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Question: Recent powerful pre-trained language models have achieved remarkable performance on most of the popular datasets for reading comprehension. It is time to introduce more challenging datasets to push the development of this field towards more comprehensive reasoning of text. In this paper, we introduce a new Reading Comprehension dataset requiring logical reasoning (ReClor) extracted from standardized graduate admission examinations. As earlier studies suggest, human-annotated datasets usually contain biases, which are often exploited by models to achieve high accuracy without truly understanding the text. In order to comprehensively evaluate the logical reasoning ability of models on ReClor, we propose to identify biased data points and separate them into EASY set while the rest as HARD set. Empirical show that the state-of-the-art models have an outstanding ability to capture biases contained in the dataset with high accuracy on EASY set. However, they struggle on HARD set with poor performance near that of random guess, indicating more research is needed to essentially enhance the logical reasoning ability of current models. Machine reading comprehension (MRC) is a fundamental task in Natural Language Processing, which requires models to understand a body of text and answer a particular question related to the context. With success of unsupervised representation learning in NLP, language pre-training based models such as GPT-2 , BERT , XLNet and RoBERTa have achieved nearly saturated performance on most of the popular MRC datasets (; ; ;). It is time to challenge state-of-the-art models with more difficult reading comprehension tasks and move a step forward to more comprehensive analysis and reasoning over text . In natural language understanding, logical reasoning is an important ability to examine, analyze and critically evaluate arguments as they occur in ordinary language according to the definition from Law School Admission Council (2019a). It is a significant component of human intelligence and is essential in negotiation, debate and writing etc. However, existing reading comprehension datasets have none or merely a small amount of data requiring logical reasoning, e.g., 0% in MCTest dataset and 1.2% in SQuAD according to. One related task is natural language inference, which requires models to label the logical relationships of sentence pairs. However, this task only considers three types of simple logical relationships and only needs reasoning at sentence-level. To push the development of models in logical reasoning from simple logical relationship classification to multiple complicated logical reasoning and from sentence-level to passage-level, it is necessary to introduce a reading comprehension dataset targeting logical reasoning. A typical example of logical reasoning questions is shown in Table 1. Similar to the format of multiple-choice reading comprehension datasets , it contains a context, a question and four options with only one right answer. To answer the question in this example, readers need to identify the logical connections between the lines to pinpoint the conflict, then understand each of the options and select an option that solves the conflict. Human minds need extensive training and practice to get used to complex reasoning, and it will take immense efforts for crowdsourcing workers to design such logical reasoning questions. Inspired by the datasets extracted from standardized examinations , we build a dataset by selecting such logical reasoning questions from standardized exams such as GMAT 1 and LSAT 2. We finally collect 6,139 pieces of logical reasoning questions, which constitute a Reading Comprehension dataset requiring logical reasoning (ReClor). Human-annotated datasets usually contain biases (; ; ; ; ;), which are often exploited by neural network models as shortcut solutions to achieve high testing accuracy. For data points whose options can be selected correctly without knowing the contexts and questions, we classify them as biased ones. In order to fully assess the logical reasoning ability of the models, we propose to identify the biased data points and group them as EASY set, and put the rest into HARD set. Based on our experiments on these separate sets, we find that even the state-of-the-art models can only perform well on EASY set and struggle on HARD set as shown in Figure 1. This phenomenon shows that current models can well capture the biases in the dataset but lack the ability to understand the text and reason based on connections between the lines. On the other hand, human beings perform similarly on both the EASY and HARD set. It is thus observed that there is still a long way to go to equip models with true logical reasoning ability. The contributions of our paper are two-fold. First, we introduce ReClor, a new reading comprehension dataset requiring logical reasoning. We use option-only-input baselines trained with different random seeds to identify the data points with biases in the testing set, and group them as EASY set, with the rest as HARD set to facilitate comprehensive evaluation. Second, we evaluate several stateof-the-art models on ReClor and find these pre-trained language models can perform well on EASY set but struggle on the HARD set. This indicates although current models are good at exploiting biases in the dataset, they are far from capable of performing real logical reasoning yet. Reading Comprehension Datasets A variety of reading comprehension datasets have been introduced to promote the development of this field. MCTest is a dataset with 2,000 multiple-choice reading comprehension questions about fictional stories in the format similar to ReClor. proposed SQuAD dataset, which contains 107,785 questionanswer pairs on 536 Wikipedia articles. The authors manually labeled 192 examples of the dataset and found that the examples mainly require reasoning of lexical or syntactic variation. In an analysis of the above-mentioned datasets, found that none of questions requiring logical reasoning in MCTest dataset and only 1.2% in SQuAD dataset . introduced RACE dataset by collecting the English exams for middle and high school Chinese students in the age range between 12 to 18. They hired crowd workers on Amazon Mechanical Turk to label the reasoning type of 500 samples in the dataset and show that around 70 % of the samples are in the category of word matching, paraphrasing or single-sentence reasoning. To encourage progress on deeper comprehension of language, more reading comprehension datasets requiring more complicated reasoning types are introduced, such as iterative reasoning about the narrative of a story (Kočiskỳ et al., 2018), multi-hop reasoning Table 1: An example in the ReClor dataset which is modified from the Law School Admission Council (2019b). across multiple sentences and multiple documents , commonsense knowledge reasoning (; ;) and numerical discrete reasoning over paragraphs . However, to the best of our knowledge, although there are some datasets targeting logical reasoning in other NLP tasks mentioned in the next section, there is no dataset targeting evaluating logical reasoning in reading comprehension task. This work introduces a new dataset to fill this gap. Logical Reasoning in NLP There are several tasks and datasets introduced to investigate logical reasoning in NLP. The task of natural language inference, also known as recognizing textual entailment (; ; ; ;) requires models to take a pair of sentence as input and classify their relationship types, i.e., ENTAILMENT, NEUTRAL, or CONTRADICTION. SNLI and MultiNLI datasets are proposed for this task. However, this task only focuses on sentence-level logical relationship reasoning and the relationships are limited to only a few types. Another task related to logical reasoning in NLP is argument reasoning comprehension task introduced by with a dataset of this task. Given an argument with a claim and a premise, this task aims to select the correct implicit warrant from two options. Although the task is on passage-level logical reasoning, it is limited to only one logical reasoning type, i.e., identifying warrants. ReClor and the proposed task integrate various logical reasoning types into reading comprehension, with the aim to promote the development of models in logical reasoning not only from sentence-level to passage-level, but also from simple logical reasoning types to the complicated diverse ones. There have been several datasets extracted from human standardized examinations in NLP, such as RACE dataset mentioned above. Besides, NTCIR QA Lab offers comparative evaluation for solving real-world university entrance exam questions; The dataset of CLEF QA Entrance Exams Task is extracted from standardized English examinations for university admission in Japan; ARC dataset consists of 7,787 science questions targeting student grade level, ranging from 3rd grade to 9th; The dialogue-based multiple-choice reading comprehension dataset DREAM contains 10,197 questions for 6,444 multi-turn multi-party dialogues from English language exams that are designed by human experts to assess the comprehension level of Chinese learners of English. Compared with these datasets, ReClor distinguishes itself by targeting logical reasoning. 3 RECLOR DATA COLLECTION AND ANALYSIS The format of data in ReClor is similar to other multiple-choice reading comprehension datasets , where a data point contains a context, a question and four answer options, among which only one option is right/most suitable. We collect reading comprehen- sion problems that require complicated logical reasoning. However, producing such data requires the ability to perform complex logical reasoning, which makes it hard for crowdsourcing workers to generate such logical questions. Fortunately, we find the reading comprehension problems in some standardized tests, such as GMAT and LSAT, are highly in line with our expectation. We construct a dataset containing 6,139 logical reasoning questions sourced from open websites and books 3. In the original problems, there are five answer options in which only one is right. To comply with fair use of law 4, we shuffle the order of answer options and randomly delete one of the wrong options for each data point, which in four options with one right option and three wrong options. Furthermore, similar to ImageNet dataset 5, we plan to offer the dataset to researchers/educators who agree to have it for non-commercial research and/or educational use only. As mentioned above, we collect 6,139 data points, in which 91.22% are from actual exams of GMAT and LSAT while others are from high-quality practice exams. They are divided into training set, validation set and testing set with 4,639, 500 and 1,000 data points respectively. The overall statistics of ReClor and comparison with other similar multiple-choice MRC datasets are summarized in Table 2. As shown, ReClor is of comparable size and relatively large vocabulary size. Compared with RACE, the length of the context of ReCor is much shorter. In RACE, there are many redundant sentences in context to answer a question. However, in ReClor, every sentence in the context passages is important, which makes this dataset focus on evaluating the logical reasoning ability of models rather than the ability to extract relevant information from a long context. The length of answer options of ReClor is largest among these datasets. We analyze and manually annotate the types of questions on the testing set and group them into 18 categories, whose percentages and descriptions are shown in Table 3. The percentages of different types of questions reflect those in the logical reasoning module of GMAT and LSAT. Some examples of different types of logical reasoning are listed in Figure 2, and more examples are listed in the Appendix C. Taking two examples, we further express how humans would solve such questions in Table 4, showing the challenge of ReClor. The dataset is collected from exams devised by experts in logical reasoning, which means it is annotated by humans and may introduce biases in the dataset. Recent studies have shown that models can utilize the biases in a dataset of natural language understanding to perform well on the task without truly understanding the text (; ; ; ; ;). It is necessary to analyze such data biases to help evaluate models. In the ReClor dataset, the common context and question are shared across the four options for each data point, so we focus on the analysis of the difference in lexical choice and sentence length of the right and wrong options without contexts and questions. We first investigate the biases of lexical choice. We lowercase the options and then use WordPiece tokenization of BERT BASE , for the tokens in options, we analyze their conditional probability of label l ∈ {right, wrong} given by the token t by p(l|t) = count(t, l)/count(t). The larger the correlation score is for a particular token, the more likely it contributes to the prediction of related option. Many neural network based models such as FastText , Bi-LSTM, GPT , GPT-2 , BERT , XLNet , If the purpose of laws is to contribute to people's happiness, we have a basis for criticizing existing laws as well as proposing new laws. Hence, if that is not the purpose, then we have no basis for the evaluation of existing laws, from which we must conclude that existing laws acquire legitimacy simply because they are the laws Question: The reasoning in the argument is flawed in that the argument Options: A. takes a sufficient condition for a state of affairs to be a necessary condition for it B. draws a about how the world actually is on the basis of claims about how it should be C. infers a causal relationship from the mere presence of a correlation D. trades on the use of a term in one sense in a premise and in a different sense in the Answer: A Reasoning Process of Humans: We may first look at the question to understand the specific task of the question -identify a flaw. We then analyze the argument in the context. The 'existing laws acquire legitimacy simply because they are the laws.' is based on the argument (purpose is NOT happiness) → (NOT basis for criticizing laws), which is obtained from the first statement: (purpose is happiness) → (basis for criticizing laws). However, we know ¬A → ¬B cannot be obtained from A → B. Therefore, we should choose option A that describes this flaw. The distractors here are different types of reasoning flaws. Prior knowledge of basic logical rules is needed to correctly answer this question. Psychologist: Phonemic awareness, or the knowledge that spoken language can be broken into component sounds, is essential for learning to read an alphabetic language. But one also needs to learn how sounds are symbolically represented by means of letters; otherwise, phonemic awareness will not translate into the ability to read an alphabetic language. Yet many children who are taught by the whole-language method, which emphasizes the ways words sound, learn to read alphabetic languages. Question: Which one of the following can be properly inferred from the psychologist's statements? A. The whole-language method invariably succeeds in teaching awareness of how spoken language can be broken into component sounds. B. Some children who are taught by the whole-language method are not prevented from learning how sounds are represented by means of letters. C. The whole-language method succeeds in teaching many children how to represent sounds symbolically by means of letters. D. When the whole-language method succeeds in teaching someone how to represent sounds by means of letters, that person acquires the ability to read an alphabetic language. Answer: B Reasoning Process of Humans: Looking at the question and we know that it is asking about implication. From the first two sentences in context, we know that there are two necessary conditions to read an alphabetic language: phonemic awareness and symbolic letters. We also learn [(NOT symbolic letters) AND (phonemic awareness)] → read an alphabetic language (denoted as Formula 1). The last sentence in the context says that many children are taught by the whole-language method to learn a language. As for option A, from the context, we only know the whole language method works for'many' children, which cannot be inferred to'invariably' works. As for option B, combing three sentences in the context, we know that the whole-language method meets the two necessary conditions to learn a language, especially the last sentence mentions'learn to read alphabetic languages'. Children learn to read alphabetic languages means that they must recognize symbolic letters that represent sound because symbolic letters is a necessary condition of read an alphabetic language; otherwise, they cannot read because of Formula 1 mentioned above. Therefore, option B is correct. As for option C, from the context we only know the whole-language method teaches phonemic awareness and read an alphabetic language. Symbolic letters may be taught by other methods, so C is wrong. As for D, similar to C, symbolic letters may be taught by other methods and we also cannot obtain: symbolic letters → read an alphabetic language. Table 4: Two examples to show how humans to solve the questions. Context: The current pattern of human consumption of resources, in which we rely on nonrenewable resources, for example metal ore, must eventually change. Since there is only so much metal ore available, ultimately, we must either do without or turn to renewable resources to take its place. A. We cannot indefinitely replace exhausted nonrenewable resources with other nonrenewable resources. B. Consumption of nonrenewable resources will not continue to increase in the near future. C. There are renewable resource replacements for all of the nonrenewable resources currently being consumed. D. Ultimately we cannot do without nonrenewable resources. Context: Some theorists argue that literary critics should strive to be value-neutral in their literary criticism. These theorists maintain that by exposing the meaning of literary works without evaluating them, critics will enable readers to make their own judgments about the works' merits. But literary criticism cannot be completely value-neutral. Thus, some theorists are mistaken about what is an appropriate goal for literary criticism. Q: The argument's follows logically if which one of the following is assumed? A. Any critic who is able to help readers make their own judgments about literary works' merits should strive to produce value-neutral criticism. B. If it is impossible to produce completely value-neutral literary criticism, then critics should not even try to be value-neutral. C. The less readers understand the meaning of a literary work, the less capable they will be of evaluating that work's merits.. D. Critics are more likely to provide criticisms of the works they like than to provide criticisms of the works they dislike. The television show Henry was not widely watched until it was scheduled for Tuesday evenings immediately after That' s Life, the most popular show on television. During the year after the move, Henry was consistently one of the ten most-watched shows on television. Since Henry' s recent move to Wednesday evenings, however, it has been watched by far fewer people. We must conclude that Henry was widely watched before the move to Wednesday evenings because it followed That' s Life and not because people especially liked it. Q: Which one of the following, if true, most strengthens the argument? A. The show that now follows That's Life on Tuesdays has double the number of viewers it had before being moved. B. Henry has been on the air for three years, but That's Life has been on the air for only two years. C. After its recent move to Wednesday, Henry was aired at the same time as the second most popular show on television. D. That's Life was not widely watched during the first year it was aired. Context: When a certain gland becomes cancerous in humans, it produces high levels of a particular protein. A blood test can determine the level of this protein well before a cancer of the gland could be detected by other means. Some doctors recommend that aggressive anticancer treatment should be begun as early as possible for anyone who is tested and is found to have high levels of the protein. Q: Which one of the following, if true, most seriously weakens the doctors' recommendation? A. The blood test for the protein has been in use for some time to monitor the condition of patients who have been diagnosed as having cancer of the gland. B. Before the blood test became available, about one third of all cases of cancer of the gland were detected in early stages. C. So far, no patients whose protein levels were found to be normal have subsequently developed cancer of the gland. D. Enlargement of the gland, a common condition infrequently associated with cancer, in high levels of the protein. Context: In a study, pairs of trained dogs were placed side by side and given a command such as "sit". After both obeyed the command, one dog was given a treat while its partner was given no reward at all. Over time, the dogs who went unrewarded began to disobey the command. This shows that dogs have an aversion to being treated unfairly. Q: Which one of the following would be most useful to know in order to evaluate the argument? A. Were dogs who were accustomed to receiving regular rewards prior to the study more inclined to obey the command? B. How many repetitions were required before the unrewarded dogs began to disobey the command? RoBERTa have achieved impressive in various NLP tasks. We challenge these neural models with ReClor to investigate how well they can perform. Details of the baseline models and implementation are shown in the Appendix A and B. As mentioned earlier, biases prevalently exist in human-annotated datasets (; ; ;), which are often exploited by models to perform well without truly understanding the text. Therefore, it is necessary to find out the biased data points in ReClor in order to evaluate models in a more comprehensive manner . To this end, we feed the five strong baseline models (GPT, GPT-2, BERT BASE, XLNet BASE and RoBERTa BASE) with ONLY THE ANSWER OPTIONS for each problem. In other words, we purposely remove the context and question in the inputs. In this way, we are able to identify those problems that can be answered correctly by merely exploiting the biases in answer options without knowing the relevant context and question. However, the setting of this task is a multiple-choice question with 4 probable options, and even a chance baseline could have 25% probability to get it right. To eliminate the effect of random guess, we set four different random seeds for each model and pick the data points that are predicted correctly in all four cases to form the EASY set. Then, the data points which are predicted correctly by the models at random could be nearly eliminated, since any data point only has a probability of (25%) 4 = 0.39% to be guessed right consecutively for four times. Then we unite the sets of data points that are consistently predicted right by each model, because intuitively different models may learn different biases of the dataset. The above process is formulated as the following expression, where C seed1 BERT denotes the set of data points which are predicted correctly by BERT BASE with seed 1, and similarly for the rest. Table 6 shows the average performance for each model trained with four different random seeds and the number of data points predicted correctly by all of them. Finally, we get 440 data points from the testing set C TEST and we denote this subset as EASY set C EASY and the other as HARD set C HARD. Table 6: Average accuracy of each model using four different random seeds with only answer options as input, and the number of their common correct predictions. Among multiple-choice reading comprehension or QA datasets from exams, although the size of ReClor is comparable to those of ACR and DREAM , it is much smaller than. Recent studies (; ; ;) have shown the effectiveness of pre-training on similar tasks or datasets then fine-tuning on the target dataset for transfer learning. find that by first training on RACE and then further fine-tuning on the target dataset, the performances of BERT BASE on multiple-choice dataset MC500 and DREAM can significantly boost from 69.5% to 81.2%, and from 63.2% to 70.2%, respectively. However, they also find that the model cannot obtain significant improvement even performs worse if it is first fine-tuned on span-based dataset like SQuAD . ReClor is a multiple-choice dataset, so we choose RACE for fine-tuning study. The performance of all tested models on the ReClor is presented in Table 7. This dataset is built on questions designed for students who apply for admission to graduate schools, thus we randomly choose 100 samples from the testing set and divide them into ten tests, which are distributed to ten different graduate students in a university. We take the average of their scores and present it as the baseline of graduate students. The data of ReClor are carefully chosen and modified from only high-quality questions from standardized graduate entrance exams. We set the ceiling performance to 100% since ambiguous questions are not included in the dataset. The performance of fastText is better than random guess, showing that word correlation could be used to help improve performance to some extent. It is difficult for Bi-LSTM to converge on this Table 7: Accuracy (%) of models and human performance. The column Input means whether to input context (C), question (Q) and answer options (A). The RACE column represents whether to first use RACE to fine-tune before training on ReClor. dataset. Transformer-based pre-training models have relatively good performance, close to the performance of graduate students. However, we find that these models only perform well on EASY set with around 70% accuracy, showing these models have an outstanding ability to capture the biases of the dataset, but they perform poorly on HARD set with only around 30% accuracy. In contrast, humans can still keep good performance on HARD set. We notice the difference in testing accuracy performed by graduate students on EASY and HARD set, but this could be due to the small number of students participated in the experiments. Therefore, we say humans perform relatively consistent on both biased and non-biased dataset. It is noticed that if the models are first trained on RACE and then fine-tuned on ReClor, they could obtain significant improvement, especially on HARD set. The overall performance of XLNet LARGE is even very close to that of graduate students. This similar phenomenon can also be observed on DREAM dataset by , which shows the potential of transfer learning for reasoning tasks. However, even after fine-tuning on RACE, the best performance of these strong baselines on HARD set is around 50%, still lower than that of graduate students and far away from ceiling performance. Experiments in different input settings are also done. Compared with the input setting of answer options only (A), the setting of questions and answer options (Q, A) can not bring significant improvement. This may be because some questions e.g., Which one of the following is an assumption required by the argument?, Which one of the following, if true, most strengthens the argument? can be used in the same reasoning types of question, which could not offer much information. Further adding context causes significant boost, showing the high informativeness of the context. We further analyze the model performance with respect to different question types of logical reasoning. Some are shown in Figure 4 and the full are shown in Figure 5, 6 and 7 in the Appendix E. Three models of BERT LARGE, XLNet LARGE and RoBERTa LARGE all perform poorly on the type of MATCH PRINCIPLES/POINTS on EASY set. On HARD set, the three models perform poorly on certain types such as STRENGTHEN, WEAKEN and ROLE which require extensive logical reasoning. However, they perform relatively better on other types, such as and SUMMARY/MAIN POINT that are more straight-forward. For the of transfer learning, we analyze XLNet LARGE in detail. Though the overall performance is significantly boosted after fine-tuning on RACE first, the histograms in the bottom of Figure 4 show that on EASY set, accuracy improves a similar amount among different question type, while on HARD set, significant improvement on some question types is observed, such as EVALUATION, and SUM-MARY/MAIN POINT. This may be because these types require less logical reasoning to some extent compared with other types, and similar question types may also be found in RACE dataset. Thus, the pre-training on RACE helps enhance the ability of logical reasoning especially of relatively simple reasoning types, but more methods are still needed to further enhance the ability especially that of relatively complex reasoning types. Figure 4: Performance of models on EASY (left) and HARD (right) testing sets and that of models. XLNet LARGE +Fine-Tune means the model is first fine-tuned on RACE before training on ReClor. In this paper, we introduce ReClor, a reading comprehension dataset requiring logical reasoning, with the aim to push research progress on logical reasoning in NLP forward from sentence-level to passage-level and from simple logical reasoning to multiple complicated one. We propose to identify biased data points and split the testing set into EASY and HARD group for biased and non-biased data separately. We further empirically study the different behaviors of state-of-the-art models on these two testing sets, and find recent powerful transformer-based pre-trained language models have an excellent ability to exploit the biases in the dataset but have difficulty in understanding and reasoning given the non-biased data with low performance close to or slightly better than random guess. These show there is a long way to equip deep learning models with real logical reasoning abilities. We hope this work would inspire more research in future to adopt similar split technique and evaluation scheme when reporting their model performance. We also show by first fine-tuning on a large-scale dataset RACE then fine-tuning on ReClor, the models could obtain significant improvement, showing the potential of transfer learning to solve reasoning tasks. fastText FastText models sentences as a bag of n-grams, and tries to predict the probability of each answer being correct independently. We choose the answer with the highest score as the prediction for the multiple-choice setting. LSTM sentence encoder A two-layer bi-LSTM is randomly initialized as a sentence encoder with GloVe word embedding . With a span of text as input, the last hidden state of the second layer is max-pooled and then fed into a fully-connected layer to compute the output score. GPT and GPT-2 GPT and GPT-2 are both transformer based models which are pre-trained using unsupervised method with a standard language modeling objective. GPT is pre-trained on BooksCorpus; GPT-2 is pre-trained using a larger dataset called WebText. Here we use the smallest model proposed in as our GPT-2 baseline. To fine-tune on ReClor, the final hidden vector corresponding to the last input token ([ classify] ) is used as the aggregate representation followed by an extra fully connected layer to compute the score. BERT BERT is also a transformer based model which is trained by using BooksCorpus and English Wikipedia in two unsupervised tasks, i.e., Masked LM (MLM) and Next Sentence Prediction (NSP). During fine-tuning, the final hidden vector corresponding to the first input token ([CLS]) is used as the aggregate representation followed by two extra fully connected layers to compute the score. XLNet XLNet The input format of different models is shown in Table 8. start Context delimiter Question || Option classify GPT-2 start Context delimiter Question || Option classify BERT [CLS] Context [SEP] Question || Option [SEP] [PAD]... XLNet <pad>... Context <sep> Question || Option <sep> <cls> RoBERTa <s> Context </s> </s> Question || Option </s> <pad>... Adam is used by all models with β 1 = 0.9, β 2 = 0.999 and = 1e − 8. For fastText, we use its python library 6 by converting ReClor to the required form, and keep the default setting of the hyper parameters. For Bi-LSTM, we use a two-layer Bidirectional LSTM with the GloVe 300d word embedding followed by max-pooling and a fully-connected layer. We train the model for 100 epochs using a batch size of 64 and learning rate of 0.1. A learning rate decay of 0.5 is also applied every 10 epochs. For pre-training models, we modify the code of PyTorch-Transformers of Hugging Face 7 to implement them on ReClor. We use a batch size of 24 and fine-tune for 50 epochs. For different models, we select the best fine-tuning learning rate among Table 9: Hyperparameters for finetuning pre-training language models on ReClor 6.25e-05, 2e-05, 1e-05, and 5e-06 on the validation set. The maximum input sequence length for all models is 256. The hyperparameters are shown in Table 9. Type: Necessary Assumptions Definition: identify the claim that must be true or is required in order for the argument to work Context: Slash-and-burn agriculture involves burning several acres of forest, leaving vegetable ash that provides ample fertilizer for three or four years of bountiful crops. On the cleared land nutrients leach out of the soil, however, and the land becomes too poor to support agriculture. New land is then cleared by burning and the process starts again. Since most farming in the tropics uses this method, forests in this region will eventually be permanently eradicated. Question: The argument depends on the assumption that Options: A. forests in the tropics do not regenerate well enough to restore themselves once they have been cleared by the slash-and-burn method B. some other methods of agriculture are not as destructive to the environment in tropical regions as the slash-and-burn method is C. forests in the tropics are naturally deficient in nutrients that are needed to support the growth of plants that are not native to those regions D. slash-and-burn agriculture is particularly suitable for farming in tropical areas Answer: A Table 10: The definition and an example of the logical reasoning type -Necessary Assumptions Type: Sufficient Assumptions Definition: identify a sufficient assumption, that is, an assumption that, if added to the argument, would make it logically valid Context: Geologist: A new method for forecasting earthquakes has reliably predicted several earthquakes. Unfortunately, this method can predict only that an earthquake will fall somewhere within a range of two and a half points on the Richter scale. Thus, since a difference of two and a half points can be the difference between a marginally perceptible shaking and a quake that causes considerable damage, the new method is unlikely to be useful. Question: Which one of the following, if assumed, enables the geologist's to be properly inferred? A. An earthquake-forecasting method is unlikely to be useful unless its predictions always differentiate earthquakes that are barely noticeable from ones that in substantial destruction. B. Several well-established methods for forecasting earthquakes can predict within much narrower ranges than two and a half points on the Richter scale. C. Even if an earthquake-forecasting method makes predictions within a very narrow range on the Richter scale, this method is not likely to be useful unless its predictions are reliable. D. An earthquake-forecasting method has not been shown to be useful until it has been used to reliably predict a large number of earthquakes. Answer: A Table 11: The definition and an example of the logical reasoning type -Sufficient Assumptions Type: Strengthen Definition: identify information that would strengthen an argument Context: Financial success does not guarantee happiness. This claim is not mere proverbial wisdom but a fact verified by statistics. In a recently concluded survey, only one-third of the respondents who claimed to have achieved financial success reported that they were happy. Question: Which one of the following, if true, most strongly supports the drawn from the survey ? Options: A. Most of the respondents who reported they were unhappy were in fact happy. B. The respondents who reported financial success were, for the most part, financially successful. C. Many of the respondents who claimed not to have achieved financial success reported that they were happy five years ago. D. Many of the respondents who failed to report financial success were in fact financially successful. Answer: B Table 12: The definition and an example of the logical reasoning type -Strengthen Type: Weaken Definition: identify information that would weaken an argument Context: "DNA fingerprinting" is a recently-introduced biochemical procedure that uses a pattern derived from a person' s genetic material to match a suspect' s genetic material against that of a specimen from a crime scene. Proponents have claimed astronomically high odds against obtaining a match by chance alone. These odds are based on an assumption that there is independence between the different characteristics represented by a single pattern. Question: Which one of the following, if true, casts the most doubt on the claim of the proponents of DNA fingerprinting? Options: A. The skill required of laboratory technicians performing the DNA fingerprinting procedure is not extraordinary. B. There is a generally accepted theoretical basis for interpreting the patterns produced by the procedure. C. In the whole population there are various different subgroups, within each of which certain sets of genetic characteristics are shared. D. In the investigation of certain genetic diseases, the techniques used in DNA fingerprinting have traced the transmission of the diseases among the living members of very large families. Answer: C Table 13: The definition and an example of the logical reasoning type -Weaken Type: Evaluation Definition: identify information that would be useful to know to evaluate an argument Context: George: Some scientists say that global warming will occur because people are releasing large amounts of carbon dioxide into the atmosphere by burning trees and fossil fuels. We can see, though, that the predicted warming is occurring already. In the middle of last winter, we had a month of springlike weather in our area, and this fall, because of unusually mild temperatures, the leaves on our town' s trees were three weeks late in turning color. Question: Which one of the following would it be most relevant to investigate in evaluating the of George's argument? Options: A. whether air pollution is causing some trees in the area to lose their leaves B. what proportion of global emissions of carbon dioxide is due to the burning of trees by humans C. whether unusually warm weather is occurring elsewhere on the globe more frequently than before D. when leaves on the trees in the town usually change color Answer: C To reduce the mosquito population in a resort area, hundreds of trees were planted that bear fruit attractive to birds. Over the years, as the trees matured, they attracted a variety of bird species and greatly increased the summer bird population in the area. As expected, the birds ate many mosquitoes. However, the planting of the fruit trees had the very opposite of its intended effect. Which one of the following, if true, most helps to explain the apparently paradoxical ? Options: A. Most of the species of birds that were attracted by the trees that were planted did not eat mosquitoes. B. Increases and decreases in mosquito populations tend to follow a cyclical pattern. C. The species of birds that were attracted in the greatest number by the fruit of the trees that were planted did not eat mosquitoes. D. The birds attracted to the area by the trees ate many more insects that prey on mosquitoes than they did mosquitoes. Answer: D Buying elaborate screensavers -programs that put moving images on a computer monitor to prevent damage -can cost a company far more in employee time than it saves in electricity and monitor protection. Employees cannot resist spending time playing with screensavers that flash interesting graphics across their screens. Which one of the following most closely conforms to the principle illustrated above? Options: A. An electronic keyboard may be cheaper to buy than a piano but more expensive to repair. B. An energy-efficient insulation system may cost more up front but will ultimately save money over the life of the house. C. The time that it takes to have a pizza delivered may be longer than it takes to cook a complete dinner. D. A complicated hotel security system may cost more in customer goodwill than it saves in losses by theft. Answer: D The museum' s night security guard maintains that the thieves who stole the portrait did not enter the museum at any point at or above ground level. Therefore, the thieves must have gained access to the museum from below ground level. The flawed pattern of reasoning in the argument above is most similar to that in which one of the following? Options: A. As had generally been expected, not all questionnaires were sent in by the official deadline. It follows that plans must have been made for the processing of questionnaires received late. B. The store's competitors claim that the store, in selling off the shirts at those prices, neither made any profit nor broke even. Consequently, the store's customers must have been able to buy shirts there at less than the store's cost. C. The product label establishes that this insecticide is safe for both humans and pets. Therefore, the insecticide must also be safe for such wild mammals as deer and rabbits. D. If the census is to be believed, the percentage of men who are married is higher than the percentage of women who are married. Thus, the census must show a higher number of men than of women overall. Answer: B It is an absurd idea that whatever artistic endeavor the government refuses to support it does not allow, as one can see by rephrasing the statement to read: No one is allowed to create art without a government subsidy. The pattern of reasoning in which one of the following is most similar to that in the argument above? " Options: A. The notion that every scientist who has been supported by a government grant will be successful is absurd, as one can see by rewording it:No scientist is allowed to do research without a government grant. B. The notion that every scientist who is supported by a government grant will be successful is absurd, as one can see by rewording it:No scientist lacking governmental support will be successful. C. The claim that any driver who is not arrested does not break the law is absurd, as one can see by rewording it: Every driver who gets arrested has broken the law. D. The claim that any driver who is not arrested does not break the law is absurd, as one can see by rewording it: Every driver who breaks the law gets arrested. Answer: D If such remote considerations were relevant, almost every other consideration would be too. But this would make determining the seriousness of an offense so difficult that it would be impossible to apply the proportionality principle. The statement that considerations as remote as what an offender did years ago are relevant to the seriousness of an offense plays which one of the following roles in the argument? A. It is an allegedly untenable consequence of a view rejected in the argument's overall . B. It is a statement the argument provides grounds to accept and from which the overall is inferred. C. It is the overall in favor of which the argument offers evidence. D. It is a premise offered in support of an intermediate of the argument. Answer: A The tidal range at a particular location is the difference in height between high tide and low tide. Tidal studies have shown that one of the greatest tidal ranges in the world is found in the Bay of Fundy and reaches more than seventeen meters. Since the only forces involved in inducing the tides are the sun' s and moon' s gravity, the magnitudes of tidal ranges also must be explained entirely by gravitational forces. Question: Which one of the following most accurately describes a flaw in the reasoning above? Options: A. It does not differentiate between the tidal effect of the sun and the tidal effect of the moon. B. It fails to consider that the size of a tidal range could be affected by the conditions in which gravitational forces act. C. It presumes, without providing warrant, that most activity within the world's oceans is a of an interplay of gravitational forces. D. It gives only one example of a tidal range. Answer: B PhishCo runs a number of farms in the arid province of Nufa, depending largely on irrigation. Now, as part of a plan to efficiently increase the farms' total production, it plans to drill down to an aquifer containing warm, slightly salty water that will be used to raise fish in ponds. The water from the ponds will later be used to supplement piped-in irrigation water for PhishCo's vegetable fields, and the ponds and accompanying vegetation should help reduce the heat in the area of the farms. Which of the following would, if true, most strongly suggest that the plan, if implemented, would increase the overall efficiency of PhishCo's farms? Options: A. Organic waste from fish in the pond water will help to fertilize fields where it is used for irrigation. B. Fish raised on PhishCo's farms are likely to be saleable in the nearest urban areas. C. Ponds will be located on low-lying land now partially occupied by grain crops. D. The government of Nufa will help to arrange loan financing to partially cover the costs of drilling. Answer: A

We introduce ReClor, a reading comprehension dataset requiring logical reasoning, and find that current state-of-the-art models struggle with real logical reasoning with poor performance near that of random guess.

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We provide insightful understanding of sequence-labeling NER and propose to use two types of cross structures, both of which bring theoretical and empirical improvements.


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Question: This paper explores the scenarios under which an attacker can claim that ‘Noise and access to the softmax layer of the model is all you need’ to steal the weights of a convolutional neural network whose architecture is already known. We were able to achieve 96% test accuracy using the stolen MNIST model and 82% accuracy using stolen KMNIST model learned using only i.i.d. Bernoulli noise inputs. We posit that this theft-susceptibility of the weights is indicative of the complexity of the dataset and propose a new metric that captures the same. The goal of this dissemination is to not just showcase how far knowing the architecture can take you in terms of model stealing, but to also draw attention to this rather idiosyncratic weight learnability aspects of CNNs spurred by i.i.d. noise input. We also disseminate some initial obtained with using the Ising probability distribution in lieu of the i.i.d. Bernoulli distribution In this paper, we consider the fate of an adamant attacker who is adamant about only using noise as input to a convolutional neural network (CNN) whose architecture is known and whose weights are the target of theft. We assume that the attacker has earned access to the softmax layer and is not restricted in terms of the number of inputs to be used to carry out the attack. At the outset, we'd like to emphasize that our goal in disseminating these is not to convince the reader on the real-world validity of the attacker-scenario described above or to showcase a novel attack. This paper contains our initial explorations after a chance discovery that we could populate the weights of an MNIST-trained CNN model by just using noise as input into the framework described below. Preliminary work. Under review by the International Conference on Machine Learning (ICML). Do not distribute. Through a set of empirical experiments, which we are duly open sourcing to aid reproducibility, we seek to draw the attention of the community on the following two issues:1. This risk of model weight theft clearly entails an interplay between the dataset as well as the architecture. Given a fixed architecture, can we use the level of susceptibility as a novel metric of complexity of the dataset?2. Given the wide variations in success attained by varying the noise distribution, how do we formally characterize the relationship between the input noise distribution being used by the attacker and the true distribution of the data, while considering a specific CNN architecture? What aspects of the true data distribution are actually important for model extraction?The rest of the paper is structured as follows:In Section 2, we provide a brief literature survey of the related work. In Section 3, we describe the methodology used to carry out the attack. In Section 4, we cover the main obtained and conclude the paper in Section 5. The art form of stealing machine learning models has received a lot of attention in the recent years. In BID9, the authors specifically targeted real-world ML-as-aservice BID7 platforms such as BigML and Amazon Machine Learning and demonstrated effective attacks that ed in extraction of machine learning models with near-perfect fidelity for several popular model classes. In BID2, the authors trained what they termed as a copycat network using Non-Problem Domain images and stolen labels to achieve impressive in the three problems of facial expression, object, and crosswalk classification. This was followed by work on Knockoff Nets BID6, where the authors demonstrated that by merely querying with random images sourced from an entirely different distribution than that of the black box target training data, one could not just train a well-performing knockoff but it was possible to achieve high accuracy even when the knockoff was constructed using a completely different architecture. This work differs from the above works in that the attacker is adamant on only using noise images as querying inputs. Intriguingly enough, the state-of-the-art CNNs are not robust enough to provide a flat (uniform) softmax output (with weight 1/number-of-classes) when we input noninput-domain noise at the input layer. This was been studied under two contexts. The first context was within the framework of fooling images. In BID5, the authors showcased how to generate synthetic images that were noiselike and completely unrecognizable to the human-eye but ones that state-of-the-art CNNs classified as one of the training classes with 99.99% confidence. The second text was with regards to what the authors in BID3 stated to be rubbish-class examples. Here, they showcased that the high levels of confident mis-predictions exuded by state-of-the-art trained on MNIST and CIFAR-10 datasets in response to isotropic Gaussian noise inputs. In this work, we focus on using Bernoulli noise-samples as inputs and using the softmax responses of the target model to siphon away the weights. We propose a framework for model extraction without possession of samples from the true dataset which the model has been trained on or the purpose of the model other than the dimensionality of the input tensors as well as the ability to access the ing class distribution from what is assumed to be a softmax activation given an input. We make the additional assumption that the architecture of the model to be extracted is known by the adversary. In our experiments, we assume that the input tensor is of dimension 28 by 28 and each pixel has values on the interval. The black box model which we attempt to extract, F (·), whose architecture is described in TAB1, is trained to convergence on a standard dataset for 12 epochs using the Adadelta optimizer with an initial learning rate of 1.0 and a minibatch size of 128 (Mni). From this point onward, this model is assumed to be a black box in which we have no access to the parameters of each layer. We procedurally generate a dataset of'stimuli' comprised of 600000 28 by 28 binary tensors where each pixel is sampled from a Bernoulli distribution with a success probability parameter p. In other words, let each image DISPLAYFORM0 28×28 where x i rand,j,k ∼ Bern(p) for i ∈ {1, ..., 600000}. We sample these tensors with probability parameters p ∈ {0.01, 0.11, ...0.91}, where each p is We train a new model, F extract (·), on the stimulus response pairs, {( DISPLAYFORM0 pairs with no regularization and evaluate on the dataset originally used to train F (·). The architecture for this model is the same as F (·), except we remove the dropout layers to encourage overfitting. We train for 50 epochs using the Adadelta optimizer with an initial learning rate of 1.0 and a minibatch size of 128. Additionally, we acknowledge a significant class imbalance in the highest probability classes in the softmax vectors y rand, so we remedy this by computing class weights according to the argmax of each softmax vector, and applying this re-weighting during the training of F extract (·). We show the full extraction algorithm in Algorithm 1 and summarize it in FIG0.We evaluate our proposed framework on four datasets from the MNIST family of datasets with identical dimensions: MNIST, KMNIST, Fashion MNIST, and notMNIST BID4 BID1 BID10 not). We evaluated the effect of sampling random data x i rand from different distributions on the performance of F extract (·) on the MNIST validation set. We used the same training procedure as found in the previously described experiments with two exceptions: we sample only 60000 procedurally generated examples and we train F extract (·) for only 10 epochs. We evaluated the use of the uniform distribution on the bounded interval, the standard normal distribution, the standard Gumbel distribution, the Bernoulli distribution with success parameter p = 0.5, and samples from an Ising model simulation with inverse temperature parameter β ∈ [0.0, 0.1, ..., 0.9] and ing values scaled to {0, 1}. The Ising prior is defined by the density BID8: DISPLAYFORM0 Examples of images sampled from the Ising model can be found in Figure 6.For this experiment, we evaluated the role of the inverse temeprature β parameter of the Ising sampler in training F extract (·). We first partition the stimulus response pairs, (X Ising, Y Ising) into 10 subsets with 7000 examples each corresponding to the different β parameters used to generate the samples, where (X Ising, Y Ising) = β∈{0.0,0.1,...,0.9} {(X Ising,β, Y Ising,β)}. We train F extract (·) for 10 epochs for each β and validate on the original dataset. We performed this experiment for MNIST, KMNIST, Fashion MNIST, and notMNIST and report the variation in performance over different values of β. We evaluate the efficacy of our framework by training F (·) on MNIST and going on to evaluate the performance of F extract (·) on MNIST after extraction. We found that F (·) achieved a validation accuracy of 99.03% and F extract (·) achieved a validation accuracy of 95.93%. The distribution of the argmax of Y rand can be found in Figure 2. The most underrepresented class according to the argmax of Y rand was class 6 represented by 198 out of 600000 random examples. Input: DISPLAYFORM0 for p in {0.01, 0.11, ..., 0.91} do for q in {0, 1, ..., numRandomExamples/10} do for j in {0, 1, ..., dim-1} do for k in {0, 1, ..., dim-1} do x sample,j,k ∼ Bern(p) end for end for X rand = X rand ∪ x sample end for end for Initialize F extract (·). DISPLAYFORM1 Our experiments with KMNIST ed in F (·) achieving a validation accuracy of 94.79% and F extract (·) achieving a validation accuracy of 81.18%. Class 8 was found to be the class with the fewest representatives according to the argmax of Y rand, which had 272 representative examples out of 600000. On the Fashion MNIST dataset, we found that F (·) achieved a validation accuracy of 92.16%, while F extract (·) achieved a validation accuracy of 75.31%. For Fashion MNIST, the most underrepresented class according to the argmax of Y rand was class 7 (sneaker) with only 12 out of 600000 random examples. Notably, the most common mispredictions according to FIG1 were incorrectly predicting class 5 (sandal) when the ground truth is class 7 (sneaker) and predicting class 5 (sandal) when the ground truth is class 9 (ankle boot). F extract (·) seems to predict the majority of examples from shoe-like classes to be of class 5 (sandal). We found that the notMNIST dataset had a more uniform class distribution according to the argmax of Y rand than the other datasets that we evaluated. The class with the fewest representatives in this sense was class 9 (the letter j) Table 2. Performance using different noise distributions. DISPLAYFORM0 70.03% BERNOULLI (p = 0.5) 76.58% ISING (β ∈ {0.0, 0.1, ..., 0.9}) 98.02%with 3950 out of 600000 examples. Despite this potential advantage, the extracted model F extract (·) failed to generalize to the notMNIST validation set, achieving an accuracy of 10.47%, and as can be seen in FIG1, F extract (·) predicts class 5 (the letter e) in the vast majority of cases. In contrast, F (·) achieved a validation accuracy of 88.62%. In evaluating the effect of sampling from different distributions to construct X rand, we found that among the uniform, standard normal, standard Gumbel, Bernoulli distributions, and the Ising model, samples from the Ising model attained the highest accuracy at 98.02% when evaluating F extract (·) on the MNIST validation set. The for each of the other distributions can be found in Figure 2. We postulate that this is due to the modelling of spatial correlations, which is a property which is lacking when sampling from the uniform, standard normal, standard Gumbel, and Bernoulli distributions, as the pixels are assumed to be i.i.d. We propose a measure of model extraction hardness ing from the dataset which the original model is trained on as the ratio of the post-extraction validation accuracy (using F extract (·)) and the pre-extraction validation accuracy (using F (·)) under our framework. We show that the ing ratios are align with the mainstream intuition regarding the general relative learnability of MNIST, KMNIST, Fashion MNIST, and notMNIST. For MNIST, we found this ratio to be 0.9687, the ratio for KMNIST was 0.8564, for Fashion MNIST we found it to be 0.8171, and notMNIST achieved a ratio of 0.1181. We found that the loss and accuracy undergo'phase transitions' as the value of β is varied. In FIG2, we see that across datasets, the losses tend to be minimized around β = 0.3, however the behavior of larger values of β varies from dataset to dataset. We postulate that this is indicative of the different distributions of the amount of spatial correlation across each dataset. We also found that accuracy is maximized at β = 0.4 for MNIST, β = 0.3 for KMNIST and Fashion MNIST, and β = 0.2 for notMNIST, where the behavior here also varies as β increases from the optimal value. We show this in FIG2. In this paper, we demonstrated a framework for extracting model parameters by training a new model on random impulse response pairs gleaned from the softmax output of the victim neural network. We went on to demonstrate the variation in model extractability based on the dataset which the original model was trained on. Finally, we proposed our framework as a method for which relative dataset complexity can be measured. This is a work in progress and we are currently working along the following three directions: In our experiments, pixels are notably i.i.d., whereas in real world settings, image data is comprised of pixels which are spatially correlated. In this vein, we intend to establish the relationship between the temperature of an Ising prior and the accuracy obtained by the stolen model. We will experiment with different architectures, specifically exploring the architecture unknown scenario where the attacker has a fixed plug-and-play swissarmy-knife architecture whose weights are learned by the noise and true-model softmax outputs. Additionally, we will explore methods for constructing X rand which gives more uniform distributions over argmax(Y rand) and evaluate the associated effect on the performance of F extract (·). TAB0 Garbage in, model out: Weight theft with just noise TAB0 Garbage in, model out: Weight theft with just noise

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Input only noise , glean the softmax outputs, steal the weights

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This work aims to provide quantitative answers to the relative importance of concepts of interest via concept activation vectors (CAV). In particular, this framework enables non-machine learning experts to express concepts of interest and test hypotheses using examples (e.g., a set of pictures that illustrate  the concept). We show that CAV can be learned given a relatively small set of examples. Hypothesis testing with CAV can answer whether a particular concept (e.g., gender) is more important in predicting a given class (e.g., doctor) than other sets of concepts. Interpreting networks with CAV does not require any retraining or modification of the network.


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Question: We propose a new form of an autoencoding model which incorporates the best properties of variational autoencoders (VAE) and generative adversarial networks (GAN). It is known that GAN can produce very realistic samples while VAE does not suffer from mode collapsing problem. Our model optimizes λ-Jeffreys divergence between the model distribution and the true data distribution. We show that it takes the best properties of VAE and GAN objectives. It consists of two parts. One of these parts can be optimized by using the standard adversarial training, and the second one is the very objective of the VAE model. However, the straightforward way of substituting the VAE loss does not work well if we use an explicit likelihood such as Gaussian or Laplace which have limited flexibility in high dimensions and are unnatural for modelling images in the space of pixels. To tackle this problem we propose a novel approach to train the VAE model with an implicit likelihood by an adversarially trained discriminator. In an extensive set of experiments on CIFAR-10 and TinyImagent datasets, we show that our model achieves the state-of-the-art generation and reconstruction quality and demonstrate how we can balance between mode-seeking and mode-covering behaviour of our model by adjusting the weight λ in our objective. Variational autoencoder (VAE) (; ; Titsias & Lázaro-) is one of the most popular approaches for modeling complex high-dimensional distributions. It has been applied successfully to many practical problems. It has several nice properties such as learning low-dimensional representations for the objects and ability to conditional generation. Due to an explicit reconstruction term in its objective, one may ensure that VAE can generate all objects from the training set. These advantages, however, come at a price. It is a known fact that VAE tends to generate unrealistic objects, e.g., blurred images. Such behaviour can be explained by the properties of a maximum likelihood estimation (MLE) which is used to fit a restricted VAE model p θ (x) in data that comes from a complex distribution p with an equiprobable mixture of two Gaussians with learnable location and scale. Plots a)-c) show pairwise comparisons of optimal log-densities, the plot d) compares optimal densities themselves. ). This way, we encourage our model to be mode-seeking while still having relatively high values of p θ (x) on all objects from a training set, thus preventing the mode-collapse. We note that J λ (p θ (x) p * (x)) is not symmetric with respect to p θ (x) and p * (x) and by the weight λ we can balance between mode-seeking and mass-covering behaviour. However, the straightforward way of substituting each KL term with GAN and VAE losses does not work well in practice if we use an explicit likelihood for object reconstruction in VAE objective. Such simple distributions as Gaussian or Laplace that are usually used in VAE have limited flexibility and are unnatural for modelling images in the space of pixels. To tackle this problem we propose a novel approach to train the VAE model in an adversarial manner. We show how we can estimate the implicit likelihood in our loss function by an adversarially trained discriminator. We theoretically analyze the introduced loss function and show that under assumptions of optimal discriminators, our model minimizes the λ-Jeffreys divergence J λ (p θ (x) p * (x)) and we call our method as Implicit λ-Jeffreys Autoencoder (λ-IJAE). In an extensive set of experiments, we evaluate the generation and reconstruction ability of our model on CIFAR10 and TinyImagenet datasets. It shows the state-of-the-art trade-off between generation and reconstruction quality. We demonstrate how we can balance between the ability of generating realistic images and the reconstruction ability by changing the weight λ in our objective. Based on our experimental study we derive a default choice for λ that establishes a reasonable compromise between mode-seeking and mass-covering behaviour of our model and this choice is consistent over these two datasets. Relation to forward KL-based methods. We can say that all VAE-based models minimize the upper bound on the forward KL D KL (p * (x) p θ ). In recent years there have been many extensions and improvements of the standard VAE. One direction of research is to inroduce the discriminator as a part of data likelihood to leverage its intermediate layers for measuring similarity between objects. However, these models do not have a sound theoretical justification about what distance between p θ (x) and p * (x) they optimize. The other way is to consider more complex variational distribution q ϕ (z|x). One can either use better variational bounds (; Maaløe et al., 2016; ; ;) or apply the adversarial training to match q ϕ (z|x) and the prior distribution p(z) or to match the marginals q ϕ (z) and p(z) . Although these methods improve approximate inference in VAE model, they remain in the scope of MLE framework. As we discussed above within this framework the model with a limited capacity is going to have the mass-covering behaviour. Relation to reverse KL-based methods. The vanilla GAN framework is equivalent under the assumption of optimal discriminator to minimization of Jensen-Shanon divergence JSD(p * (x) p θ (x)) . With a minor modification of a generator loss we can obtain the equivalence to minimization of the reverse KL D KL (p θ (x) p * (x)) . There have been proposed many autoencoder models which utilize one of these two divergences JSD(p One approach is to minimize the divergence between joint distributions p * (x)q(z|x) and p θ (x|z)p(z) in a GAN framework . ALICE model introduces an additional entropy loss for dealing with the non-identifiability issues in previous works. Other methods (; a; ; ;) use the reverse KL D KL (p θ (x) p * (x)) as an additional term to encourage mode-seeking behaviour. Relation to Jeffreys divergence-based methods. To the best of our knowledge, there are only two other autoencoder models which minimize λ-Jeffreys divergence for λ = 0.5. It is an important case when λ-Jeffreys divergence equals symmetric KL divergence. These methods are AS-VAE (a) and SVAE and they are most closely related to our work. AS-VAE is a special case of SVAE method therefore further we will consider only SVAE. There are two most crucial differences between SVAE and λ-IJAE models. The first one is that SVAE minimizes ) between marginal distributions p * (x) and p θ (x) for arbitrary λ. The second difference is that the SVAE's loss J λ (p * (x)q(z|x) p θ (x|z)p(z)) solely did not give good reconstructions in experiments. Therefore, authors introduced additional data-fit terms E p * (x)q θ (z|x) log p θ (x|z) + E p θ (x|z)p(z) log q ϕ (z|x) where p θ (x|z) and q ϕ (z|x) are explicit densities. In contrast, λ-IJAE model achieves good generation and reconstruction quality as it is and allows training implicit p θ (x|z) and q ϕ (z|x) distributions. These two differences make SVAE and λ-IJAE models significantly distinct, and we observe it in practice. Consider samples x ∈ X from the true data distribution p * (x). The aim of generative models is to fit a model distribution p θ (x) to p * (x). Most popular models are GAN and VAE. In practice, we observe that they have significantly different properties. VAE tends to cover all modes of p * (x) at the cost of capturing low probability regions as well. As a , it often generates unspecific and/or blurry images. On the other hand, GAN is highly mode-seeking, i.e. it tends to concentrate most of its probability mass in a small number of modes of p * (x). Therefore it may not cover significant part of p * (x) which is also known as a mode collapse problem. Such radical contrast between VAE and GAN can be explained by the fact that they optimize different divergences between p θ (x) and p * (x). Variational Inference. VAE is trained by MLE: max θ E p * (x) log p θ (x). The distribution p θ (x) is defined as an integral over a latent variable z: p θ (x) = p θ (x|z)p(z)dz, and in practice it is typically intractable. Variational inference sidesteps this issue by introducing an encoder model (also known as a variational distribution) q ϕ (z|x) and replacing the intractable log p θ (x) with a tractable evidence lower bound (ELBO): Then we maximize ELBO L ELBO (θ, ϕ) with respect to θ and ϕ. One can easily derive that MLE is equivalent to optimizing the forward KL D KL (p * p θ): Adversarial Training. The adversarial framework is based on a game between a generator G θ (z) and a discriminator D ψ (x) which classifies objects from p * (x) versus ones from p θ (x): showed that the loss of the generator is equivalent to the Jensen-Shanon It is easy to recognize this as an instance of classification-based Density Ratio Estimation (DRE) . Following this framework, one can consider different generator's objectives while keeping the same objective for the discriminator. DRE relies on the fact that. By this approach we can obtain a likelihood-free estimator for the reverse D KL (p θ p *) : (Un)Biased Gradients in Adversarial Training. Since in practice the discriminator D ψ (x) is only trained to work for one particular set of generator parameters θ, we need to be cautious regarding validity of gradients obtained by DRE approach. For example, consider the forward KL D KL (p * p θ). If we apply DRE, we will arrive at E p * (x) log. However, we can notice that in practice this expression does not depend on θ in any way, i.e.. This is because the forward KL depends on θ only through the ratio of densities, which is replaced by a point estimate using a discriminator which has no idea regarding p θ's local behaviour. This shows we need to be careful when designing adversarial learning objective as to ensure unbiased gradients. Luckily, JSD(p θ p *) and D KL (p θ p *) are not affected by this problem: for any x. Then Proof. Given in Appendix, section A. VAE provide a theoretically sound way to learn generative models with a natural and coherent encoder. However, they are known to generate blurry and unspecific samples that have inferior perceptual quality compared to generative models based on adversarial learning. The main cause for that is that the root principle VAEs are built upon -MLE framework -is equivalent to minimization of the forward KL D KL (p * p θ). While D KL (p * p θ) recovers the true data-generating process p * (x) if the model p θ (x) has enough capacity, in a more realistic case of an insufficiently expressive model p θ (x) it is known to be mass-covering. As a , the model is forced to cover all modes of p * (x) even at the cost of covering low-probability regions as well. This in turn might lead to blurry samples as the model does not have the capacity to concentrate inside the modes. On the other hand, the reverse KL D KL (p θ p *) has mode-seeking behavior that penalizes covering low-probability regions and thus the model p θ (x) tends to cover only a few of the modes of p * (x). Following this reasoning, we propose a more balanced divergence -one that seeks modes, but still does a decent job covering all modes of p * (x) to prevent mode collapse. We chose λ-Jeffreys divergence We illustrate the advantage of λ-Jeffreys divergence for λ = 0.5 over Forward KL, Reverse KL and JSD divergences in the case of a model with limited capacity in Figure 1. In this figure we compared divergences in a simple task (see Appendix, section B) of approximating a mixture of 4 Gaussians with a mixture of just two: both Reverse KL and JSD exhibit mode-seeking behavior, completely dropping side modes, whereas the Forward KL assigns much more probability to tails and does poor job capturing the central modes. On a contrast, λ-Jeffreys divergence uses one mixture component to capture the most probable mode, and the other to ensure mass-covering. The optimization of λ-Jeffreys divergence consists of two parts. The first one is the minimization of the reverse KL D KL (p θ (x) p * (x)) which can be implemented as a standard GAN optimization as we discussed in Section 3. The second part is the optimization of the forward KL D KL (p * (x) p θ (x)) and we tackle it by maximization of the ELBO L ELBO (θ, ϕ) as in VAE. So, we obtain an upper bound on λ-Jeffreys divergence by incorporating GAN and VAE objectives: The ELBO term L ELBO (θ, ϕ) can be decomposed into two parts: (i) a reconstruction term. While both terms are easy to deal with in cases of explicit p(x|z) and q(z|x), an implicit formulation poses some challenges. In the next two sections we address them. Typically to optimize the reconstruction term E p * (x) E qϕ(z|x) log p θ (x|z) the conditional likelihood p θ (x|z) is defined explicitly as a fully factorized Gaussian or Laplace distribution (; ; Titsias & Lázaro-; b; ; ;). While convenient, such choice might limit the expressivity of the generator G θ (z). As we discussed previously, optimization of the forward KL(p * p θ) leads to a mass-covering behavior. The undesired properties of this behavior such as sampling unrealistic and/or blurry images can be more significant if a capacity of our model p θ (x) is limited. Therefore we propose a technique which allows to extend the class of possible likelihoods for p θ (x|z) to implicit ones. We note that typically in VAE the decoder p θ (x|z) first maps the latent code z to the space X, which is then used to parametrize the distribution of z's decodings x|z. For example, this is the case for N (x|G θ (z), σI) or Laplace(x|G θ (z), σI). We also use the output of the generator G θ (z) ∈ X to parametrize an implicit likelihood. In particular, we assume p θ (x|z) = r(x|G θ (z)) for some symmetric likelihood r(x|y): (ii) r(x = a|y = b) has a mode at a = b. While the Gaussian and Laplace likelihoods are symmetric and explicit, in general we do not require r(x|y) to be explicit, only being able to generate samples from r(x|y) is required. The idea is to introduce a discriminator D τ (x, z, y) which classifies two types of triplets: • fake class: (x, z, y) ∼ p * (x)q ϕ (z|x)r (y|G θ (z)). We note that r(y|x) and r (y|x) can be different and we will utilize this possibility in practice. Then we train the discriminator D τ (x, z, y) using the standard binary cross-entropy objective: If we apply the straightforward way to obtain an objective for the generator G θ (z) we will derive that we should minimize E p * (x)qϕ(z|x) log Dτ (x,z,x) 1−Dτ (x,z,x). Indeed, given an optimal discriminator for D τ * (x, z, y) = r(y|x) r(y|x)+r (y|G θ (z)), we obtain: Dτ (x,z,x) given the optimal D τ * (x, z, y) is equivalent to maximizing the reconstruction term with p θ (x|z) = r (x|G θ (z)). However, we face in practice the same issue as we discussed in Section 3 that ∇ θ E p * (x)qϕ(z|x) log Dτ (x,z,x) 1−Dτ (x,z,x) = 0 because D τ (x, z, x) does not depend on θ explicitly even for optimal τ = τ *. We can overcome this issue by exploiting the properties of symmetric likelihoods if we minimize a slightly different loss for the generator The following theorem guarantees the gradients will be unbiased in the optimal discriminator case: Theorem 1. Let D τ * (x, z, y) be the optimal solution for the objective and r(y|x) and r (y|x) are symmetric likelihoods. Then Proof. Given in Appendix, section A. So, we obtain that we can maximize the reconstruction term E p * (x) E qϕ(z|x) log r(x|G θ (z)) by minimizing −E p * (x) E qϕ(z|x) log Dτ (x,z,G θ (z)) 1−Dτ (x,z,G θ (z)) and optimize it using gradient based methods. We note again that we do not require an access to an analytic form of r(y|G θ (z)). It is an open question what is the best choice for the r(y|G θ (z)). Our expectations from r(y|G θ (z)) are that it should encourage realistic reconstructions and highly penalize for visually distorted images. In experiments, as r(y|x) we use a distribution over cyclic shifts in all directions of an image x. This distribution is symmetric with respect to all directions and has a mode in x, therefore it is the symmetric likelihood (see Definition 1 for details). Although in practice we use r(y|x) which has an explicit form due to non-optimality of D τ (x, z, y) (that is always the case when training on finite datasets) the ratio log Dτ (x,z,G θ (z)) 1−Dτ (x,z,G θ (z)) sets implicit likelihood of reconstructions. We can think of the non-optimality of D τ (x, z, y) as a form of regularization that allows us to convert explicit function r(y|x) into implicit likelihood that has desirable properties, i.e. encourages realistic reconstructions of x and penalizes unrealistic ones. Implicit Encoder. The KL term from L ELBO (θ, ϕ) can be computed either analytically, using the Monte Carlo estimation or by the adversarial manner. We chose the latter approach proposed by because it enables implicit variational distribution q ϕ (z|x) defined by a neural sampler (encoder) E ϕ (x, ξ) where ξ ∼ N (·|0, I). For this purpose we should train a discriminator D ζ (x, z) which tries to distinguish pairs (x, z) from p * (x)q ϕ (z|x) versus the ones from p ) is a reverse KL with respect to parameters ϕ, therefore we can substitute it by the expression −E qϕ(z|x) log Putting it all together we arrive at the following objective: In practice, discriminators are not optimal therefore we train our model by alternating gradients. We maximize objectives,, In experiments, we evaluate generation and reconstruction ability of our model on datasets CIFAR-10 and TinyImageNet. We used a standard ResNet architecture for the encoder E ϕ (x, ξ), the generator G θ (z) and for all three discriminators D ψ (x), D τ (x, z, y), D ζ (x, z). The complete architecture description for all networks and hyperparameters used in λ-IJAE can be found in Appendix, section D. To compare our method to other autoencoding methods in the best way, we also used official and publicly available code for baselines. For AGE 1 we use a pretrained model. For SVAE 2, TwoStage-VAE (2SVAE) 3 we report metrics reproduced using officially provided code and hyperparameters. For α-GAN we also use public implementation 4 with same architecture as in λ-IJAE. In experiments, for symmetric likelihoods r(y|x) and r (y|x) we use the following: r(y|x) is a continuous distribution over cyclic shifts in all directions of an image x. In practice, we discretize this distribution. To sample from it: (i) we sample one of four directions (top, bottom, right, left) equally probable; (ii) then sample the size of a shift (maximum size S = 5 pixels) from 0 to S (iii) as a , we shift an image x to the selected direction in a size which is sampled. For r (y|x) in practice we observe that the best choice is when r (y|x) is close to a delta function δ x (y). Therefore, we use r (y|x) = N (y|x, σI) which is clearly a symmetric likelihood. We set σ = 10 −8. For r(y|x) as an implicit likelihood we also studied a distribution over small rotations of x, however, we observed that cyclic shifts achieve better . Evaluation. We evaluate our model on both generation and reconstruction tasks. The quality of the former is assessed using Inception Score (IS) . To calculate these metrics we used the official implementation provided in tensorflow 1.13 . The reconstruction quality is evaluated using LPIPS, proposed by . LPIPS compares images based on high-level features obtained by the pre-trained network. It was show by that LPIPS is a good metric which captures perceptual similarity between images. We use the official implementation (LPIPS github) to compute LPIPS. Ablation Study. To show the importance of the implicit conditional likelihood r(y|x) we compare λ-IJAE with its modification which has instead of implicit r(y|x) a standard Gaussian or Laplace distribution. We call such models λ-IJAE-L 2 and λ-IJAE-L 1 respectively. In Figure 2 we compare λ-IJAE with λ-IJAE-L 2 and λ-IJAE-L 1 in terms of IS (generation quality) and LPIPS (reconstruction quality). We see that λ-IJAE significantly outperforms these baselines and allows to achieve paretooptimal for different choice of λ. Comparison with Baselines. We assess generation and reconstruction quality of λ-IJAE on CIFAR-10 and TinyImageNet datasets. We compare the to closest baselines with publicly available code. We provide visual in Appendix, section C. Quantitative are given in Figure 3 and in Table 1. In Figure 3 we compare the methods with respect to IS and LPIPS. Considering both metrics λ-IJAE achieves a better trade-off between reconstruction and generation quality within these Reconstruction and generation quality on CIFAR10 and TinyImagenet for models that allow reconstructions. Baseline models were trained using publicly available code, if possible, to fill reconstruction quality metrics. ↓ -lower is better, ↑ -higher is better, best is marked with bold. WAE 4.18 ± 0.04 ALI ) 5.34 ± 0.04 ALICE 6.02 ± 0.03 AS-VAE (b) 6.3 VAE (resnet) 3.45 ± 0.02 0.09 ± 0.03 2Stage-VAE 3.85 ± 0.03 0.06 ± 0.03 α-GAN 5.20 ± 0.08 0.04 ± 0.02 AGE 5.90 ± 0.04 0.06 ± 0.02 SVAE 6.56 ± 0.07 0.19 ± 0.08 6.98 ± 0.1 0.07 ± 0.03 TinyImagenet AGE 6.75 ± 0.09 0.27 ± 0.09 SVAE 5.09 ± 0.05 0.28 ± 0.08 2Stage-VAE 4.22 ± 0.05 6.87 ± 0.09 0.09 ± 0.03 datasets. We see that small values of λ give a good IS score while remain the decent reconstruction quality in terms of LPIPS. However, if decrease λ further LPIPS will start to degrade. Therefore, we chose the λ = 0.3 as a reasonable trade-off between generation and reconstruction ability of λ-IJAE. For this choice of λ we compute the for Table 1. From these Table 1 we see that λ-IJAE achieves the state-of-the-art trade-off between generation and reconstruction quality. It confirms our justification about λ-Jeffreys divergence that it takes the best properties of both KL divergences. In the paper, we considered a fusion of VAE and GAN models that takes the best of two worlds: it has sharp and coherent samples and can encode observations into low-dimensional representations. We provide a theoretical analysis of our objective and show that it is equivalent to the Jeffreys divergence. In experiments, we demonstrate that our model achieves a good balance between generation and reconstruction quality. It confirms our assumption that the Jeffreys divergence is the right choice for learning complex high-dimensional distributions in the case of the limited capacity of the model. Proof. Now we will show the second term is equal to zero given our assumptions: Where we have used the and properties of the likelihoods r(x|y) (Definition 1): To generate the plot 1 we considered the following setup: a target distribution was a mixture: While the model as an equiprobable mixture of two learnable Gaussians: The optimal θ was found by making 10,000 stochastic gradient descent iterations on Monte Carlo estimations of the corresponding divergences with a batch size of 1000. We did 50 independent runs for each method to explore different local optima and chose the best one based on a divergence estimate with 100,000 samples Monte Carlo samples.

We propose a new form of an autoencoding model which incorporates the best properties of variational autoencoders (VAE) and generative adversarial networks (GAN)

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An empirical study of variational inference based on chi-square divergence minimization, showing that minimizing the CUBO is trickier than maximizing the ELBO