et al., 2014), training the interface with large data volumes collected over a period of several months to achieve robustness against future changes in neural recordings (Sussillo et al., 2016), and a semi-unsupervised approach based on aligning the PDF of newly predicted movements to a previously established PDF of typical movements (Dyer et al., 2017). ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Other approaches, more similar to ours, are based on the assumption that the relationship between latent dynamics and movement intent will remain stable despite changes in the recorded neural signals. Recent studies reveal the potential of latent dynamics for BMI stability. Kao et al. (2017) use past information about population dynamics to partially stabilize a BMI even under severe loss of recorded neurons, by aligning the remaining neurons to previously learned dynamics. Pandarinath et al. (2018) extract a single latent space from concatenating neural recordings over five months, and show that a predictor of movement kinematics based on these latent signals is reliable across all the recorded sessions. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A male rhesus monkey (Macaca mulatta) sat in a primate chair with the forearm restrained and its hand secured into a padded, custom fit box. A torque cell with six degrees of freedom was mounted onto the box. The monkey was trained to generate isometric torques that controlled a computer cursor displayed on a screen placed at eye-level, and performed a 2D center-out task in which the cursor moved from a central target to one of eight peripheral targets equally spaced along a circle centered on the central target (Figure 1A) (Oby et al., 2012). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To record neural activity, we implanted a 96-channel microelectrode array (Blackrock Microsystems, Salt Lake City, Utah) into the hand area of primary motor cortex (M1). Prior to implanting the array, we intraoperatively identified the hand area of M1 through sulcal landmarks, and by stimulating the surface of the cortex to elicit twitches of the wrist and hand muscles. We also implanted electrodes in 14 muscles of the forearm and hand, allowing us to record the electromyograms (EMGs) that quantify the level of activity in each of the muscles. Data was collected in five experimental sessions spanning 16 days. All methods were approved by Northwestern Universitys IACUC and carried out in accordance with the Guide for the Care and Use of Laboratory Animals. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 METHODS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "4.1 COMPUTATIONAL INTERFACE ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Our goal is to reliably predict the actual patterns of muscle activity during task execution, based on the recorded neural signals. Similar real-time predictions of kinematics are the basis of BMIs used to provide control of a computer cursor or a robotic limb to a paralyzed person (Taylor et al., 2002; Hochberg et al., 2006; Collinger et al., 2013). Predictions of muscle activity have been used to control the intensity of electrical stimulation of muscles that are temporarily paralyzed by a pharmacological peripheral nerve block (Ethier et al., 2012), a procedure that effectively bypasses the spinal cord to restore voluntary control of the paralyzed muscles. Similar methods have been attempted recently in humans (Bouton et al., 2016; Ajiboye et al., 2017). ",
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+    {
+        "type": "text",
+        "text": "The BMI is a computational interface that transforms neural signals into command signals for movement control, in this case the EMG patterns. Here we propose an interface that consists of two components, a neural autoencoder (AE) and an EMG predictor (Figure 1B). The AE is a fully connected multilayer network consisting of an input layer, five layers of hidden units, and an output layer. The reconstruction aims at minimizing the mean square error (MSE) between input and output signals. Units in the latent and output layers implement linear readouts of the activity of the preceding layer. Units in the remaining hidden layers implement a linear readout followed by an exponential nonlinearity. To provide inputs to the AE, we start with neural data $\\mathbf { } _  \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf { }$ consisting of the spike trains recorded from $n$ electrodes at time $t$ . We bin neural spikes at $5 0 ~ \\mathrm { m s }$ intervals and apply a Gaussian filter with a standard deviation of $1 2 5 ~ \\mathrm { m s }$ to the binned spike counts to obtain $n$ -dimensional smoothed firing rates $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ . The output layer provides $n$ -dimensional estimates $\\widehat { \\mathbf { x } } _ { t }$ of the inputs $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ . The latent layer activity ${ \\boldsymbol { z } } _ { t }$ consist of $l$ latent variables, with $l < n$ . ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The EMG data ${ \\mathbf { } } _ { \\pmb { y } _ { t } }$ is the envelope of the muscle activity recorded from $m$ muscles, with $m < n$ . The $l$ -dimensional latent activity ${ \\boldsymbol { z } } _ { t }$ is mapped onto the $m$ -dimensional EMGs through a long-short term memory (LSTM) (Hochreiter & Schmidhuber, 1997) layer with $m$ units, followed by a linear layer $\\widehat { \\pmb { y } } _ { t } = \\dot { \\pmb { W } } ^ { T } L S T M ( \\pmb { z } _ { t } )$ . To train the model, we minimize a loss function $\\mathcal { L }$ that simultaneously baccounts for two losses: $\\mathcal { L } ^ { \\dot { \\pmb { x } } } : \\mathbb { R } ^ { n }  \\mathbb { R } ^ { + }$ is the MSE of the reconstruction of the smooth firing rates, and $\\mathcal { L } ^ { y } : \\mathbb { R } ^ { m }  \\mathbb { R } ^ { + }$ is the MSE of the EMG predictions: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/23f2fcbb2a9abf6e8140397cadccbf6074b3c7f79c3c639d940fb1c0f58939c9.jpg",
+        "text": "$$\n\\mathcal { L } = \\lambda \\mathcal { L } ^ { \\pmb { x } } + \\mathcal { L } ^ { \\pmb { y } } = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\left( \\lambda | | \\widehat { \\pmb x } _ { t } - { \\pmb x } _ { t } | | ^ { 2 } + | | \\widehat { \\pmb y } _ { t } - { \\pmb y } _ { t } | | ^ { 2 } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here $T$ is the number of time samples, and $t$ labels the time-ordered data points that constitute the training set. The factor $\\lambda$ that multiplies the AE loss adjusts for the different units and different value ranges of firing rates and EMGs; it equalizes the contributions of the two terms in the loss function so that the learning algorithm does not prioritize one over the other. The value of $\\lambda$ is updated for each new training iteration; it is computed as the ratio $\\begin{array} { r } { \\lambda = \\frac { \\mathcal { L } ^ { y } } { \\mathcal { L } ^ { x } } } \\end{array}$ of the respective losses at the end of the preceding iteration. Once the neural AE and the EMG predictor networks have been trained on the data acquired on the first recording session, indicated as day-0, their weights remain fixed. ",
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+    {
+        "type": "text",
+        "text": "4.2 DOMAIN ADAPTATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "To stabilize a fixed BMI, we need to align the latent space of later days to that of the first day, when the fixed interface was initially built. This step is necessary to provide statistically stationary inputs to the EMG predictor. We first use two different approaches to align latent variables across days: Canonical Correlation Analysis (CCA) between latent trajectories and Kullback-Leibler (KL) ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/cfbcb111a80d57cb5ed1486983e8932baa2c61db93b5efacaab21f3a690f7d2b.jpg",
+        "image_caption": [
+            "Figure 1: Experimental setup and methods. A. The isometric wrist center-out task with its eight targets, color coded. BMI schematics: recorded neural activity predicts muscle activity. B. The BMI consists of two networks: a neural AE and an EMG predictor. Recorded neural activity is binned and smoothed to provide an input to the AE. The activity of the low-dimensional latent space provides an input to the predictor of muscle activity. C. The ADAN architecture that aligns the firing rates of day- $k$ to those of day-0, when the BMI was built. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "divergence minimization between latent distributions. We then use an ADAN to align the distributions of the residuals of the reconstructed neural data. This procedure results in the alignment of the distributions of the neural data and of their latent representation across days. ",
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+    {
+        "type": "text",
+        "text": "4.2.1 CANONICAL CORRELATION ANALYSIS (CCA) ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Consider the latent activities $Z _ { 0 }$ corresponding to day-0 and $Z _ { k }$ corresponding to a later day- $k$ ; the AE is fixed after being trained with day-0 data. Both $Z _ { 0 }$ and $Z _ { k }$ are matrices of dimension $l$ by $8 \\tau$ , where $l$ is the dimensionality of the latent space and $\\tau$ is the average time duration of each trial; the factor of 8 arises from concatenating the averaged latent activities for each of the eight targets. The goal of CCA is to find a linear transformation of the latent variables $Z _ { k }$ so that they are maximally correlated with a linear transformation of the latent variables $Z _ { 0 }$ (Bach & Jordan, 2002). This well established approach, involving only linear algebra, has been successfully applied to the analysis of M1 neural data (Sussillo et al., 2015; Gallego et al., 2018; Russo et al., 2018). In summary, the analysis starts with a $Q R$ decomposition of the transposed latent activity matrices, $Z _ { 0 } ^ { T } = \\dot { Q _ { 0 } } R _ { 0 }$ , $Z _ { k } ^ { T } = Q _ { k } R _ { k }$ . Next, we construct the inner product matrix of $Q _ { 0 }$ and $Q _ { k }$ , and perform a singular value decomposition to obtain $Q _ { 0 } ^ { T } Q _ { k } = U S V ^ { T }$ . The new latent space directions along which correlations are maximized are given by $M _ { 0 } = R _ { 0 } ^ { - 1 } U$ , and $M _ { k } = R _ { k } ^ { - 1 } V$ , respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The implementation of CCA requires a one-to-one correspondence between data points in the two sets; this restricts its application to neural data that can be matched in time across days. Matching is achieved here through the repeated execution of highly stereotypic movements; the correspondence is then established by pairing average trials to a given target across different days. In a real-life scenario, motor behaviors are not structured and moment-to-moment movement intent is less clear, interfering with the possibility of establishing such a correspondence. Alignment using CCA requires a supervised calibration through the repetition of stereotyped tasks, but ideally the alignment would be achieved based on data obtained during natural, voluntary movements. A successful unsupervised approach to the alignment problem is thus highly desirable. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2.2 KULLBACK-LEIBLER DIVERGENCE MINIMIZATION (KLDM) ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For the unsupervised approach, we seek to match the probability distribution of the latent variables of day- $k$ to that of day-0, without a need for the point-to-point correspondence provided in CCA by their time sequence. We use the fixed AE trained on day-0 data to map the neural data of day-0 and day- $k$ onto two sets of $l$ -dimensional latent variables, $\\{ z _ { 0 } \\}$ and $\\left\\{ z _ { k } \\right\\}$ , respectively. We then compute the mean and covariance matrices for each of these two empirical distributions, and capture their first and second order statistics by approximating these two distributions by multivariate Gaussians: $p _ { 0 } ( z _ { 0 } ) \\sim \\mathcal { N } ( z _ { 0 } ; \\pmb { \\mu } _ { 0 } , \\pmb { \\Sigma } _ { 0 } )$ and $\\dot { p _ { k } } ( \\bar { z } _ { k } ^ { - } ) \\sim \\mathcal { N } ( z _ { k } ; \\mu _ { k } , \\Sigma _ { k } )$ . We then minimize the KL divergence between them, ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2107aa93d500f3ea1c473b7d979aa1ac6f71d31c8a92754ccde252d44a72eeeb.jpg",
+        "text": "$$\nD _ { \\mathrm { K L } } ( p _ { k } ( z _ { k } ) | | p _ { 0 } ( z _ { 0 } ) ) = { \\frac { 1 } { 2 } } { \\Big ( } t r ( \\Sigma _ { 0 } ^ { - 1 } \\Sigma _ { k } ) + ( \\mu _ { 0 } - \\mu _ { k } ) ^ { T } \\Sigma _ { 0 } ^ { - 1 } ( \\mu _ { 0 } - \\mu _ { k } ) ) - l + l n { \\frac { | \\Sigma _ { 0 } | } { | \\Sigma _ { k } | } } { \\Big ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To minimize the KL divergence, we implemented a map from neural activity to latent activity using a network with the same architecture as the encoder section of the BMI’s AE. This network was initialized with the weights obtained after training the BMI’s AE on the day-0 data. Subsequent training was driven by the gradient of the cost function of equation 2, evaluated on a training set provided by day- $k$ recordings of neural activity. This process aligns the day- $k$ latent PDF to that of day-0 through two global linear operations: a translation through the match of the means, and a rotation through the match of the eigenvectors of the covariance matrices; a nonuniform scaling follows from the match of the eigenvalues of the covariance matrices. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To improve on the Gaussian assumption for the distribution of latent variables, we have trained an alternative BMI in which the AE (Figure 1B) is replaced by a Variational AE (Kingma & Welling, 2013). We train the VAE by adding to the interface loss function (equation 1) a regularizer term: the Kullback-Leibler (KL) divergence $D _ { \\mathrm { K L } } ( p _ { 0 } ( z _ { 0 } ) \\vert \\vert q ( z _ { 0 } ) )$ between the probability distribution $p _ { 0 } ( z _ { 0 } )$ of the latent activity on day-0 and $q ( z _ { 0 } ) = \\mathcal { N } ( z _ { 0 } ; 0 , I )$ . The latent variables of the VAE are thus subject to the additional soft constraint of conforming to a normal distribution. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.2.3 ADVERSARIAL DOMAIN ADAPTATION NETWORK (ADAN) ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In addition to matching the probability distributions of latent variables of day- $k$ to those of day-0, we seek an alternative approach: to match the probability distributions of the residuals of the reconstructed firing rates (Zhao et al., 2016), as a proxy for matching the distributions of the neural recordings and their corresponding latent variables. To this end, we train an ADAN whose architecture is very similar to that of a Generative Adversarial Network (GAN): it consists of two deep neural networks, a distribution alignment module and a discriminator module (Figure 1C). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The discriminator is an AE (Zhao et al., 2016) with the same architecture as the one used for the BMI (Figure 1B). The discriminator parameters $\\theta _ { D }$ are initialized with the weights of the BMI AE, trained on the day-0 neural data. The goal of the discriminator is to maximize the difference between the neural reconstruction losses of day- $k$ and day-0. The great dissimilarity between the probability distribution of day-0 residuals and that of day- $k$ residuals obtained with the discriminator in its initialized state results in a strong signal that facilitates subsequent discriminator training. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The distribution alignment module works as an adversary to the discriminator by minimizing the neural reconstruction losses of day- $k$ (Warde-Farley & Bengio, 2017). It consists of a hidden layer with exponential units and a linear readout layer, each with $n$ fully connected units. The aligner parameters $\\theta _ { A }$ , the weights of the $n$ by $n$ connectivity matrices from input to hidden and from hidden to output, are initialized as the corresponding identity matrices. The aligner module receives as inputs the firing rates $X _ { k }$ of day- $k$ . During training, the gradients through the discriminator bring the output $A ( X _ { k } )$ of the aligner closer to $X _ { 0 }$ . The adversarial mechanism provided by the discriminator allows us to achieve this alignment in an unsupervised manner. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To train the ADAN, we need to quantify the reconstruction losses. Given input data $\\boldsymbol { X }$ , the discriminator outputs $\\hat { \\pmb X } = \\hat { \\pmb X } ( \\pmb X , \\bar { \\theta } _ { D } )$ , with residuals $R ( X , \\theta _ { D } ) = \\left( X - \\hat { X } ( X , \\theta _ { D } ) \\right)$ . Consider the scalar reconstruction losses $\\mathbf { r }$ obtained by taking the $L ^ { 1 }$ norm of each column of $\\pmb { R }$ . Let $\\rho _ { 0 }$ and $\\rho _ { k }$ be the distributions of the scalar losses for day-0 and day- $k$ , respectively, and let $\\mu _ { 0 }$ and $\\mu _ { k }$ be their corresponding means. We measure the dissimilarity between these two distributions by a lower bound to the Wasserstein distance (Arjovsky et al., 2017), provided by the absolute value of the difference between the means: $W ( \\rho _ { 0 } , \\rho _ { k } ) \\geq | \\mu _ { 0 } - \\mu _ { k } |$ (Berthelot et al., 2017). The discriminator is trained to learn features that discriminate between the two distributions by maximizing the corresponding Wasserstein distance. The discriminator initialization implies $\\mu _ { k } > \\mu _ { 0 }$ when the ADAN training begins. By maximizing $\\left( \\mu _ { k } - \\mu _ { 0 } \\right)$ , equivalent to minimizing $\\left( \\mu _ { 0 } - \\mu _ { k } \\right)$ , this relation is maintained during training. Since scalar residuals and their means are nonnegative, the maximization of $W ( \\rho _ { 0 } , \\rho _ { k } )$ is achieved by decreasing $\\mu _ { 0 }$ while increasing $\\mu _ { k }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Given discriminator and aligner parameters $\\theta _ { D }$ and $\\theta _ { A }$ , respectively, the discriminator and aligner loss functions $\\mathcal { L } _ { D }$ and $\\mathcal { L } _ { A }$ to be minimized can be expressed as ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "equation",
+        "img_path": "images/079ad6f825be8ffa79a9b8f686cfbae77dd77df1b434333978646a6eaaea1e0d.jpg",
+        "text": "$$\n\\left\\{ \\begin{array} { l l } { \\mathcal { L } _ { D } = \\mu _ { 0 } ( X _ { 0 } ; \\theta _ { D } ) - \\mu _ { k } ( A ( X _ { k } ; \\theta _ { A } ) ; \\theta _ { D } ) } & { \\mathrm { ~ f o r } \\theta _ { D } } \\\\ { \\mathcal { L } _ { A } = \\mu _ { k } ( A ( X _ { k } ; \\theta _ { A } ) ; \\theta _ { D } ) } & { \\mathrm { ~ f o r } \\theta _ { A } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "5 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Figure 2A illustrates the firing rates $n = 9 6$ ), the latent variables $( l = 1 0 )$ ), the reconstructed firing rates, and the actual and predicted muscle activity for two representative muscles, a wrist flexor and an extensor, over a set of eight trials (one trial per target location) of a test data set randomly selected from day-0 data. The overall performance of the interface is summarized in Figure 2B, quantified using the percentage of the variance accounted for $( \\% \\mathrm { { V A F } ) }$ for five-fold cross-validated data. The blue bar shows the accuracy of EMG predictions directly obtained from the $n$ -dimensional firing rates, without the dimensionality reduction step; this EMG predictor consists of an LSTM layer with $n$ units, followed by a linear layer. The green bar shows the accuracy of EMG predictions when the interface is trained sequentially: first the AE is trained in an unsupervised manner, and then the EMG predictor is trained with the resulting $l$ -dimensional latent variables as input. The performance of the sequentially trained interface is worse than that of an EMG predictor trained directly on neural activity as input; the difference is small but significant (paired t-test, $\\scriptstyle \\mathrm { p = 0 . 0 0 6 } )$ ). In contrast, when the EMG predictor is trained simultaneously with the AE (red bar), there is no significant difference (paired t-test, $\\mathrm { p { = } 0 . 9 7 1 }$ ) in performance between EMG predictions based on the $n$ -dimensional neural activity and the $l$ -dimensional latent variables. In simultaneous training, the AE is trained using the joint loss function of equation 1 that includes not only the unsupervised neural reconstruction loss but also a supervised regression loss that quantifies the quality of EMG predictions. Therefore, the supervision of the dimensionality reduction step through the integration of relevant movement information leads to a latent representation that better captures neural variability related to movement intent. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For the implementation of the domain adaptation techniques, the interface was trained using only the data of day-0 and remained fixed afterward. Both CCA and KLDM were designed to match latent variables across days. Therefore, when using these methods, we first use the encoder part of the fixed AE to map neural activity of subsequent days onto latent activity, we then apply CCA or KLDM to align day- $k$ latent activity to that of day-0, and finally use the fixed EMG predictor to predict EMGs from the aligned latent activity. While KLDM explicitly seeks to match first and second order statistics of the latent variables through a Gaussian approximation, CCA aligns the latent variables using a point-to point correspondence across days provided by the latent trajectories. The effect of CCA alignment is illustrated in Figure 3A, where we show 2D t-SNE visualizations of 10D latent trajectories. Each trajectory is an average over all trials for a given target. The differences between day-16 and day-0 latent trajectories reflect the impact of turnover in the recorded neurons. Comparison between these two sets of trajectories reveals a variety of transformations, including nonuniform rotation, scaling, and skewing. In spite of the complexity of these transformations, the available point-to-point correspondence along these trajectories allows CCA to achieve a good alignment. The mechanism underlying KLDM alignment is illustrated in 3B, where we show the empirical probability distribution of the latent variables along a randomly chosen, representative direction within the 10D latent space. Results are shown for day-0 (blue), for day-16 (red), and for day-16 after alignment with KLDM (yellow). The effects of using a BMI based on a VAE instead of the AE are shown in Supplementary Figure S1. ",
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+    {
+        "type": "image",
+        "img_path": "images/00ee1987bc8b1cd5138a59515ec4b8f0b0f5780b00d2a81e39f0a42f74026cd3.jpg",
+        "image_caption": [
+            "Figure 2: Neural to muscle BMI. A. Example firing rates recorded from the hand area of primary motor cortex while the monkey performed the isometric wrist task; we also show latent variables and reconstructed firing rates. The quality of EMG predictions is illustrated by comparison to actual EMGs for two representative muscles, for each of the eight target directions. B. Performance comparison between EMG predictions from $n$ -dimensional firing rates (blue) and EMG predictions from $l$ -dimensional latent variables, obtained either by training the predictor sequentially (green) or simultaneously (red) with the neural AE. Error bars represent standard deviation of the mean. "
+        ],
+        "image_footnote": [],
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+        "text": "In contrast to CCA and KLDM, ADAN is designed to match the high-dimensional neural recordings across days via the $L ^ { 1 }$ norm of their residuals. In the ADAN architecture, the discriminator module is an AE that receives as inputs the neural activity of day-0 or day- $k$ and outputs their reconstructions. The residuals of the reconstruction of neural signals follow from the difference between the discriminator’s inputs and outputs. The ADAN aligns the day- $k$ residual statistics to those of day-0 by focusing on the $L ^ { 1 }$ norm of the residuals and minimizing the distance between these scalar PDFs across days; this procedure results in the alignment of the neural recordings and consequently their latent representation. The aligner module of ADAN aligns day- $k$ neural activity to that of day-0; this aligned neural activity is then used as input to the fixed BMI. Figure 3C shows the 1D distribution of the $L ^ { 1 }$ norm of the vector residuals, and in Figure 3D a 2D t-SNE (Maaten & Hinton, 2008) visualization of the vector residuals based on 1000 randomly sampled data points. Residuals correspond to the errors in firing rate reconstructions using the day-0 fixed AE for both day-0 data (blue) and day-16 data (red). Residuals for the day-16 data after alignment with ADAN are shown in yellow. Figure 3E shows the empirical probability distribution of the latent variables along the same representative, randomly chosen dimension within the 10D latent space used in Figure 3B. Results are shown for latent variables on day-0 using the fixed AE (blue), for the latent variables on day16 along the same dimension using the same, fixed AE (red), and for day-16 latent variables after alignment with ADAN (yellow). A 2D t-SNE visualization of latent variables aligned with ADAN is shown in comparison to the results of a simple center-and-scale alignment in Supplementary Figure S2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The performance of the BMI before and after domain adaptation with CCA, KLDM, and ADAN is summarized in Figure 4A and quantified using the $\\% \\mathrm { V A F }$ in EMG predictions. We report mean and standard deviation for five-fold cross-validated data. Blue symbols indicate the performance of an interface that is updated on each day; this provides an upper bound for the potential benefits of neural domain adaptation. Red symbols illustrate the natural deterioration in the performance of a fixed interface due to the gradual deterioration of neural recordings. Green, orange, and purple symbols indicate the extent to which the performance of a fixed interface improves after alignment using CCA, KLDM, and ADAN, respectively. The comparable performance of CCA and KLDM reflects that both methods achieve alignment based on latent statistics; the use of ADAN directly for latent space alignment does not produce better results than these two methods. In contrast, when ADAN is used for alignment based on residual statistics, interface stability improves. This ADAN provides a better alignment because the residuals amplify the mismatch that results when a fixed day-0 AE is applied to later-day data (see Figures 3C and D). Although the improvement achieved for day-16 with ADAN over its competitors is small, about $6 \\%$ , it is statistically significant (one-way ANOVA with Tukey’s test, $p < 0 . 0 1 $ ). We have been unable to achieve this degree of improvement with any of the many other domain adaptation approaches we tried. This improvement is even more remarkable given that domain adaptation with ADAN requires a surprisingly small amount of data. Figure 4B shows the percentage improvement in EMG predictions as a function of the amount of training data. Subsequent symbols are obtained by adding 6s of data (120 samples) to the training set, and computing the average percentage improvement for the entire day $2 0 \\mathrm { m i n }$ recordings), for all days after day-0. EMG prediction accuracy saturates at ${ \\sim } 1 \\mathrm { m i n }$ ; this need for a small training set is ideal for practical applications. ",
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+        "image_caption": [
+            "Figure 3: Domain adaptation. A. 2D t-SNE visualization of the averaged 10D latent neural trajectories as the monkey performed the isometric wrist task for day-0, day-16 before alignment, and day-16 after alignment with CCA. B. Probability distribution of the 10D latent variables along a randomly chosen, representative direction. We show the distribution at day-0, and the distribution at day-16 before and after alignment using KLDM. C. Probability distribution of the $L ^ { 1 }$ norm of the vector residuals of the reconstructed firing rates for day-0 and day-16, before and after adversarial alignment using ADAN. D. 2D t-SNE visualization of the vector residuals of the reconstructed firing rates for day-0 and day-16, before and after adversarial alignment using ADAN. E. Same as $\\mathbf { B }$ , but for alignment using ADAN. "
+        ],
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+    {
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+        "img_path": "images/f9ec77dbb1b97dbbacc163c530a6470f7cafb2507eb877dceb142bd58fac66de.jpg",
+        "image_caption": [
+            "Figure 4: A. EMG prediction performance using a fixed BMI decoder. Blue symbols represent the sustained performance of interfaces retrained on a daily basis. Red symbols illustrate the deterioration in performance of a fixed interface without domain adaptation. The performance of a fixed interface after domain adaptation is shown for CCA (green), KLDM (orange), and ADAN (purple). Error bars represent standard deviation of the mean. B. Average improvements in EMG prediction performance for alignment using ADAN as a function of the amount of training data needed for domain adaptation at the beginning of each day, averaged over all days after day-0. Shading represents standard deviation of the mean. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We address the problem of stabilizing a fixed Brain-Machine Interface against performance deterioration due to the loss and turnover of recorded neural signals. We introduce a new approach to extracting a low-dimensional latent representation of the neural signals while simultaneously inferring movement intent. We then implement various domain adaption methods to stabilize the latent representation over time, including Canonical Correlation Analysis and the minimization of a Kullback-Leibler divergence. These two methods provide comparable improvement in the performance of the interface. We find that an Adversarial Domain Adaptation Network trained to match the empirical probability distribution of the residuals of the reconstructed neural recordings restores the latent representation of neural trajectories and outperforms the two methods based on latent variables, while requiring remarkably little data to solve the domain adaptation problem. In addition, ADAN solves the domain adaptation problem in a manner that is not task specific, and thus is potentially applicable to unconstrained movements. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Here we report on improvements in interface stability obtained offline, without a user in the loop. Online, closed-loop performance is not particularly well correlated with offline accuracy; in an online evaluation of performance, the user’s ability to adapt at an unknown rate and to an unknown extent to an imperfect BMI obscures the performance improvements obtained with domain adaptation. Although the open-loop performance improvement demonstrated here is encouraging, additional experiments, both open and closed-loop, with additional animals and involving additional tasks, are required to fully validate our results and to establish that the improvements demonstrated here facilitate the sustained use of a brain-machine interface. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "These results are based upon work supported the National Institute of Neurological Disorders and Stroke (NINDS) under Grant No. 5R01NS053603-12. We thank F. Mussa-Ivaldi, E. Perreault, X. Ma, K. Bodkin, E. Altan, and M. Medina for discussions and comments. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
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+        "image_caption": [
+            "Figure S1: Variational autoencoder. A. EMG prediction performance using a different BMI, based on a VAE decoder, in comparison to the performance of a BMI based on the traditional autoencoder. Blue symbols represent the sustained performance of an interface retrained on a daily basis. Red symbols illustrate the deterioration in the performance of a fixed interface without domain adaptation. Orange symbols represent the performance of a fixed interface when the latent variables are aligned using KLDM. For each of these three cases, solid lines represent the performance of an AE-based BMI, and dashed lines that of a VAE-based BMI. Error bars represent standard deviation of the mean. B. Probability distribution of the 10D latent variables along the same dimension used in Figure 3B, now obtained with the fixed VAE trained on day-0. We show the distribution at day-0, and the distribution at day-16 before and after alignment using KLDM. In comparison to Figure 3B, the use of a VAE greatly improves the Gaussian nature of the latent variables’ distribution. However, this additional constraint in the autoencoder results in a slight deterioration of the BMI’s ability to predict EMGs, as shown in A. "
+        ],
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+        "image_caption": [
+            "Figure S2: Visualization of the probability distribution of latent variables in 2D using t-SNE. A. Latent variables on day-0. B. Latent variables on day-16, before alignment. C. Latent variables on day-16, after alignment to those of day-0 using T&S: a global translation to match the respective means followed by a global scaling to match the respective variances (yellow). Also shown, latent variables on day-0 (blue) on the same projection. D. Latent variables on day-16, after alignment to those of day-0 using ADAN (yellow). Also shown, latent variables on day-0 (blue) on the same projection. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 13
+    }
+]

parse/train/dBE8OI8_ZOa/dBE8OI8_ZOa.md

@@ -0,0 +1,269 @@
+# Dataset Distillation with Infinitely Wide Convolutional Networks
+
+# Timothy Nguyen† ∗
+
+Roman Novak♠
+
+Lechao Xiao♠
+
+DeepMind† Google Research, Brain Team♠ timothycnguyen@deepmind.com {romann, xlc, jaehlee}@google.com
+
+# Abstract
+
+The effectiveness of machine learning algorithms arises from being able to extract useful features from large amounts of data. As model and dataset sizes increase, dataset distillation methods that compress large datasets into significantly smaller yet highly performant ones will become valuable in terms of training efficiency and useful feature extraction. To that end, we apply a novel distributed kernel-based meta-learning framework to achieve state-of-the-art results for dataset distillation using infinitely wide convolutional neural networks. For instance, using only 10 datapoints $0 . 0 2 \%$ of original dataset), we obtain over $65 \%$ test accuracy on CIFAR10 image classification task, a dramatic improvement over the previous best test accuracy of $40 \%$ . Our state-of-the-art results extend across many other settings for MNIST, Fashion-MNIST, CIFAR-10, CIFAR-100, and SVHN. Furthermore, we perform some preliminary analyses of our distilled datasets to shed light on how they differ from naturally occurring data.
+
+# 1 Introduction
+
+Deep learning has become extraordinarily successful in a wide variety of settings through the availability of large datasets [Krizhevsky et al., 2012, Devlin et al., 2018, Brown et al., 2020, Dosovitskiy et al., 2020]. Such large datasets enable a neural network to learn useful representations of the data that are adapted to solving tasks of interest. Unfortunately, it can be prohibitively costly to acquire such large datasets and train a neural network for the requisite amount of time.
+
+One way to mitigate this problem is by constructing smaller datasets that are nevertheless informative. Some direct approaches to this include choosing a representative subset of the dataset (i.e. a coreset) or else performing a low-dimensional projection that reduces the number of features. However, such methods typically introduce a tradeoff between performance and dataset size, since what they produce is a coarse approximation of the full dataset. By contrast, the approach of dataset distillation is to synthesize datasets that are more informative than their natural counterparts when equalizing for dataset size [Wang et al., 2018, Bohdal et al., 2020, Nguyen et al., 2021, Zhao and Bilen, 2021]. Such resulting datasets will not arise from the distribution of natural images but will nevertheless capture features useful to a neural network, a capability which remains mysterious and is far from being well-understood [Ilyas et al., 2019, Huh et al., 2016, Hermann and Lampinen, 2020].
+
+The applications of such smaller, distilled datasets are diverse. For nonparametric methods that scale poorly with the training dataset (e.g. nearest-neighbors or kernel-ridge regression), having a reduced dataset decreases the associated memory and inference costs. For the training of neural networks, such distilled datasets have found several applications in the literature, including increasing the effectiveness of replay methods in continual learning [Borsos et al., 2020] and helping to accelerate neural architecture search [Zhao et al., 2021, Zhao and Bilen, 2021].
+
+In this paper, we perform a large-scale extension of the methods of Nguyen et al. [2021] to obtain new state-of-the-art (SOTA) dataset distillation results. Specifically, we apply the algorithms KIP (Kernel Inducing Points) and LS (Label Solve), first developed in Nguyen et al. [2021], to infinitely wide convolutional networks by implementing a novel, distributed meta-learning framework that draws upon hundreds of accelerators per training. The need for such resources is necessitated by the computational costs of using infinitely wide neural networks built out of components occurring in modern image classification models: convolutional and pooling layers (see $\ S$ for details). The consequence is that we obtain distilled datasets that are effective for both kernel ridge-regression and neural network training.
+
+Additionally, we initiate a preliminary study of the images and labels which KIP learns. We provide a visual and quantitative analysis of the data learned and find some surprising results concerning their interpretability and their dimensional and spectral properties. Given the efficacy of KIP and LS learned data, we believe a better understanding of them would aid in the understanding of feature learning in neural networks.
+
+To summarize, our contributions are as follows:
+
+1. We achieve SOTA dataset distillation results on a wide variety of datasets (MNIST, FashionMNIST, SVHN, CIFAR-10, CIFAR-100) for both kernel ridge-regression and neural network training. In several instances, our results achieve an impressively wide margin over prior art, including over $2 5 \%$ and $37 \%$ absolute gain in accuracy on CIFAR-10 and SVHN image classification, respectively, when using only 10 images (Tables 1, 2, A11).   
+2. We develop a novel, distributed meta-learning framework specifically tailored to the computational burdens of sophisticated neural kernels (§2.1).   
+3. We highlight and analyze some of the peculiar features of the distilled datasets we obtain, illustrating how they differ from natural data (§4).   
+4. We open source the distilled datasets, which used thousands of GPU hours, for the research community to further investigate at https://github.com/google-research/ google-research/tree/master/kip.
+
+# 2 Setup
+
+Background on infinitely wide convolutional networks. Recent literature has established that Bayesian and gradient-descent trained neural networks converge to Gaussian Processes (GP) as the number of hidden units in intermediary layers approaches infinity (see $\ S 5$ ). These results hold for many different architectures, including convolutional networks, which converge to a particular GP in the limit of infinite channels [Novak et al., 2019, Garriga-Alonso et al., 2019, Arora et al., 2019]. Bayesian networks are described by the Neural Network Gaussian Process (NNGP) kernel, while gradient descent networks are described by the Neural Tangent Kernel (NTK). Since we are interested in synthesizing datasets that can be used with both kernel methods and common gradient-descent trained neural networks, we focus on NTK in this work.
+
+Infinitely wide networks have been shown to achieve SOTA (among non-parametric kernels) results on image classification tasks [Novak et al., 2019, Arora et al., 2019, Li et al., 2019, Shankar et al., 2020, Bietti, 2021] and even rival finite-width networks in certain settings [Arora et al., 2020, Lee et al., 2020]. This makes such kernels especially suitable for our task. As convolutional models, they encode useful inductive biases of locality and translation invariance [Novak et al., 2019], which enable good generalization. Moreover, flexible and efficient computation of these kernels are possible due to the Neural Tangent library [Novak et al., 2020].
+
+Specific models considered. The central neural network (and corresponding infinite-width model) we consider is a simple 4-layer convolutional network with average pooling layers that we refer to as ConvNet throughout the text. This architecture is a slightly modified version of the default model used by Zhao and Bilen [2021], Zhao et al. [2021] and was chosen for ease of baselining (see $\ S \mathrm { A }$ for details). In several other settings we also consider convolutional networks without pooling layers ConvVec,2 and networks with no convolutions and only fully-connected layers FC. Depth of architecture (as measured by number of hidden layers) is indicated by an integer suffix.
+
+Background on algorithms. We review the Kernel Inducing Points (KIP) and Label Solve (LS) algorithms introduced by Nguyen et al. [2021]. Given a kernel $K$ , the kernel ridge-regression (KRR) loss function trained on a support dataset $( X _ { s } , y _ { s } )$ and evaluated on a target dataset $( X _ { t } , y _ { t } )$ is
+
+$$
+L ( X _ { s } , y _ { s } ) = \frac { 1 } { 2 } \left\| y _ { t } - K _ { X _ { t } X _ { s } } ( K _ { X _ { s } X _ { s } } + \lambda I ) ^ { - 1 } y _ { s } \right\| _ { 2 } ^ { 2 } ,
+$$
+
+where if $U$ and $V$ are sets, $K _ { U V }$ is the matrix of kernel elements $( K ( u , v ) ) _ { u \in U , v \in V }$ . Here $\lambda > 0$ is a fixed regularization parameter. The KIP algorithm consists of minimizing (1) with respect to the support set (either just the $X _ { s }$ or along with the labels $y _ { s }$ ). Here, we sample $( X _ { t } , y _ { t } )$ from a target dataset $\mathcal { D }$ at every (meta)step, and update the support set using gradient-based methods. Additional variations include augmenting the $X _ { t }$ or sampling a different kernel $K$ (from a fixed family of kernels) at each step.
+
+The Label Solve algorithm consists of solving for the least-norm minimizer of (1) with respect to $y _ { s }$ This yields the labels
+
+$$
+y _ { s } ^ { * } = \Big ( K _ { X _ { t } X _ { s } } ( K _ { X _ { s } X _ { s } } + \lambda I ) ^ { - 1 } \Big ) ^ { + } y _ { t } ,
+$$
+
+where $A ^ { + }$ denotes the pseudo-inverse of the matrix $A$ . Note that here $( X _ { t } , y _ { t } ) = \mathcal { D }$ , i.e. the labels are solved using the whole target set.
+
+In our applications of KIP and Label Solve, the target dataset $\mathcal { D }$ is always significantly larger than the support set $( X _ { s } , y _ { s } )$ . Hence, the learned support set or solved labels can be regarded as distilled versions of their respective targets. We also initialize our support images to be a subset of natural images, though they could also be initialized randomly.
+
+Based on the infinite-width correspondence outlined above and in $\ S 5$ , dataset distillation using KIP or LS that is optimized for KRR should extend to the corresponding finite-width neural network training. Our experimental results in $\ S 3$ validate this expectation across many settings.
+
+# 2.1 Client-Server Distributed Workflow
+
+We invoke a client-server model of distributed computation3, in which a server distributes independent workloads to a large pool of client workers that share a queue for receiving and sending work. Our distributed implementation of the KIP algorithm has two distinct stages:
+
+Forward pass: In this step, we compute the support-support and target-support matrices $K ( X _ { s } , X _ { s } )$ and $K ( X _ { t } , X _ { s } )$ . To do so, we partition $X _ { s } \times X _ { s }$ and $X _ { t } \times X _ { s }$ into pairs of images $( x , x ^ { \prime } )$ , each with batch size $B$ . We send such pairs to workers compute the respective matrix block $K ( \boldsymbol { x } , \boldsymbol { x } ^ { \prime } )$ . The server aggregates all these blocks to obtain the $K ( \bar { X } _ { s } , X _ { s } )$ and $K ( X _ { t } , X _ { s } )$ matrices.
+
+Backward pass: In this step, we need to compute the gradient of the loss $L$ (1) with respect to the support set $X _ { s }$ . We need only consider $\partial L / \partial { \bar { X } } _ { s }$ since $\partial L / \partial y _ { s }$ is cheap to compute. By the chain rule, we can write
+
+$$
+\frac { \partial L } { \partial X _ { s } } = \frac { \partial L } { \partial ( K ( X _ { s } , X _ { s } ) ) } \frac { \partial K ( X _ { s } , X _ { s } ) } { \partial X _ { s } } + \frac { \partial L } { \partial ( K ( X _ { t } , X _ { s } ) ) } \frac { \partial K ( X _ { t } , X _ { s } ) } { \partial X _ { s } } .
+$$
+
+The derivatives of $L$ with respect to the kernel matrices are inexpensive, since $L$ depends in a simple way on them (matrix multiplication and inversion). What is expensive to compute is the derivative of the kernel matrices with respect to the inputs. Each kernel element is an independent function of the inputs and a naive computation of the derivative of a block would require forwardmode differentiation, infeasible due to the size of the input images and the cost to compute the individual kernel elements. Thus our main novelty is to divide up the gradient computation into backward differentiation sub-computations, specifically by using the built-in function ${ \mathrm { j a x . v j p } }$ in JAX [Bradbury et al., 2018]. Denoting $K = K \bar { ( } X _ { s } , X _ { s } )$ or $K ( X _ { t } , X _ { s } )$ for short-hand, we divide the matrix $\partial L / \partial K$ , computed on the server, into $B \times B$ blocks corresponding to $\partial L / \partial K ( x , x ^ { ' } )$ , where $x$ and $x ^ { \prime }$ each have batch size $B$ . We send each such block, along with the corresponding block of image data $( x , x ^ { \prime } )$ , to a worker. The worker then treats the $\partial L / \partial K ( x , x ^ { \prime } )$ it receives as the cotangent vector argument of ${ \mathrm { j a x . v j p } }$ that, via contraction, converts the derivative of $K ( x , x ^ { \prime } )$ with respect to $x$ into a scalar. The server aggregates all these partial gradient computations performed by the workers, over all possible $B \times B$ blocks, to compute the total gradient $\partial L / \partial X _ { s }$ used to update $X _ { s }$ .
+
+Table 1: Comparison with other methods. The left group consists of neural network based methods. The right group consists of kernel ridge-regression. All settings for KIP involve the use of label-learning. Grayscale datasets use standard channel-wise preprocessing while RGB datasets use regularized ZCA preprocessing.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Imgs/ Class</td><td rowspan="2">DC1</td><td rowspan="2">DSA1</td><td rowspan="2">KIP FC1 aug</td><td rowspan="2">LS ConvNet2,3</td><td colspan="2"> KIP ConvNet2</td></tr><tr><td>no aug</td><td>aug</td></tr><tr><td rowspan="3">MNIST</td><td>1</td><td>91.7±0.5</td><td>88.7±0.6</td><td>85.5±0.1</td><td>73.4</td><td>97.3±0.1</td><td>96.5±0.1</td></tr><tr><td>10</td><td>97.4±0.2</td><td>97.8±0.1</td><td>97.2±0.2</td><td>96.4</td><td>99.1±0.1</td><td>99.1±0.1</td></tr><tr><td>50</td><td>98.8±0.1</td><td>99.2±0.1</td><td>98.4±0.1</td><td>98.3</td><td>99.4±0.1</td><td>99.5±0.1</td></tr><tr><td rowspan="3">Fashion- MNIST</td><td>1</td><td>70.5±0.6</td><td>70.6±0.6</td><td></td><td>65.3</td><td>82.9±0.2</td><td>76.7±0.2</td></tr><tr><td>10</td><td>82.3±0.4</td><td>84.6±0.3</td><td></td><td>80.8</td><td>91.0±0.1</td><td>88.8±0.1</td></tr><tr><td>50</td><td>83.6±0.4</td><td>88.7±0.2</td><td></td><td>86.9</td><td>92.4±0.1</td><td>91.0±0.1</td></tr><tr><td rowspan="3">SVHN</td><td>1</td><td>31.2±1.4</td><td>27.5±1.4</td><td></td><td>23.9</td><td>62.4±0.2</td><td>64.3±0.4</td></tr><tr><td>10</td><td>76.1±0.6</td><td>79.2±0.5</td><td></td><td>52.8</td><td>79.3±0.1</td><td>81.1±0.5</td></tr><tr><td>50</td><td>82.3±0.3</td><td>84.4±0.4</td><td>■</td><td>76.8</td><td>82.0±0.1</td><td>84.3±0.1</td></tr><tr><td rowspan="3">CIFAR-10</td><td>1</td><td>28.3±0.5</td><td>28.8±0.7</td><td>40.5±0.4</td><td>26.1</td><td>64.7±0.2</td><td>63.4±0.1</td></tr><tr><td>10</td><td>44.9±0.5</td><td>52.1±0.5</td><td>53.1±0.5</td><td>53.6</td><td>75.6±0.2</td><td>75.5±0.1</td></tr><tr><td>50</td><td>53.9±0.5</td><td>60.6±0.5</td><td>58.6±0.4</td><td>65.9</td><td>78.2±0.2</td><td>80.6±0.1</td></tr><tr><td rowspan="2">CIFAR-100</td><td>1</td><td>12.8±0.3</td><td>13.9±0.3</td><td>-</td><td>23.8</td><td>34.9±0.1</td><td>33.3±0.3</td></tr><tr><td>10</td><td>25.2±0.3</td><td>32.3±0.3</td><td>=</td><td>39.2</td><td>47.9±0.2</td><td>49.5±0.3</td></tr></table>
+
+1 DC [Zhao et al., 2021], DSA [Zhao and Bilen, 2021], KIP FC [Nguyen et al., 2021]. 2 Ours. 3 LD [Bohdal et al., 2020] is another baseline which distills only labels using the AlexNet architecture. Our LS achieves higher test accuracy than theirs in every dataset category.
+
+# 3 Experimental Results
+
+# 3.1 Kernel Distillation Results
+
+We apply the KIP and LS algorithms using the ConvNet architecture on the datasets MNIST [LeCun et al., 2010], Fashion MNIST [Xiao et al., 2017], SVHN [Netzer et al., 2011], CIFAR-10 [Krizhevsky, 2009], and CIFAR-100. Here, the goal is to condense the train dataset down to a learned dataset of size 1, 10, or 50 images per class. We consider a variety of hyperparameter settings (image preprocessing method, whether to augment target data, and whether to train the support labels for KIP), the full details of which are described in $\ S \mathrm { A }$ . For space reasons, we show a subset of our results in Table 1, with results corresponding to the remaining set of hyperparameters left to Tables A3-A10. We highlight here that a crucial ingredient for our strong results in the RGB dataset setting is the use of regularized ZCA preprocessing. Note the variable effect that our augmentations have on performance (see the last two columns of Table 1): they typically only provides a benefit for a sufficiently large support set. This result is consistent with Zhao et al. [2021], Zhao and Bilen [2021], in which gains from augmentations are also generally obtained from larger support sets. We tried varying the fraction (0.25 and 0.5) of each target batch that is augmented at each training step and found that while 10 images still did best without augmentations, 100 and 500 images typically did slightly better with some partial augmentations (versus none or full). For instance, for 500 images on CIFAR-10, we obtained $8 1 . 1 \%$ test accuracy using augmentation rate 0.5. Thus, our observation is that given that larger support sets can distill larger target datasets, as the former increases in size, the latter can be augmented more aggressively for obtaining optimal generalization performance.
+
+![](images/148a563acf6ae0ac3256cedf7abae07a8148ed52736016b4de5e15cdee20844b.jpg)  
+Figure 1: KIP with kernel sampling vs individual kernels. Left: Evaluation of three kernels, ConvNet, Conv-Vec3, Conv-Vec8 for KRR with respect to four train settings: sampling KIP (“all”) which uses all the kernels or else KIP trained with the individual kernels. For all three kernels, “all” is a close second place, outperformed only if the kernel used for training is exactly the same as the one used for testing. Right: We take the learned images of the four train settings described above and transfer them to finite-width neural networks corresponding to ConvNet, Conv-Vec3, Conv-Vec8. Each point is a neural network trained on a specified KIP learned checkpoint. Top row is sampling KIP images and bottom row is the baseline using just ConvNet for KIP. These plots indicate that sampling KIP improves performance across the architectures that are sampled, for both MSE and cross entropy loss. Settings: CIFAR-10, 100 images, no augmentations, no ZCA, no label learning.
+
+Remarkably, our results in Table 1 outperform all prior baselines across all dataset settings. Our results are especially strong in the small support size regime, with our 1 image per class results for KRR outperforming over 100 times as many natural images (see Table A1). We also obtain a significant margin over prior art across all datasets, with our largest margin being a $37 \%$ absolute gain in test accuracy for the SVHN dataset.
+
+# 3.2 Kernel Transfer
+
+In $\ S 3 . 1$ , we focused on obtaining state-of-the-art dataset distillation results for image classification using a specific kernel (ConvNet). Here, we consider the variation of KIP in which we sample from a family of kernels (which we call sampling KIP). We validate that sampling KIP adds robustness to the learned images in that they perform well for the family of kernels sampled during training.
+
+In Figure 1, we plot test performance of sampling KIP when using the kernels ConvNet, Conv-Vec3, and Conv-Vec8 (denoted by “all”) alongside KIP trained with just the individual kernels. Sampling KIP performs well at test time when using any of the three kernels, whereas datasets trained using a single kernel have a significant performance drop when using a different kernel.
+
+# 3.3 Neural Network Transfer
+
+In this section, we study how our distilled datasets optimized using KIP and LS transfer to the setting of finite-width neural networks. The main results are shown in Table 2. The third column shows the best neural network performance obtained by training on a KIP dataset of the corresponding size with respect to some choice of KIP and neural network training hyperparameters (see $\ S$ for details). Since the datasets are optimized for kernel ridge-regression and not for neural network training itself, we expect some performance loss when transferring to finite-width networks, which we record in the fourth column. Remarkably, the drop due to this transfer is quite moderate or small and sometimes the transfer can even lead to gain in performance (see LS for SVHN dataset with 10 images per class).
+
+Overall, our transfer to finite-width networks outperforms prior art based on DC/DSA [Zhao et al., 2021, Zhao and Bilen, 2021] in the 1 image per class setting for all the RGB datasets (SVHN, CIFAR-10, CIFAR-100). Moreover, for CIFAR-10, we outperform DC/DSA in all settings.
+
+Table 2: Transfer of KIP and LS to neural network training. Datasets obtained from KIP and LS using the ConvNet kernel are optimized for kernel ridge-regression and thus have reduced performance when used for training the corresponding finite-width ConvNet neural network. Remarkably, the loss in performance is mostly moderate and even small in many instances. Grayscale datasets use standard channel-wise preprocessing while RGB datasets use regularized ZCA preprocessing. The KIP datasets used here can have augmentations or no augmentations and, unlike those in Table 1, can have either fixed or learned labels. $^ *$ denotes best chosen transfer is obtained with learned labels.   
+
+<table><tr><td></td><td>Imgs/Class</td><td>DC/DSA</td><td>KIP to NN</td><td>Perf. change</td><td>LS to NN</td><td>Perf. change</td></tr><tr><td rowspan="3">MNIST</td><td>1</td><td>91.7±0.5</td><td>90.1±0.1</td><td>-5.5</td><td>71.0±0.2</td><td>-2.4</td></tr><tr><td>10</td><td>97.8±0.1</td><td>97.5±0.0</td><td>-1.1</td><td>95.2±0.1</td><td>-1.2</td></tr><tr><td>50</td><td>99.2±0.1</td><td>98.3±0.1</td><td>-0.8</td><td>97.9±0.0</td><td>-0.4</td></tr><tr><td rowspan="3">Fashion-MNIST</td><td>1</td><td>70.6±0.6</td><td>73.5±0.5*</td><td>-9.8</td><td>61.2±0.1</td><td>-4.1</td></tr><tr><td>10</td><td>84.6±0.3</td><td>86.8±0.1</td><td>-1.3</td><td>79.7±0.1</td><td>-1.2</td></tr><tr><td>50</td><td>88.7±0.2</td><td>88.0±0.1*</td><td>-4.5</td><td>85.0±0.1</td><td>-1.8</td></tr><tr><td rowspan="3">SVHN</td><td>1</td><td>31.2±1.4</td><td>57.3±0.1*</td><td>-8.3</td><td>23.8±0.2</td><td>-0.2</td></tr><tr><td>10</td><td>79.2±0.5</td><td>75.0±0.1</td><td>-1.6</td><td>53.2±0.3</td><td>0.4</td></tr><tr><td>50</td><td>84.4±0.4</td><td>80.5±0.1</td><td>-1.0</td><td>76.5±0.3</td><td>-0.4</td></tr><tr><td rowspan="3">CIFAR-10</td><td>1</td><td>28.8±0.7</td><td>49.9±0.2</td><td>-9.2</td><td>24.7±0.1</td><td>-1.4</td></tr><tr><td>10</td><td>52.1±0.5</td><td>62.7±0.3</td><td>-4.6</td><td>49.3±0.1</td><td>-4.3</td></tr><tr><td>50</td><td>60.6±0.5</td><td>68.6±0.2</td><td>-4.5</td><td>62.0±0.2</td><td>-3.9</td></tr><tr><td rowspan="2">CIFAR-100</td><td>1</td><td>13.9±0.3</td><td>15.7±0.2*</td><td>-18.1</td><td>11.8±0.2</td><td>-12.0</td></tr><tr><td>10</td><td>32.3±0.3</td><td>28.3±0.1</td><td>-17.4</td><td>25.0±0.1</td><td>-14.2</td></tr></table>
+
+Figures 2 and 3 provide a closer look at KIP transfer changes under various settings. The first of these tracks how transfer performance changes when adding additional layers as function of the number of KIP training steps used. The normalization layers appear to harm performance for MSE loss, which can be anticipated from their absence in the KIP and LS optimization procedures. However they appear to provide some benefit for cross entropy loss. For Figure 3, we observe that as KIP training progresses, the downstream finite-width network’s performance also improves in general. A notable exception is observed when learning the labels in KIP, where longer training steps lead to deterioration of information useful to training finite-width neural networks. We also observe that as predicted by infinite-width theory [Jacot et al., 2018, Lee et al., 2019], the overall gap between KIP or LS performance and finite-width neural network decreases as the width increases. While our best performing transfer is obtained with width 1024, Figure 3 (middle) suggest that even with modest width of 64, our transfer can outperform prior art of $6 0 . 6 \%$ by Zhao and Bilen [2021].
+
+Finally, in Figure 4, we investigate the performance of KIP images over the course of training of a single run, as compared to natural images, over a range of hyperparameters. We find the outperformance of KIP images above natural images consistent across hyperparameters and checkpoints. This suggests that our KIP images may also be effective for accelerated hyperparameter search, an application of dataset distillation explored in Zhao et al. [2021], Zhao and Bilen [2021].
+
+Altogether, we find our neural network training results encouraging. First, it validates the applicability of infinite-width methods to the setting of finite width [Huang and Yau, 2020, Dyer and Gur-Ari, 2020, Andreassen and Dyer, 2020, Yaida, 2020, Lee et al., 2020]. Second, we find some of the transfer results quite surprising, including efficacy of label solve and the use of cross-entropy for certain settings (see $\ S \mathrm { A }$ for the full details).
+
+# 4 Understanding KIP Images and Labels
+
+A natural question to consider is what causes KIP to improve generalization performance. Does it simplify support images, removing noise and minor sources of variation while keeping only the core features shared between many target images of a given class? Or does it make them more complex by producing outputs that combine characteristics of many samples in a single resulting collage? While properly answering this question is subject to precise definitions of simplicity and complexity, we find that KIP tends to increase the complexity of pictures based on the following experiments:
+
+![](images/4b2f59db73df265ffd78c4beb7e58efaa0c701708ef3304ac1729acf17834d3d.jpg)
+
+![](images/d7101d842d66d8536c0fe0794798318cf8448c9d5fec3295db42110b30f393b0.jpg)  
+Figure 2: Robustness to neural network variations. KIP ConvNet images (trained with fixed labels) are tested on variations of the ConvNet neural network, including those which have various normalization layers (layer, instance, batch). A similar architecture to ConvNet, the Myrtle5 architecture (without normalization layers) [Shankar et al., 2020], which differs from the ConvNet architecture by having an additional convolutional layer at the bottom and a global average pooling that replaces the final local average pooling at the top, is also tested. Finally, mean-square error is compared with cross-entropy loss (left versus right). Settings: CIFAR-10, 500 images, ZCA, no label learning.
+
+![](images/d4bafe9124596e9d37dbd9d82f76504e27d351417c7bcb6711299bb95919161d.jpg)  
+Figure 3: Variations for neural network transfer. Left: Plot of transfer performance as a function of KIP training steps across various train settings. Here (a) denotes augmentations used during KIP training and $\left( \mathrm { a } { + } \mathrm { l } \right)$ denotes that additionally the labels were learned. MSE and XENT denote mean-square-error and cross entropy loss for the neural network, where for the case of XENT and $\left( \mathrm { a } \mathrm { + } \mathrm { l } \right)$ , the labels for the neural network are the argmax of the learned labels. Middle: Exploring the effect of width on transferability of vanilla KIP data. Right: The effect of width on the transferability of label solved data. Settings: CIFAR-10, 500 images, ZCA.   
+Figure 4: Hyperparameter robustness. In the above, KIP images across eight different checkpoints are used to train the ConvNet neural network. Each point in each plot is a neural network training with a different hyperparameter, and its location records the final test accuracy when training on natural images versus the KIP images obtained from initializing from such images. For both MSE and cross entropy loss, KIP images consistently exceed natural images across many hyperparameters. Settings: CIFAR-10, 500 images, ZCA, no augmentations, no label learning.
+
+![](images/e6c595174262ae29110808481345bc35e97724fca04c12c87c4596c09e29c9e0.jpg)  
+Figure 5: Examples of learned images. Images are initialized from natural images in the top row (Init) and converge to images in the bottom row (Trained). Settings: 100 images distilled, no ZCA, no label training, augmentations.
+
+• Visual analysis: qualitatively, KIP images tend to be richer in textures and contours. • Dimensional analysis: KIP produces images of higher intrinsic dimensionality (i.e. an estimate of the dimensionality of the manifold on which the images lie) than natural images. • Spectral analysis: unlike natural images, for which the bulk of generalization performance is explained by a small number of top few eigendirections, KIP images leverage the whole spectrum much more evenly.
+
+Combined, these results let us conclude that KIP usually increases complexity, integrating features from many target images into much fewer support images.
+
+Visual Analysis. A visual inspection of our learned data leads to intriguing observations in terms of interpretability. Figure 5 shows examples of KIP learned images from CIFAR-100. The resulting images are heterogeneous in terms of how they can be interpreted as distilling the data. For instance, the distilled apple image seems to consist of many apples nested within a possibly larger apple, whereas the distilled bottle image starts off as two bottles and before transforming into one, while other classes (like the beaver) are altogether visually indistinct. Investigating and quantifying aspects that make these images generalize so well is a promising avenue for future work. We show examples from other datasets in Figure A2.
+
+In Figure 6, we compare MNIST KIP data learned with and without label learning. For the latter case with images and labels optimized jointly, while labels become more informative, encoding richer inter-class information, the images become less interpretable. This behavior consistently leads to superior KRR results, but appears to not be leveraged as efficiently in the neural network transfer setting (Table 2). Experimental details can be found in $\ S \mathrm { A }$ .
+
+Dimensional Analysis. We study the intrinsic dimension of KIP images and find that they tend to grow. Intrinsic dimension (ID) was first defined by Bennett [1969] as “the number of free parameters required in a hypothetical signal generator capable of producing a close approximation to each signal in the collection”. In our context, it is the dimensionality of the manifold embedded into the image space which contains all the support images. Intuitively, simple datasets have a low ID as they can be described by a small number of coordinates on the low-dimensional data manifold.
+
+ID can be defined and estimated differently based on assumptions on the manifold structure and the probability density function of the images on this manifold (see [Camastra and Staiano, 2016] for review). We use the “Two-NN” method developed by [Facco et al., 2017], which makes relatively few assumptions and allows to estimate ID only from two nearest-neighbor distances for each datapoint.
+
+Figure 7 shows that the ID is increasing for the learned KIP images as a function of the training step across a variety of configurations (training with or without augmentations/label learning) and datasets. One might expect that a distillation procedure should decrease dimensionality. On the other hand, Ansuini et al. [2019] showed that ID increases in the earlier layers of a trained neural network. It remains to be understood if this latter observation has any relationship with our increased ID. Note that ZCA preprocessing, which played an important role for getting the best performance for our RGB datasets, increases dimensionality of the underlying data (see Figure A4).
+
+![](images/744ef5e48025d1576b73820fcc92a143cc5fbf893ece6f8a392eca2b6478b9c4.jpg)  
+Figure 6: Dataset distillation with trainable and non-trainable labels. Top row: initialization of support images and labels. Middle row: trained images if labels remain fixed. Bottom row: trained images and labels, jointly optimized. Settings: 100 images distilled, no augmentations.
+
+![](images/a06942204ec114b405c74d18fc52c11993ea495a87514472cc9da54909bdfd25.jpg)  
+Figure 7: Intrinsic dimension of a learned datasets grows during training. As training progresses, intrinsic dimension of the learned dataset grows, indicating that training non-trivially transforms the data manifold. See Figures A2 and 6 for visual examples of learned images, and Figures A3 and A4 for similar observations using other metrics and settings. Settings: 500 images distilled, no ZCA.
+
+Spectral Analysis. Another distinguishing property of KIP images is how their spectral components contribute to performance. In Figure 8, we spot how different spectral bands of KIP learned images affect test performance as compared to their initial natural images. Here, we use the FC2, Conv-Vec8, and ConvNet architectures. We note that for natural images (light bars), most of their performance is captured by the top $20 \%$ of eigenvalues. For KIP images, the performance is either more evenly distributed across the bands (FC and Conv-Vec8) or else is skewed towards the tail (ConvNet).
+
+# 5 Related Work
+
+Dataset distillation was first studied in Wang et al. [2018]. The work of Sucholutsky and Schonlau [2019], Bohdal et al. [2020] build upon it by distilling labels. Zhao et al. [2021] proposes condensing a training set by harnessing a gradient matching condition. Zhao and Bilen [2021] takes this idea further by applying a suitable augmentation strategy. Note that while Zhao and Bilen [2021] is limited in augmentation expressiveness (they have to apply a single augmentation per training iteration), we can sample augmentations independently per image in our target set per train step. Our work together with Nguyen et al. [2021] are, to the best of our knowledge, the only works using kernel-based methods for dataset distillation on image classification datasets.
+
+![](images/4c290bf4e301cc2883615e72ab385da78ab75a5398ad90dbbe6ea01579e15c20.jpg)  
+Figure 8: Spectral contribution to test accuracy shifts to the tail. By setting the ridge-parameter to zero in kernel-ridge regression and composing $K _ { X _ { s } X _ { s } } ^ { - 1 }$ with the spectral projection onto various eigenspaces, we can explore how different spectral bands affect test accuracy of kernel ridgeregression. We plot the relative change in test accuracy using contiguous bands of $20 \%$ of the eigenvalues. Settings: CIFAR-10, 500 images. Further details in $\ S$ .
+
+Our use of kernels stems from the correspondence between infinitely-wide neural networks and kernel methods [Neal, 1994, Lee et al., 2018, Matthews et al., 2018, Jacot et al., 2018, Novak et al., 2019, Garriga-Alonso et al., 2019, Arora et al., 2019, Yang, 2019a,b, Hron et al., 2020] and the extended correspondence with the finite width corrections [Dyer and Gur-Ari, 2020, Huang and Yau, 2020, Yaida, 2020]. These correspondences underlie the transferability of our KRR results to neural networks , and have been utilized in understanding trainability [Xiao et al., 2020], generalizations [Adlam and Pennington, 2020], training dynamics [Lewkowycz et al., 2020, Lewkowycz and Gur-Ari, 2020], uncertainty [Adlam et al., 2021], and demonstrated their effectiveness for smaller datasets [Arora et al., 2020] and neural architecture search [Park et al., 2020, Chen et al., 2021].
+
+# 6 Conclusion
+
+We performed an extensive study of dataset distillation using the KIP and LS algorithms applied to convolutional architectures, obtaining SOTA results on a variety of image classification datasets. In some cases, our learned datasets were more effective than a natural dataset two orders of magnitude larger in size. There are many interesting followup directions and questions from our work:
+
+First, integrating efficient kernel-approximation methods into our algorithms, such as those of Zandieh et al. [2021], will reduce computational burden and enable scaling up to larger datasets. In this direction, the understanding of how various resources (e.g. data, parameter count, compute) scale when optimizing for neural network performance has received significant attention as machine learning models continue to stretch computational limits [Hestness et al., 2017, Rosenfeld et al., 2020, Kaplan et al., 2020, Bahri et al., 2021]. Developing our understanding of how to harness smaller, yet more useful representations data would aid in such endeavors. In particular, it would be especially interesting to explore how well datasets can be compressed as they scale up in size.
+
+Second, LS and KIP with label learning shows that optimizing labels is a very powerful tool for dataset distillation. The labels we obtain are quite far away from standard, interpretable labels and we feel their effectiveness suggests that understanding of how to optimally label data warrants further study.
+
+Finally, the novel features obtained by our learned datasets, and those of dataset distillation methods in general, may reveal insights into interpretability and the nature of sample-efficient representations. For instance, observe the bee and bicycle images in Figure 5: the bee class distills into what appears to be spurious visual features (e.g. pollen), while the bicycle class distills to the essential contours of a typical bicycle. Additional analyses and explorations of this type could offer insights into the perennial question of how neural networks learn and generalize.
+
+Acknowledgments We would like to acknowledge special thanks to Samuel S. Schoenholz, who proposed and helped develop the overall strategy for our distributed KIP learning methodology. We are also grateful to Ekin Dogus Cubuk and Manuel Kroiss for helpful discussions.
+
+# References
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+        "text": "Dataset Distillation with Infinitely Wide Convolutional Networks ",
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+        "text": "Timothy Nguyen† ∗ ",
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+        "text": "DeepMind† Google Research, Brain Team♠ timothycnguyen@deepmind.com {romann, xlc, jaehlee}@google.com ",
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+        "type": "text",
+        "text": "Abstract ",
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+        "text": "The effectiveness of machine learning algorithms arises from being able to extract useful features from large amounts of data. As model and dataset sizes increase, dataset distillation methods that compress large datasets into significantly smaller yet highly performant ones will become valuable in terms of training efficiency and useful feature extraction. To that end, we apply a novel distributed kernel-based meta-learning framework to achieve state-of-the-art results for dataset distillation using infinitely wide convolutional neural networks. For instance, using only 10 datapoints $0 . 0 2 \\%$ of original dataset), we obtain over $65 \\%$ test accuracy on CIFAR10 image classification task, a dramatic improvement over the previous best test accuracy of $40 \\%$ . Our state-of-the-art results extend across many other settings for MNIST, Fashion-MNIST, CIFAR-10, CIFAR-100, and SVHN. Furthermore, we perform some preliminary analyses of our distilled datasets to shed light on how they differ from naturally occurring data. ",
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+        "text": "1 Introduction ",
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+        "text": "Deep learning has become extraordinarily successful in a wide variety of settings through the availability of large datasets [Krizhevsky et al., 2012, Devlin et al., 2018, Brown et al., 2020, Dosovitskiy et al., 2020]. Such large datasets enable a neural network to learn useful representations of the data that are adapted to solving tasks of interest. Unfortunately, it can be prohibitively costly to acquire such large datasets and train a neural network for the requisite amount of time. ",
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+        "text": "One way to mitigate this problem is by constructing smaller datasets that are nevertheless informative. Some direct approaches to this include choosing a representative subset of the dataset (i.e. a coreset) or else performing a low-dimensional projection that reduces the number of features. However, such methods typically introduce a tradeoff between performance and dataset size, since what they produce is a coarse approximation of the full dataset. By contrast, the approach of dataset distillation is to synthesize datasets that are more informative than their natural counterparts when equalizing for dataset size [Wang et al., 2018, Bohdal et al., 2020, Nguyen et al., 2021, Zhao and Bilen, 2021]. Such resulting datasets will not arise from the distribution of natural images but will nevertheless capture features useful to a neural network, a capability which remains mysterious and is far from being well-understood [Ilyas et al., 2019, Huh et al., 2016, Hermann and Lampinen, 2020]. ",
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+        "text": "The applications of such smaller, distilled datasets are diverse. For nonparametric methods that scale poorly with the training dataset (e.g. nearest-neighbors or kernel-ridge regression), having a reduced dataset decreases the associated memory and inference costs. For the training of neural networks, such distilled datasets have found several applications in the literature, including increasing the effectiveness of replay methods in continual learning [Borsos et al., 2020] and helping to accelerate neural architecture search [Zhao et al., 2021, Zhao and Bilen, 2021]. ",
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+        "text": "In this paper, we perform a large-scale extension of the methods of Nguyen et al. [2021] to obtain new state-of-the-art (SOTA) dataset distillation results. Specifically, we apply the algorithms KIP (Kernel Inducing Points) and LS (Label Solve), first developed in Nguyen et al. [2021], to infinitely wide convolutional networks by implementing a novel, distributed meta-learning framework that draws upon hundreds of accelerators per training. The need for such resources is necessitated by the computational costs of using infinitely wide neural networks built out of components occurring in modern image classification models: convolutional and pooling layers (see $\\ S$ for details). The consequence is that we obtain distilled datasets that are effective for both kernel ridge-regression and neural network training. ",
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+        "text": "Additionally, we initiate a preliminary study of the images and labels which KIP learns. We provide a visual and quantitative analysis of the data learned and find some surprising results concerning their interpretability and their dimensional and spectral properties. Given the efficacy of KIP and LS learned data, we believe a better understanding of them would aid in the understanding of feature learning in neural networks. ",
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+        "text": "To summarize, our contributions are as follows: ",
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+        "text": "1. We achieve SOTA dataset distillation results on a wide variety of datasets (MNIST, FashionMNIST, SVHN, CIFAR-10, CIFAR-100) for both kernel ridge-regression and neural network training. In several instances, our results achieve an impressively wide margin over prior art, including over $2 5 \\%$ and $37 \\%$ absolute gain in accuracy on CIFAR-10 and SVHN image classification, respectively, when using only 10 images (Tables 1, 2, A11).   \n2. We develop a novel, distributed meta-learning framework specifically tailored to the computational burdens of sophisticated neural kernels (§2.1).   \n3. We highlight and analyze some of the peculiar features of the distilled datasets we obtain, illustrating how they differ from natural data (§4).   \n4. We open source the distilled datasets, which used thousands of GPU hours, for the research community to further investigate at https://github.com/google-research/ google-research/tree/master/kip. ",
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+        "text": "Background on infinitely wide convolutional networks. Recent literature has established that Bayesian and gradient-descent trained neural networks converge to Gaussian Processes (GP) as the number of hidden units in intermediary layers approaches infinity (see $\\ S 5$ ). These results hold for many different architectures, including convolutional networks, which converge to a particular GP in the limit of infinite channels [Novak et al., 2019, Garriga-Alonso et al., 2019, Arora et al., 2019]. Bayesian networks are described by the Neural Network Gaussian Process (NNGP) kernel, while gradient descent networks are described by the Neural Tangent Kernel (NTK). Since we are interested in synthesizing datasets that can be used with both kernel methods and common gradient-descent trained neural networks, we focus on NTK in this work. ",
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+    {
+        "type": "text",
+        "text": "Infinitely wide networks have been shown to achieve SOTA (among non-parametric kernels) results on image classification tasks [Novak et al., 2019, Arora et al., 2019, Li et al., 2019, Shankar et al., 2020, Bietti, 2021] and even rival finite-width networks in certain settings [Arora et al., 2020, Lee et al., 2020]. This makes such kernels especially suitable for our task. As convolutional models, they encode useful inductive biases of locality and translation invariance [Novak et al., 2019], which enable good generalization. Moreover, flexible and efficient computation of these kernels are possible due to the Neural Tangent library [Novak et al., 2020]. ",
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+        "text": "Specific models considered. The central neural network (and corresponding infinite-width model) we consider is a simple 4-layer convolutional network with average pooling layers that we refer to as ConvNet throughout the text. This architecture is a slightly modified version of the default model used by Zhao and Bilen [2021], Zhao et al. [2021] and was chosen for ease of baselining (see $\\ S \\mathrm { A }$ for details). In several other settings we also consider convolutional networks without pooling layers ConvVec,2 and networks with no convolutions and only fully-connected layers FC. Depth of architecture (as measured by number of hidden layers) is indicated by an integer suffix. ",
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+        "text": "Background on algorithms. We review the Kernel Inducing Points (KIP) and Label Solve (LS) algorithms introduced by Nguyen et al. [2021]. Given a kernel $K$ , the kernel ridge-regression (KRR) loss function trained on a support dataset $( X _ { s } , y _ { s } )$ and evaluated on a target dataset $( X _ { t } , y _ { t } )$ is ",
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+        "img_path": "images/4d09c5178edafa32325041f77ec541c226a9c85f24857929893b13d7a3119d0d.jpg",
+        "text": "$$\nL ( X _ { s } , y _ { s } ) = \\frac { 1 } { 2 } \\left\\| y _ { t } - K _ { X _ { t } X _ { s } } ( K _ { X _ { s } X _ { s } } + \\lambda I ) ^ { - 1 } y _ { s } \\right\\| _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
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+        "text": "where if $U$ and $V$ are sets, $K _ { U V }$ is the matrix of kernel elements $( K ( u , v ) ) _ { u \\in U , v \\in V }$ . Here $\\lambda > 0$ is a fixed regularization parameter. The KIP algorithm consists of minimizing (1) with respect to the support set (either just the $X _ { s }$ or along with the labels $y _ { s }$ ). Here, we sample $( X _ { t } , y _ { t } )$ from a target dataset $\\mathcal { D }$ at every (meta)step, and update the support set using gradient-based methods. Additional variations include augmenting the $X _ { t }$ or sampling a different kernel $K$ (from a fixed family of kernels) at each step. ",
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+        "type": "text",
+        "text": "The Label Solve algorithm consists of solving for the least-norm minimizer of (1) with respect to $y _ { s }$ This yields the labels ",
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+    {
+        "type": "equation",
+        "img_path": "images/6400c985b95cd910d952871747652b49009ec08e5b543a0d2f75cb6d00886af7.jpg",
+        "text": "$$\ny _ { s } ^ { * } = \\Big ( K _ { X _ { t } X _ { s } } ( K _ { X _ { s } X _ { s } } + \\lambda I ) ^ { - 1 } \\Big ) ^ { + } y _ { t } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $A ^ { + }$ denotes the pseudo-inverse of the matrix $A$ . Note that here $( X _ { t } , y _ { t } ) = \\mathcal { D }$ , i.e. the labels are solved using the whole target set. ",
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+        "text": "In our applications of KIP and Label Solve, the target dataset $\\mathcal { D }$ is always significantly larger than the support set $( X _ { s } , y _ { s } )$ . Hence, the learned support set or solved labels can be regarded as distilled versions of their respective targets. We also initialize our support images to be a subset of natural images, though they could also be initialized randomly. ",
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+        "text": "Based on the infinite-width correspondence outlined above and in $\\ S 5$ , dataset distillation using KIP or LS that is optimized for KRR should extend to the corresponding finite-width neural network training. Our experimental results in $\\ S 3$ validate this expectation across many settings. ",
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+        "text": "2.1 Client-Server Distributed Workflow ",
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+        "text": "We invoke a client-server model of distributed computation3, in which a server distributes independent workloads to a large pool of client workers that share a queue for receiving and sending work. Our distributed implementation of the KIP algorithm has two distinct stages: ",
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+        "text": "Forward pass: In this step, we compute the support-support and target-support matrices $K ( X _ { s } , X _ { s } )$ and $K ( X _ { t } , X _ { s } )$ . To do so, we partition $X _ { s } \\times X _ { s }$ and $X _ { t } \\times X _ { s }$ into pairs of images $( x , x ^ { \\prime } )$ , each with batch size $B$ . We send such pairs to workers compute the respective matrix block $K ( \\boldsymbol { x } , \\boldsymbol { x } ^ { \\prime } )$ . The server aggregates all these blocks to obtain the $K ( \\bar { X } _ { s } , X _ { s } )$ and $K ( X _ { t } , X _ { s } )$ matrices. ",
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+        "type": "text",
+        "text": "Backward pass: In this step, we need to compute the gradient of the loss $L$ (1) with respect to the support set $X _ { s }$ . We need only consider $\\partial L / \\partial { \\bar { X } } _ { s }$ since $\\partial L / \\partial y _ { s }$ is cheap to compute. By the chain rule, we can write ",
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+        "img_path": "images/a0223d220bd86139fdd8c7685f0a6f5ec74cfdd88e27573f6f19316bb7fbcc1f.jpg",
+        "text": "$$\n\\frac { \\partial L } { \\partial X _ { s } } = \\frac { \\partial L } { \\partial ( K ( X _ { s } , X _ { s } ) ) } \\frac { \\partial K ( X _ { s } , X _ { s } ) } { \\partial X _ { s } } + \\frac { \\partial L } { \\partial ( K ( X _ { t } , X _ { s } ) ) } \\frac { \\partial K ( X _ { t } , X _ { s } ) } { \\partial X _ { s } } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The derivatives of $L$ with respect to the kernel matrices are inexpensive, since $L$ depends in a simple way on them (matrix multiplication and inversion). What is expensive to compute is the derivative of the kernel matrices with respect to the inputs. Each kernel element is an independent function of the inputs and a naive computation of the derivative of a block would require forwardmode differentiation, infeasible due to the size of the input images and the cost to compute the individual kernel elements. Thus our main novelty is to divide up the gradient computation into backward differentiation sub-computations, specifically by using the built-in function ${ \\mathrm { j a x . v j p } }$ in JAX [Bradbury et al., 2018]. Denoting $K = K \\bar { ( } X _ { s } , X _ { s } )$ or $K ( X _ { t } , X _ { s } )$ for short-hand, we divide the matrix $\\partial L / \\partial K$ , computed on the server, into $B \\times B$ blocks corresponding to $\\partial L / \\partial K ( x , x ^ { ' } )$ , where $x$ and $x ^ { \\prime }$ each have batch size $B$ . We send each such block, along with the corresponding block of image data $( x , x ^ { \\prime } )$ , to a worker. The worker then treats the $\\partial L / \\partial K ( x , x ^ { \\prime } )$ it receives as the cotangent vector argument of ${ \\mathrm { j a x . v j p } }$ that, via contraction, converts the derivative of $K ( x , x ^ { \\prime } )$ with respect to $x$ into a scalar. The server aggregates all these partial gradient computations performed by the workers, over all possible $B \\times B$ blocks, to compute the total gradient $\\partial L / \\partial X _ { s }$ used to update $X _ { s }$ . ",
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+    {
+        "type": "table",
+        "img_path": "images/93a161c49cc38c7dc021cdfc3acc93e546f582a7e6e2e85d3f67088d33ea5aef.jpg",
+        "table_caption": [
+            "Table 1: Comparison with other methods. The left group consists of neural network based methods. The right group consists of kernel ridge-regression. All settings for KIP involve the use of label-learning. Grayscale datasets use standard channel-wise preprocessing while RGB datasets use regularized ZCA preprocessing. "
+        ],
+        "table_footnote": [
+            "1 DC [Zhao et al., 2021], DSA [Zhao and Bilen, 2021], KIP FC [Nguyen et al., 2021]. 2 Ours. 3 LD [Bohdal et al., 2020] is another baseline which distills only labels using the AlexNet architecture. Our LS achieves higher test accuracy than theirs in every dataset category. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Imgs/ Class</td><td rowspan=\"2\">DC1</td><td rowspan=\"2\">DSA1</td><td rowspan=\"2\">KIP FC1 aug</td><td rowspan=\"2\">LS ConvNet2,3</td><td colspan=\"2\"> KIP ConvNet2</td></tr><tr><td>no aug</td><td>aug</td></tr><tr><td rowspan=\"3\">MNIST</td><td>1</td><td>91.7±0.5</td><td>88.7±0.6</td><td>85.5±0.1</td><td>73.4</td><td>97.3±0.1</td><td>96.5±0.1</td></tr><tr><td>10</td><td>97.4±0.2</td><td>97.8±0.1</td><td>97.2±0.2</td><td>96.4</td><td>99.1±0.1</td><td>99.1±0.1</td></tr><tr><td>50</td><td>98.8±0.1</td><td>99.2±0.1</td><td>98.4±0.1</td><td>98.3</td><td>99.4±0.1</td><td>99.5±0.1</td></tr><tr><td rowspan=\"3\">Fashion- MNIST</td><td>1</td><td>70.5±0.6</td><td>70.6±0.6</td><td></td><td>65.3</td><td>82.9±0.2</td><td>76.7±0.2</td></tr><tr><td>10</td><td>82.3±0.4</td><td>84.6±0.3</td><td></td><td>80.8</td><td>91.0±0.1</td><td>88.8±0.1</td></tr><tr><td>50</td><td>83.6±0.4</td><td>88.7±0.2</td><td></td><td>86.9</td><td>92.4±0.1</td><td>91.0±0.1</td></tr><tr><td rowspan=\"3\">SVHN</td><td>1</td><td>31.2±1.4</td><td>27.5±1.4</td><td></td><td>23.9</td><td>62.4±0.2</td><td>64.3±0.4</td></tr><tr><td>10</td><td>76.1±0.6</td><td>79.2±0.5</td><td></td><td>52.8</td><td>79.3±0.1</td><td>81.1±0.5</td></tr><tr><td>50</td><td>82.3±0.3</td><td>84.4±0.4</td><td>■</td><td>76.8</td><td>82.0±0.1</td><td>84.3±0.1</td></tr><tr><td rowspan=\"3\">CIFAR-10</td><td>1</td><td>28.3±0.5</td><td>28.8±0.7</td><td>40.5±0.4</td><td>26.1</td><td>64.7±0.2</td><td>63.4±0.1</td></tr><tr><td>10</td><td>44.9±0.5</td><td>52.1±0.5</td><td>53.1±0.5</td><td>53.6</td><td>75.6±0.2</td><td>75.5±0.1</td></tr><tr><td>50</td><td>53.9±0.5</td><td>60.6±0.5</td><td>58.6±0.4</td><td>65.9</td><td>78.2±0.2</td><td>80.6±0.1</td></tr><tr><td rowspan=\"2\">CIFAR-100</td><td>1</td><td>12.8±0.3</td><td>13.9±0.3</td><td>-</td><td>23.8</td><td>34.9±0.1</td><td>33.3±0.3</td></tr><tr><td>10</td><td>25.2±0.3</td><td>32.3±0.3</td><td>=</td><td>39.2</td><td>47.9±0.2</td><td>49.5±0.3</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3 Experimental Results ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 Kernel Distillation Results ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We apply the KIP and LS algorithms using the ConvNet architecture on the datasets MNIST [LeCun et al., 2010], Fashion MNIST [Xiao et al., 2017], SVHN [Netzer et al., 2011], CIFAR-10 [Krizhevsky, 2009], and CIFAR-100. Here, the goal is to condense the train dataset down to a learned dataset of size 1, 10, or 50 images per class. We consider a variety of hyperparameter settings (image preprocessing method, whether to augment target data, and whether to train the support labels for KIP), the full details of which are described in $\\ S \\mathrm { A }$ . For space reasons, we show a subset of our results in Table 1, with results corresponding to the remaining set of hyperparameters left to Tables A3-A10. We highlight here that a crucial ingredient for our strong results in the RGB dataset setting is the use of regularized ZCA preprocessing. Note the variable effect that our augmentations have on performance (see the last two columns of Table 1): they typically only provides a benefit for a sufficiently large support set. This result is consistent with Zhao et al. [2021], Zhao and Bilen [2021], in which gains from augmentations are also generally obtained from larger support sets. We tried varying the fraction (0.25 and 0.5) of each target batch that is augmented at each training step and found that while 10 images still did best without augmentations, 100 and 500 images typically did slightly better with some partial augmentations (versus none or full). For instance, for 500 images on CIFAR-10, we obtained $8 1 . 1 \\%$ test accuracy using augmentation rate 0.5. Thus, our observation is that given that larger support sets can distill larger target datasets, as the former increases in size, the latter can be augmented more aggressively for obtaining optimal generalization performance. ",
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+    {
+        "type": "image",
+        "img_path": "images/148a563acf6ae0ac3256cedf7abae07a8148ed52736016b4de5e15cdee20844b.jpg",
+        "image_caption": [
+            "Figure 1: KIP with kernel sampling vs individual kernels. Left: Evaluation of three kernels, ConvNet, Conv-Vec3, Conv-Vec8 for KRR with respect to four train settings: sampling KIP (“all”) which uses all the kernels or else KIP trained with the individual kernels. For all three kernels, “all” is a close second place, outperformed only if the kernel used for training is exactly the same as the one used for testing. Right: We take the learned images of the four train settings described above and transfer them to finite-width neural networks corresponding to ConvNet, Conv-Vec3, Conv-Vec8. Each point is a neural network trained on a specified KIP learned checkpoint. Top row is sampling KIP images and bottom row is the baseline using just ConvNet for KIP. These plots indicate that sampling KIP improves performance across the architectures that are sampled, for both MSE and cross entropy loss. Settings: CIFAR-10, 100 images, no augmentations, no ZCA, no label learning. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "Remarkably, our results in Table 1 outperform all prior baselines across all dataset settings. Our results are especially strong in the small support size regime, with our 1 image per class results for KRR outperforming over 100 times as many natural images (see Table A1). We also obtain a significant margin over prior art across all datasets, with our largest margin being a $37 \\%$ absolute gain in test accuracy for the SVHN dataset. ",
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+        "text": "3.2 Kernel Transfer ",
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+        "text": "In $\\ S 3 . 1$ , we focused on obtaining state-of-the-art dataset distillation results for image classification using a specific kernel (ConvNet). Here, we consider the variation of KIP in which we sample from a family of kernels (which we call sampling KIP). We validate that sampling KIP adds robustness to the learned images in that they perform well for the family of kernels sampled during training. ",
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+        "type": "text",
+        "text": "In Figure 1, we plot test performance of sampling KIP when using the kernels ConvNet, Conv-Vec3, and Conv-Vec8 (denoted by “all”) alongside KIP trained with just the individual kernels. Sampling KIP performs well at test time when using any of the three kernels, whereas datasets trained using a single kernel have a significant performance drop when using a different kernel. ",
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+    {
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+        "text": "3.3 Neural Network Transfer ",
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+        "text": "In this section, we study how our distilled datasets optimized using KIP and LS transfer to the setting of finite-width neural networks. The main results are shown in Table 2. The third column shows the best neural network performance obtained by training on a KIP dataset of the corresponding size with respect to some choice of KIP and neural network training hyperparameters (see $\\ S$ for details). Since the datasets are optimized for kernel ridge-regression and not for neural network training itself, we expect some performance loss when transferring to finite-width networks, which we record in the fourth column. Remarkably, the drop due to this transfer is quite moderate or small and sometimes the transfer can even lead to gain in performance (see LS for SVHN dataset with 10 images per class). ",
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+        "text": "Overall, our transfer to finite-width networks outperforms prior art based on DC/DSA [Zhao et al., 2021, Zhao and Bilen, 2021] in the 1 image per class setting for all the RGB datasets (SVHN, CIFAR-10, CIFAR-100). Moreover, for CIFAR-10, we outperform DC/DSA in all settings. ",
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+        "table_caption": [
+            "Table 2: Transfer of KIP and LS to neural network training. Datasets obtained from KIP and LS using the ConvNet kernel are optimized for kernel ridge-regression and thus have reduced performance when used for training the corresponding finite-width ConvNet neural network. Remarkably, the loss in performance is mostly moderate and even small in many instances. Grayscale datasets use standard channel-wise preprocessing while RGB datasets use regularized ZCA preprocessing. The KIP datasets used here can have augmentations or no augmentations and, unlike those in Table 1, can have either fixed or learned labels. $^ *$ denotes best chosen transfer is obtained with learned labels. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Imgs/Class</td><td>DC/DSA</td><td>KIP to NN</td><td>Perf. change</td><td>LS to NN</td><td>Perf. change</td></tr><tr><td rowspan=\"3\">MNIST</td><td>1</td><td>91.7±0.5</td><td>90.1±0.1</td><td>-5.5</td><td>71.0±0.2</td><td>-2.4</td></tr><tr><td>10</td><td>97.8±0.1</td><td>97.5±0.0</td><td>-1.1</td><td>95.2±0.1</td><td>-1.2</td></tr><tr><td>50</td><td>99.2±0.1</td><td>98.3±0.1</td><td>-0.8</td><td>97.9±0.0</td><td>-0.4</td></tr><tr><td rowspan=\"3\">Fashion-MNIST</td><td>1</td><td>70.6±0.6</td><td>73.5±0.5*</td><td>-9.8</td><td>61.2±0.1</td><td>-4.1</td></tr><tr><td>10</td><td>84.6±0.3</td><td>86.8±0.1</td><td>-1.3</td><td>79.7±0.1</td><td>-1.2</td></tr><tr><td>50</td><td>88.7±0.2</td><td>88.0±0.1*</td><td>-4.5</td><td>85.0±0.1</td><td>-1.8</td></tr><tr><td rowspan=\"3\">SVHN</td><td>1</td><td>31.2±1.4</td><td>57.3±0.1*</td><td>-8.3</td><td>23.8±0.2</td><td>-0.2</td></tr><tr><td>10</td><td>79.2±0.5</td><td>75.0±0.1</td><td>-1.6</td><td>53.2±0.3</td><td>0.4</td></tr><tr><td>50</td><td>84.4±0.4</td><td>80.5±0.1</td><td>-1.0</td><td>76.5±0.3</td><td>-0.4</td></tr><tr><td rowspan=\"3\">CIFAR-10</td><td>1</td><td>28.8±0.7</td><td>49.9±0.2</td><td>-9.2</td><td>24.7±0.1</td><td>-1.4</td></tr><tr><td>10</td><td>52.1±0.5</td><td>62.7±0.3</td><td>-4.6</td><td>49.3±0.1</td><td>-4.3</td></tr><tr><td>50</td><td>60.6±0.5</td><td>68.6±0.2</td><td>-4.5</td><td>62.0±0.2</td><td>-3.9</td></tr><tr><td rowspan=\"2\">CIFAR-100</td><td>1</td><td>13.9±0.3</td><td>15.7±0.2*</td><td>-18.1</td><td>11.8±0.2</td><td>-12.0</td></tr><tr><td>10</td><td>32.3±0.3</td><td>28.3±0.1</td><td>-17.4</td><td>25.0±0.1</td><td>-14.2</td></tr></table>",
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+        "type": "text",
+        "text": "Figures 2 and 3 provide a closer look at KIP transfer changes under various settings. The first of these tracks how transfer performance changes when adding additional layers as function of the number of KIP training steps used. The normalization layers appear to harm performance for MSE loss, which can be anticipated from their absence in the KIP and LS optimization procedures. However they appear to provide some benefit for cross entropy loss. For Figure 3, we observe that as KIP training progresses, the downstream finite-width network’s performance also improves in general. A notable exception is observed when learning the labels in KIP, where longer training steps lead to deterioration of information useful to training finite-width neural networks. We also observe that as predicted by infinite-width theory [Jacot et al., 2018, Lee et al., 2019], the overall gap between KIP or LS performance and finite-width neural network decreases as the width increases. While our best performing transfer is obtained with width 1024, Figure 3 (middle) suggest that even with modest width of 64, our transfer can outperform prior art of $6 0 . 6 \\%$ by Zhao and Bilen [2021]. ",
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+        "text": "Finally, in Figure 4, we investigate the performance of KIP images over the course of training of a single run, as compared to natural images, over a range of hyperparameters. We find the outperformance of KIP images above natural images consistent across hyperparameters and checkpoints. This suggests that our KIP images may also be effective for accelerated hyperparameter search, an application of dataset distillation explored in Zhao et al. [2021], Zhao and Bilen [2021]. ",
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+        "text": "Altogether, we find our neural network training results encouraging. First, it validates the applicability of infinite-width methods to the setting of finite width [Huang and Yau, 2020, Dyer and Gur-Ari, 2020, Andreassen and Dyer, 2020, Yaida, 2020, Lee et al., 2020]. Second, we find some of the transfer results quite surprising, including efficacy of label solve and the use of cross-entropy for certain settings (see $\\ S \\mathrm { A }$ for the full details). ",
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+        "text": "4 Understanding KIP Images and Labels ",
+        "text_level": 1,
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+        "text": "A natural question to consider is what causes KIP to improve generalization performance. Does it simplify support images, removing noise and minor sources of variation while keeping only the core features shared between many target images of a given class? Or does it make them more complex by producing outputs that combine characteristics of many samples in a single resulting collage? While properly answering this question is subject to precise definitions of simplicity and complexity, we find that KIP tends to increase the complexity of pictures based on the following experiments: ",
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+            "Figure 2: Robustness to neural network variations. KIP ConvNet images (trained with fixed labels) are tested on variations of the ConvNet neural network, including those which have various normalization layers (layer, instance, batch). A similar architecture to ConvNet, the Myrtle5 architecture (without normalization layers) [Shankar et al., 2020], which differs from the ConvNet architecture by having an additional convolutional layer at the bottom and a global average pooling that replaces the final local average pooling at the top, is also tested. Finally, mean-square error is compared with cross-entropy loss (left versus right). Settings: CIFAR-10, 500 images, ZCA, no label learning. "
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+        "image_caption": [
+            "Figure 3: Variations for neural network transfer. Left: Plot of transfer performance as a function of KIP training steps across various train settings. Here (a) denotes augmentations used during KIP training and $\\left( \\mathrm { a } { + } \\mathrm { l } \\right)$ denotes that additionally the labels were learned. MSE and XENT denote mean-square-error and cross entropy loss for the neural network, where for the case of XENT and $\\left( \\mathrm { a } \\mathrm { + } \\mathrm { l } \\right)$ , the labels for the neural network are the argmax of the learned labels. Middle: Exploring the effect of width on transferability of vanilla KIP data. Right: The effect of width on the transferability of label solved data. Settings: CIFAR-10, 500 images, ZCA. ",
+            "Figure 4: Hyperparameter robustness. In the above, KIP images across eight different checkpoints are used to train the ConvNet neural network. Each point in each plot is a neural network training with a different hyperparameter, and its location records the final test accuracy when training on natural images versus the KIP images obtained from initializing from such images. For both MSE and cross entropy loss, KIP images consistently exceed natural images across many hyperparameters. Settings: CIFAR-10, 500 images, ZCA, no augmentations, no label learning. "
+        ],
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+        "image_caption": [
+            "Figure 5: Examples of learned images. Images are initialized from natural images in the top row (Init) and converge to images in the bottom row (Trained). Settings: 100 images distilled, no ZCA, no label training, augmentations. "
+        ],
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+    {
+        "type": "text",
+        "text": "• Visual analysis: qualitatively, KIP images tend to be richer in textures and contours. • Dimensional analysis: KIP produces images of higher intrinsic dimensionality (i.e. an estimate of the dimensionality of the manifold on which the images lie) than natural images. • Spectral analysis: unlike natural images, for which the bulk of generalization performance is explained by a small number of top few eigendirections, KIP images leverage the whole spectrum much more evenly. ",
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+    {
+        "type": "text",
+        "text": "Combined, these results let us conclude that KIP usually increases complexity, integrating features from many target images into much fewer support images. ",
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+        "type": "text",
+        "text": "Visual Analysis. A visual inspection of our learned data leads to intriguing observations in terms of interpretability. Figure 5 shows examples of KIP learned images from CIFAR-100. The resulting images are heterogeneous in terms of how they can be interpreted as distilling the data. For instance, the distilled apple image seems to consist of many apples nested within a possibly larger apple, whereas the distilled bottle image starts off as two bottles and before transforming into one, while other classes (like the beaver) are altogether visually indistinct. Investigating and quantifying aspects that make these images generalize so well is a promising avenue for future work. We show examples from other datasets in Figure A2. ",
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+    {
+        "type": "text",
+        "text": "In Figure 6, we compare MNIST KIP data learned with and without label learning. For the latter case with images and labels optimized jointly, while labels become more informative, encoding richer inter-class information, the images become less interpretable. This behavior consistently leads to superior KRR results, but appears to not be leveraged as efficiently in the neural network transfer setting (Table 2). Experimental details can be found in $\\ S \\mathrm { A }$ . ",
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+    {
+        "type": "text",
+        "text": "Dimensional Analysis. We study the intrinsic dimension of KIP images and find that they tend to grow. Intrinsic dimension (ID) was first defined by Bennett [1969] as “the number of free parameters required in a hypothetical signal generator capable of producing a close approximation to each signal in the collection”. In our context, it is the dimensionality of the manifold embedded into the image space which contains all the support images. Intuitively, simple datasets have a low ID as they can be described by a small number of coordinates on the low-dimensional data manifold. ",
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+    {
+        "type": "text",
+        "text": "ID can be defined and estimated differently based on assumptions on the manifold structure and the probability density function of the images on this manifold (see [Camastra and Staiano, 2016] for review). We use the “Two-NN” method developed by [Facco et al., 2017], which makes relatively few assumptions and allows to estimate ID only from two nearest-neighbor distances for each datapoint. ",
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+    {
+        "type": "text",
+        "text": "Figure 7 shows that the ID is increasing for the learned KIP images as a function of the training step across a variety of configurations (training with or without augmentations/label learning) and datasets. One might expect that a distillation procedure should decrease dimensionality. On the other hand, Ansuini et al. [2019] showed that ID increases in the earlier layers of a trained neural network. It remains to be understood if this latter observation has any relationship with our increased ID. Note that ZCA preprocessing, which played an important role for getting the best performance for our RGB datasets, increases dimensionality of the underlying data (see Figure A4). ",
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+    {
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+        "img_path": "images/744ef5e48025d1576b73820fcc92a143cc5fbf893ece6f8a392eca2b6478b9c4.jpg",
+        "image_caption": [
+            "Figure 6: Dataset distillation with trainable and non-trainable labels. Top row: initialization of support images and labels. Middle row: trained images if labels remain fixed. Bottom row: trained images and labels, jointly optimized. Settings: 100 images distilled, no augmentations. "
+        ],
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+    {
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+        "img_path": "images/a06942204ec114b405c74d18fc52c11993ea495a87514472cc9da54909bdfd25.jpg",
+        "image_caption": [
+            "Figure 7: Intrinsic dimension of a learned datasets grows during training. As training progresses, intrinsic dimension of the learned dataset grows, indicating that training non-trivially transforms the data manifold. See Figures A2 and 6 for visual examples of learned images, and Figures A3 and A4 for similar observations using other metrics and settings. Settings: 500 images distilled, no ZCA. "
+        ],
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+    {
+        "type": "text",
+        "text": "Spectral Analysis. Another distinguishing property of KIP images is how their spectral components contribute to performance. In Figure 8, we spot how different spectral bands of KIP learned images affect test performance as compared to their initial natural images. Here, we use the FC2, Conv-Vec8, and ConvNet architectures. We note that for natural images (light bars), most of their performance is captured by the top $20 \\%$ of eigenvalues. For KIP images, the performance is either more evenly distributed across the bands (FC and Conv-Vec8) or else is skewed towards the tail (ConvNet). ",
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+    {
+        "type": "text",
+        "text": "5 Related Work ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Dataset distillation was first studied in Wang et al. [2018]. The work of Sucholutsky and Schonlau [2019], Bohdal et al. [2020] build upon it by distilling labels. Zhao et al. [2021] proposes condensing a training set by harnessing a gradient matching condition. Zhao and Bilen [2021] takes this idea further by applying a suitable augmentation strategy. Note that while Zhao and Bilen [2021] is limited in augmentation expressiveness (they have to apply a single augmentation per training iteration), we can sample augmentations independently per image in our target set per train step. Our work together with Nguyen et al. [2021] are, to the best of our knowledge, the only works using kernel-based methods for dataset distillation on image classification datasets. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/4c290bf4e301cc2883615e72ab385da78ab75a5398ad90dbbe6ea01579e15c20.jpg",
+        "image_caption": [
+            "Figure 8: Spectral contribution to test accuracy shifts to the tail. By setting the ridge-parameter to zero in kernel-ridge regression and composing $K _ { X _ { s } X _ { s } } ^ { - 1 }$ with the spectral projection onto various eigenspaces, we can explore how different spectral bands affect test accuracy of kernel ridgeregression. We plot the relative change in test accuracy using contiguous bands of $20 \\%$ of the eigenvalues. Settings: CIFAR-10, 500 images. Further details in $\\ S$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "Our use of kernels stems from the correspondence between infinitely-wide neural networks and kernel methods [Neal, 1994, Lee et al., 2018, Matthews et al., 2018, Jacot et al., 2018, Novak et al., 2019, Garriga-Alonso et al., 2019, Arora et al., 2019, Yang, 2019a,b, Hron et al., 2020] and the extended correspondence with the finite width corrections [Dyer and Gur-Ari, 2020, Huang and Yau, 2020, Yaida, 2020]. These correspondences underlie the transferability of our KRR results to neural networks , and have been utilized in understanding trainability [Xiao et al., 2020], generalizations [Adlam and Pennington, 2020], training dynamics [Lewkowycz et al., 2020, Lewkowycz and Gur-Ari, 2020], uncertainty [Adlam et al., 2021], and demonstrated their effectiveness for smaller datasets [Arora et al., 2020] and neural architecture search [Park et al., 2020, Chen et al., 2021]. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We performed an extensive study of dataset distillation using the KIP and LS algorithms applied to convolutional architectures, obtaining SOTA results on a variety of image classification datasets. In some cases, our learned datasets were more effective than a natural dataset two orders of magnitude larger in size. There are many interesting followup directions and questions from our work: ",
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+    {
+        "type": "text",
+        "text": "First, integrating efficient kernel-approximation methods into our algorithms, such as those of Zandieh et al. [2021], will reduce computational burden and enable scaling up to larger datasets. In this direction, the understanding of how various resources (e.g. data, parameter count, compute) scale when optimizing for neural network performance has received significant attention as machine learning models continue to stretch computational limits [Hestness et al., 2017, Rosenfeld et al., 2020, Kaplan et al., 2020, Bahri et al., 2021]. Developing our understanding of how to harness smaller, yet more useful representations data would aid in such endeavors. In particular, it would be especially interesting to explore how well datasets can be compressed as they scale up in size. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Second, LS and KIP with label learning shows that optimizing labels is a very powerful tool for dataset distillation. The labels we obtain are quite far away from standard, interpretable labels and we feel their effectiveness suggests that understanding of how to optimally label data warrants further study. ",
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+    {
+        "type": "text",
+        "text": "Finally, the novel features obtained by our learned datasets, and those of dataset distillation methods in general, may reveal insights into interpretability and the nature of sample-efficient representations. For instance, observe the bee and bicycle images in Figure 5: the bee class distills into what appears to be spurious visual features (e.g. pollen), while the bicycle class distills to the essential contours of a typical bicycle. Additional analyses and explorations of this type could offer insights into the perennial question of how neural networks learn and generalize. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments We would like to acknowledge special thanks to Samuel S. Schoenholz, who proposed and helped develop the overall strategy for our distributed KIP learning methodology. We are also grateful to Ekin Dogus Cubuk and Manuel Kroiss for helpful discussions. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
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